Research ArticleRobust Production Planning in Fashion Apparel Industry underDemand Uncertainty via Conditional Value at Risk

Abderrahim Ait-Alla1 Michael Teucke1 Michael Luumltjen1

Samaneh Beheshti-Kashi1 and Hamid Reza Karimi2

1 Bremer Institut fur Produktion und Logistik GmbH at the University of Bremen (BIBA) 28359 Bremen Germany2Department of Engineering Faculty of Technology and Science University of Agder 4898 Grimstad Norway

Correspondence should be addressed to Michael Teucke tckbibauni-bremende

Received 31 January 2014 Accepted 25 March 2014 Published 29 April 2014

Academic Editor Michael Freitag

Copyright copy 2014 Abderrahim Ait-Alla et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents a mathematical model for robust production planning The model helps fashion apparel suppliers in makingdecisions concerning allocation of production orders to different production plants characterized by different lead times andproduction costs and in proper time scheduling and sequencing of these production orders The model aims at optimizingthese decisions concerning objectives of minimal production costs and minimal tardiness It considers several factors such as thestochastic nature of customer demand differences in production and transport costs and transport times between productionplants in different regions Finally the model is applied to a case study The results of numerical computations are presented Theimplications of the model results on different fashion related product types and delivery strategies as well as the modelrsquos limitationsand potentials for expansion are discussed Results indicate that the production planning model using conditional value at risk(CVaR) as the risk measure performs robustly and provides flexibility in decision analysis between different scenarios

1 Introduction

This contribution deals with production planning problemsof fashion apparel products Fashion apparel products belongto the most important consumer goods Global retail rev-enues amounted to $1032 billion in 2009 and are expectedto grow to $1163 billion by 2016 [1]

Production planning for fashion apparel products has tocope with demand uncertainties Accordingly the uncertainnature of the customer demand has to be taken into consider-ation by generating the production plan and in particular theproduction quantities in order to meet uncertain customerdemand in the best way possible and maximize the profit byminimizing production costs

In this case production planning in particular thecorrect placement of production orders concerning placeor region of production as well as time scheduling andsequencing of production orders is of high economic impor-tance for fashion apparel suppliers However at the timeof generating the production plan the predicted customer

demands are largely uncertain Therefore it is crucial toproduce a robust production plan which canmanage the riskresulting from the demand forecastThis risk trade-off can beachieved by constraining the objective function or problemlimits with CVaR Indeed CVaR intends to protect againstundesirable realization of uncertain parameters beyond theexpected evaluation due to the uncertainty of system param-eters [2]

Existing papers dealing with the robust optimization infashion apparel do not take into account the risk of losingmore than an acceptable level of profits due to write-offscaused by an overly optimistic demand forecast or earningless than a desired target profit due to an overly pessimisticdemand forecast This paper is the first study to address thisproblem for the fashion apparel industry

To deal with this uncertainty in customer demand inthe apparel industry we propose a risk-constrained profit-expected maximization model This model considers thestochastic nature of customer demand and generates a pro-duction plan which indicates the quantities of each product

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 901861 10 pageshttpdxdoiorg1011552014901861

2 Mathematical Problems in Engineering

that should be produced the start of the production of eachproduct and the facility in which the products have to beproduced The objective is to maximize the total profit ofproductions by means of CVaR

Consideration of risk in the optimization problem has acrucial role in optimization under uncertainty particularlywhen the optimization problem has to deal with the lossesthat might be incurred under conditions of unfavorabledemand To consider the risk of erroneous demand forecastsa loss function 119891(119883 119904) will be defined where 119883 representsa decision vector and 119904 is a vector representing uncertaintyrelated to the future values of a number of stochastic parame-ters of the problemThese stochastic parameters presented bythe vector 119904 are governed by probability distribution119875

119904 Once

the loss function 119891 is defined we denote the distributionfunction of 119891 by 120593(119883 120573) = 119875119904 | 119891(119883 119904) le 120573 whichcorresponds to the probability that the value of the lossfunction for each realisation of a scenario of the vector 119904 andfor a fixed 119883 does not exceed the value 120573 In this contextfor a specified confidence value 120572 isin [0 1[ that could beequal to 090 or 095 in some applications 120572-VaR denotesthe probability that the expected value of the loss functionexceeds 120573 only in (1 minus 120572) sdot 100 for all possible realisationsof stochastic parameters which can be seen as the worst-case scenario Based on the definition of value at risk (VaR)conditional value at risk (CVaR) is defined as the mean of thetail distribution exceeding VaR [3]

The model is applied to a problem based on a case studyin the apparel industry Different factories located in differentcountries manufacture a number of product types and sellthese products to customers in Europe The orders consist ofthe type of products delivery date and quantity Usually thequantity demand of a product 119894 is of stochastic nature andhaving a probability distribution 119901(119894) After collecting ordersthe production plan has to be generated for the next seasonby considering the manufacturing capacity production costand transport cost while satisfying a CVaR constraint in orderto maximize the expected total profit

In order to generate different possible demand realisa-tions for the mathematical model different scenarios of Ωare generated by means of Monte Carlo simulation using thedemand probability distribution of each product within theset of products

The paper is structured as follows the next chapterwill provide some additional information on productionplanning and demand uncertainty in the apparel industryThe subsequent chapter studies some of the literature onrobust production planning under demand uncertainty Afterthat our model will be described and subsequently appliedto a case study that includes three scenarios (a pessimisticscenario a normal scenario and an optimistic scenario) dif-fering in customer demand These three scenarios have beenproposed in order to cover different potential realizationsof customer demand which can occur but are not knownat the time of planning Thereby the customer demand canbe estimated as low (pessimistic estimate) normal (normalestimate) or high (optimistic estimate) based on historicdata and the recommendations of experts The paper endswith a conclusion and an outlook on further research

2 Production Planning and DemandUncertainty in Apparel Industry

Manufacturing and finishing of ready-to-wear garments area complex process involving often a large number of collabo-rating parties within a production network [4]These includegarment manufacturers and their raw material suppliersprocurement agencies logistic service providers and retailers[5]

Due to limited potential for automation and price impor-tance in competition production of fashion apparel productshas been largely outsourced to low wage countries oftensituated either in East Asia the Near East in particularin China Turkey Bangladesh India and Vietnam Due tothe absence of technical constraints and availability of sub-contractors production can be shifted with few constraintsbetween different regions However the large majority ofthe products are still sold in Europe and North Americaeven though rising domestic demand in Asian countriessuch as China has started to increasingly compete for localproduction resources [6ndash8] In this context many Europeangarments suppliers have reduced the depth of their ownproduction or dropped it and have adapted their roletowards planning and coordinating activities within apparelsupply chains integrating garment manufacturers and rawmaterials suppliers logistic service providers and retailers[9]The wide geographic distribution of supply chains resultsin considerable lead times of apparel production due to thetime needed for procurement of raw materials and the timeneeded for transport of products from the production plantsto the customers [10 11]

In addition to low product costs quality and reliabilityof logistic services are crucial for economic success in theapparel branch Fashion apparel products are characterizedby rapid product obsolescenceThe sales seasons for a periodof a few months or even weeks are short in comparison totheir long production lead and delivery times This increasesthe importance of adherence to delivery dates On average95 of stock keeping units change for every new sales seasonat least twice or four times a year [5 12] The productioncycle is characterized by fixed seasonal cycles with fixeddates for product offers orders and deliveries which arerepeated production volumes based on aggregation of theretail preorders However as in practice production anddistribution lead times usually exceed the length of thedelivery times expected by customers production planningand procurement of raw materials have to start before theend of the preorder period using forecasts of total demandbased on the orders arrived so far Products may also beproduced in larger quantities than if only based on preorderswith the excess offered directly from the warehouses to calledpostorders as long as stocks last [13]

Once their sales season has ended articles which havenot been sold so far can be sold only for reduced prices andthuswith reduced profit in the next seasonThus at the end ofa sales season the stock levels of the products should ideallybe close to zero On the other hand stocks should not run outbefore the end of the sales period Reproduction of successfulproducts during a sales season is not possible due to the long

Mathematical Problems in Engineering 3

delivery times [12] For these reasons demand forecasting hasto be considered an important input of production planningand supply chain planning models Generally forecasting ofpotential sales permeates all aspects of business operations[14] Customer demand for fashion products is volatile andmay vary broadly for different variants of the same productDemand fluctuations are difficult to predict at the time ofproduction planning Optimistic forecasts may lead to over-stocks and higher production costs and cause markdownsPessimistic forecasts may lead to loss of opportunity revenuesfor potential sales that could not be realized and to customerdissatisfaction with later brand changes

On the other hand traditional statistical forecastingmethods are difficult to apply [15] Specifically developeddemand forecasting models often use soft computing meth-ods such as artificial neural networks [16] fuzzy logic andevolutionary procedures [12 17] In addition the integrationof preorder information as well as expert judgments hasproduced accurate forecasts [18] Most of these models havenot been integrated into standard business software and aredifficult to use with sometimesmixed quality of the forecastsOrder fulfilment and product delivery failures such as stock-outs or increased storage costs and product depreciationwrite-offs continue to affect profits in the apparel branch ina very unfavourable way It is estimated that due to the longlead times and demand forecasting difficulties mentioned inthe second section of this chapter up to 30ndash40 of high-fashion articles cannot be sold during the short sales seasonbefore the life expectancy of the article expires [19] Theneed to cope with this situation and the impetus to realizeQuick Response concepts have resulted in vertical integrationof apparel suppliers and retailers in a number of cases[20] Vertically integrated suppliers generally achieve betterreaction to market changes due to shortened reaction timesand more closely integrated information flows Howeverthese advantages cannot easily be copied by most traditionalsuppliers which are often small or medium sized enterprizesStill it is estimated that they have to write off up to 15ndash20of their high-fashion articles [19]

3 Literature Review on Problem RelatedProduction Planning

Production and distribution operations are two key functionsin the supply chain In general the supply chainmanagementprocedures emphasize production scheduling more than thedistribution scheduling because distribution ismore flexibleWhen integrated supply chain management of productionand distribution is realized the resources are used moreefficiently [21] In a comprehensive review of integratedscheduling approachesMula et al showed that over 44 papershave been published between 1989 and 2009 [22]

In general master tactical and operational planninglevels can be separated The purpose of most models is tominimize the total cost of the supply chain Some work at themaster planning level integrating procurement in addition toproduction and distribution scheduling is described in [23]Most of the models focus on tactical planning considering

different demand levels costs and capacities and use cen-tralized scheduling approaches Scholz-Reiter et al showed ascalable graph-based scheduling approach which is focusedon the operational level and real-time scheduling [24] Alarge variety of OR techniques have been applied comprisingmathematical optimisation heuristics and metaheuristicsbut it could be observed that most of the papers focus ondeterministic solutions [22]

The special conditions of production planning for fashionproducts have received some studies as well The work in[25] described a networked production planning processin a fashion oriented apparel supply chain The productionplanning process was found to be an important area ofimprovement for a network in a time-based logic shorteningthe production planning period in fact significantly affectsthe weighted average delivery anticipation It leads howeverto increase setup and transportation costs due to the greaternumber of jobs generated during a campaign Scheduling ofintegrated production and distribution systems in dynamicenvironments holds a potential to improve efficiency butalso poses a challenging planning task due to its compu-tational complexity Long lead times external and internalperturbations in productive processes unstable businessenvironments and contextual differences (eg institutionaleconomic and cultural) emphasize the relevance to theargued integration [26] Three types of aggregate produc-tion planning methods for the apparel industry have beenproposed in [27] These allow changing a production modelseasonally according to actual demands or maintaining thesame production model for several seasons with productionfor inventoryThe mentioned work on integrated productionand distribution planning does not deal with stochastic inputdata such as demands The work in [28] studied empiricallythe impact of the subsidy policy on total factor productivityfor the example of Chines cotton production

Modern large and widely distributed production net-works are subject to many forms of disturbances KarimiDuffy and Dashkovsky et al performed research on howto better deal with such dynamic influences One possiblesolution is to introduce autonomous control that is to allowsome parts of a large network to make their own decisionsbased on local situation and available information Howeverstability of the network and robustness with respect toexternal and internal disturbances and time delays in signalsmust be assured to guarantee a reasonable performance andvitality of the whole system For this purpose their workproposed an approach for controller design for large scaleautonomous work systems capable of coping with time delaysand explains its implementation and advantages on a concreteexample [29ndash31]

Robust optimization models try to formulate productionplanning problems in a way that cost or wastage effectsof uncertainty or risks are minimized or expected profitis maximized A robust optimization model for a multisiteproduction medium-term planning problem is developed in[32] based on the problems facing a multinational lingeriecompany with production sites in East Asia It generates acost minimal production plan for an uncertain environmentwith associated probabilities of different economic growth

4 Mathematical Problems in Engineering

scenarios The cost minimal production plan is less sensitiveto changes in the noisy and uncertain data The work in [33]dealt with a portfolio selection model in which the method-ologies of robust optimization are used for the minimizationof the conditional value at risk of a portfolio of shares Thework in [34] proposed a method for robust self-schedulingbased on CVaRThe proposedmethod is based on a security-constrained optimal power flow (SCOPF) program thatexplicitly treats the trade-off between risk and reward Thework in [35] proposed a fuzzy mathematical programmingmodel for supply chain planning which considers supplydemand and process uncertainties The model has beenformulated as a fuzzy mixed-integer linear programmingmodel where data are ill-known and modelled by triangularfuzzy numbers The fuzzy model provides the decisionmaker with alternative decision plans for different degreesof satisfaction This proposal is tested by using data from areal automobile supply chain A supply chain design problemfor a new market opportunity with uncertain demand inan agile manufacturing setting is considered in [36] Thismodel integrates the design of supply chain and productionplanning for the supply chainrsquos members and develops arobust optimization formulation A novel framework basedon conditional value at risk theory has been applied in [37] tothe problem of operational planning for large-scale industrialbatch plants under demand due date and amount uncertaintyThe objective of the proposed model is to provide a dailyproduction profile that not only is a tight upper bound onthe production capacity of the plant but also is immune tothe various forms of demand uncertainty In further workThe work in [38] applied a robust optimization frameworkas well as CVAR theory to the multisite operational plan-ning problem under multiple forms of system uncertaintyThey considered different forms of system uncertainty suchas demand due date demand amount and transportationtime uncertainty in the model Their objective is to ensurethe maximization of customer satisfaction along with theminimization of resource misallocation

A robust multiobjective mixed integer nonlinear pro-gramming model is proposed by [39] to deal with a multisitemultiperiod multiproduct aggregate production planning(APP) problem under uncertainty considering two conflict-ing objectives simultaneously as well as the uncertain natureof the supply chain Their proposed model is solved as asingle-objective mixed integer programming model applyingthe LP-metrics method A two-stage real world capacitatedproduction system with lead time and setup decisions uncer-tain production costs and customer demand is studied in[40] using a robust optimization approach A mixed-integerprogramming (MIP) model is developed to minimize totalproduction costs

4 Description of the Model

Our model maximizes expected profits in scenarios ofstochastic customer demand The model generates produc-tion plans containing the production quantities of eachproduct of a given product program the production start of

each product and the production plants where the productsshould be produced

Notation A set 119868 of 119899 products has to be manufacturedunder restriction of several time and resource constraintsThe forecast demand of each product to be manufactured isuncertain and a probabilistic distribution is used to character-ize this uncertainty The manufacturing of different productshas to take place in just one of119898 different facilities located indifferent countries

A facility 119895 isin 119869 has a fixed production cost CP119894119895to produce

the product 119894 a fixed transport cost CT119895 and a production

capacity CAP119894119895

of products which can be manufacturedduring one period

A product 119894 isin 119868 has a delivery due date DD119894 and if it is

supplied earlier than the recommendedDD119894that would cause

extra holding costs CH119894 and if it is supplied later than the due

date DD119894that would cause penalty costs PEC

119894

The objective is to find a feasible plan which determinesthe quantity of each product to be produced and indicatesthe start of production of each product 119894 manufactured fromfacility 119895 to fulfil the stochastic demand and maximize thetotal production profit

Parameters In the following the parameters for themodel aredefined

119879 is set of planning periods where 119896 = |119879|

119905 is index of the planning period 119905 isin 1 119896119869 is set of facilities where 119898 = |119869|119895 is index of facility 119895 isin 1 119898119868 is set of products where 119899 = |119868|119894 is index of product 119894 isin 1 119899Ω is set of all possible scenarios of stochastic variablesΩ = 1 2 119878119904 is index of scenario res Realisation of stochasticvariables 119904 isin [1 119878] where 119878 = |Ω|

119863119904119894is the demand of product 119894 under scenario 119904

119875119904is probability of scenario 119904

SP119894is the unit selling price for product 119894

CP119894119895is the unit production cost for product 119894 manu-

factured in facility 119895CT119895is transport cost from the facility 119895 in terms of

number of periodsCH119894is the unit inventory holding cost for product 119894 at

the end of productionSL119894is salvage value per unit for product 119894

SR119894is shortage penalty per unit for product 119894

TT119895is transport time from the facility 119895 in terms of

number of periodsDD119894is delivery due date of product 119894 in terms of

number of periods

Mathematical Problems in Engineering 5

PF119894is penalty cost incurred to each late delivery for

product 119894 when it is supplied later than the deliverydate DD

119894

CAP119894119895is production capacity (units) for product 119894

during a period in facility 119895120572 is confidence level of risk parameter CVaR where120572 isin [0 1[120573 is risk parameter120582 is weight presented on solution variance120583 is weight placed on model infeasibility whichcontrols the trade-off between solution and modelrobustness

Variables Consider the following

119883119894is quantity of product 119894 that shall be produced

(decision variable)119884119894119895is binary variable which indicates whether prod-

uct 119894 is produced in facility 119895 (decision variable)STP119894is start period of the production of product 119894

(decision integer variable)TP119894is time required in periods to produce 119883

119894of

product 119894 TP119894= sum119898

119895=1119884119894119895(119883119894CAP

119894119895)

ExpEnd119875119894is expected delivery date of product 119894 in

period unit ExpEnd119875119894= STP

119894+ TP119894+ sum119898

119895=1119884119894119895sdot TT119895

MIN119904119894is quantity of product 119894 effectively sold under

scenario 119904 MIN119904119894= min(119883

119894 119863119904119894)

WAS119904119894is wastage quantity of product 119894 under scenario

119904 WAS119904119894= max(0 119883

119894minus 119863119904119894)

SHO119904119894is shortage quantity of product 119894 under scenario

119904 SHO119904119894= max(0 119863119904

119894minus 119883119894)

119868119894is binary variable which indicates whether the

product 119894 has to be held or not after its productionand transportation

119868119894=

1 if ExpEnd119875119894lt DD

119894

0 else(1)

Objective Function The objective function has the followingcomponents

(1) sales revenue(2) production cost(3) transport cost(4) holding cost(5) penalty cost due to a late delivery(6) risk cost incurred for possible wastage cost(7) risk cost incurred for possible shortage cost

The above components have been considered due to the factthat they are the most important costs which can be mainlyaffected by the decision variable of the model

The objective is to maximize the total profit revenueconsisting of sales revenue gained by selling the productsproduction (PC) transport (TC) penalty holding (HC) andother risk costs which can be incurred for possible wastage orshortage quantities

SR119904 (sales revenue) =119899

sum119894=1

MIN119904119894sdot SP119894 (2)

PC (production cost) =119899

sum119894=1

119898

sum119895=1

119884119894119895X119894CP119894119895 (3)

TC (transport cost) =119899

sum119894=1

119898

sum119895=1

119884119894119895119883119894CT119895 (4)

HC (holding cost) =119899

sum119894=1

119868119894sdot (DD

119894minus ExpEnd119875

119894) sdot 119883119894CH119894

(5)

PEC (penalty cost)

=119899

sum119894=1

(1 minus 119868119894) (ExpEnd119875

119894minus DD

119894)119883119894PF119894

(6)

WC119904 (wastage cost) =119899

sum119894=1

WAS119904119894sdot SL119894 (7)

SC119904 (shortage cost) =119899

sum119894=1

SHO119904119894sdot SR119894 (8)

Equation (2) represents the total sales revenue due to sellingthe set of products 119868 Equation (3) is the total productioncost of manufacturing the set of products 119868 Equation (4) isthe total transport cost of the set products 119868 Equation (5)corresponds to the total holding cost of units of productswhich has to be stored in the warehouses for a determinedholding period Equation (6) represents the total penalty costof supplying the set of products 119868 later than the appropriatedelivery date Equation (7) represents the wastage cost toliquidate the overstocks set of products 119868 Equation (8)represents an artificial penalty for the demand dissatisfactionand subsequently to loss of opportunity revenue

The following formulation of robust problem is definedaccording to Leung et al [32]

We denote the profit function for each scenario 119904 with119865119904= [SR119904 + WC119904 minus SC119904 minus PC minus TC minus HC minus PEC] (9)

The objective function which represents the maximizingexpected profit of the production planning problem with thedemand uncertainty is formulated as follows

max119864 [profit]

= maxΩ

sum119904=1

119901119904119865119904+ 120582Ω

sum119904=1

119901119904[119865119904minusΩ

sum1199041015840=1

11990111990410158401198651199041015840]

+120583Ω

sum119904=1

119901119904120579119904

(10)

6 Mathematical Problems in Engineering

The first term of (10) is the mean value of the total profitThesecond term of (10) denotes the measure solution robustnessof the model The third term in (10) represents the modelrobustness and is used to penalize model infeasibility

Constraints First we define the loss function as follows

forall119904 isin Ω 119891 (119883 119904) =119899

sum119894=1

119891 (119883119894 119904) = WC119904 + SC119904 minus 120573 sdot SR119904

(11)

The objective function is subject to the following constraints

forall119894 isin 119868 ExpEnd119875119894le 119896 (12)

forall119894 isin 119868119898

sum119895=1

119884119894119895

= 1 (13)

forall119894 isin 119868 STP119894gt 0 (14)

forall119894 1198941015840 isin 119868 if exist120583 isin 119869 so that 119884119894120583

= 1198841198941015840120583= 1

STP119894lt STP

1198941015840 then STP

1198941015840 ge STP

119894+ TP119894

(15)

forall119904 isin Ω 119875 [119904 | 119891 (119883 119904) le 0] le 120572 (16)

Constraint (12) ensures that the expected delivery date forall products should not exceed the length of the planninghorizon 119896 = |119879| Constraint (13) guarantees that theproduction of a product 119894 will take place only in one facility119895 Constraint (14) indicates that the start of production ofa product 119894 is positive and integer Constraint (15) ensuresthat the production of more than one product cannot becarried out in a parallel manner Constraint (16) restricts theprobability of the loss function to be negative for 120572 sdot 100which means that the undesirable realization of uncertaintiescan restrict the loss function to be positive in only (1 minus 120572) sdot100 As an example for 120572 = 095 the constraint (16) has tobe satisfied for 95 for all possible realizations of demand119863119904

119894

for all 119894 isin 119868 and for all 119904 isin Ω

5 Experimental Results for a Case Studywith Three Scenarios

In order to cover different potential realizations of customerdemand three scenarios have been selected including a ldquopes-simisticrdquo scenario characterized by assumed low customerdemand a ldquonormalrdquo scenario characterized by average cus-tomer demand and an ldquooptimisticrdquo scenario characterized byrelatively high customer demand These three scenarios takeinto account the insecurity concerning customer demand forthe products Different values have been chosen based on datataken from a real case study

In this section we analyze the behavior of the modelby means of a case study based scenario The input dataparameters of the model are summarized in Tables 1 and 2

Three different scenarios are considered as a basis forapplication of the model and its numerical results Thefirst scenario ldquopessimistic scenariordquo illustrates low demand

Table 1 Input parameter used to solve the model

Parameters Inputs119896 = |119879| 10119898 = |119869| 2119899 = |119868| 4Transport cost from facility 1(China)

300 (pro 1000product units)

Transport cost from facility 2(Turkey)

100 (pro 1000product units)

Transport time from facility 1(China) 3

Transport time from facility 2(Turkey) 1

120572 095120573 01

Table 2 Unit production parameters

Product 1 Product 2 Product 3 Product 4Production

Minimum value 10000 7500 5000 8000Maximum value 40000 30750 20750 31600

Production costFacility 1 1 1 2 2Facility 2 3 2 3 5

CapacityFacility 1 20000 20000 20000 20000Facility 2 10000 10000 150000 10000

Penalty factor PF119894

01 02 08 05Delivery due date DD

1198946 6 6 6

Shortage cost unit 5 4 5 9Salvage cost unit 05 05 1 1Unit selling price 8 6 8 14

forecasts for the next season The second scenario ldquonormalscenariordquo represents the case when the customer demandis normal The third scenario ldquooptimistic scenariordquo assumeshigh customer demand To characterize the uncertaintiesof demand forecast of different products a distributionprobability is applied Figures 1 and 2 show an example ofthe demand distribution respectively of the products underpessimistic scenario and demand distribution of the product1 for the three scenarios

The risk to be considered from solving the model canoccur only because of wastage or shortage possibility There-fore according to our definition of the loss function119891(119883 119904) =sum119899

119894=1119891(119883119894 119904) = WC119904 + SC119904 minus 01 sdot SR119904 the constraint (16)

for this case study ensures that the probability that the riskcosts incurring by respectively wastage and shortage costs donot exceed the 10 (120573 = 01) of the expected profit revenuein 95 (120572 = 095) of all possible realization of stochasticparameters

The model has been efficiently solved by means offrontline risk solver which supports the robust optimizationthrough CVaR

Mathematical Problems in Engineering 7

025

02

015

01

005

0

11

12

135 15

17

18

195 21

22

235 25

265

275 29

31

32

Product 1

Product 1

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

6 8 9

105

115

125

135

145

155

165

175

185 20

21

22

23

24

Product 2

Product 2025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

55 6 7

75

85 9

95

105 11

12

125 13

14

145

155 16

Product 3

Product 3

(c)

025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

9

10

11

12

13

14

155 17

175 19

20

21

22

235

245 26

Product 4

Product 4

(d)

Figure 1 Demand distribution of the products in a pessimistic scenario

11

12

135 15

165 18

19

21

22

235 25

265

275 29

31

32

Pessimistic scenario

Pessimistic scenario025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

(Values in thousands)

Rela

tive p

roba

bilit

y

12

13

15

16

175 19

21

225 24

25

27

28

30

315 33

34

36

Normal scenario

Normal scenario

012

01

008

006

004

002

0

014

(b)

(Values in thousands)

Rela

tive p

roba

bilit

y

02

018

016

014

012

01

008

006

004

002

0

19

205

215

225 24

25

265 28

29

305 32

33

345

355 37

38

395

Optimistic scenario

Optimistic scenario

(c)

Figure 2 Demand distribution of product 1 in different scenarios

51 Pessimistic Scenario Table 3 lists the production plan ina pessimistic scenario The expected average profit of theproduction for the next season is about 219000 Moreoverthe solution can guarantee that the loss function will benegative in 96 cases of all possible realizations of stochasticparameters (Figure 3)

52 Normal Scenario In the normal scenario the demandof product 1 and product 4 respectively has nearly doubledwhereas the demand of product 2 and product 3 has beenslightly increased This is due to the low penalty factorof product 1 and product 4 which can be incurred foreach delayed delivery and for the capacity restriction of

8 Mathematical Problems in Engineering

012

01

008

006

004

002

0

120

130

150

160

180

190

200

220

230

240

260

270

280

300

310

320

(Values in thousands)

Rela

tive p

roba

bilit

y

Objective fct pessimistic

Objective fct pessimistic

(a)

(Values in thousands)

Rela

tive p

roba

bilit

y

016

014

012

01

008

006

004

002

0

minus25

minus23

minus21

minus19

minus17

minus15

minus13

minus11

minus8

minus6

minus5 3 0 2 3 5

Loss fct pessimistic

Loss fct pessimistic

(b)

Figure 3 Profit and loss function distribution in a pessimistic scenario

018

016

014

012

01

008

006

004

002

0

100

150

170

220

240

270

300

330

370

390

430

450

470

530

550

580

Objective fct normal

Objective fct normal

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

016

014

012

01

008

006

004

002

0

minus40

minus35

minus30

minus27

minus23

minus17

minus13

minus10

minus5 0 3 7

13

17

22

27

Loss fct normal

Loss fct normal

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

Figure 4 Profit and loss function distribution in a normal scenario

Table 3 Production plan with the demand quantities in a pes-simistic scenario

Product 1 Product 2 Product 3 Product 4Demand nextseason 17969 16549 10169 14945

Start of production 3 2 1 1Expected deliverydate 6 6 5 4

Produced InFacility 1 0 1 1 1Facility 2 1 0 0 0

the facilities (Table 4) In this case the expected averageprofit is about 450000 and the loss function is negative in975 of all possible realizations of the stochastic parameters(Figure 4)

53 Optimistic Scenario Since the optimistic scenario pre-dicts a boom economic scenario the production reaches its

Table 4 Production plan with the demand quantities in a normalscenario

Product 1 Product 2 Product 3 Product 4Demand next season 32355 19682 14415 26284Start of production 1 4 3 1Expected delivery date 6 8 7 6Produced in

Facility 1 0 1 1 1Facility 2 1 0 0 0

maximum (Table 5) In this case the average profit is about665000 and the loss function is satisfied with 991 of allpossible realizations of the stochastic parameters (Figure 5)

6 Conclusion and Outlook

In this paper we have presented a risk-constrained profit-expected maximization model for a textile industry scenario

Mathematical Problems in Engineering 9

045

04

035

03

025

02

015

01

005

0

225

250

275

325

350

400

425

475

500

525

575

600

625

675

625

750

Objective fct optimistic

Objective fct optimistic

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

04

035

03

025

02

015

01

005

0

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

minus2 2 7

11

16

20

Loss fct optimistic

Loss fct optimistic

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

Figure 5 Profit and loss function distribution in an optimistic scenario

Table 5 Production plan with the demand quantities in an opti-mistic scenario

Product 1 Product 2 Product 3 Product 4Demand next season 37933 29312 19410 31600Start of production 1 4 3 1Expected delivery date 6 9 7 6Produced in

Facility 1 0 1 1 1Facility 2 1 0 0 0

The forecast demand uncertainty has been explicitly con-sidered in the model by means of the robust optimizationand conditional value at risk theory by introducing andrestricting a loss function The robust optimization modeloriginally proposed by [32] was adopted as the benchmarkformulation of uncertainty considered in this paper A casestudy has been used to demonstrate the viability of theproposed mathematical model Results indicate that theproduction planning model using CVaR as the risk measureperforms robustly and provides flexibility in decision analysisbetween different scenarios The next step will be the furtherdevelopment of the profit maximization model to considerother types of uncertainties for example concerning theavailability of the production plants transportation meansand demand due dates

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Datamonitor Global Apparel Retail 2010 httpwwwdata-monitorcom

[2] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of Banking and Finance vol26 no 7 pp 1443ndash1471 2002

[3] S Zhu and M Fukushima ldquoWorst-case conditional value-at-risk with application to robust portfolio managementrdquo Opera-tions Research vol 57 no 5 pp 1155ndash1168 2009

[4] A Bruckner and S Muller Supply Chain Management in derBekleidungsindustrie Forschungsstelle der Bekleidungsindus-trie Cologne Germany 2003

[5] J Fissahn Marktorientierte Beschaffung in der Bekleidungsin-dustrie Munster University Munster Germany edition 2001

[6] A Schneider Internationalisierungsstrategien in der deutschenTextil-und Bekleidungsindustrie-eine empirische Untersuchung[MS dissertation] RWTH Aachen University Aachen Ger-many 2003

[7] G Gereffi and O Memedovic The Global Apparel ValueChainmdashWhat Prospects for Upgrading by Developing CountriesUNIDO Vienna Austria 2003

[8] MGromling and JMatthesGlobalisierung und Strukturwandelder Deutschen Textil-und Bekleidungsindustrie Dt Inst-VerlagColougne Germany 2003

[9] I Langenhorst Shop-logistik in der bekleidungswirtschaft eineanalyse der anforderungen herstellerinitiierter shop-systeme andie logistikprozesse der bekleidungsindustrie [PhD thesis]Munster University 1999

[10] H-C Pfohl M Gomm and X Shen China Textil-undBekleidungs-Supply Chain Zwischen Deutschland und ChinaFree Beratung Korschenbroich Germany in H Wolf-Kluthausen Jahrbuch der Logistik 2007

[11] M Teucke and B Scholz-Reiter ldquoImproving order allocationin fashion supply chains using radio frequency identification(RFID) technologiesrdquo in Fashion Supply Chain ManagementUsing Radio Frequency Identification (RFID) Technologies WK Wong and Z X Guo Eds vol 152 of Woodhead PublishingSeries in Textiles Woodhead Publishing Cambridge UK 2014

[12] S Thomassey ldquoSales forecasts in clothing industry the keysuccess factor of the supply chain managementrdquo InternationalJournal of Production Economics vol 128 no 2 pp 470ndash4832010

[13] D Ahlert and G Dieckheuer Marktorientierte Beschaffung inder Bekleidungsindustrie FATM Munster Germany 2001

[14] J T Mentzer C C Bienstock and K B Kahn ldquoBenchmarkingsales forecastingmanagementrdquo Business Horizons vol 42 no 3pp 48ndash56 1999

10 Mathematical Problems in Engineering

[15] J Quick M Rinis C Schmidt and B Walber SupplyTexmdashErfolgreiches Supply-Management in KMU Der Textil- UndBekleidungsindustrie Forschungsinstitut fur Rationalisierung(FIR) eV an der Universitat Aachen Aachen Germany 2010

[16] Z-L Sun T-M Choi K-F Au and Y Yu ldquoSales forecastingusing extreme learning machine with applications in fashionretailingrdquo Decision Support Systems vol 46 no 1 pp 411ndash4192008

[17] S Thomassey M Happiette and J-M Castelain ldquoA globalforecasting support system adapted to textile distributionrdquoInternational Journal of Production Economics vol 96 no 1 pp81ndash95 2005

[18] J Mostard R Teunter and R de Koster ldquoForecasting demandfor single-period products a case study in the apparel industryrdquoEuropean Journal of Operational Research vol 211 no 1 pp 139ndash147 2011

[19] K Hoyndorff S Hulsmann D Spee and M ten HompelFashion Logistics Grundlagen Uber Prozesse und IT Entlang derSupply Chain Munich Huss-Verlag GmbH 2010

[20] J Richardson ldquoVertical integration and rapid response infashion apparelrdquoOrganization Science vol 7 no 4 pp 400ndash4121996

[21] Z-L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[22] J Mula D Peidro M Dıaz-Madronero and E Vicens ldquoMathe-matical programming models for supply chain production andtransport planningrdquo European Journal of Operational Researchvol 204 no 3 pp 377ndash390 2010

[23] S A Torabi and E Hassini ldquoAn interactive possibilistic pro-gramming approach for multiple objective supply chain masterplanningrdquo Fuzzy Sets and Systems vol 159 no 2 pp 193ndash2142008

[24] B Scholz-Reiter J Hartmann T Makuschewitz and E MFrazzon ldquoA generic approach for the graph-based integratedproduction and intermodal transport scheduling with capacityrestrictionsrdquo in Procedia CIRP vol 7 pp 109ndash114 2013

[25] A D Toni andAMeneghetti ldquoProduction planning process fora network of firms in the textile-apparel industryrdquo InternationalJournal of Production Economics vol 65 no 1 pp 17ndash32 2000

[26] E M Frazzon A G N Novaes B Scholz-Reiter J Hartmannand T Makuschewitz ldquoAn approach for the rescheduling ofintegrated production and transport systemsrdquo in Proceedingsof the 4th World Conference Production amp Operations Manage-ment Serving the World Production amp Operations ManagementSociety p 10 Amsterdam The Netherlands 2012

[27] H Ishikura ldquoStudy on the production planning of apparel prod-ucts determining optimal production times and quantitiesrdquoComputers and Industrial Engineering vol 27 no 1ndash4 pp 19ndash22 1994

[28] Y W Tan J B Guan and H R Karimi ldquoThe Impact ofthe subsidy policy on total factor productivity an empiricalanalysis of Chinarsquos cotton productionrdquo Mathematical Problemsin Engineering vol 2013 Article ID 248537 8 pages 2013

[29] H R Karimi N A Duffie and S Dashkovskiy ldquoLocalcapacity Hinfin control for production networks of autonomouswork systems with time-varying delaysrdquo IEEE Transactions onAutomation Science and Engineering vol 7 no 4 pp 849ndash8572010

[30] S Dashkovskiy H R Karimi and M Kosmykov ldquoStabilityanalysis of logistics networks with time-delaysrdquo Journal ofProduction Planning amp Control vol 24 no 7 pp 567ndash574 2013

[31] A Mehrsai H R Karimi and B Scholz-Reiter ldquoApplication oflearning pallets for real-time scheduling by the use of radialbasis function networkrdquo Neurocomputing vol 101 pp 82ndash932013

[32] S C H Leung S O S Tsang W L Ng and Y Wu ldquoArobust optimization model for multi-site production planningproblem in an uncertain environmentrdquo European Journal ofOperational Research vol 181 no 1 pp 224ndash238 2007

[33] A-G Quaranta and A Zaffaroni ldquoRobust optimization ofconditional value at risk and portfolio selectionrdquo Journal ofBanking and Finance vol 32 no 10 pp 2046ndash2056 2008

[34] R A Jabr ldquoRobust self-scheduling under price uncertaintyusing conditional Value-at-Riskrdquo IEEE Transactions on PowerSystems vol 20 no 4 pp 1852ndash1858 2005

[35] D Peidro J Mula R Poler and J-L Verdegay ldquoFuzzy opti-mization for supply chain planning under supply demand andprocess uncertaintiesrdquo Fuzzy Sets and Systems vol 160 no 18pp 2640ndash2657 2009

[36] F Pan and R Nagi ldquoRobust supply chain design underuncertain demand in agile manufacturingrdquo Computers andOperations Research vol 37 no 4 pp 668ndash683 2010

[37] P M Verderame and C A Floudas ldquoOperational planning oflarge-scale industrial batch plants under demand due date andamount uncertainty II Conditional value-at-risk frameworkrdquoIndustrial and Engineering Chemistry Research vol 49 no 1 pp260ndash275 2010

[38] P M Verderame and C A Floudas ldquoMultisite planning underdemand and transportation time uncertainty Robust optimiza-tion and conditional value-at-risk frameworksrdquo Industrial andEngineering Chemistry Research vol 50 no 9 pp 4959ndash49822011

[39] S M J Mirzapour Al-E-Hashem H Malekly and M BAryanezhad ldquoA multi-objective robust optimization modelfor multi-product multi-site aggregate production planningin a supply chain under uncertaintyrdquo International Journal ofProduction Economics vol 134 no 1 pp 28ndash42 2011

[40] D Rahmani R Ramezanian P Fattahi and M HeydarildquoA robust optimization model for multi-product two-stagecapacitated production planning under uncertaintyrdquo AppliedMathematical Modelling vol 37 no 20-21 pp 8957ndash8971 2013

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

that should be produced the start of the production of eachproduct and the facility in which the products have to beproduced The objective is to maximize the total profit ofproductions by means of CVaR

Consideration of risk in the optimization problem has acrucial role in optimization under uncertainty particularlywhen the optimization problem has to deal with the lossesthat might be incurred under conditions of unfavorabledemand To consider the risk of erroneous demand forecastsa loss function 119891(119883 119904) will be defined where 119883 representsa decision vector and 119904 is a vector representing uncertaintyrelated to the future values of a number of stochastic parame-ters of the problemThese stochastic parameters presented bythe vector 119904 are governed by probability distribution119875

119904 Once

the loss function 119891 is defined we denote the distributionfunction of 119891 by 120593(119883 120573) = 119875119904 | 119891(119883 119904) le 120573 whichcorresponds to the probability that the value of the lossfunction for each realisation of a scenario of the vector 119904 andfor a fixed 119883 does not exceed the value 120573 In this contextfor a specified confidence value 120572 isin [0 1[ that could beequal to 090 or 095 in some applications 120572-VaR denotesthe probability that the expected value of the loss functionexceeds 120573 only in (1 minus 120572) sdot 100 for all possible realisationsof stochastic parameters which can be seen as the worst-case scenario Based on the definition of value at risk (VaR)conditional value at risk (CVaR) is defined as the mean of thetail distribution exceeding VaR [3]

The model is applied to a problem based on a case studyin the apparel industry Different factories located in differentcountries manufacture a number of product types and sellthese products to customers in Europe The orders consist ofthe type of products delivery date and quantity Usually thequantity demand of a product 119894 is of stochastic nature andhaving a probability distribution 119901(119894) After collecting ordersthe production plan has to be generated for the next seasonby considering the manufacturing capacity production costand transport cost while satisfying a CVaR constraint in orderto maximize the expected total profit

In order to generate different possible demand realisa-tions for the mathematical model different scenarios of Ωare generated by means of Monte Carlo simulation using thedemand probability distribution of each product within theset of products

The paper is structured as follows the next chapterwill provide some additional information on productionplanning and demand uncertainty in the apparel industryThe subsequent chapter studies some of the literature onrobust production planning under demand uncertainty Afterthat our model will be described and subsequently appliedto a case study that includes three scenarios (a pessimisticscenario a normal scenario and an optimistic scenario) dif-fering in customer demand These three scenarios have beenproposed in order to cover different potential realizationsof customer demand which can occur but are not knownat the time of planning Thereby the customer demand canbe estimated as low (pessimistic estimate) normal (normalestimate) or high (optimistic estimate) based on historicdata and the recommendations of experts The paper endswith a conclusion and an outlook on further research

2 Production Planning and DemandUncertainty in Apparel Industry

Manufacturing and finishing of ready-to-wear garments area complex process involving often a large number of collabo-rating parties within a production network [4]These includegarment manufacturers and their raw material suppliersprocurement agencies logistic service providers and retailers[5]

Due to limited potential for automation and price impor-tance in competition production of fashion apparel productshas been largely outsourced to low wage countries oftensituated either in East Asia the Near East in particularin China Turkey Bangladesh India and Vietnam Due tothe absence of technical constraints and availability of sub-contractors production can be shifted with few constraintsbetween different regions However the large majority ofthe products are still sold in Europe and North Americaeven though rising domestic demand in Asian countriessuch as China has started to increasingly compete for localproduction resources [6ndash8] In this context many Europeangarments suppliers have reduced the depth of their ownproduction or dropped it and have adapted their roletowards planning and coordinating activities within apparelsupply chains integrating garment manufacturers and rawmaterials suppliers logistic service providers and retailers[9]The wide geographic distribution of supply chains resultsin considerable lead times of apparel production due to thetime needed for procurement of raw materials and the timeneeded for transport of products from the production plantsto the customers [10 11]

In addition to low product costs quality and reliabilityof logistic services are crucial for economic success in theapparel branch Fashion apparel products are characterizedby rapid product obsolescenceThe sales seasons for a periodof a few months or even weeks are short in comparison totheir long production lead and delivery times This increasesthe importance of adherence to delivery dates On average95 of stock keeping units change for every new sales seasonat least twice or four times a year [5 12] The productioncycle is characterized by fixed seasonal cycles with fixeddates for product offers orders and deliveries which arerepeated production volumes based on aggregation of theretail preorders However as in practice production anddistribution lead times usually exceed the length of thedelivery times expected by customers production planningand procurement of raw materials have to start before theend of the preorder period using forecasts of total demandbased on the orders arrived so far Products may also beproduced in larger quantities than if only based on preorderswith the excess offered directly from the warehouses to calledpostorders as long as stocks last [13]

Once their sales season has ended articles which havenot been sold so far can be sold only for reduced prices andthuswith reduced profit in the next seasonThus at the end ofa sales season the stock levels of the products should ideallybe close to zero On the other hand stocks should not run outbefore the end of the sales period Reproduction of successfulproducts during a sales season is not possible due to the long

Mathematical Problems in Engineering 3

delivery times [12] For these reasons demand forecasting hasto be considered an important input of production planningand supply chain planning models Generally forecasting ofpotential sales permeates all aspects of business operations[14] Customer demand for fashion products is volatile andmay vary broadly for different variants of the same productDemand fluctuations are difficult to predict at the time ofproduction planning Optimistic forecasts may lead to over-stocks and higher production costs and cause markdownsPessimistic forecasts may lead to loss of opportunity revenuesfor potential sales that could not be realized and to customerdissatisfaction with later brand changes

On the other hand traditional statistical forecastingmethods are difficult to apply [15] Specifically developeddemand forecasting models often use soft computing meth-ods such as artificial neural networks [16] fuzzy logic andevolutionary procedures [12 17] In addition the integrationof preorder information as well as expert judgments hasproduced accurate forecasts [18] Most of these models havenot been integrated into standard business software and aredifficult to use with sometimesmixed quality of the forecastsOrder fulfilment and product delivery failures such as stock-outs or increased storage costs and product depreciationwrite-offs continue to affect profits in the apparel branch ina very unfavourable way It is estimated that due to the longlead times and demand forecasting difficulties mentioned inthe second section of this chapter up to 30ndash40 of high-fashion articles cannot be sold during the short sales seasonbefore the life expectancy of the article expires [19] Theneed to cope with this situation and the impetus to realizeQuick Response concepts have resulted in vertical integrationof apparel suppliers and retailers in a number of cases[20] Vertically integrated suppliers generally achieve betterreaction to market changes due to shortened reaction timesand more closely integrated information flows Howeverthese advantages cannot easily be copied by most traditionalsuppliers which are often small or medium sized enterprizesStill it is estimated that they have to write off up to 15ndash20of their high-fashion articles [19]

3 Literature Review on Problem RelatedProduction Planning

Production and distribution operations are two key functionsin the supply chain In general the supply chainmanagementprocedures emphasize production scheduling more than thedistribution scheduling because distribution ismore flexibleWhen integrated supply chain management of productionand distribution is realized the resources are used moreefficiently [21] In a comprehensive review of integratedscheduling approachesMula et al showed that over 44 papershave been published between 1989 and 2009 [22]

In general master tactical and operational planninglevels can be separated The purpose of most models is tominimize the total cost of the supply chain Some work at themaster planning level integrating procurement in addition toproduction and distribution scheduling is described in [23]Most of the models focus on tactical planning considering

different demand levels costs and capacities and use cen-tralized scheduling approaches Scholz-Reiter et al showed ascalable graph-based scheduling approach which is focusedon the operational level and real-time scheduling [24] Alarge variety of OR techniques have been applied comprisingmathematical optimisation heuristics and metaheuristicsbut it could be observed that most of the papers focus ondeterministic solutions [22]

The special conditions of production planning for fashionproducts have received some studies as well The work in[25] described a networked production planning processin a fashion oriented apparel supply chain The productionplanning process was found to be an important area ofimprovement for a network in a time-based logic shorteningthe production planning period in fact significantly affectsthe weighted average delivery anticipation It leads howeverto increase setup and transportation costs due to the greaternumber of jobs generated during a campaign Scheduling ofintegrated production and distribution systems in dynamicenvironments holds a potential to improve efficiency butalso poses a challenging planning task due to its compu-tational complexity Long lead times external and internalperturbations in productive processes unstable businessenvironments and contextual differences (eg institutionaleconomic and cultural) emphasize the relevance to theargued integration [26] Three types of aggregate produc-tion planning methods for the apparel industry have beenproposed in [27] These allow changing a production modelseasonally according to actual demands or maintaining thesame production model for several seasons with productionfor inventoryThe mentioned work on integrated productionand distribution planning does not deal with stochastic inputdata such as demands The work in [28] studied empiricallythe impact of the subsidy policy on total factor productivityfor the example of Chines cotton production

Modern large and widely distributed production net-works are subject to many forms of disturbances KarimiDuffy and Dashkovsky et al performed research on howto better deal with such dynamic influences One possiblesolution is to introduce autonomous control that is to allowsome parts of a large network to make their own decisionsbased on local situation and available information Howeverstability of the network and robustness with respect toexternal and internal disturbances and time delays in signalsmust be assured to guarantee a reasonable performance andvitality of the whole system For this purpose their workproposed an approach for controller design for large scaleautonomous work systems capable of coping with time delaysand explains its implementation and advantages on a concreteexample [29ndash31]

Robust optimization models try to formulate productionplanning problems in a way that cost or wastage effectsof uncertainty or risks are minimized or expected profitis maximized A robust optimization model for a multisiteproduction medium-term planning problem is developed in[32] based on the problems facing a multinational lingeriecompany with production sites in East Asia It generates acost minimal production plan for an uncertain environmentwith associated probabilities of different economic growth

4 Mathematical Problems in Engineering

scenarios The cost minimal production plan is less sensitiveto changes in the noisy and uncertain data The work in [33]dealt with a portfolio selection model in which the method-ologies of robust optimization are used for the minimizationof the conditional value at risk of a portfolio of shares Thework in [34] proposed a method for robust self-schedulingbased on CVaRThe proposedmethod is based on a security-constrained optimal power flow (SCOPF) program thatexplicitly treats the trade-off between risk and reward Thework in [35] proposed a fuzzy mathematical programmingmodel for supply chain planning which considers supplydemand and process uncertainties The model has beenformulated as a fuzzy mixed-integer linear programmingmodel where data are ill-known and modelled by triangularfuzzy numbers The fuzzy model provides the decisionmaker with alternative decision plans for different degreesof satisfaction This proposal is tested by using data from areal automobile supply chain A supply chain design problemfor a new market opportunity with uncertain demand inan agile manufacturing setting is considered in [36] Thismodel integrates the design of supply chain and productionplanning for the supply chainrsquos members and develops arobust optimization formulation A novel framework basedon conditional value at risk theory has been applied in [37] tothe problem of operational planning for large-scale industrialbatch plants under demand due date and amount uncertaintyThe objective of the proposed model is to provide a dailyproduction profile that not only is a tight upper bound onthe production capacity of the plant but also is immune tothe various forms of demand uncertainty In further workThe work in [38] applied a robust optimization frameworkas well as CVAR theory to the multisite operational plan-ning problem under multiple forms of system uncertaintyThey considered different forms of system uncertainty suchas demand due date demand amount and transportationtime uncertainty in the model Their objective is to ensurethe maximization of customer satisfaction along with theminimization of resource misallocation

A robust multiobjective mixed integer nonlinear pro-gramming model is proposed by [39] to deal with a multisitemultiperiod multiproduct aggregate production planning(APP) problem under uncertainty considering two conflict-ing objectives simultaneously as well as the uncertain natureof the supply chain Their proposed model is solved as asingle-objective mixed integer programming model applyingthe LP-metrics method A two-stage real world capacitatedproduction system with lead time and setup decisions uncer-tain production costs and customer demand is studied in[40] using a robust optimization approach A mixed-integerprogramming (MIP) model is developed to minimize totalproduction costs

4 Description of the Model

Our model maximizes expected profits in scenarios ofstochastic customer demand The model generates produc-tion plans containing the production quantities of eachproduct of a given product program the production start of

each product and the production plants where the productsshould be produced

Notation A set 119868 of 119899 products has to be manufacturedunder restriction of several time and resource constraintsThe forecast demand of each product to be manufactured isuncertain and a probabilistic distribution is used to character-ize this uncertainty The manufacturing of different productshas to take place in just one of119898 different facilities located indifferent countries

A facility 119895 isin 119869 has a fixed production cost CP119894119895to produce

the product 119894 a fixed transport cost CT119895 and a production

capacity CAP119894119895

of products which can be manufacturedduring one period

A product 119894 isin 119868 has a delivery due date DD119894 and if it is

supplied earlier than the recommendedDD119894that would cause

extra holding costs CH119894 and if it is supplied later than the due

date DD119894that would cause penalty costs PEC

119894

The objective is to find a feasible plan which determinesthe quantity of each product to be produced and indicatesthe start of production of each product 119894 manufactured fromfacility 119895 to fulfil the stochastic demand and maximize thetotal production profit

Parameters In the following the parameters for themodel aredefined

119879 is set of planning periods where 119896 = |119879|

119905 is index of the planning period 119905 isin 1 119896119869 is set of facilities where 119898 = |119869|119895 is index of facility 119895 isin 1 119898119868 is set of products where 119899 = |119868|119894 is index of product 119894 isin 1 119899Ω is set of all possible scenarios of stochastic variablesΩ = 1 2 119878119904 is index of scenario res Realisation of stochasticvariables 119904 isin [1 119878] where 119878 = |Ω|

119863119904119894is the demand of product 119894 under scenario 119904

119875119904is probability of scenario 119904

SP119894is the unit selling price for product 119894

CP119894119895is the unit production cost for product 119894 manu-

factured in facility 119895CT119895is transport cost from the facility 119895 in terms of

number of periodsCH119894is the unit inventory holding cost for product 119894 at

the end of productionSL119894is salvage value per unit for product 119894

SR119894is shortage penalty per unit for product 119894

TT119895is transport time from the facility 119895 in terms of

number of periodsDD119894is delivery due date of product 119894 in terms of

number of periods

Mathematical Problems in Engineering 5

PF119894is penalty cost incurred to each late delivery for

product 119894 when it is supplied later than the deliverydate DD

119894

CAP119894119895is production capacity (units) for product 119894

during a period in facility 119895120572 is confidence level of risk parameter CVaR where120572 isin [0 1[120573 is risk parameter120582 is weight presented on solution variance120583 is weight placed on model infeasibility whichcontrols the trade-off between solution and modelrobustness

Variables Consider the following

119883119894is quantity of product 119894 that shall be produced

(decision variable)119884119894119895is binary variable which indicates whether prod-

uct 119894 is produced in facility 119895 (decision variable)STP119894is start period of the production of product 119894

(decision integer variable)TP119894is time required in periods to produce 119883

119894of

product 119894 TP119894= sum119898

119895=1119884119894119895(119883119894CAP

119894119895)

ExpEnd119875119894is expected delivery date of product 119894 in

period unit ExpEnd119875119894= STP

119894+ TP119894+ sum119898

119895=1119884119894119895sdot TT119895

MIN119904119894is quantity of product 119894 effectively sold under

scenario 119904 MIN119904119894= min(119883

119894 119863119904119894)

WAS119904119894is wastage quantity of product 119894 under scenario

119904 WAS119904119894= max(0 119883

119894minus 119863119904119894)

SHO119904119894is shortage quantity of product 119894 under scenario

119904 SHO119904119894= max(0 119863119904

119894minus 119883119894)

119868119894is binary variable which indicates whether the

product 119894 has to be held or not after its productionand transportation

119868119894=

1 if ExpEnd119875119894lt DD

119894

0 else(1)

Objective Function The objective function has the followingcomponents

(1) sales revenue(2) production cost(3) transport cost(4) holding cost(5) penalty cost due to a late delivery(6) risk cost incurred for possible wastage cost(7) risk cost incurred for possible shortage cost

The above components have been considered due to the factthat they are the most important costs which can be mainlyaffected by the decision variable of the model

The objective is to maximize the total profit revenueconsisting of sales revenue gained by selling the productsproduction (PC) transport (TC) penalty holding (HC) andother risk costs which can be incurred for possible wastage orshortage quantities

SR119904 (sales revenue) =119899

sum119894=1

MIN119904119894sdot SP119894 (2)

PC (production cost) =119899

sum119894=1

119898

sum119895=1

119884119894119895X119894CP119894119895 (3)

TC (transport cost) =119899

sum119894=1

119898

sum119895=1

119884119894119895119883119894CT119895 (4)

HC (holding cost) =119899

sum119894=1

119868119894sdot (DD

119894minus ExpEnd119875

119894) sdot 119883119894CH119894

(5)

PEC (penalty cost)

=119899

sum119894=1

(1 minus 119868119894) (ExpEnd119875

119894minus DD

119894)119883119894PF119894

(6)

WC119904 (wastage cost) =119899

sum119894=1

WAS119904119894sdot SL119894 (7)

SC119904 (shortage cost) =119899

sum119894=1

SHO119904119894sdot SR119894 (8)

Equation (2) represents the total sales revenue due to sellingthe set of products 119868 Equation (3) is the total productioncost of manufacturing the set of products 119868 Equation (4) isthe total transport cost of the set products 119868 Equation (5)corresponds to the total holding cost of units of productswhich has to be stored in the warehouses for a determinedholding period Equation (6) represents the total penalty costof supplying the set of products 119868 later than the appropriatedelivery date Equation (7) represents the wastage cost toliquidate the overstocks set of products 119868 Equation (8)represents an artificial penalty for the demand dissatisfactionand subsequently to loss of opportunity revenue

The following formulation of robust problem is definedaccording to Leung et al [32]

We denote the profit function for each scenario 119904 with119865119904= [SR119904 + WC119904 minus SC119904 minus PC minus TC minus HC minus PEC] (9)

The objective function which represents the maximizingexpected profit of the production planning problem with thedemand uncertainty is formulated as follows

max119864 [profit]

= maxΩ

sum119904=1

119901119904119865119904+ 120582Ω

sum119904=1

119901119904[119865119904minusΩ

sum1199041015840=1

11990111990410158401198651199041015840]

+120583Ω

sum119904=1

119901119904120579119904

(10)

6 Mathematical Problems in Engineering

The first term of (10) is the mean value of the total profitThesecond term of (10) denotes the measure solution robustnessof the model The third term in (10) represents the modelrobustness and is used to penalize model infeasibility

Constraints First we define the loss function as follows

forall119904 isin Ω 119891 (119883 119904) =119899

sum119894=1

119891 (119883119894 119904) = WC119904 + SC119904 minus 120573 sdot SR119904

(11)

The objective function is subject to the following constraints

forall119894 isin 119868 ExpEnd119875119894le 119896 (12)

forall119894 isin 119868119898

sum119895=1

119884119894119895

= 1 (13)

forall119894 isin 119868 STP119894gt 0 (14)

forall119894 1198941015840 isin 119868 if exist120583 isin 119869 so that 119884119894120583

= 1198841198941015840120583= 1

STP119894lt STP

1198941015840 then STP

1198941015840 ge STP

119894+ TP119894

(15)

forall119904 isin Ω 119875 [119904 | 119891 (119883 119904) le 0] le 120572 (16)

Constraint (12) ensures that the expected delivery date forall products should not exceed the length of the planninghorizon 119896 = |119879| Constraint (13) guarantees that theproduction of a product 119894 will take place only in one facility119895 Constraint (14) indicates that the start of production ofa product 119894 is positive and integer Constraint (15) ensuresthat the production of more than one product cannot becarried out in a parallel manner Constraint (16) restricts theprobability of the loss function to be negative for 120572 sdot 100which means that the undesirable realization of uncertaintiescan restrict the loss function to be positive in only (1 minus 120572) sdot100 As an example for 120572 = 095 the constraint (16) has tobe satisfied for 95 for all possible realizations of demand119863119904

119894

for all 119894 isin 119868 and for all 119904 isin Ω

5 Experimental Results for a Case Studywith Three Scenarios

In order to cover different potential realizations of customerdemand three scenarios have been selected including a ldquopes-simisticrdquo scenario characterized by assumed low customerdemand a ldquonormalrdquo scenario characterized by average cus-tomer demand and an ldquooptimisticrdquo scenario characterized byrelatively high customer demand These three scenarios takeinto account the insecurity concerning customer demand forthe products Different values have been chosen based on datataken from a real case study

In this section we analyze the behavior of the modelby means of a case study based scenario The input dataparameters of the model are summarized in Tables 1 and 2

Three different scenarios are considered as a basis forapplication of the model and its numerical results Thefirst scenario ldquopessimistic scenariordquo illustrates low demand

Table 1 Input parameter used to solve the model

Parameters Inputs119896 = |119879| 10119898 = |119869| 2119899 = |119868| 4Transport cost from facility 1(China)

300 (pro 1000product units)

Transport cost from facility 2(Turkey)

100 (pro 1000product units)

Transport time from facility 1(China) 3

Transport time from facility 2(Turkey) 1

120572 095120573 01

Table 2 Unit production parameters

Product 1 Product 2 Product 3 Product 4Production

Minimum value 10000 7500 5000 8000Maximum value 40000 30750 20750 31600

Production costFacility 1 1 1 2 2Facility 2 3 2 3 5

CapacityFacility 1 20000 20000 20000 20000Facility 2 10000 10000 150000 10000

Penalty factor PF119894

01 02 08 05Delivery due date DD

1198946 6 6 6

Shortage cost unit 5 4 5 9Salvage cost unit 05 05 1 1Unit selling price 8 6 8 14

forecasts for the next season The second scenario ldquonormalscenariordquo represents the case when the customer demandis normal The third scenario ldquooptimistic scenariordquo assumeshigh customer demand To characterize the uncertaintiesof demand forecast of different products a distributionprobability is applied Figures 1 and 2 show an example ofthe demand distribution respectively of the products underpessimistic scenario and demand distribution of the product1 for the three scenarios

The risk to be considered from solving the model canoccur only because of wastage or shortage possibility There-fore according to our definition of the loss function119891(119883 119904) =sum119899

119894=1119891(119883119894 119904) = WC119904 + SC119904 minus 01 sdot SR119904 the constraint (16)

for this case study ensures that the probability that the riskcosts incurring by respectively wastage and shortage costs donot exceed the 10 (120573 = 01) of the expected profit revenuein 95 (120572 = 095) of all possible realization of stochasticparameters

The model has been efficiently solved by means offrontline risk solver which supports the robust optimizationthrough CVaR

Mathematical Problems in Engineering 7

025

02

015

01

005

0

11

12

135 15

17

18

195 21

22

235 25

265

275 29

31

32

Product 1

Product 1

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

6 8 9

105

115

125

135

145

155

165

175

185 20

21

22

23

24

Product 2

Product 2025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

55 6 7

75

85 9

95

105 11

12

125 13

14

145

155 16

Product 3

Product 3

(c)

025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

9

10

11

12

13

14

155 17

175 19

20

21

22

235

245 26

Product 4

Product 4

(d)

Figure 1 Demand distribution of the products in a pessimistic scenario

11

12

135 15

165 18

19

21

22

235 25

265

275 29

31

32

Pessimistic scenario

Pessimistic scenario025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

(Values in thousands)

Rela

tive p

roba

bilit

y

12

13

15

16

175 19

21

225 24

25

27

28

30

315 33

34

36

Normal scenario

Normal scenario

012

01

008

006

004

002

0

014

(b)

(Values in thousands)

Rela

tive p

roba

bilit

y

02

018

016

014

012

01

008

006

004

002

0

19

205

215

225 24

25

265 28

29

305 32

33

345

355 37

38

395

Optimistic scenario

Optimistic scenario

(c)

Figure 2 Demand distribution of product 1 in different scenarios

51 Pessimistic Scenario Table 3 lists the production plan ina pessimistic scenario The expected average profit of theproduction for the next season is about 219000 Moreoverthe solution can guarantee that the loss function will benegative in 96 cases of all possible realizations of stochasticparameters (Figure 3)

52 Normal Scenario In the normal scenario the demandof product 1 and product 4 respectively has nearly doubledwhereas the demand of product 2 and product 3 has beenslightly increased This is due to the low penalty factorof product 1 and product 4 which can be incurred foreach delayed delivery and for the capacity restriction of

8 Mathematical Problems in Engineering

012

01

008

006

004

002

0

120

130

150

160

180

190

200

220

230

240

260

270

280

300

310

320

(Values in thousands)

Rela

tive p

roba

bilit

y

Objective fct pessimistic

Objective fct pessimistic

(a)

(Values in thousands)

Rela

tive p

roba

bilit

y

016

014

012

01

008

006

004

002

0

minus25

minus23

minus21

minus19

minus17

minus15

minus13

minus11

minus8

minus6

minus5 3 0 2 3 5

Loss fct pessimistic

Loss fct pessimistic

(b)

Figure 3 Profit and loss function distribution in a pessimistic scenario

018

016

014

012

01

008

006

004

002

0

100

150

170

220

240

270

300

330

370

390

430

450

470

530

550

580

Objective fct normal

Objective fct normal

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

016

014

012

01

008

006

004

002

0

minus40

minus35

minus30

minus27

minus23

minus17

minus13

minus10

minus5 0 3 7

13

17

22

27

Loss fct normal

Loss fct normal

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

Figure 4 Profit and loss function distribution in a normal scenario

Table 3 Production plan with the demand quantities in a pes-simistic scenario

Product 1 Product 2 Product 3 Product 4Demand nextseason 17969 16549 10169 14945

Start of production 3 2 1 1Expected deliverydate 6 6 5 4

Produced InFacility 1 0 1 1 1Facility 2 1 0 0 0

the facilities (Table 4) In this case the expected averageprofit is about 450000 and the loss function is negative in975 of all possible realizations of the stochastic parameters(Figure 4)

53 Optimistic Scenario Since the optimistic scenario pre-dicts a boom economic scenario the production reaches its

Table 4 Production plan with the demand quantities in a normalscenario

Product 1 Product 2 Product 3 Product 4Demand next season 32355 19682 14415 26284Start of production 1 4 3 1Expected delivery date 6 8 7 6Produced in

Facility 1 0 1 1 1Facility 2 1 0 0 0

maximum (Table 5) In this case the average profit is about665000 and the loss function is satisfied with 991 of allpossible realizations of the stochastic parameters (Figure 5)

6 Conclusion and Outlook

In this paper we have presented a risk-constrained profit-expected maximization model for a textile industry scenario

Mathematical Problems in Engineering 9

045

04

035

03

025

02

015

01

005

0

225

250

275

325

350

400

425

475

500

525

575

600

625

675

625

750

Objective fct optimistic

Objective fct optimistic

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

04

035

03

025

02

015

01

005

0

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

minus2 2 7

11

16

20

Loss fct optimistic

Loss fct optimistic

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

Figure 5 Profit and loss function distribution in an optimistic scenario

Table 5 Production plan with the demand quantities in an opti-mistic scenario

Product 1 Product 2 Product 3 Product 4Demand next season 37933 29312 19410 31600Start of production 1 4 3 1Expected delivery date 6 9 7 6Produced in

Facility 1 0 1 1 1Facility 2 1 0 0 0

The forecast demand uncertainty has been explicitly con-sidered in the model by means of the robust optimizationand conditional value at risk theory by introducing andrestricting a loss function The robust optimization modeloriginally proposed by [32] was adopted as the benchmarkformulation of uncertainty considered in this paper A casestudy has been used to demonstrate the viability of theproposed mathematical model Results indicate that theproduction planning model using CVaR as the risk measureperforms robustly and provides flexibility in decision analysisbetween different scenarios The next step will be the furtherdevelopment of the profit maximization model to considerother types of uncertainties for example concerning theavailability of the production plants transportation meansand demand due dates

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Datamonitor Global Apparel Retail 2010 httpwwwdata-monitorcom

[2] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of Banking and Finance vol26 no 7 pp 1443ndash1471 2002

[3] S Zhu and M Fukushima ldquoWorst-case conditional value-at-risk with application to robust portfolio managementrdquo Opera-tions Research vol 57 no 5 pp 1155ndash1168 2009

[4] A Bruckner and S Muller Supply Chain Management in derBekleidungsindustrie Forschungsstelle der Bekleidungsindus-trie Cologne Germany 2003

[5] J Fissahn Marktorientierte Beschaffung in der Bekleidungsin-dustrie Munster University Munster Germany edition 2001

[6] A Schneider Internationalisierungsstrategien in der deutschenTextil-und Bekleidungsindustrie-eine empirische Untersuchung[MS dissertation] RWTH Aachen University Aachen Ger-many 2003

[7] G Gereffi and O Memedovic The Global Apparel ValueChainmdashWhat Prospects for Upgrading by Developing CountriesUNIDO Vienna Austria 2003

[8] MGromling and JMatthesGlobalisierung und Strukturwandelder Deutschen Textil-und Bekleidungsindustrie Dt Inst-VerlagColougne Germany 2003

[9] I Langenhorst Shop-logistik in der bekleidungswirtschaft eineanalyse der anforderungen herstellerinitiierter shop-systeme andie logistikprozesse der bekleidungsindustrie [PhD thesis]Munster University 1999

[10] H-C Pfohl M Gomm and X Shen China Textil-undBekleidungs-Supply Chain Zwischen Deutschland und ChinaFree Beratung Korschenbroich Germany in H Wolf-Kluthausen Jahrbuch der Logistik 2007

[11] M Teucke and B Scholz-Reiter ldquoImproving order allocationin fashion supply chains using radio frequency identification(RFID) technologiesrdquo in Fashion Supply Chain ManagementUsing Radio Frequency Identification (RFID) Technologies WK Wong and Z X Guo Eds vol 152 of Woodhead PublishingSeries in Textiles Woodhead Publishing Cambridge UK 2014

[12] S Thomassey ldquoSales forecasts in clothing industry the keysuccess factor of the supply chain managementrdquo InternationalJournal of Production Economics vol 128 no 2 pp 470ndash4832010

[13] D Ahlert and G Dieckheuer Marktorientierte Beschaffung inder Bekleidungsindustrie FATM Munster Germany 2001

[14] J T Mentzer C C Bienstock and K B Kahn ldquoBenchmarkingsales forecastingmanagementrdquo Business Horizons vol 42 no 3pp 48ndash56 1999

10 Mathematical Problems in Engineering

[15] J Quick M Rinis C Schmidt and B Walber SupplyTexmdashErfolgreiches Supply-Management in KMU Der Textil- UndBekleidungsindustrie Forschungsinstitut fur Rationalisierung(FIR) eV an der Universitat Aachen Aachen Germany 2010

[16] Z-L Sun T-M Choi K-F Au and Y Yu ldquoSales forecastingusing extreme learning machine with applications in fashionretailingrdquo Decision Support Systems vol 46 no 1 pp 411ndash4192008

[17] S Thomassey M Happiette and J-M Castelain ldquoA globalforecasting support system adapted to textile distributionrdquoInternational Journal of Production Economics vol 96 no 1 pp81ndash95 2005

[18] J Mostard R Teunter and R de Koster ldquoForecasting demandfor single-period products a case study in the apparel industryrdquoEuropean Journal of Operational Research vol 211 no 1 pp 139ndash147 2011

[19] K Hoyndorff S Hulsmann D Spee and M ten HompelFashion Logistics Grundlagen Uber Prozesse und IT Entlang derSupply Chain Munich Huss-Verlag GmbH 2010

[20] J Richardson ldquoVertical integration and rapid response infashion apparelrdquoOrganization Science vol 7 no 4 pp 400ndash4121996

[21] Z-L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[22] J Mula D Peidro M Dıaz-Madronero and E Vicens ldquoMathe-matical programming models for supply chain production andtransport planningrdquo European Journal of Operational Researchvol 204 no 3 pp 377ndash390 2010

[23] S A Torabi and E Hassini ldquoAn interactive possibilistic pro-gramming approach for multiple objective supply chain masterplanningrdquo Fuzzy Sets and Systems vol 159 no 2 pp 193ndash2142008

[24] B Scholz-Reiter J Hartmann T Makuschewitz and E MFrazzon ldquoA generic approach for the graph-based integratedproduction and intermodal transport scheduling with capacityrestrictionsrdquo in Procedia CIRP vol 7 pp 109ndash114 2013

[25] A D Toni andAMeneghetti ldquoProduction planning process fora network of firms in the textile-apparel industryrdquo InternationalJournal of Production Economics vol 65 no 1 pp 17ndash32 2000

[26] E M Frazzon A G N Novaes B Scholz-Reiter J Hartmannand T Makuschewitz ldquoAn approach for the rescheduling ofintegrated production and transport systemsrdquo in Proceedingsof the 4th World Conference Production amp Operations Manage-ment Serving the World Production amp Operations ManagementSociety p 10 Amsterdam The Netherlands 2012

[27] H Ishikura ldquoStudy on the production planning of apparel prod-ucts determining optimal production times and quantitiesrdquoComputers and Industrial Engineering vol 27 no 1ndash4 pp 19ndash22 1994

[28] Y W Tan J B Guan and H R Karimi ldquoThe Impact ofthe subsidy policy on total factor productivity an empiricalanalysis of Chinarsquos cotton productionrdquo Mathematical Problemsin Engineering vol 2013 Article ID 248537 8 pages 2013

[29] H R Karimi N A Duffie and S Dashkovskiy ldquoLocalcapacity Hinfin control for production networks of autonomouswork systems with time-varying delaysrdquo IEEE Transactions onAutomation Science and Engineering vol 7 no 4 pp 849ndash8572010

[30] S Dashkovskiy H R Karimi and M Kosmykov ldquoStabilityanalysis of logistics networks with time-delaysrdquo Journal ofProduction Planning amp Control vol 24 no 7 pp 567ndash574 2013

[31] A Mehrsai H R Karimi and B Scholz-Reiter ldquoApplication oflearning pallets for real-time scheduling by the use of radialbasis function networkrdquo Neurocomputing vol 101 pp 82ndash932013

[32] S C H Leung S O S Tsang W L Ng and Y Wu ldquoArobust optimization model for multi-site production planningproblem in an uncertain environmentrdquo European Journal ofOperational Research vol 181 no 1 pp 224ndash238 2007

[33] A-G Quaranta and A Zaffaroni ldquoRobust optimization ofconditional value at risk and portfolio selectionrdquo Journal ofBanking and Finance vol 32 no 10 pp 2046ndash2056 2008

[34] R A Jabr ldquoRobust self-scheduling under price uncertaintyusing conditional Value-at-Riskrdquo IEEE Transactions on PowerSystems vol 20 no 4 pp 1852ndash1858 2005

[35] D Peidro J Mula R Poler and J-L Verdegay ldquoFuzzy opti-mization for supply chain planning under supply demand andprocess uncertaintiesrdquo Fuzzy Sets and Systems vol 160 no 18pp 2640ndash2657 2009

[36] F Pan and R Nagi ldquoRobust supply chain design underuncertain demand in agile manufacturingrdquo Computers andOperations Research vol 37 no 4 pp 668ndash683 2010

[37] P M Verderame and C A Floudas ldquoOperational planning oflarge-scale industrial batch plants under demand due date andamount uncertainty II Conditional value-at-risk frameworkrdquoIndustrial and Engineering Chemistry Research vol 49 no 1 pp260ndash275 2010

[38] P M Verderame and C A Floudas ldquoMultisite planning underdemand and transportation time uncertainty Robust optimiza-tion and conditional value-at-risk frameworksrdquo Industrial andEngineering Chemistry Research vol 50 no 9 pp 4959ndash49822011

[39] S M J Mirzapour Al-E-Hashem H Malekly and M BAryanezhad ldquoA multi-objective robust optimization modelfor multi-product multi-site aggregate production planningin a supply chain under uncertaintyrdquo International Journal ofProduction Economics vol 134 no 1 pp 28ndash42 2011

[40] D Rahmani R Ramezanian P Fattahi and M HeydarildquoA robust optimization model for multi-product two-stagecapacitated production planning under uncertaintyrdquo AppliedMathematical Modelling vol 37 no 20-21 pp 8957ndash8971 2013

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Decision SciencesAdvances in

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

delivery times [12] For these reasons demand forecasting hasto be considered an important input of production planningand supply chain planning models Generally forecasting ofpotential sales permeates all aspects of business operations[14] Customer demand for fashion products is volatile andmay vary broadly for different variants of the same productDemand fluctuations are difficult to predict at the time ofproduction planning Optimistic forecasts may lead to over-stocks and higher production costs and cause markdownsPessimistic forecasts may lead to loss of opportunity revenuesfor potential sales that could not be realized and to customerdissatisfaction with later brand changes

On the other hand traditional statistical forecastingmethods are difficult to apply [15] Specifically developeddemand forecasting models often use soft computing meth-ods such as artificial neural networks [16] fuzzy logic andevolutionary procedures [12 17] In addition the integrationof preorder information as well as expert judgments hasproduced accurate forecasts [18] Most of these models havenot been integrated into standard business software and aredifficult to use with sometimesmixed quality of the forecastsOrder fulfilment and product delivery failures such as stock-outs or increased storage costs and product depreciationwrite-offs continue to affect profits in the apparel branch ina very unfavourable way It is estimated that due to the longlead times and demand forecasting difficulties mentioned inthe second section of this chapter up to 30ndash40 of high-fashion articles cannot be sold during the short sales seasonbefore the life expectancy of the article expires [19] Theneed to cope with this situation and the impetus to realizeQuick Response concepts have resulted in vertical integrationof apparel suppliers and retailers in a number of cases[20] Vertically integrated suppliers generally achieve betterreaction to market changes due to shortened reaction timesand more closely integrated information flows Howeverthese advantages cannot easily be copied by most traditionalsuppliers which are often small or medium sized enterprizesStill it is estimated that they have to write off up to 15ndash20of their high-fashion articles [19]

3 Literature Review on Problem RelatedProduction Planning

Production and distribution operations are two key functionsin the supply chain In general the supply chainmanagementprocedures emphasize production scheduling more than thedistribution scheduling because distribution ismore flexibleWhen integrated supply chain management of productionand distribution is realized the resources are used moreefficiently [21] In a comprehensive review of integratedscheduling approachesMula et al showed that over 44 papershave been published between 1989 and 2009 [22]

In general master tactical and operational planninglevels can be separated The purpose of most models is tominimize the total cost of the supply chain Some work at themaster planning level integrating procurement in addition toproduction and distribution scheduling is described in [23]Most of the models focus on tactical planning considering

different demand levels costs and capacities and use cen-tralized scheduling approaches Scholz-Reiter et al showed ascalable graph-based scheduling approach which is focusedon the operational level and real-time scheduling [24] Alarge variety of OR techniques have been applied comprisingmathematical optimisation heuristics and metaheuristicsbut it could be observed that most of the papers focus ondeterministic solutions [22]

The special conditions of production planning for fashionproducts have received some studies as well The work in[25] described a networked production planning processin a fashion oriented apparel supply chain The productionplanning process was found to be an important area ofimprovement for a network in a time-based logic shorteningthe production planning period in fact significantly affectsthe weighted average delivery anticipation It leads howeverto increase setup and transportation costs due to the greaternumber of jobs generated during a campaign Scheduling ofintegrated production and distribution systems in dynamicenvironments holds a potential to improve efficiency butalso poses a challenging planning task due to its compu-tational complexity Long lead times external and internalperturbations in productive processes unstable businessenvironments and contextual differences (eg institutionaleconomic and cultural) emphasize the relevance to theargued integration [26] Three types of aggregate produc-tion planning methods for the apparel industry have beenproposed in [27] These allow changing a production modelseasonally according to actual demands or maintaining thesame production model for several seasons with productionfor inventoryThe mentioned work on integrated productionand distribution planning does not deal with stochastic inputdata such as demands The work in [28] studied empiricallythe impact of the subsidy policy on total factor productivityfor the example of Chines cotton production

Modern large and widely distributed production net-works are subject to many forms of disturbances KarimiDuffy and Dashkovsky et al performed research on howto better deal with such dynamic influences One possiblesolution is to introduce autonomous control that is to allowsome parts of a large network to make their own decisionsbased on local situation and available information Howeverstability of the network and robustness with respect toexternal and internal disturbances and time delays in signalsmust be assured to guarantee a reasonable performance andvitality of the whole system For this purpose their workproposed an approach for controller design for large scaleautonomous work systems capable of coping with time delaysand explains its implementation and advantages on a concreteexample [29ndash31]

Robust optimization models try to formulate productionplanning problems in a way that cost or wastage effectsof uncertainty or risks are minimized or expected profitis maximized A robust optimization model for a multisiteproduction medium-term planning problem is developed in[32] based on the problems facing a multinational lingeriecompany with production sites in East Asia It generates acost minimal production plan for an uncertain environmentwith associated probabilities of different economic growth

4 Mathematical Problems in Engineering

scenarios The cost minimal production plan is less sensitiveto changes in the noisy and uncertain data The work in [33]dealt with a portfolio selection model in which the method-ologies of robust optimization are used for the minimizationof the conditional value at risk of a portfolio of shares Thework in [34] proposed a method for robust self-schedulingbased on CVaRThe proposedmethod is based on a security-constrained optimal power flow (SCOPF) program thatexplicitly treats the trade-off between risk and reward Thework in [35] proposed a fuzzy mathematical programmingmodel for supply chain planning which considers supplydemand and process uncertainties The model has beenformulated as a fuzzy mixed-integer linear programmingmodel where data are ill-known and modelled by triangularfuzzy numbers The fuzzy model provides the decisionmaker with alternative decision plans for different degreesof satisfaction This proposal is tested by using data from areal automobile supply chain A supply chain design problemfor a new market opportunity with uncertain demand inan agile manufacturing setting is considered in [36] Thismodel integrates the design of supply chain and productionplanning for the supply chainrsquos members and develops arobust optimization formulation A novel framework basedon conditional value at risk theory has been applied in [37] tothe problem of operational planning for large-scale industrialbatch plants under demand due date and amount uncertaintyThe objective of the proposed model is to provide a dailyproduction profile that not only is a tight upper bound onthe production capacity of the plant but also is immune tothe various forms of demand uncertainty In further workThe work in [38] applied a robust optimization frameworkas well as CVAR theory to the multisite operational plan-ning problem under multiple forms of system uncertaintyThey considered different forms of system uncertainty suchas demand due date demand amount and transportationtime uncertainty in the model Their objective is to ensurethe maximization of customer satisfaction along with theminimization of resource misallocation

A robust multiobjective mixed integer nonlinear pro-gramming model is proposed by [39] to deal with a multisitemultiperiod multiproduct aggregate production planning(APP) problem under uncertainty considering two conflict-ing objectives simultaneously as well as the uncertain natureof the supply chain Their proposed model is solved as asingle-objective mixed integer programming model applyingthe LP-metrics method A two-stage real world capacitatedproduction system with lead time and setup decisions uncer-tain production costs and customer demand is studied in[40] using a robust optimization approach A mixed-integerprogramming (MIP) model is developed to minimize totalproduction costs

4 Description of the Model

Our model maximizes expected profits in scenarios ofstochastic customer demand The model generates produc-tion plans containing the production quantities of eachproduct of a given product program the production start of

each product and the production plants where the productsshould be produced

Notation A set 119868 of 119899 products has to be manufacturedunder restriction of several time and resource constraintsThe forecast demand of each product to be manufactured isuncertain and a probabilistic distribution is used to character-ize this uncertainty The manufacturing of different productshas to take place in just one of119898 different facilities located indifferent countries

A facility 119895 isin 119869 has a fixed production cost CP119894119895to produce

the product 119894 a fixed transport cost CT119895 and a production

capacity CAP119894119895

of products which can be manufacturedduring one period

A product 119894 isin 119868 has a delivery due date DD119894 and if it is

supplied earlier than the recommendedDD119894that would cause

extra holding costs CH119894 and if it is supplied later than the due

date DD119894that would cause penalty costs PEC

119894

The objective is to find a feasible plan which determinesthe quantity of each product to be produced and indicatesthe start of production of each product 119894 manufactured fromfacility 119895 to fulfil the stochastic demand and maximize thetotal production profit

Parameters In the following the parameters for themodel aredefined

119879 is set of planning periods where 119896 = |119879|

119905 is index of the planning period 119905 isin 1 119896119869 is set of facilities where 119898 = |119869|119895 is index of facility 119895 isin 1 119898119868 is set of products where 119899 = |119868|119894 is index of product 119894 isin 1 119899Ω is set of all possible scenarios of stochastic variablesΩ = 1 2 119878119904 is index of scenario res Realisation of stochasticvariables 119904 isin [1 119878] where 119878 = |Ω|

119863119904119894is the demand of product 119894 under scenario 119904

119875119904is probability of scenario 119904

SP119894is the unit selling price for product 119894

CP119894119895is the unit production cost for product 119894 manu-

factured in facility 119895CT119895is transport cost from the facility 119895 in terms of

number of periodsCH119894is the unit inventory holding cost for product 119894 at

the end of productionSL119894is salvage value per unit for product 119894

SR119894is shortage penalty per unit for product 119894

TT119895is transport time from the facility 119895 in terms of

number of periodsDD119894is delivery due date of product 119894 in terms of

number of periods

Mathematical Problems in Engineering 5

PF119894is penalty cost incurred to each late delivery for

product 119894 when it is supplied later than the deliverydate DD

119894

CAP119894119895is production capacity (units) for product 119894

during a period in facility 119895120572 is confidence level of risk parameter CVaR where120572 isin [0 1[120573 is risk parameter120582 is weight presented on solution variance120583 is weight placed on model infeasibility whichcontrols the trade-off between solution and modelrobustness

Variables Consider the following

119883119894is quantity of product 119894 that shall be produced

(decision variable)119884119894119895is binary variable which indicates whether prod-

uct 119894 is produced in facility 119895 (decision variable)STP119894is start period of the production of product 119894

(decision integer variable)TP119894is time required in periods to produce 119883

119894of

product 119894 TP119894= sum119898

119895=1119884119894119895(119883119894CAP

119894119895)

ExpEnd119875119894is expected delivery date of product 119894 in

period unit ExpEnd119875119894= STP

119894+ TP119894+ sum119898

119895=1119884119894119895sdot TT119895

MIN119904119894is quantity of product 119894 effectively sold under

scenario 119904 MIN119904119894= min(119883

119894 119863119904119894)

WAS119904119894is wastage quantity of product 119894 under scenario

119904 WAS119904119894= max(0 119883

119894minus 119863119904119894)

SHO119904119894is shortage quantity of product 119894 under scenario

119904 SHO119904119894= max(0 119863119904

119894minus 119883119894)

119868119894is binary variable which indicates whether the

product 119894 has to be held or not after its productionand transportation

119868119894=

1 if ExpEnd119875119894lt DD

119894

0 else(1)

Objective Function The objective function has the followingcomponents

(1) sales revenue(2) production cost(3) transport cost(4) holding cost(5) penalty cost due to a late delivery(6) risk cost incurred for possible wastage cost(7) risk cost incurred for possible shortage cost

The above components have been considered due to the factthat they are the most important costs which can be mainlyaffected by the decision variable of the model

The objective is to maximize the total profit revenueconsisting of sales revenue gained by selling the productsproduction (PC) transport (TC) penalty holding (HC) andother risk costs which can be incurred for possible wastage orshortage quantities

SR119904 (sales revenue) =119899

sum119894=1

MIN119904119894sdot SP119894 (2)

PC (production cost) =119899

sum119894=1

119898

sum119895=1

119884119894119895X119894CP119894119895 (3)

TC (transport cost) =119899

sum119894=1

119898

sum119895=1

119884119894119895119883119894CT119895 (4)

HC (holding cost) =119899

sum119894=1

119868119894sdot (DD

119894minus ExpEnd119875

119894) sdot 119883119894CH119894

(5)

PEC (penalty cost)

=119899

sum119894=1

(1 minus 119868119894) (ExpEnd119875

119894minus DD

119894)119883119894PF119894

(6)

WC119904 (wastage cost) =119899

sum119894=1

WAS119904119894sdot SL119894 (7)

SC119904 (shortage cost) =119899

sum119894=1

SHO119904119894sdot SR119894 (8)

Equation (2) represents the total sales revenue due to sellingthe set of products 119868 Equation (3) is the total productioncost of manufacturing the set of products 119868 Equation (4) isthe total transport cost of the set products 119868 Equation (5)corresponds to the total holding cost of units of productswhich has to be stored in the warehouses for a determinedholding period Equation (6) represents the total penalty costof supplying the set of products 119868 later than the appropriatedelivery date Equation (7) represents the wastage cost toliquidate the overstocks set of products 119868 Equation (8)represents an artificial penalty for the demand dissatisfactionand subsequently to loss of opportunity revenue

The following formulation of robust problem is definedaccording to Leung et al [32]

We denote the profit function for each scenario 119904 with119865119904= [SR119904 + WC119904 minus SC119904 minus PC minus TC minus HC minus PEC] (9)

The objective function which represents the maximizingexpected profit of the production planning problem with thedemand uncertainty is formulated as follows

max119864 [profit]

= maxΩ

sum119904=1

119901119904119865119904+ 120582Ω

sum119904=1

119901119904[119865119904minusΩ

sum1199041015840=1

11990111990410158401198651199041015840]

+120583Ω

sum119904=1

119901119904120579119904

(10)

6 Mathematical Problems in Engineering

The first term of (10) is the mean value of the total profitThesecond term of (10) denotes the measure solution robustnessof the model The third term in (10) represents the modelrobustness and is used to penalize model infeasibility

Constraints First we define the loss function as follows

forall119904 isin Ω 119891 (119883 119904) =119899

sum119894=1

119891 (119883119894 119904) = WC119904 + SC119904 minus 120573 sdot SR119904

(11)

The objective function is subject to the following constraints

forall119894 isin 119868 ExpEnd119875119894le 119896 (12)

forall119894 isin 119868119898

sum119895=1

119884119894119895

= 1 (13)

forall119894 isin 119868 STP119894gt 0 (14)

forall119894 1198941015840 isin 119868 if exist120583 isin 119869 so that 119884119894120583

= 1198841198941015840120583= 1

STP119894lt STP

1198941015840 then STP

1198941015840 ge STP

119894+ TP119894

(15)

forall119904 isin Ω 119875 [119904 | 119891 (119883 119904) le 0] le 120572 (16)

Constraint (12) ensures that the expected delivery date forall products should not exceed the length of the planninghorizon 119896 = |119879| Constraint (13) guarantees that theproduction of a product 119894 will take place only in one facility119895 Constraint (14) indicates that the start of production ofa product 119894 is positive and integer Constraint (15) ensuresthat the production of more than one product cannot becarried out in a parallel manner Constraint (16) restricts theprobability of the loss function to be negative for 120572 sdot 100which means that the undesirable realization of uncertaintiescan restrict the loss function to be positive in only (1 minus 120572) sdot100 As an example for 120572 = 095 the constraint (16) has tobe satisfied for 95 for all possible realizations of demand119863119904

119894

for all 119894 isin 119868 and for all 119904 isin Ω

5 Experimental Results for a Case Studywith Three Scenarios

In order to cover different potential realizations of customerdemand three scenarios have been selected including a ldquopes-simisticrdquo scenario characterized by assumed low customerdemand a ldquonormalrdquo scenario characterized by average cus-tomer demand and an ldquooptimisticrdquo scenario characterized byrelatively high customer demand These three scenarios takeinto account the insecurity concerning customer demand forthe products Different values have been chosen based on datataken from a real case study

In this section we analyze the behavior of the modelby means of a case study based scenario The input dataparameters of the model are summarized in Tables 1 and 2

Three different scenarios are considered as a basis forapplication of the model and its numerical results Thefirst scenario ldquopessimistic scenariordquo illustrates low demand

Table 1 Input parameter used to solve the model

Parameters Inputs119896 = |119879| 10119898 = |119869| 2119899 = |119868| 4Transport cost from facility 1(China)

300 (pro 1000product units)

Transport cost from facility 2(Turkey)

100 (pro 1000product units)

Transport time from facility 1(China) 3

Transport time from facility 2(Turkey) 1

120572 095120573 01

Table 2 Unit production parameters

Product 1 Product 2 Product 3 Product 4Production

Minimum value 10000 7500 5000 8000Maximum value 40000 30750 20750 31600

Production costFacility 1 1 1 2 2Facility 2 3 2 3 5

CapacityFacility 1 20000 20000 20000 20000Facility 2 10000 10000 150000 10000

Penalty factor PF119894

01 02 08 05Delivery due date DD

1198946 6 6 6

Shortage cost unit 5 4 5 9Salvage cost unit 05 05 1 1Unit selling price 8 6 8 14

forecasts for the next season The second scenario ldquonormalscenariordquo represents the case when the customer demandis normal The third scenario ldquooptimistic scenariordquo assumeshigh customer demand To characterize the uncertaintiesof demand forecast of different products a distributionprobability is applied Figures 1 and 2 show an example ofthe demand distribution respectively of the products underpessimistic scenario and demand distribution of the product1 for the three scenarios

The risk to be considered from solving the model canoccur only because of wastage or shortage possibility There-fore according to our definition of the loss function119891(119883 119904) =sum119899

119894=1119891(119883119894 119904) = WC119904 + SC119904 minus 01 sdot SR119904 the constraint (16)

for this case study ensures that the probability that the riskcosts incurring by respectively wastage and shortage costs donot exceed the 10 (120573 = 01) of the expected profit revenuein 95 (120572 = 095) of all possible realization of stochasticparameters

The model has been efficiently solved by means offrontline risk solver which supports the robust optimizationthrough CVaR

Mathematical Problems in Engineering 7

025

02

015

01

005

0

11

12

135 15

17

18

195 21

22

235 25

265

275 29

31

32

Product 1

Product 1

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

6 8 9

105

115

125

135

145

155

165

175

185 20

21

22

23

24

Product 2

Product 2025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

55 6 7

75

85 9

95

105 11

12

125 13

14

145

155 16

Product 3

Product 3

(c)

025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

9

10

11

12

13

14

155 17

175 19

20

21

22

235

245 26

Product 4

Product 4

(d)

Figure 1 Demand distribution of the products in a pessimistic scenario

11

12

135 15

165 18

19

21

22

235 25

265

275 29

31

32

Pessimistic scenario

Pessimistic scenario025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

(Values in thousands)

Rela

tive p

roba

bilit

y

12

13

15

16

175 19

21

225 24

25

27

28

30

315 33

34

36

Normal scenario

Normal scenario

012

01

008

006

004

002

0

014

(b)

(Values in thousands)

Rela

tive p

roba

bilit

y

02

018

016

014

012

01

008

006

004

002

0

19

205

215

225 24

25

265 28

29

305 32

33

345

355 37

38

395

Optimistic scenario

Optimistic scenario

(c)

Figure 2 Demand distribution of product 1 in different scenarios

51 Pessimistic Scenario Table 3 lists the production plan ina pessimistic scenario The expected average profit of theproduction for the next season is about 219000 Moreoverthe solution can guarantee that the loss function will benegative in 96 cases of all possible realizations of stochasticparameters (Figure 3)

52 Normal Scenario In the normal scenario the demandof product 1 and product 4 respectively has nearly doubledwhereas the demand of product 2 and product 3 has beenslightly increased This is due to the low penalty factorof product 1 and product 4 which can be incurred foreach delayed delivery and for the capacity restriction of

8 Mathematical Problems in Engineering

012

01

008

006

004

002

0

120

130

150

160

180

190

200

220

230

240

260

270

280

300

310

320

(Values in thousands)

Rela

tive p

roba

bilit

y

Objective fct pessimistic

Objective fct pessimistic

(a)

(Values in thousands)

Rela

tive p

roba

bilit

y

016

014

012

01

008

006

004

002

0

minus25

minus23

minus21

minus19

minus17

minus15

minus13

minus11

minus8

minus6

minus5 3 0 2 3 5

Loss fct pessimistic

Loss fct pessimistic

(b)

Figure 3 Profit and loss function distribution in a pessimistic scenario

018

016

014

012

01

008

006

004

002

0

100

150

170

220

240

270

300

330

370

390

430

450

470

530

550

580

Objective fct normal

Objective fct normal

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

016

014

012

01

008

006

004

002

0

minus40

minus35

minus30

minus27

minus23

minus17

minus13

minus10

minus5 0 3 7

13

17

22

27

Loss fct normal

Loss fct normal

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

Figure 4 Profit and loss function distribution in a normal scenario

Table 3 Production plan with the demand quantities in a pes-simistic scenario

Product 1 Product 2 Product 3 Product 4Demand nextseason 17969 16549 10169 14945

Start of production 3 2 1 1Expected deliverydate 6 6 5 4

Produced InFacility 1 0 1 1 1Facility 2 1 0 0 0

the facilities (Table 4) In this case the expected averageprofit is about 450000 and the loss function is negative in975 of all possible realizations of the stochastic parameters(Figure 4)

53 Optimistic Scenario Since the optimistic scenario pre-dicts a boom economic scenario the production reaches its

Table 4 Production plan with the demand quantities in a normalscenario

Product 1 Product 2 Product 3 Product 4Demand next season 32355 19682 14415 26284Start of production 1 4 3 1Expected delivery date 6 8 7 6Produced in

Facility 1 0 1 1 1Facility 2 1 0 0 0

maximum (Table 5) In this case the average profit is about665000 and the loss function is satisfied with 991 of allpossible realizations of the stochastic parameters (Figure 5)

6 Conclusion and Outlook

In this paper we have presented a risk-constrained profit-expected maximization model for a textile industry scenario

Mathematical Problems in Engineering 9

045

04

035

03

025

02

015

01

005

0

225

250

275

325

350

400

425

475

500

525

575

600

625

675

625

750

Objective fct optimistic

Objective fct optimistic

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

04

035

03

025

02

015

01

005

0

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

minus2 2 7

11

16

20

Loss fct optimistic

Loss fct optimistic

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

Figure 5 Profit and loss function distribution in an optimistic scenario

Table 5 Production plan with the demand quantities in an opti-mistic scenario

Product 1 Product 2 Product 3 Product 4Demand next season 37933 29312 19410 31600Start of production 1 4 3 1Expected delivery date 6 9 7 6Produced in

Facility 1 0 1 1 1Facility 2 1 0 0 0

The forecast demand uncertainty has been explicitly con-sidered in the model by means of the robust optimizationand conditional value at risk theory by introducing andrestricting a loss function The robust optimization modeloriginally proposed by [32] was adopted as the benchmarkformulation of uncertainty considered in this paper A casestudy has been used to demonstrate the viability of theproposed mathematical model Results indicate that theproduction planning model using CVaR as the risk measureperforms robustly and provides flexibility in decision analysisbetween different scenarios The next step will be the furtherdevelopment of the profit maximization model to considerother types of uncertainties for example concerning theavailability of the production plants transportation meansand demand due dates

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Datamonitor Global Apparel Retail 2010 httpwwwdata-monitorcom

[2] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of Banking and Finance vol26 no 7 pp 1443ndash1471 2002

[3] S Zhu and M Fukushima ldquoWorst-case conditional value-at-risk with application to robust portfolio managementrdquo Opera-tions Research vol 57 no 5 pp 1155ndash1168 2009

[4] A Bruckner and S Muller Supply Chain Management in derBekleidungsindustrie Forschungsstelle der Bekleidungsindus-trie Cologne Germany 2003

[5] J Fissahn Marktorientierte Beschaffung in der Bekleidungsin-dustrie Munster University Munster Germany edition 2001

[6] A Schneider Internationalisierungsstrategien in der deutschenTextil-und Bekleidungsindustrie-eine empirische Untersuchung[MS dissertation] RWTH Aachen University Aachen Ger-many 2003

[7] G Gereffi and O Memedovic The Global Apparel ValueChainmdashWhat Prospects for Upgrading by Developing CountriesUNIDO Vienna Austria 2003

[8] MGromling and JMatthesGlobalisierung und Strukturwandelder Deutschen Textil-und Bekleidungsindustrie Dt Inst-VerlagColougne Germany 2003

[9] I Langenhorst Shop-logistik in der bekleidungswirtschaft eineanalyse der anforderungen herstellerinitiierter shop-systeme andie logistikprozesse der bekleidungsindustrie [PhD thesis]Munster University 1999

[10] H-C Pfohl M Gomm and X Shen China Textil-undBekleidungs-Supply Chain Zwischen Deutschland und ChinaFree Beratung Korschenbroich Germany in H Wolf-Kluthausen Jahrbuch der Logistik 2007

[11] M Teucke and B Scholz-Reiter ldquoImproving order allocationin fashion supply chains using radio frequency identification(RFID) technologiesrdquo in Fashion Supply Chain ManagementUsing Radio Frequency Identification (RFID) Technologies WK Wong and Z X Guo Eds vol 152 of Woodhead PublishingSeries in Textiles Woodhead Publishing Cambridge UK 2014

[12] S Thomassey ldquoSales forecasts in clothing industry the keysuccess factor of the supply chain managementrdquo InternationalJournal of Production Economics vol 128 no 2 pp 470ndash4832010

[13] D Ahlert and G Dieckheuer Marktorientierte Beschaffung inder Bekleidungsindustrie FATM Munster Germany 2001

[14] J T Mentzer C C Bienstock and K B Kahn ldquoBenchmarkingsales forecastingmanagementrdquo Business Horizons vol 42 no 3pp 48ndash56 1999

10 Mathematical Problems in Engineering

[15] J Quick M Rinis C Schmidt and B Walber SupplyTexmdashErfolgreiches Supply-Management in KMU Der Textil- UndBekleidungsindustrie Forschungsinstitut fur Rationalisierung(FIR) eV an der Universitat Aachen Aachen Germany 2010

[16] Z-L Sun T-M Choi K-F Au and Y Yu ldquoSales forecastingusing extreme learning machine with applications in fashionretailingrdquo Decision Support Systems vol 46 no 1 pp 411ndash4192008

[17] S Thomassey M Happiette and J-M Castelain ldquoA globalforecasting support system adapted to textile distributionrdquoInternational Journal of Production Economics vol 96 no 1 pp81ndash95 2005

[18] J Mostard R Teunter and R de Koster ldquoForecasting demandfor single-period products a case study in the apparel industryrdquoEuropean Journal of Operational Research vol 211 no 1 pp 139ndash147 2011

[19] K Hoyndorff S Hulsmann D Spee and M ten HompelFashion Logistics Grundlagen Uber Prozesse und IT Entlang derSupply Chain Munich Huss-Verlag GmbH 2010

[20] J Richardson ldquoVertical integration and rapid response infashion apparelrdquoOrganization Science vol 7 no 4 pp 400ndash4121996

[21] Z-L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[22] J Mula D Peidro M Dıaz-Madronero and E Vicens ldquoMathe-matical programming models for supply chain production andtransport planningrdquo European Journal of Operational Researchvol 204 no 3 pp 377ndash390 2010

[23] S A Torabi and E Hassini ldquoAn interactive possibilistic pro-gramming approach for multiple objective supply chain masterplanningrdquo Fuzzy Sets and Systems vol 159 no 2 pp 193ndash2142008

[24] B Scholz-Reiter J Hartmann T Makuschewitz and E MFrazzon ldquoA generic approach for the graph-based integratedproduction and intermodal transport scheduling with capacityrestrictionsrdquo in Procedia CIRP vol 7 pp 109ndash114 2013

[25] A D Toni andAMeneghetti ldquoProduction planning process fora network of firms in the textile-apparel industryrdquo InternationalJournal of Production Economics vol 65 no 1 pp 17ndash32 2000

[26] E M Frazzon A G N Novaes B Scholz-Reiter J Hartmannand T Makuschewitz ldquoAn approach for the rescheduling ofintegrated production and transport systemsrdquo in Proceedingsof the 4th World Conference Production amp Operations Manage-ment Serving the World Production amp Operations ManagementSociety p 10 Amsterdam The Netherlands 2012

[27] H Ishikura ldquoStudy on the production planning of apparel prod-ucts determining optimal production times and quantitiesrdquoComputers and Industrial Engineering vol 27 no 1ndash4 pp 19ndash22 1994

[28] Y W Tan J B Guan and H R Karimi ldquoThe Impact ofthe subsidy policy on total factor productivity an empiricalanalysis of Chinarsquos cotton productionrdquo Mathematical Problemsin Engineering vol 2013 Article ID 248537 8 pages 2013

[29] H R Karimi N A Duffie and S Dashkovskiy ldquoLocalcapacity Hinfin control for production networks of autonomouswork systems with time-varying delaysrdquo IEEE Transactions onAutomation Science and Engineering vol 7 no 4 pp 849ndash8572010

[30] S Dashkovskiy H R Karimi and M Kosmykov ldquoStabilityanalysis of logistics networks with time-delaysrdquo Journal ofProduction Planning amp Control vol 24 no 7 pp 567ndash574 2013

[31] A Mehrsai H R Karimi and B Scholz-Reiter ldquoApplication oflearning pallets for real-time scheduling by the use of radialbasis function networkrdquo Neurocomputing vol 101 pp 82ndash932013

[32] S C H Leung S O S Tsang W L Ng and Y Wu ldquoArobust optimization model for multi-site production planningproblem in an uncertain environmentrdquo European Journal ofOperational Research vol 181 no 1 pp 224ndash238 2007

[33] A-G Quaranta and A Zaffaroni ldquoRobust optimization ofconditional value at risk and portfolio selectionrdquo Journal ofBanking and Finance vol 32 no 10 pp 2046ndash2056 2008

[34] R A Jabr ldquoRobust self-scheduling under price uncertaintyusing conditional Value-at-Riskrdquo IEEE Transactions on PowerSystems vol 20 no 4 pp 1852ndash1858 2005

[35] D Peidro J Mula R Poler and J-L Verdegay ldquoFuzzy opti-mization for supply chain planning under supply demand andprocess uncertaintiesrdquo Fuzzy Sets and Systems vol 160 no 18pp 2640ndash2657 2009

[36] F Pan and R Nagi ldquoRobust supply chain design underuncertain demand in agile manufacturingrdquo Computers andOperations Research vol 37 no 4 pp 668ndash683 2010

[37] P M Verderame and C A Floudas ldquoOperational planning oflarge-scale industrial batch plants under demand due date andamount uncertainty II Conditional value-at-risk frameworkrdquoIndustrial and Engineering Chemistry Research vol 49 no 1 pp260ndash275 2010

[38] P M Verderame and C A Floudas ldquoMultisite planning underdemand and transportation time uncertainty Robust optimiza-tion and conditional value-at-risk frameworksrdquo Industrial andEngineering Chemistry Research vol 50 no 9 pp 4959ndash49822011

[39] S M J Mirzapour Al-E-Hashem H Malekly and M BAryanezhad ldquoA multi-objective robust optimization modelfor multi-product multi-site aggregate production planningin a supply chain under uncertaintyrdquo International Journal ofProduction Economics vol 134 no 1 pp 28ndash42 2011

[40] D Rahmani R Ramezanian P Fattahi and M HeydarildquoA robust optimization model for multi-product two-stagecapacitated production planning under uncertaintyrdquo AppliedMathematical Modelling vol 37 no 20-21 pp 8957ndash8971 2013

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

scenarios The cost minimal production plan is less sensitiveto changes in the noisy and uncertain data The work in [33]dealt with a portfolio selection model in which the method-ologies of robust optimization are used for the minimizationof the conditional value at risk of a portfolio of shares Thework in [34] proposed a method for robust self-schedulingbased on CVaRThe proposedmethod is based on a security-constrained optimal power flow (SCOPF) program thatexplicitly treats the trade-off between risk and reward Thework in [35] proposed a fuzzy mathematical programmingmodel for supply chain planning which considers supplydemand and process uncertainties The model has beenformulated as a fuzzy mixed-integer linear programmingmodel where data are ill-known and modelled by triangularfuzzy numbers The fuzzy model provides the decisionmaker with alternative decision plans for different degreesof satisfaction This proposal is tested by using data from areal automobile supply chain A supply chain design problemfor a new market opportunity with uncertain demand inan agile manufacturing setting is considered in [36] Thismodel integrates the design of supply chain and productionplanning for the supply chainrsquos members and develops arobust optimization formulation A novel framework basedon conditional value at risk theory has been applied in [37] tothe problem of operational planning for large-scale industrialbatch plants under demand due date and amount uncertaintyThe objective of the proposed model is to provide a dailyproduction profile that not only is a tight upper bound onthe production capacity of the plant but also is immune tothe various forms of demand uncertainty In further workThe work in [38] applied a robust optimization frameworkas well as CVAR theory to the multisite operational plan-ning problem under multiple forms of system uncertaintyThey considered different forms of system uncertainty suchas demand due date demand amount and transportationtime uncertainty in the model Their objective is to ensurethe maximization of customer satisfaction along with theminimization of resource misallocation

A robust multiobjective mixed integer nonlinear pro-gramming model is proposed by [39] to deal with a multisitemultiperiod multiproduct aggregate production planning(APP) problem under uncertainty considering two conflict-ing objectives simultaneously as well as the uncertain natureof the supply chain Their proposed model is solved as asingle-objective mixed integer programming model applyingthe LP-metrics method A two-stage real world capacitatedproduction system with lead time and setup decisions uncer-tain production costs and customer demand is studied in[40] using a robust optimization approach A mixed-integerprogramming (MIP) model is developed to minimize totalproduction costs

4 Description of the Model

Our model maximizes expected profits in scenarios ofstochastic customer demand The model generates produc-tion plans containing the production quantities of eachproduct of a given product program the production start of

each product and the production plants where the productsshould be produced

Notation A set 119868 of 119899 products has to be manufacturedunder restriction of several time and resource constraintsThe forecast demand of each product to be manufactured isuncertain and a probabilistic distribution is used to character-ize this uncertainty The manufacturing of different productshas to take place in just one of119898 different facilities located indifferent countries

A facility 119895 isin 119869 has a fixed production cost CP119894119895to produce

the product 119894 a fixed transport cost CT119895 and a production

capacity CAP119894119895

of products which can be manufacturedduring one period

A product 119894 isin 119868 has a delivery due date DD119894 and if it is

supplied earlier than the recommendedDD119894that would cause

extra holding costs CH119894 and if it is supplied later than the due

date DD119894that would cause penalty costs PEC

119894

The objective is to find a feasible plan which determinesthe quantity of each product to be produced and indicatesthe start of production of each product 119894 manufactured fromfacility 119895 to fulfil the stochastic demand and maximize thetotal production profit

Parameters In the following the parameters for themodel aredefined

119879 is set of planning periods where 119896 = |119879|

119905 is index of the planning period 119905 isin 1 119896119869 is set of facilities where 119898 = |119869|119895 is index of facility 119895 isin 1 119898119868 is set of products where 119899 = |119868|119894 is index of product 119894 isin 1 119899Ω is set of all possible scenarios of stochastic variablesΩ = 1 2 119878119904 is index of scenario res Realisation of stochasticvariables 119904 isin [1 119878] where 119878 = |Ω|

119863119904119894is the demand of product 119894 under scenario 119904

119875119904is probability of scenario 119904

SP119894is the unit selling price for product 119894

CP119894119895is the unit production cost for product 119894 manu-

factured in facility 119895CT119895is transport cost from the facility 119895 in terms of

number of periodsCH119894is the unit inventory holding cost for product 119894 at

the end of productionSL119894is salvage value per unit for product 119894

SR119894is shortage penalty per unit for product 119894

TT119895is transport time from the facility 119895 in terms of

number of periodsDD119894is delivery due date of product 119894 in terms of

number of periods

Mathematical Problems in Engineering 5

PF119894is penalty cost incurred to each late delivery for

product 119894 when it is supplied later than the deliverydate DD

119894

CAP119894119895is production capacity (units) for product 119894

during a period in facility 119895120572 is confidence level of risk parameter CVaR where120572 isin [0 1[120573 is risk parameter120582 is weight presented on solution variance120583 is weight placed on model infeasibility whichcontrols the trade-off between solution and modelrobustness

Variables Consider the following

119883119894is quantity of product 119894 that shall be produced

(decision variable)119884119894119895is binary variable which indicates whether prod-

uct 119894 is produced in facility 119895 (decision variable)STP119894is start period of the production of product 119894

(decision integer variable)TP119894is time required in periods to produce 119883

119894of

product 119894 TP119894= sum119898

119895=1119884119894119895(119883119894CAP

119894119895)

ExpEnd119875119894is expected delivery date of product 119894 in

period unit ExpEnd119875119894= STP

119894+ TP119894+ sum119898

119895=1119884119894119895sdot TT119895

MIN119904119894is quantity of product 119894 effectively sold under

scenario 119904 MIN119904119894= min(119883

119894 119863119904119894)

WAS119904119894is wastage quantity of product 119894 under scenario

119904 WAS119904119894= max(0 119883

119894minus 119863119904119894)

SHO119904119894is shortage quantity of product 119894 under scenario

119904 SHO119904119894= max(0 119863119904

119894minus 119883119894)

119868119894is binary variable which indicates whether the

product 119894 has to be held or not after its productionand transportation

119868119894=

1 if ExpEnd119875119894lt DD

119894

0 else(1)

Objective Function The objective function has the followingcomponents

(1) sales revenue(2) production cost(3) transport cost(4) holding cost(5) penalty cost due to a late delivery(6) risk cost incurred for possible wastage cost(7) risk cost incurred for possible shortage cost

The above components have been considered due to the factthat they are the most important costs which can be mainlyaffected by the decision variable of the model

The objective is to maximize the total profit revenueconsisting of sales revenue gained by selling the productsproduction (PC) transport (TC) penalty holding (HC) andother risk costs which can be incurred for possible wastage orshortage quantities

SR119904 (sales revenue) =119899

sum119894=1

MIN119904119894sdot SP119894 (2)

PC (production cost) =119899

sum119894=1

119898

sum119895=1

119884119894119895X119894CP119894119895 (3)

TC (transport cost) =119899

sum119894=1

119898

sum119895=1

119884119894119895119883119894CT119895 (4)

HC (holding cost) =119899

sum119894=1

119868119894sdot (DD

119894minus ExpEnd119875

119894) sdot 119883119894CH119894

(5)

PEC (penalty cost)

=119899

sum119894=1

(1 minus 119868119894) (ExpEnd119875

119894minus DD

119894)119883119894PF119894

(6)

WC119904 (wastage cost) =119899

sum119894=1

WAS119904119894sdot SL119894 (7)

SC119904 (shortage cost) =119899

sum119894=1

SHO119904119894sdot SR119894 (8)

Equation (2) represents the total sales revenue due to sellingthe set of products 119868 Equation (3) is the total productioncost of manufacturing the set of products 119868 Equation (4) isthe total transport cost of the set products 119868 Equation (5)corresponds to the total holding cost of units of productswhich has to be stored in the warehouses for a determinedholding period Equation (6) represents the total penalty costof supplying the set of products 119868 later than the appropriatedelivery date Equation (7) represents the wastage cost toliquidate the overstocks set of products 119868 Equation (8)represents an artificial penalty for the demand dissatisfactionand subsequently to loss of opportunity revenue

The following formulation of robust problem is definedaccording to Leung et al [32]

We denote the profit function for each scenario 119904 with119865119904= [SR119904 + WC119904 minus SC119904 minus PC minus TC minus HC minus PEC] (9)

The objective function which represents the maximizingexpected profit of the production planning problem with thedemand uncertainty is formulated as follows

max119864 [profit]

= maxΩ

sum119904=1

119901119904119865119904+ 120582Ω

sum119904=1

119901119904[119865119904minusΩ

sum1199041015840=1

11990111990410158401198651199041015840]

+120583Ω

sum119904=1

119901119904120579119904

(10)

6 Mathematical Problems in Engineering

The first term of (10) is the mean value of the total profitThesecond term of (10) denotes the measure solution robustnessof the model The third term in (10) represents the modelrobustness and is used to penalize model infeasibility

Constraints First we define the loss function as follows

forall119904 isin Ω 119891 (119883 119904) =119899

sum119894=1

119891 (119883119894 119904) = WC119904 + SC119904 minus 120573 sdot SR119904

(11)

The objective function is subject to the following constraints

forall119894 isin 119868 ExpEnd119875119894le 119896 (12)

forall119894 isin 119868119898

sum119895=1

119884119894119895

= 1 (13)

forall119894 isin 119868 STP119894gt 0 (14)

forall119894 1198941015840 isin 119868 if exist120583 isin 119869 so that 119884119894120583

= 1198841198941015840120583= 1

STP119894lt STP

1198941015840 then STP

1198941015840 ge STP

119894+ TP119894

(15)

forall119904 isin Ω 119875 [119904 | 119891 (119883 119904) le 0] le 120572 (16)

Constraint (12) ensures that the expected delivery date forall products should not exceed the length of the planninghorizon 119896 = |119879| Constraint (13) guarantees that theproduction of a product 119894 will take place only in one facility119895 Constraint (14) indicates that the start of production ofa product 119894 is positive and integer Constraint (15) ensuresthat the production of more than one product cannot becarried out in a parallel manner Constraint (16) restricts theprobability of the loss function to be negative for 120572 sdot 100which means that the undesirable realization of uncertaintiescan restrict the loss function to be positive in only (1 minus 120572) sdot100 As an example for 120572 = 095 the constraint (16) has tobe satisfied for 95 for all possible realizations of demand119863119904

119894

for all 119894 isin 119868 and for all 119904 isin Ω

5 Experimental Results for a Case Studywith Three Scenarios

In order to cover different potential realizations of customerdemand three scenarios have been selected including a ldquopes-simisticrdquo scenario characterized by assumed low customerdemand a ldquonormalrdquo scenario characterized by average cus-tomer demand and an ldquooptimisticrdquo scenario characterized byrelatively high customer demand These three scenarios takeinto account the insecurity concerning customer demand forthe products Different values have been chosen based on datataken from a real case study

In this section we analyze the behavior of the modelby means of a case study based scenario The input dataparameters of the model are summarized in Tables 1 and 2

Three different scenarios are considered as a basis forapplication of the model and its numerical results Thefirst scenario ldquopessimistic scenariordquo illustrates low demand

Table 1 Input parameter used to solve the model

Parameters Inputs119896 = |119879| 10119898 = |119869| 2119899 = |119868| 4Transport cost from facility 1(China)

300 (pro 1000product units)

Transport cost from facility 2(Turkey)

100 (pro 1000product units)

Transport time from facility 1(China) 3

Transport time from facility 2(Turkey) 1

120572 095120573 01

Table 2 Unit production parameters

Product 1 Product 2 Product 3 Product 4Production

Minimum value 10000 7500 5000 8000Maximum value 40000 30750 20750 31600

Production costFacility 1 1 1 2 2Facility 2 3 2 3 5

CapacityFacility 1 20000 20000 20000 20000Facility 2 10000 10000 150000 10000

Penalty factor PF119894

01 02 08 05Delivery due date DD

1198946 6 6 6

Shortage cost unit 5 4 5 9Salvage cost unit 05 05 1 1Unit selling price 8 6 8 14

forecasts for the next season The second scenario ldquonormalscenariordquo represents the case when the customer demandis normal The third scenario ldquooptimistic scenariordquo assumeshigh customer demand To characterize the uncertaintiesof demand forecast of different products a distributionprobability is applied Figures 1 and 2 show an example ofthe demand distribution respectively of the products underpessimistic scenario and demand distribution of the product1 for the three scenarios

The risk to be considered from solving the model canoccur only because of wastage or shortage possibility There-fore according to our definition of the loss function119891(119883 119904) =sum119899

119894=1119891(119883119894 119904) = WC119904 + SC119904 minus 01 sdot SR119904 the constraint (16)

for this case study ensures that the probability that the riskcosts incurring by respectively wastage and shortage costs donot exceed the 10 (120573 = 01) of the expected profit revenuein 95 (120572 = 095) of all possible realization of stochasticparameters

The model has been efficiently solved by means offrontline risk solver which supports the robust optimizationthrough CVaR

Mathematical Problems in Engineering 7

025

02

015

01

005

0

11

12

135 15

17

18

195 21

22

235 25

265

275 29

31

32

Product 1

Product 1

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

6 8 9

105

115

125

135

145

155

165

175

185 20

21

22

23

24

Product 2

Product 2025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

55 6 7

75

85 9

95

105 11

12

125 13

14

145

155 16

Product 3

Product 3

(c)

025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

9

10

11

12

13

14

155 17

175 19

20

21

22

235

245 26

Product 4

Product 4

(d)

Figure 1 Demand distribution of the products in a pessimistic scenario

11

12

135 15

165 18

19

21

22

235 25

265

275 29

31

32

Pessimistic scenario

Pessimistic scenario025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

(Values in thousands)

Rela

tive p

roba

bilit

y

12

13

15

16

175 19

21

225 24

25

27

28

30

315 33

34

36

Normal scenario

Normal scenario

012

01

008

006

004

002

0

014

(b)

(Values in thousands)

Rela

tive p

roba

bilit

y

02

018

016

014

012

01

008

006

004

002

0

19

205

215

225 24

25

265 28

29

305 32

33

345

355 37

38

395

Optimistic scenario

Optimistic scenario

(c)

Figure 2 Demand distribution of product 1 in different scenarios

51 Pessimistic Scenario Table 3 lists the production plan ina pessimistic scenario The expected average profit of theproduction for the next season is about 219000 Moreoverthe solution can guarantee that the loss function will benegative in 96 cases of all possible realizations of stochasticparameters (Figure 3)

52 Normal Scenario In the normal scenario the demandof product 1 and product 4 respectively has nearly doubledwhereas the demand of product 2 and product 3 has beenslightly increased This is due to the low penalty factorof product 1 and product 4 which can be incurred foreach delayed delivery and for the capacity restriction of

8 Mathematical Problems in Engineering

012

01

008

006

004

002

0

120

130

150

160

180

190

200

220

230

240

260

270

280

300

310

320

(Values in thousands)

Rela

tive p

roba

bilit

y

Objective fct pessimistic

Objective fct pessimistic

(a)

(Values in thousands)

Rela

tive p

roba

bilit

y

016

014

012

01

008

006

004

002

0

minus25

minus23

minus21

minus19

minus17

minus15

minus13

minus11

minus8

minus6

minus5 3 0 2 3 5

Loss fct pessimistic

Loss fct pessimistic

(b)

Figure 3 Profit and loss function distribution in a pessimistic scenario

018

016

014

012

01

008

006

004

002

0

100

150

170

220

240

270

300

330

370

390

430

450

470

530

550

580

Objective fct normal

Objective fct normal

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

016

014

012

01

008

006

004

002

0

minus40

minus35

minus30

minus27

minus23

minus17

minus13

minus10

minus5 0 3 7

13

17

22

27

Loss fct normal

Loss fct normal

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

Figure 4 Profit and loss function distribution in a normal scenario

Table 3 Production plan with the demand quantities in a pes-simistic scenario

Product 1 Product 2 Product 3 Product 4Demand nextseason 17969 16549 10169 14945

Start of production 3 2 1 1Expected deliverydate 6 6 5 4

Produced InFacility 1 0 1 1 1Facility 2 1 0 0 0

the facilities (Table 4) In this case the expected averageprofit is about 450000 and the loss function is negative in975 of all possible realizations of the stochastic parameters(Figure 4)

53 Optimistic Scenario Since the optimistic scenario pre-dicts a boom economic scenario the production reaches its

Table 4 Production plan with the demand quantities in a normalscenario

Product 1 Product 2 Product 3 Product 4Demand next season 32355 19682 14415 26284Start of production 1 4 3 1Expected delivery date 6 8 7 6Produced in

Facility 1 0 1 1 1Facility 2 1 0 0 0

maximum (Table 5) In this case the average profit is about665000 and the loss function is satisfied with 991 of allpossible realizations of the stochastic parameters (Figure 5)

6 Conclusion and Outlook

In this paper we have presented a risk-constrained profit-expected maximization model for a textile industry scenario

Mathematical Problems in Engineering 9

045

04

035

03

025

02

015

01

005

0

225

250

275

325

350

400

425

475

500

525

575

600

625

675

625

750

Objective fct optimistic

Objective fct optimistic

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

04

035

03

025

02

015

01

005

0

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

minus2 2 7

11

16

20

Loss fct optimistic

Loss fct optimistic

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

Figure 5 Profit and loss function distribution in an optimistic scenario

Table 5 Production plan with the demand quantities in an opti-mistic scenario

Product 1 Product 2 Product 3 Product 4Demand next season 37933 29312 19410 31600Start of production 1 4 3 1Expected delivery date 6 9 7 6Produced in

Facility 1 0 1 1 1Facility 2 1 0 0 0

The forecast demand uncertainty has been explicitly con-sidered in the model by means of the robust optimizationand conditional value at risk theory by introducing andrestricting a loss function The robust optimization modeloriginally proposed by [32] was adopted as the benchmarkformulation of uncertainty considered in this paper A casestudy has been used to demonstrate the viability of theproposed mathematical model Results indicate that theproduction planning model using CVaR as the risk measureperforms robustly and provides flexibility in decision analysisbetween different scenarios The next step will be the furtherdevelopment of the profit maximization model to considerother types of uncertainties for example concerning theavailability of the production plants transportation meansand demand due dates

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Datamonitor Global Apparel Retail 2010 httpwwwdata-monitorcom

[2] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of Banking and Finance vol26 no 7 pp 1443ndash1471 2002

[3] S Zhu and M Fukushima ldquoWorst-case conditional value-at-risk with application to robust portfolio managementrdquo Opera-tions Research vol 57 no 5 pp 1155ndash1168 2009

[4] A Bruckner and S Muller Supply Chain Management in derBekleidungsindustrie Forschungsstelle der Bekleidungsindus-trie Cologne Germany 2003

[5] J Fissahn Marktorientierte Beschaffung in der Bekleidungsin-dustrie Munster University Munster Germany edition 2001

[6] A Schneider Internationalisierungsstrategien in der deutschenTextil-und Bekleidungsindustrie-eine empirische Untersuchung[MS dissertation] RWTH Aachen University Aachen Ger-many 2003

[7] G Gereffi and O Memedovic The Global Apparel ValueChainmdashWhat Prospects for Upgrading by Developing CountriesUNIDO Vienna Austria 2003

[8] MGromling and JMatthesGlobalisierung und Strukturwandelder Deutschen Textil-und Bekleidungsindustrie Dt Inst-VerlagColougne Germany 2003

[9] I Langenhorst Shop-logistik in der bekleidungswirtschaft eineanalyse der anforderungen herstellerinitiierter shop-systeme andie logistikprozesse der bekleidungsindustrie [PhD thesis]Munster University 1999

[10] H-C Pfohl M Gomm and X Shen China Textil-undBekleidungs-Supply Chain Zwischen Deutschland und ChinaFree Beratung Korschenbroich Germany in H Wolf-Kluthausen Jahrbuch der Logistik 2007

[11] M Teucke and B Scholz-Reiter ldquoImproving order allocationin fashion supply chains using radio frequency identification(RFID) technologiesrdquo in Fashion Supply Chain ManagementUsing Radio Frequency Identification (RFID) Technologies WK Wong and Z X Guo Eds vol 152 of Woodhead PublishingSeries in Textiles Woodhead Publishing Cambridge UK 2014

[12] S Thomassey ldquoSales forecasts in clothing industry the keysuccess factor of the supply chain managementrdquo InternationalJournal of Production Economics vol 128 no 2 pp 470ndash4832010

[13] D Ahlert and G Dieckheuer Marktorientierte Beschaffung inder Bekleidungsindustrie FATM Munster Germany 2001

[14] J T Mentzer C C Bienstock and K B Kahn ldquoBenchmarkingsales forecastingmanagementrdquo Business Horizons vol 42 no 3pp 48ndash56 1999

10 Mathematical Problems in Engineering

[15] J Quick M Rinis C Schmidt and B Walber SupplyTexmdashErfolgreiches Supply-Management in KMU Der Textil- UndBekleidungsindustrie Forschungsinstitut fur Rationalisierung(FIR) eV an der Universitat Aachen Aachen Germany 2010

[16] Z-L Sun T-M Choi K-F Au and Y Yu ldquoSales forecastingusing extreme learning machine with applications in fashionretailingrdquo Decision Support Systems vol 46 no 1 pp 411ndash4192008

[17] S Thomassey M Happiette and J-M Castelain ldquoA globalforecasting support system adapted to textile distributionrdquoInternational Journal of Production Economics vol 96 no 1 pp81ndash95 2005

[18] J Mostard R Teunter and R de Koster ldquoForecasting demandfor single-period products a case study in the apparel industryrdquoEuropean Journal of Operational Research vol 211 no 1 pp 139ndash147 2011

[19] K Hoyndorff S Hulsmann D Spee and M ten HompelFashion Logistics Grundlagen Uber Prozesse und IT Entlang derSupply Chain Munich Huss-Verlag GmbH 2010

[20] J Richardson ldquoVertical integration and rapid response infashion apparelrdquoOrganization Science vol 7 no 4 pp 400ndash4121996

[21] Z-L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[22] J Mula D Peidro M Dıaz-Madronero and E Vicens ldquoMathe-matical programming models for supply chain production andtransport planningrdquo European Journal of Operational Researchvol 204 no 3 pp 377ndash390 2010

[23] S A Torabi and E Hassini ldquoAn interactive possibilistic pro-gramming approach for multiple objective supply chain masterplanningrdquo Fuzzy Sets and Systems vol 159 no 2 pp 193ndash2142008

[24] B Scholz-Reiter J Hartmann T Makuschewitz and E MFrazzon ldquoA generic approach for the graph-based integratedproduction and intermodal transport scheduling with capacityrestrictionsrdquo in Procedia CIRP vol 7 pp 109ndash114 2013

[25] A D Toni andAMeneghetti ldquoProduction planning process fora network of firms in the textile-apparel industryrdquo InternationalJournal of Production Economics vol 65 no 1 pp 17ndash32 2000

[26] E M Frazzon A G N Novaes B Scholz-Reiter J Hartmannand T Makuschewitz ldquoAn approach for the rescheduling ofintegrated production and transport systemsrdquo in Proceedingsof the 4th World Conference Production amp Operations Manage-ment Serving the World Production amp Operations ManagementSociety p 10 Amsterdam The Netherlands 2012

[27] H Ishikura ldquoStudy on the production planning of apparel prod-ucts determining optimal production times and quantitiesrdquoComputers and Industrial Engineering vol 27 no 1ndash4 pp 19ndash22 1994

[28] Y W Tan J B Guan and H R Karimi ldquoThe Impact ofthe subsidy policy on total factor productivity an empiricalanalysis of Chinarsquos cotton productionrdquo Mathematical Problemsin Engineering vol 2013 Article ID 248537 8 pages 2013

[29] H R Karimi N A Duffie and S Dashkovskiy ldquoLocalcapacity Hinfin control for production networks of autonomouswork systems with time-varying delaysrdquo IEEE Transactions onAutomation Science and Engineering vol 7 no 4 pp 849ndash8572010

[30] S Dashkovskiy H R Karimi and M Kosmykov ldquoStabilityanalysis of logistics networks with time-delaysrdquo Journal ofProduction Planning amp Control vol 24 no 7 pp 567ndash574 2013

[31] A Mehrsai H R Karimi and B Scholz-Reiter ldquoApplication oflearning pallets for real-time scheduling by the use of radialbasis function networkrdquo Neurocomputing vol 101 pp 82ndash932013

[32] S C H Leung S O S Tsang W L Ng and Y Wu ldquoArobust optimization model for multi-site production planningproblem in an uncertain environmentrdquo European Journal ofOperational Research vol 181 no 1 pp 224ndash238 2007

[33] A-G Quaranta and A Zaffaroni ldquoRobust optimization ofconditional value at risk and portfolio selectionrdquo Journal ofBanking and Finance vol 32 no 10 pp 2046ndash2056 2008

[34] R A Jabr ldquoRobust self-scheduling under price uncertaintyusing conditional Value-at-Riskrdquo IEEE Transactions on PowerSystems vol 20 no 4 pp 1852ndash1858 2005

[35] D Peidro J Mula R Poler and J-L Verdegay ldquoFuzzy opti-mization for supply chain planning under supply demand andprocess uncertaintiesrdquo Fuzzy Sets and Systems vol 160 no 18pp 2640ndash2657 2009

[36] F Pan and R Nagi ldquoRobust supply chain design underuncertain demand in agile manufacturingrdquo Computers andOperations Research vol 37 no 4 pp 668ndash683 2010

[37] P M Verderame and C A Floudas ldquoOperational planning oflarge-scale industrial batch plants under demand due date andamount uncertainty II Conditional value-at-risk frameworkrdquoIndustrial and Engineering Chemistry Research vol 49 no 1 pp260ndash275 2010

[38] P M Verderame and C A Floudas ldquoMultisite planning underdemand and transportation time uncertainty Robust optimiza-tion and conditional value-at-risk frameworksrdquo Industrial andEngineering Chemistry Research vol 50 no 9 pp 4959ndash49822011

[39] S M J Mirzapour Al-E-Hashem H Malekly and M BAryanezhad ldquoA multi-objective robust optimization modelfor multi-product multi-site aggregate production planningin a supply chain under uncertaintyrdquo International Journal ofProduction Economics vol 134 no 1 pp 28ndash42 2011

[40] D Rahmani R Ramezanian P Fattahi and M HeydarildquoA robust optimization model for multi-product two-stagecapacitated production planning under uncertaintyrdquo AppliedMathematical Modelling vol 37 no 20-21 pp 8957ndash8971 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

PF119894is penalty cost incurred to each late delivery for

product 119894 when it is supplied later than the deliverydate DD

119894

CAP119894119895is production capacity (units) for product 119894

during a period in facility 119895120572 is confidence level of risk parameter CVaR where120572 isin [0 1[120573 is risk parameter120582 is weight presented on solution variance120583 is weight placed on model infeasibility whichcontrols the trade-off between solution and modelrobustness

Variables Consider the following

119883119894is quantity of product 119894 that shall be produced

(decision variable)119884119894119895is binary variable which indicates whether prod-

uct 119894 is produced in facility 119895 (decision variable)STP119894is start period of the production of product 119894

(decision integer variable)TP119894is time required in periods to produce 119883

119894of

product 119894 TP119894= sum119898

119895=1119884119894119895(119883119894CAP

119894119895)

ExpEnd119875119894is expected delivery date of product 119894 in

period unit ExpEnd119875119894= STP

119894+ TP119894+ sum119898

119895=1119884119894119895sdot TT119895

MIN119904119894is quantity of product 119894 effectively sold under

scenario 119904 MIN119904119894= min(119883

119894 119863119904119894)

WAS119904119894is wastage quantity of product 119894 under scenario

119904 WAS119904119894= max(0 119883

119894minus 119863119904119894)

SHO119904119894is shortage quantity of product 119894 under scenario

119904 SHO119904119894= max(0 119863119904

119894minus 119883119894)

119868119894is binary variable which indicates whether the

product 119894 has to be held or not after its productionand transportation

119868119894=

1 if ExpEnd119875119894lt DD

119894

0 else(1)

Objective Function The objective function has the followingcomponents

(1) sales revenue(2) production cost(3) transport cost(4) holding cost(5) penalty cost due to a late delivery(6) risk cost incurred for possible wastage cost(7) risk cost incurred for possible shortage cost

The above components have been considered due to the factthat they are the most important costs which can be mainlyaffected by the decision variable of the model

The objective is to maximize the total profit revenueconsisting of sales revenue gained by selling the productsproduction (PC) transport (TC) penalty holding (HC) andother risk costs which can be incurred for possible wastage orshortage quantities

SR119904 (sales revenue) =119899

sum119894=1

MIN119904119894sdot SP119894 (2)

PC (production cost) =119899

sum119894=1

119898

sum119895=1

119884119894119895X119894CP119894119895 (3)

TC (transport cost) =119899

sum119894=1

119898

sum119895=1

119884119894119895119883119894CT119895 (4)

HC (holding cost) =119899

sum119894=1

119868119894sdot (DD

119894minus ExpEnd119875

119894) sdot 119883119894CH119894

(5)

PEC (penalty cost)

=119899

sum119894=1

(1 minus 119868119894) (ExpEnd119875

119894minus DD

119894)119883119894PF119894

(6)

WC119904 (wastage cost) =119899

sum119894=1

WAS119904119894sdot SL119894 (7)

SC119904 (shortage cost) =119899

sum119894=1

SHO119904119894sdot SR119894 (8)

Equation (2) represents the total sales revenue due to sellingthe set of products 119868 Equation (3) is the total productioncost of manufacturing the set of products 119868 Equation (4) isthe total transport cost of the set products 119868 Equation (5)corresponds to the total holding cost of units of productswhich has to be stored in the warehouses for a determinedholding period Equation (6) represents the total penalty costof supplying the set of products 119868 later than the appropriatedelivery date Equation (7) represents the wastage cost toliquidate the overstocks set of products 119868 Equation (8)represents an artificial penalty for the demand dissatisfactionand subsequently to loss of opportunity revenue

The following formulation of robust problem is definedaccording to Leung et al [32]

We denote the profit function for each scenario 119904 with119865119904= [SR119904 + WC119904 minus SC119904 minus PC minus TC minus HC minus PEC] (9)

The objective function which represents the maximizingexpected profit of the production planning problem with thedemand uncertainty is formulated as follows

max119864 [profit]

= maxΩ

sum119904=1

119901119904119865119904+ 120582Ω

sum119904=1

119901119904[119865119904minusΩ

sum1199041015840=1

11990111990410158401198651199041015840]

+120583Ω

sum119904=1

119901119904120579119904

(10)

6 Mathematical Problems in Engineering

The first term of (10) is the mean value of the total profitThesecond term of (10) denotes the measure solution robustnessof the model The third term in (10) represents the modelrobustness and is used to penalize model infeasibility

Constraints First we define the loss function as follows

forall119904 isin Ω 119891 (119883 119904) =119899

sum119894=1

119891 (119883119894 119904) = WC119904 + SC119904 minus 120573 sdot SR119904

(11)

The objective function is subject to the following constraints

forall119894 isin 119868 ExpEnd119875119894le 119896 (12)

forall119894 isin 119868119898

sum119895=1

119884119894119895

= 1 (13)

forall119894 isin 119868 STP119894gt 0 (14)

forall119894 1198941015840 isin 119868 if exist120583 isin 119869 so that 119884119894120583

= 1198841198941015840120583= 1

STP119894lt STP

1198941015840 then STP

1198941015840 ge STP

119894+ TP119894

(15)

forall119904 isin Ω 119875 [119904 | 119891 (119883 119904) le 0] le 120572 (16)

Constraint (12) ensures that the expected delivery date forall products should not exceed the length of the planninghorizon 119896 = |119879| Constraint (13) guarantees that theproduction of a product 119894 will take place only in one facility119895 Constraint (14) indicates that the start of production ofa product 119894 is positive and integer Constraint (15) ensuresthat the production of more than one product cannot becarried out in a parallel manner Constraint (16) restricts theprobability of the loss function to be negative for 120572 sdot 100which means that the undesirable realization of uncertaintiescan restrict the loss function to be positive in only (1 minus 120572) sdot100 As an example for 120572 = 095 the constraint (16) has tobe satisfied for 95 for all possible realizations of demand119863119904

119894

for all 119894 isin 119868 and for all 119904 isin Ω

5 Experimental Results for a Case Studywith Three Scenarios

In order to cover different potential realizations of customerdemand three scenarios have been selected including a ldquopes-simisticrdquo scenario characterized by assumed low customerdemand a ldquonormalrdquo scenario characterized by average cus-tomer demand and an ldquooptimisticrdquo scenario characterized byrelatively high customer demand These three scenarios takeinto account the insecurity concerning customer demand forthe products Different values have been chosen based on datataken from a real case study

In this section we analyze the behavior of the modelby means of a case study based scenario The input dataparameters of the model are summarized in Tables 1 and 2

Three different scenarios are considered as a basis forapplication of the model and its numerical results Thefirst scenario ldquopessimistic scenariordquo illustrates low demand

Table 1 Input parameter used to solve the model

Parameters Inputs119896 = |119879| 10119898 = |119869| 2119899 = |119868| 4Transport cost from facility 1(China)

300 (pro 1000product units)

Transport cost from facility 2(Turkey)

100 (pro 1000product units)

Transport time from facility 1(China) 3

Transport time from facility 2(Turkey) 1

120572 095120573 01

Table 2 Unit production parameters

Product 1 Product 2 Product 3 Product 4Production

Minimum value 10000 7500 5000 8000Maximum value 40000 30750 20750 31600

Production costFacility 1 1 1 2 2Facility 2 3 2 3 5

CapacityFacility 1 20000 20000 20000 20000Facility 2 10000 10000 150000 10000

Penalty factor PF119894

01 02 08 05Delivery due date DD

1198946 6 6 6

Shortage cost unit 5 4 5 9Salvage cost unit 05 05 1 1Unit selling price 8 6 8 14

forecasts for the next season The second scenario ldquonormalscenariordquo represents the case when the customer demandis normal The third scenario ldquooptimistic scenariordquo assumeshigh customer demand To characterize the uncertaintiesof demand forecast of different products a distributionprobability is applied Figures 1 and 2 show an example ofthe demand distribution respectively of the products underpessimistic scenario and demand distribution of the product1 for the three scenarios

The risk to be considered from solving the model canoccur only because of wastage or shortage possibility There-fore according to our definition of the loss function119891(119883 119904) =sum119899

119894=1119891(119883119894 119904) = WC119904 + SC119904 minus 01 sdot SR119904 the constraint (16)

for this case study ensures that the probability that the riskcosts incurring by respectively wastage and shortage costs donot exceed the 10 (120573 = 01) of the expected profit revenuein 95 (120572 = 095) of all possible realization of stochasticparameters

The model has been efficiently solved by means offrontline risk solver which supports the robust optimizationthrough CVaR

Mathematical Problems in Engineering 7

025

02

015

01

005

0

11

12

135 15

17

18

195 21

22

235 25

265

275 29

31

32

Product 1

Product 1

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

6 8 9

105

115

125

135

145

155

165

175

185 20

21

22

23

24

Product 2

Product 2025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

55 6 7

75

85 9

95

105 11

12

125 13

14

145

155 16

Product 3

Product 3

(c)

025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

9

10

11

12

13

14

155 17

175 19

20

21

22

235

245 26

Product 4

Product 4

(d)

Figure 1 Demand distribution of the products in a pessimistic scenario

11

12

135 15

165 18

19

21

22

235 25

265

275 29

31

32

Pessimistic scenario

Pessimistic scenario025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

(Values in thousands)

Rela

tive p

roba

bilit

y

12

13

15

16

175 19

21

225 24

25

27

28

30

315 33

34

36

Normal scenario

Normal scenario

012

01

008

006

004

002

0

014

(b)

(Values in thousands)

Rela

tive p

roba

bilit

y

02

018

016

014

012

01

008

006

004

002

0

19

205

215

225 24

25

265 28

29

305 32

33

345

355 37

38

395

Optimistic scenario

Optimistic scenario

(c)

Figure 2 Demand distribution of product 1 in different scenarios

51 Pessimistic Scenario Table 3 lists the production plan ina pessimistic scenario The expected average profit of theproduction for the next season is about 219000 Moreoverthe solution can guarantee that the loss function will benegative in 96 cases of all possible realizations of stochasticparameters (Figure 3)

52 Normal Scenario In the normal scenario the demandof product 1 and product 4 respectively has nearly doubledwhereas the demand of product 2 and product 3 has beenslightly increased This is due to the low penalty factorof product 1 and product 4 which can be incurred foreach delayed delivery and for the capacity restriction of

8 Mathematical Problems in Engineering

012

01

008

006

004

002

0

120

130

150

160

180

190

200

220

230

240

260

270

280

300

310

320

(Values in thousands)

Rela

tive p

roba

bilit

y

Objective fct pessimistic

Objective fct pessimistic

(a)

(Values in thousands)

Rela

tive p

roba

bilit

y

016

014

012

01

008

006

004

002

0

minus25

minus23

minus21

minus19

minus17

minus15

minus13

minus11

minus8

minus6

minus5 3 0 2 3 5

Loss fct pessimistic

Loss fct pessimistic

(b)

Figure 3 Profit and loss function distribution in a pessimistic scenario

018

016

014

012

01

008

006

004

002

0

100

150

170

220

240

270

300

330

370

390

430

450

470

530

550

580

Objective fct normal

Objective fct normal

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

016

014

012

01

008

006

004

002

0

minus40

minus35

minus30

minus27

minus23

minus17

minus13

minus10

minus5 0 3 7

13

17

22

27

Loss fct normal

Loss fct normal

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

Figure 4 Profit and loss function distribution in a normal scenario

Table 3 Production plan with the demand quantities in a pes-simistic scenario

Product 1 Product 2 Product 3 Product 4Demand nextseason 17969 16549 10169 14945

Start of production 3 2 1 1Expected deliverydate 6 6 5 4

Produced InFacility 1 0 1 1 1Facility 2 1 0 0 0

the facilities (Table 4) In this case the expected averageprofit is about 450000 and the loss function is negative in975 of all possible realizations of the stochastic parameters(Figure 4)

53 Optimistic Scenario Since the optimistic scenario pre-dicts a boom economic scenario the production reaches its

Table 4 Production plan with the demand quantities in a normalscenario

Product 1 Product 2 Product 3 Product 4Demand next season 32355 19682 14415 26284Start of production 1 4 3 1Expected delivery date 6 8 7 6Produced in

Facility 1 0 1 1 1Facility 2 1 0 0 0

maximum (Table 5) In this case the average profit is about665000 and the loss function is satisfied with 991 of allpossible realizations of the stochastic parameters (Figure 5)

6 Conclusion and Outlook

In this paper we have presented a risk-constrained profit-expected maximization model for a textile industry scenario

Mathematical Problems in Engineering 9

045

04

035

03

025

02

015

01

005

0

225

250

275

325

350

400

425

475

500

525

575

600

625

675

625

750

Objective fct optimistic

Objective fct optimistic

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

04

035

03

025

02

015

01

005

0

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

minus2 2 7

11

16

20

Loss fct optimistic

Loss fct optimistic

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

Figure 5 Profit and loss function distribution in an optimistic scenario

Table 5 Production plan with the demand quantities in an opti-mistic scenario

Product 1 Product 2 Product 3 Product 4Demand next season 37933 29312 19410 31600Start of production 1 4 3 1Expected delivery date 6 9 7 6Produced in

Facility 1 0 1 1 1Facility 2 1 0 0 0

The forecast demand uncertainty has been explicitly con-sidered in the model by means of the robust optimizationand conditional value at risk theory by introducing andrestricting a loss function The robust optimization modeloriginally proposed by [32] was adopted as the benchmarkformulation of uncertainty considered in this paper A casestudy has been used to demonstrate the viability of theproposed mathematical model Results indicate that theproduction planning model using CVaR as the risk measureperforms robustly and provides flexibility in decision analysisbetween different scenarios The next step will be the furtherdevelopment of the profit maximization model to considerother types of uncertainties for example concerning theavailability of the production plants transportation meansand demand due dates

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Datamonitor Global Apparel Retail 2010 httpwwwdata-monitorcom

[2] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of Banking and Finance vol26 no 7 pp 1443ndash1471 2002

[3] S Zhu and M Fukushima ldquoWorst-case conditional value-at-risk with application to robust portfolio managementrdquo Opera-tions Research vol 57 no 5 pp 1155ndash1168 2009

[4] A Bruckner and S Muller Supply Chain Management in derBekleidungsindustrie Forschungsstelle der Bekleidungsindus-trie Cologne Germany 2003

[5] J Fissahn Marktorientierte Beschaffung in der Bekleidungsin-dustrie Munster University Munster Germany edition 2001

[6] A Schneider Internationalisierungsstrategien in der deutschenTextil-und Bekleidungsindustrie-eine empirische Untersuchung[MS dissertation] RWTH Aachen University Aachen Ger-many 2003

[7] G Gereffi and O Memedovic The Global Apparel ValueChainmdashWhat Prospects for Upgrading by Developing CountriesUNIDO Vienna Austria 2003

[8] MGromling and JMatthesGlobalisierung und Strukturwandelder Deutschen Textil-und Bekleidungsindustrie Dt Inst-VerlagColougne Germany 2003

[9] I Langenhorst Shop-logistik in der bekleidungswirtschaft eineanalyse der anforderungen herstellerinitiierter shop-systeme andie logistikprozesse der bekleidungsindustrie [PhD thesis]Munster University 1999

[10] H-C Pfohl M Gomm and X Shen China Textil-undBekleidungs-Supply Chain Zwischen Deutschland und ChinaFree Beratung Korschenbroich Germany in H Wolf-Kluthausen Jahrbuch der Logistik 2007

[11] M Teucke and B Scholz-Reiter ldquoImproving order allocationin fashion supply chains using radio frequency identification(RFID) technologiesrdquo in Fashion Supply Chain ManagementUsing Radio Frequency Identification (RFID) Technologies WK Wong and Z X Guo Eds vol 152 of Woodhead PublishingSeries in Textiles Woodhead Publishing Cambridge UK 2014

[12] S Thomassey ldquoSales forecasts in clothing industry the keysuccess factor of the supply chain managementrdquo InternationalJournal of Production Economics vol 128 no 2 pp 470ndash4832010

[13] D Ahlert and G Dieckheuer Marktorientierte Beschaffung inder Bekleidungsindustrie FATM Munster Germany 2001

[14] J T Mentzer C C Bienstock and K B Kahn ldquoBenchmarkingsales forecastingmanagementrdquo Business Horizons vol 42 no 3pp 48ndash56 1999

10 Mathematical Problems in Engineering

[15] J Quick M Rinis C Schmidt and B Walber SupplyTexmdashErfolgreiches Supply-Management in KMU Der Textil- UndBekleidungsindustrie Forschungsinstitut fur Rationalisierung(FIR) eV an der Universitat Aachen Aachen Germany 2010

[16] Z-L Sun T-M Choi K-F Au and Y Yu ldquoSales forecastingusing extreme learning machine with applications in fashionretailingrdquo Decision Support Systems vol 46 no 1 pp 411ndash4192008

[17] S Thomassey M Happiette and J-M Castelain ldquoA globalforecasting support system adapted to textile distributionrdquoInternational Journal of Production Economics vol 96 no 1 pp81ndash95 2005

[18] J Mostard R Teunter and R de Koster ldquoForecasting demandfor single-period products a case study in the apparel industryrdquoEuropean Journal of Operational Research vol 211 no 1 pp 139ndash147 2011

[19] K Hoyndorff S Hulsmann D Spee and M ten HompelFashion Logistics Grundlagen Uber Prozesse und IT Entlang derSupply Chain Munich Huss-Verlag GmbH 2010

[20] J Richardson ldquoVertical integration and rapid response infashion apparelrdquoOrganization Science vol 7 no 4 pp 400ndash4121996

[21] Z-L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[22] J Mula D Peidro M Dıaz-Madronero and E Vicens ldquoMathe-matical programming models for supply chain production andtransport planningrdquo European Journal of Operational Researchvol 204 no 3 pp 377ndash390 2010

[23] S A Torabi and E Hassini ldquoAn interactive possibilistic pro-gramming approach for multiple objective supply chain masterplanningrdquo Fuzzy Sets and Systems vol 159 no 2 pp 193ndash2142008

[24] B Scholz-Reiter J Hartmann T Makuschewitz and E MFrazzon ldquoA generic approach for the graph-based integratedproduction and intermodal transport scheduling with capacityrestrictionsrdquo in Procedia CIRP vol 7 pp 109ndash114 2013

[25] A D Toni andAMeneghetti ldquoProduction planning process fora network of firms in the textile-apparel industryrdquo InternationalJournal of Production Economics vol 65 no 1 pp 17ndash32 2000

[26] E M Frazzon A G N Novaes B Scholz-Reiter J Hartmannand T Makuschewitz ldquoAn approach for the rescheduling ofintegrated production and transport systemsrdquo in Proceedingsof the 4th World Conference Production amp Operations Manage-ment Serving the World Production amp Operations ManagementSociety p 10 Amsterdam The Netherlands 2012

[27] H Ishikura ldquoStudy on the production planning of apparel prod-ucts determining optimal production times and quantitiesrdquoComputers and Industrial Engineering vol 27 no 1ndash4 pp 19ndash22 1994

[28] Y W Tan J B Guan and H R Karimi ldquoThe Impact ofthe subsidy policy on total factor productivity an empiricalanalysis of Chinarsquos cotton productionrdquo Mathematical Problemsin Engineering vol 2013 Article ID 248537 8 pages 2013

[29] H R Karimi N A Duffie and S Dashkovskiy ldquoLocalcapacity Hinfin control for production networks of autonomouswork systems with time-varying delaysrdquo IEEE Transactions onAutomation Science and Engineering vol 7 no 4 pp 849ndash8572010

[30] S Dashkovskiy H R Karimi and M Kosmykov ldquoStabilityanalysis of logistics networks with time-delaysrdquo Journal ofProduction Planning amp Control vol 24 no 7 pp 567ndash574 2013

[31] A Mehrsai H R Karimi and B Scholz-Reiter ldquoApplication oflearning pallets for real-time scheduling by the use of radialbasis function networkrdquo Neurocomputing vol 101 pp 82ndash932013

[32] S C H Leung S O S Tsang W L Ng and Y Wu ldquoArobust optimization model for multi-site production planningproblem in an uncertain environmentrdquo European Journal ofOperational Research vol 181 no 1 pp 224ndash238 2007

[33] A-G Quaranta and A Zaffaroni ldquoRobust optimization ofconditional value at risk and portfolio selectionrdquo Journal ofBanking and Finance vol 32 no 10 pp 2046ndash2056 2008

[34] R A Jabr ldquoRobust self-scheduling under price uncertaintyusing conditional Value-at-Riskrdquo IEEE Transactions on PowerSystems vol 20 no 4 pp 1852ndash1858 2005

[35] D Peidro J Mula R Poler and J-L Verdegay ldquoFuzzy opti-mization for supply chain planning under supply demand andprocess uncertaintiesrdquo Fuzzy Sets and Systems vol 160 no 18pp 2640ndash2657 2009

[36] F Pan and R Nagi ldquoRobust supply chain design underuncertain demand in agile manufacturingrdquo Computers andOperations Research vol 37 no 4 pp 668ndash683 2010

[37] P M Verderame and C A Floudas ldquoOperational planning oflarge-scale industrial batch plants under demand due date andamount uncertainty II Conditional value-at-risk frameworkrdquoIndustrial and Engineering Chemistry Research vol 49 no 1 pp260ndash275 2010

[38] P M Verderame and C A Floudas ldquoMultisite planning underdemand and transportation time uncertainty Robust optimiza-tion and conditional value-at-risk frameworksrdquo Industrial andEngineering Chemistry Research vol 50 no 9 pp 4959ndash49822011

[39] S M J Mirzapour Al-E-Hashem H Malekly and M BAryanezhad ldquoA multi-objective robust optimization modelfor multi-product multi-site aggregate production planningin a supply chain under uncertaintyrdquo International Journal ofProduction Economics vol 134 no 1 pp 28ndash42 2011

[40] D Rahmani R Ramezanian P Fattahi and M HeydarildquoA robust optimization model for multi-product two-stagecapacitated production planning under uncertaintyrdquo AppliedMathematical Modelling vol 37 no 20-21 pp 8957ndash8971 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

The first term of (10) is the mean value of the total profitThesecond term of (10) denotes the measure solution robustnessof the model The third term in (10) represents the modelrobustness and is used to penalize model infeasibility

Constraints First we define the loss function as follows

forall119904 isin Ω 119891 (119883 119904) =119899

sum119894=1

119891 (119883119894 119904) = WC119904 + SC119904 minus 120573 sdot SR119904

(11)

The objective function is subject to the following constraints

forall119894 isin 119868 ExpEnd119875119894le 119896 (12)

forall119894 isin 119868119898

sum119895=1

119884119894119895

= 1 (13)

forall119894 isin 119868 STP119894gt 0 (14)

forall119894 1198941015840 isin 119868 if exist120583 isin 119869 so that 119884119894120583

= 1198841198941015840120583= 1

STP119894lt STP

1198941015840 then STP

1198941015840 ge STP

119894+ TP119894

(15)

forall119904 isin Ω 119875 [119904 | 119891 (119883 119904) le 0] le 120572 (16)

Constraint (12) ensures that the expected delivery date forall products should not exceed the length of the planninghorizon 119896 = |119879| Constraint (13) guarantees that theproduction of a product 119894 will take place only in one facility119895 Constraint (14) indicates that the start of production ofa product 119894 is positive and integer Constraint (15) ensuresthat the production of more than one product cannot becarried out in a parallel manner Constraint (16) restricts theprobability of the loss function to be negative for 120572 sdot 100which means that the undesirable realization of uncertaintiescan restrict the loss function to be positive in only (1 minus 120572) sdot100 As an example for 120572 = 095 the constraint (16) has tobe satisfied for 95 for all possible realizations of demand119863119904

119894

for all 119894 isin 119868 and for all 119904 isin Ω

5 Experimental Results for a Case Studywith Three Scenarios

In order to cover different potential realizations of customerdemand three scenarios have been selected including a ldquopes-simisticrdquo scenario characterized by assumed low customerdemand a ldquonormalrdquo scenario characterized by average cus-tomer demand and an ldquooptimisticrdquo scenario characterized byrelatively high customer demand These three scenarios takeinto account the insecurity concerning customer demand forthe products Different values have been chosen based on datataken from a real case study

In this section we analyze the behavior of the modelby means of a case study based scenario The input dataparameters of the model are summarized in Tables 1 and 2

Three different scenarios are considered as a basis forapplication of the model and its numerical results Thefirst scenario ldquopessimistic scenariordquo illustrates low demand

Table 1 Input parameter used to solve the model

Parameters Inputs119896 = |119879| 10119898 = |119869| 2119899 = |119868| 4Transport cost from facility 1(China)

300 (pro 1000product units)

Transport cost from facility 2(Turkey)

100 (pro 1000product units)

Transport time from facility 1(China) 3

Transport time from facility 2(Turkey) 1

120572 095120573 01

Table 2 Unit production parameters

Product 1 Product 2 Product 3 Product 4Production

Minimum value 10000 7500 5000 8000Maximum value 40000 30750 20750 31600

Production costFacility 1 1 1 2 2Facility 2 3 2 3 5

CapacityFacility 1 20000 20000 20000 20000Facility 2 10000 10000 150000 10000

Penalty factor PF119894

01 02 08 05Delivery due date DD

1198946 6 6 6

Shortage cost unit 5 4 5 9Salvage cost unit 05 05 1 1Unit selling price 8 6 8 14

forecasts for the next season The second scenario ldquonormalscenariordquo represents the case when the customer demandis normal The third scenario ldquooptimistic scenariordquo assumeshigh customer demand To characterize the uncertaintiesof demand forecast of different products a distributionprobability is applied Figures 1 and 2 show an example ofthe demand distribution respectively of the products underpessimistic scenario and demand distribution of the product1 for the three scenarios

The risk to be considered from solving the model canoccur only because of wastage or shortage possibility There-fore according to our definition of the loss function119891(119883 119904) =sum119899

119894=1119891(119883119894 119904) = WC119904 + SC119904 minus 01 sdot SR119904 the constraint (16)

for this case study ensures that the probability that the riskcosts incurring by respectively wastage and shortage costs donot exceed the 10 (120573 = 01) of the expected profit revenuein 95 (120572 = 095) of all possible realization of stochasticparameters

The model has been efficiently solved by means offrontline risk solver which supports the robust optimizationthrough CVaR

Mathematical Problems in Engineering 7

025

02

015

01

005

0

11

12

135 15

17

18

195 21

22

235 25

265

275 29

31

32

Product 1

Product 1

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

6 8 9

105

115

125

135

145

155

165

175

185 20

21

22

23

24

Product 2

Product 2025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

55 6 7

75

85 9

95

105 11

12

125 13

14

145

155 16

Product 3

Product 3

(c)

025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

9

10

11

12

13

14

155 17

175 19

20

21

22

235

245 26

Product 4

Product 4

(d)

Figure 1 Demand distribution of the products in a pessimistic scenario

11

12

135 15

165 18

19

21

22

235 25

265

275 29

31

32

Pessimistic scenario

Pessimistic scenario025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

(Values in thousands)

Rela

tive p

roba

bilit

y

12

13

15

16

175 19

21

225 24

25

27

28

30

315 33

34

36

Normal scenario

Normal scenario

012

01

008

006

004

002

0

014

(b)

(Values in thousands)

Rela

tive p

roba

bilit

y

02

018

016

014

012

01

008

006

004

002

0

19

205

215

225 24

25

265 28

29

305 32

33

345

355 37

38

395

Optimistic scenario

Optimistic scenario

(c)

Figure 2 Demand distribution of product 1 in different scenarios

51 Pessimistic Scenario Table 3 lists the production plan ina pessimistic scenario The expected average profit of theproduction for the next season is about 219000 Moreoverthe solution can guarantee that the loss function will benegative in 96 cases of all possible realizations of stochasticparameters (Figure 3)

52 Normal Scenario In the normal scenario the demandof product 1 and product 4 respectively has nearly doubledwhereas the demand of product 2 and product 3 has beenslightly increased This is due to the low penalty factorof product 1 and product 4 which can be incurred foreach delayed delivery and for the capacity restriction of

8 Mathematical Problems in Engineering

012

01

008

006

004

002

0

120

130

150

160

180

190

200

220

230

240

260

270

280

300

310

320

(Values in thousands)

Rela

tive p

roba

bilit

y

Objective fct pessimistic

Objective fct pessimistic

(a)

(Values in thousands)

Rela

tive p

roba

bilit

y

016

014

012

01

008

006

004

002

0

minus25

minus23

minus21

minus19

minus17

minus15

minus13

minus11

minus8

minus6

minus5 3 0 2 3 5

Loss fct pessimistic

Loss fct pessimistic

(b)

Figure 3 Profit and loss function distribution in a pessimistic scenario

018

016

014

012

01

008

006

004

002

0

100

150

170

220

240

270

300

330

370

390

430

450

470

530

550

580

Objective fct normal

Objective fct normal

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

016

014

012

01

008

006

004

002

0

minus40

minus35

minus30

minus27

minus23

minus17

minus13

minus10

minus5 0 3 7

13

17

22

27

Loss fct normal

Loss fct normal

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

Figure 4 Profit and loss function distribution in a normal scenario

Table 3 Production plan with the demand quantities in a pes-simistic scenario

Product 1 Product 2 Product 3 Product 4Demand nextseason 17969 16549 10169 14945

Start of production 3 2 1 1Expected deliverydate 6 6 5 4

Produced InFacility 1 0 1 1 1Facility 2 1 0 0 0

the facilities (Table 4) In this case the expected averageprofit is about 450000 and the loss function is negative in975 of all possible realizations of the stochastic parameters(Figure 4)

53 Optimistic Scenario Since the optimistic scenario pre-dicts a boom economic scenario the production reaches its

Table 4 Production plan with the demand quantities in a normalscenario

Product 1 Product 2 Product 3 Product 4Demand next season 32355 19682 14415 26284Start of production 1 4 3 1Expected delivery date 6 8 7 6Produced in

Facility 1 0 1 1 1Facility 2 1 0 0 0

maximum (Table 5) In this case the average profit is about665000 and the loss function is satisfied with 991 of allpossible realizations of the stochastic parameters (Figure 5)

6 Conclusion and Outlook

In this paper we have presented a risk-constrained profit-expected maximization model for a textile industry scenario

Mathematical Problems in Engineering 9

045

04

035

03

025

02

015

01

005

0

225

250

275

325

350

400

425

475

500

525

575

600

625

675

625

750

Objective fct optimistic

Objective fct optimistic

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

04

035

03

025

02

015

01

005

0

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

minus2 2 7

11

16

20

Loss fct optimistic

Loss fct optimistic

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

Figure 5 Profit and loss function distribution in an optimistic scenario

Table 5 Production plan with the demand quantities in an opti-mistic scenario

Product 1 Product 2 Product 3 Product 4Demand next season 37933 29312 19410 31600Start of production 1 4 3 1Expected delivery date 6 9 7 6Produced in

Facility 1 0 1 1 1Facility 2 1 0 0 0

The forecast demand uncertainty has been explicitly con-sidered in the model by means of the robust optimizationand conditional value at risk theory by introducing andrestricting a loss function The robust optimization modeloriginally proposed by [32] was adopted as the benchmarkformulation of uncertainty considered in this paper A casestudy has been used to demonstrate the viability of theproposed mathematical model Results indicate that theproduction planning model using CVaR as the risk measureperforms robustly and provides flexibility in decision analysisbetween different scenarios The next step will be the furtherdevelopment of the profit maximization model to considerother types of uncertainties for example concerning theavailability of the production plants transportation meansand demand due dates

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Datamonitor Global Apparel Retail 2010 httpwwwdata-monitorcom

[2] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of Banking and Finance vol26 no 7 pp 1443ndash1471 2002

[3] S Zhu and M Fukushima ldquoWorst-case conditional value-at-risk with application to robust portfolio managementrdquo Opera-tions Research vol 57 no 5 pp 1155ndash1168 2009

[4] A Bruckner and S Muller Supply Chain Management in derBekleidungsindustrie Forschungsstelle der Bekleidungsindus-trie Cologne Germany 2003

[5] J Fissahn Marktorientierte Beschaffung in der Bekleidungsin-dustrie Munster University Munster Germany edition 2001

[6] A Schneider Internationalisierungsstrategien in der deutschenTextil-und Bekleidungsindustrie-eine empirische Untersuchung[MS dissertation] RWTH Aachen University Aachen Ger-many 2003

[7] G Gereffi and O Memedovic The Global Apparel ValueChainmdashWhat Prospects for Upgrading by Developing CountriesUNIDO Vienna Austria 2003

[8] MGromling and JMatthesGlobalisierung und Strukturwandelder Deutschen Textil-und Bekleidungsindustrie Dt Inst-VerlagColougne Germany 2003

[9] I Langenhorst Shop-logistik in der bekleidungswirtschaft eineanalyse der anforderungen herstellerinitiierter shop-systeme andie logistikprozesse der bekleidungsindustrie [PhD thesis]Munster University 1999

[10] H-C Pfohl M Gomm and X Shen China Textil-undBekleidungs-Supply Chain Zwischen Deutschland und ChinaFree Beratung Korschenbroich Germany in H Wolf-Kluthausen Jahrbuch der Logistik 2007

[11] M Teucke and B Scholz-Reiter ldquoImproving order allocationin fashion supply chains using radio frequency identification(RFID) technologiesrdquo in Fashion Supply Chain ManagementUsing Radio Frequency Identification (RFID) Technologies WK Wong and Z X Guo Eds vol 152 of Woodhead PublishingSeries in Textiles Woodhead Publishing Cambridge UK 2014

[12] S Thomassey ldquoSales forecasts in clothing industry the keysuccess factor of the supply chain managementrdquo InternationalJournal of Production Economics vol 128 no 2 pp 470ndash4832010

[13] D Ahlert and G Dieckheuer Marktorientierte Beschaffung inder Bekleidungsindustrie FATM Munster Germany 2001

[14] J T Mentzer C C Bienstock and K B Kahn ldquoBenchmarkingsales forecastingmanagementrdquo Business Horizons vol 42 no 3pp 48ndash56 1999

10 Mathematical Problems in Engineering

[15] J Quick M Rinis C Schmidt and B Walber SupplyTexmdashErfolgreiches Supply-Management in KMU Der Textil- UndBekleidungsindustrie Forschungsinstitut fur Rationalisierung(FIR) eV an der Universitat Aachen Aachen Germany 2010

[16] Z-L Sun T-M Choi K-F Au and Y Yu ldquoSales forecastingusing extreme learning machine with applications in fashionretailingrdquo Decision Support Systems vol 46 no 1 pp 411ndash4192008

[17] S Thomassey M Happiette and J-M Castelain ldquoA globalforecasting support system adapted to textile distributionrdquoInternational Journal of Production Economics vol 96 no 1 pp81ndash95 2005

[18] J Mostard R Teunter and R de Koster ldquoForecasting demandfor single-period products a case study in the apparel industryrdquoEuropean Journal of Operational Research vol 211 no 1 pp 139ndash147 2011

[19] K Hoyndorff S Hulsmann D Spee and M ten HompelFashion Logistics Grundlagen Uber Prozesse und IT Entlang derSupply Chain Munich Huss-Verlag GmbH 2010

[20] J Richardson ldquoVertical integration and rapid response infashion apparelrdquoOrganization Science vol 7 no 4 pp 400ndash4121996

[21] Z-L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[22] J Mula D Peidro M Dıaz-Madronero and E Vicens ldquoMathe-matical programming models for supply chain production andtransport planningrdquo European Journal of Operational Researchvol 204 no 3 pp 377ndash390 2010

[23] S A Torabi and E Hassini ldquoAn interactive possibilistic pro-gramming approach for multiple objective supply chain masterplanningrdquo Fuzzy Sets and Systems vol 159 no 2 pp 193ndash2142008

[24] B Scholz-Reiter J Hartmann T Makuschewitz and E MFrazzon ldquoA generic approach for the graph-based integratedproduction and intermodal transport scheduling with capacityrestrictionsrdquo in Procedia CIRP vol 7 pp 109ndash114 2013

[25] A D Toni andAMeneghetti ldquoProduction planning process fora network of firms in the textile-apparel industryrdquo InternationalJournal of Production Economics vol 65 no 1 pp 17ndash32 2000

[26] E M Frazzon A G N Novaes B Scholz-Reiter J Hartmannand T Makuschewitz ldquoAn approach for the rescheduling ofintegrated production and transport systemsrdquo in Proceedingsof the 4th World Conference Production amp Operations Manage-ment Serving the World Production amp Operations ManagementSociety p 10 Amsterdam The Netherlands 2012

[27] H Ishikura ldquoStudy on the production planning of apparel prod-ucts determining optimal production times and quantitiesrdquoComputers and Industrial Engineering vol 27 no 1ndash4 pp 19ndash22 1994

[28] Y W Tan J B Guan and H R Karimi ldquoThe Impact ofthe subsidy policy on total factor productivity an empiricalanalysis of Chinarsquos cotton productionrdquo Mathematical Problemsin Engineering vol 2013 Article ID 248537 8 pages 2013

[29] H R Karimi N A Duffie and S Dashkovskiy ldquoLocalcapacity Hinfin control for production networks of autonomouswork systems with time-varying delaysrdquo IEEE Transactions onAutomation Science and Engineering vol 7 no 4 pp 849ndash8572010

[30] S Dashkovskiy H R Karimi and M Kosmykov ldquoStabilityanalysis of logistics networks with time-delaysrdquo Journal ofProduction Planning amp Control vol 24 no 7 pp 567ndash574 2013

[31] A Mehrsai H R Karimi and B Scholz-Reiter ldquoApplication oflearning pallets for real-time scheduling by the use of radialbasis function networkrdquo Neurocomputing vol 101 pp 82ndash932013

[32] S C H Leung S O S Tsang W L Ng and Y Wu ldquoArobust optimization model for multi-site production planningproblem in an uncertain environmentrdquo European Journal ofOperational Research vol 181 no 1 pp 224ndash238 2007

[33] A-G Quaranta and A Zaffaroni ldquoRobust optimization ofconditional value at risk and portfolio selectionrdquo Journal ofBanking and Finance vol 32 no 10 pp 2046ndash2056 2008

[34] R A Jabr ldquoRobust self-scheduling under price uncertaintyusing conditional Value-at-Riskrdquo IEEE Transactions on PowerSystems vol 20 no 4 pp 1852ndash1858 2005

[35] D Peidro J Mula R Poler and J-L Verdegay ldquoFuzzy opti-mization for supply chain planning under supply demand andprocess uncertaintiesrdquo Fuzzy Sets and Systems vol 160 no 18pp 2640ndash2657 2009

[36] F Pan and R Nagi ldquoRobust supply chain design underuncertain demand in agile manufacturingrdquo Computers andOperations Research vol 37 no 4 pp 668ndash683 2010

[37] P M Verderame and C A Floudas ldquoOperational planning oflarge-scale industrial batch plants under demand due date andamount uncertainty II Conditional value-at-risk frameworkrdquoIndustrial and Engineering Chemistry Research vol 49 no 1 pp260ndash275 2010

[38] P M Verderame and C A Floudas ldquoMultisite planning underdemand and transportation time uncertainty Robust optimiza-tion and conditional value-at-risk frameworksrdquo Industrial andEngineering Chemistry Research vol 50 no 9 pp 4959ndash49822011

[39] S M J Mirzapour Al-E-Hashem H Malekly and M BAryanezhad ldquoA multi-objective robust optimization modelfor multi-product multi-site aggregate production planningin a supply chain under uncertaintyrdquo International Journal ofProduction Economics vol 134 no 1 pp 28ndash42 2011

[40] D Rahmani R Ramezanian P Fattahi and M HeydarildquoA robust optimization model for multi-product two-stagecapacitated production planning under uncertaintyrdquo AppliedMathematical Modelling vol 37 no 20-21 pp 8957ndash8971 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

025

02

015

01

005

0

11

12

135 15

17

18

195 21

22

235 25

265

275 29

31

32

Product 1

Product 1

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

6 8 9

105

115

125

135

145

155

165

175

185 20

21

22

23

24

Product 2

Product 2025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

55 6 7

75

85 9

95

105 11

12

125 13

14

145

155 16

Product 3

Product 3

(c)

025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

9

10

11

12

13

14

155 17

175 19

20

21

22

235

245 26

Product 4

Product 4

(d)

Figure 1 Demand distribution of the products in a pessimistic scenario

11

12

135 15

165 18

19

21

22

235 25

265

275 29

31

32

Pessimistic scenario

Pessimistic scenario025

02

015

01

005

0

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

(Values in thousands)

Rela

tive p

roba

bilit

y

12

13

15

16

175 19

21

225 24

25

27

28

30

315 33

34

36

Normal scenario

Normal scenario

012

01

008

006

004

002

0

014

(b)

(Values in thousands)

Rela

tive p

roba

bilit

y

02

018

016

014

012

01

008

006

004

002

0

19

205

215

225 24

25

265 28

29

305 32

33

345

355 37

38

395

Optimistic scenario

Optimistic scenario

(c)

Figure 2 Demand distribution of product 1 in different scenarios

51 Pessimistic Scenario Table 3 lists the production plan ina pessimistic scenario The expected average profit of theproduction for the next season is about 219000 Moreoverthe solution can guarantee that the loss function will benegative in 96 cases of all possible realizations of stochasticparameters (Figure 3)

52 Normal Scenario In the normal scenario the demandof product 1 and product 4 respectively has nearly doubledwhereas the demand of product 2 and product 3 has beenslightly increased This is due to the low penalty factorof product 1 and product 4 which can be incurred foreach delayed delivery and for the capacity restriction of

8 Mathematical Problems in Engineering

012

01

008

006

004

002

0

120

130

150

160

180

190

200

220

230

240

260

270

280

300

310

320

(Values in thousands)

Rela

tive p

roba

bilit

y

Objective fct pessimistic

Objective fct pessimistic

(a)

(Values in thousands)

Rela

tive p

roba

bilit

y

016

014

012

01

008

006

004

002

0

minus25

minus23

minus21

minus19

minus17

minus15

minus13

minus11

minus8

minus6

minus5 3 0 2 3 5

Loss fct pessimistic

Loss fct pessimistic

(b)

Figure 3 Profit and loss function distribution in a pessimistic scenario

018

016

014

012

01

008

006

004

002

0

100

150

170

220

240

270

300

330

370

390

430

450

470

530

550

580

Objective fct normal

Objective fct normal

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

016

014

012

01

008

006

004

002

0

minus40

minus35

minus30

minus27

minus23

minus17

minus13

minus10

minus5 0 3 7

13

17

22

27

Loss fct normal

Loss fct normal

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

Figure 4 Profit and loss function distribution in a normal scenario

Table 3 Production plan with the demand quantities in a pes-simistic scenario

Product 1 Product 2 Product 3 Product 4Demand nextseason 17969 16549 10169 14945

Start of production 3 2 1 1Expected deliverydate 6 6 5 4

Produced InFacility 1 0 1 1 1Facility 2 1 0 0 0

the facilities (Table 4) In this case the expected averageprofit is about 450000 and the loss function is negative in975 of all possible realizations of the stochastic parameters(Figure 4)

53 Optimistic Scenario Since the optimistic scenario pre-dicts a boom economic scenario the production reaches its

Table 4 Production plan with the demand quantities in a normalscenario

Product 1 Product 2 Product 3 Product 4Demand next season 32355 19682 14415 26284Start of production 1 4 3 1Expected delivery date 6 8 7 6Produced in

Facility 1 0 1 1 1Facility 2 1 0 0 0

maximum (Table 5) In this case the average profit is about665000 and the loss function is satisfied with 991 of allpossible realizations of the stochastic parameters (Figure 5)

6 Conclusion and Outlook

In this paper we have presented a risk-constrained profit-expected maximization model for a textile industry scenario

Mathematical Problems in Engineering 9

045

04

035

03

025

02

015

01

005

0

225

250

275

325

350

400

425

475

500

525

575

600

625

675

625

750

Objective fct optimistic

Objective fct optimistic

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

04

035

03

025

02

015

01

005

0

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

minus2 2 7

11

16

20

Loss fct optimistic

Loss fct optimistic

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

Figure 5 Profit and loss function distribution in an optimistic scenario

Table 5 Production plan with the demand quantities in an opti-mistic scenario

Product 1 Product 2 Product 3 Product 4Demand next season 37933 29312 19410 31600Start of production 1 4 3 1Expected delivery date 6 9 7 6Produced in

Facility 1 0 1 1 1Facility 2 1 0 0 0

The forecast demand uncertainty has been explicitly con-sidered in the model by means of the robust optimizationand conditional value at risk theory by introducing andrestricting a loss function The robust optimization modeloriginally proposed by [32] was adopted as the benchmarkformulation of uncertainty considered in this paper A casestudy has been used to demonstrate the viability of theproposed mathematical model Results indicate that theproduction planning model using CVaR as the risk measureperforms robustly and provides flexibility in decision analysisbetween different scenarios The next step will be the furtherdevelopment of the profit maximization model to considerother types of uncertainties for example concerning theavailability of the production plants transportation meansand demand due dates

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Datamonitor Global Apparel Retail 2010 httpwwwdata-monitorcom

[2] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of Banking and Finance vol26 no 7 pp 1443ndash1471 2002

[3] S Zhu and M Fukushima ldquoWorst-case conditional value-at-risk with application to robust portfolio managementrdquo Opera-tions Research vol 57 no 5 pp 1155ndash1168 2009

[4] A Bruckner and S Muller Supply Chain Management in derBekleidungsindustrie Forschungsstelle der Bekleidungsindus-trie Cologne Germany 2003

[5] J Fissahn Marktorientierte Beschaffung in der Bekleidungsin-dustrie Munster University Munster Germany edition 2001

[6] A Schneider Internationalisierungsstrategien in der deutschenTextil-und Bekleidungsindustrie-eine empirische Untersuchung[MS dissertation] RWTH Aachen University Aachen Ger-many 2003

[7] G Gereffi and O Memedovic The Global Apparel ValueChainmdashWhat Prospects for Upgrading by Developing CountriesUNIDO Vienna Austria 2003

[8] MGromling and JMatthesGlobalisierung und Strukturwandelder Deutschen Textil-und Bekleidungsindustrie Dt Inst-VerlagColougne Germany 2003

[9] I Langenhorst Shop-logistik in der bekleidungswirtschaft eineanalyse der anforderungen herstellerinitiierter shop-systeme andie logistikprozesse der bekleidungsindustrie [PhD thesis]Munster University 1999

[10] H-C Pfohl M Gomm and X Shen China Textil-undBekleidungs-Supply Chain Zwischen Deutschland und ChinaFree Beratung Korschenbroich Germany in H Wolf-Kluthausen Jahrbuch der Logistik 2007

[11] M Teucke and B Scholz-Reiter ldquoImproving order allocationin fashion supply chains using radio frequency identification(RFID) technologiesrdquo in Fashion Supply Chain ManagementUsing Radio Frequency Identification (RFID) Technologies WK Wong and Z X Guo Eds vol 152 of Woodhead PublishingSeries in Textiles Woodhead Publishing Cambridge UK 2014

[12] S Thomassey ldquoSales forecasts in clothing industry the keysuccess factor of the supply chain managementrdquo InternationalJournal of Production Economics vol 128 no 2 pp 470ndash4832010

[13] D Ahlert and G Dieckheuer Marktorientierte Beschaffung inder Bekleidungsindustrie FATM Munster Germany 2001

[14] J T Mentzer C C Bienstock and K B Kahn ldquoBenchmarkingsales forecastingmanagementrdquo Business Horizons vol 42 no 3pp 48ndash56 1999

10 Mathematical Problems in Engineering

[15] J Quick M Rinis C Schmidt and B Walber SupplyTexmdashErfolgreiches Supply-Management in KMU Der Textil- UndBekleidungsindustrie Forschungsinstitut fur Rationalisierung(FIR) eV an der Universitat Aachen Aachen Germany 2010

[16] Z-L Sun T-M Choi K-F Au and Y Yu ldquoSales forecastingusing extreme learning machine with applications in fashionretailingrdquo Decision Support Systems vol 46 no 1 pp 411ndash4192008

[17] S Thomassey M Happiette and J-M Castelain ldquoA globalforecasting support system adapted to textile distributionrdquoInternational Journal of Production Economics vol 96 no 1 pp81ndash95 2005

[18] J Mostard R Teunter and R de Koster ldquoForecasting demandfor single-period products a case study in the apparel industryrdquoEuropean Journal of Operational Research vol 211 no 1 pp 139ndash147 2011

[19] K Hoyndorff S Hulsmann D Spee and M ten HompelFashion Logistics Grundlagen Uber Prozesse und IT Entlang derSupply Chain Munich Huss-Verlag GmbH 2010

[20] J Richardson ldquoVertical integration and rapid response infashion apparelrdquoOrganization Science vol 7 no 4 pp 400ndash4121996

[21] Z-L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[22] J Mula D Peidro M Dıaz-Madronero and E Vicens ldquoMathe-matical programming models for supply chain production andtransport planningrdquo European Journal of Operational Researchvol 204 no 3 pp 377ndash390 2010

[23] S A Torabi and E Hassini ldquoAn interactive possibilistic pro-gramming approach for multiple objective supply chain masterplanningrdquo Fuzzy Sets and Systems vol 159 no 2 pp 193ndash2142008

[24] B Scholz-Reiter J Hartmann T Makuschewitz and E MFrazzon ldquoA generic approach for the graph-based integratedproduction and intermodal transport scheduling with capacityrestrictionsrdquo in Procedia CIRP vol 7 pp 109ndash114 2013

[25] A D Toni andAMeneghetti ldquoProduction planning process fora network of firms in the textile-apparel industryrdquo InternationalJournal of Production Economics vol 65 no 1 pp 17ndash32 2000

[26] E M Frazzon A G N Novaes B Scholz-Reiter J Hartmannand T Makuschewitz ldquoAn approach for the rescheduling ofintegrated production and transport systemsrdquo in Proceedingsof the 4th World Conference Production amp Operations Manage-ment Serving the World Production amp Operations ManagementSociety p 10 Amsterdam The Netherlands 2012

[27] H Ishikura ldquoStudy on the production planning of apparel prod-ucts determining optimal production times and quantitiesrdquoComputers and Industrial Engineering vol 27 no 1ndash4 pp 19ndash22 1994

[28] Y W Tan J B Guan and H R Karimi ldquoThe Impact ofthe subsidy policy on total factor productivity an empiricalanalysis of Chinarsquos cotton productionrdquo Mathematical Problemsin Engineering vol 2013 Article ID 248537 8 pages 2013

[29] H R Karimi N A Duffie and S Dashkovskiy ldquoLocalcapacity Hinfin control for production networks of autonomouswork systems with time-varying delaysrdquo IEEE Transactions onAutomation Science and Engineering vol 7 no 4 pp 849ndash8572010

[30] S Dashkovskiy H R Karimi and M Kosmykov ldquoStabilityanalysis of logistics networks with time-delaysrdquo Journal ofProduction Planning amp Control vol 24 no 7 pp 567ndash574 2013

[31] A Mehrsai H R Karimi and B Scholz-Reiter ldquoApplication oflearning pallets for real-time scheduling by the use of radialbasis function networkrdquo Neurocomputing vol 101 pp 82ndash932013

[32] S C H Leung S O S Tsang W L Ng and Y Wu ldquoArobust optimization model for multi-site production planningproblem in an uncertain environmentrdquo European Journal ofOperational Research vol 181 no 1 pp 224ndash238 2007

[33] A-G Quaranta and A Zaffaroni ldquoRobust optimization ofconditional value at risk and portfolio selectionrdquo Journal ofBanking and Finance vol 32 no 10 pp 2046ndash2056 2008

[34] R A Jabr ldquoRobust self-scheduling under price uncertaintyusing conditional Value-at-Riskrdquo IEEE Transactions on PowerSystems vol 20 no 4 pp 1852ndash1858 2005

[35] D Peidro J Mula R Poler and J-L Verdegay ldquoFuzzy opti-mization for supply chain planning under supply demand andprocess uncertaintiesrdquo Fuzzy Sets and Systems vol 160 no 18pp 2640ndash2657 2009

[36] F Pan and R Nagi ldquoRobust supply chain design underuncertain demand in agile manufacturingrdquo Computers andOperations Research vol 37 no 4 pp 668ndash683 2010

[37] P M Verderame and C A Floudas ldquoOperational planning oflarge-scale industrial batch plants under demand due date andamount uncertainty II Conditional value-at-risk frameworkrdquoIndustrial and Engineering Chemistry Research vol 49 no 1 pp260ndash275 2010

[38] P M Verderame and C A Floudas ldquoMultisite planning underdemand and transportation time uncertainty Robust optimiza-tion and conditional value-at-risk frameworksrdquo Industrial andEngineering Chemistry Research vol 50 no 9 pp 4959ndash49822011

[39] S M J Mirzapour Al-E-Hashem H Malekly and M BAryanezhad ldquoA multi-objective robust optimization modelfor multi-product multi-site aggregate production planningin a supply chain under uncertaintyrdquo International Journal ofProduction Economics vol 134 no 1 pp 28ndash42 2011

[40] D Rahmani R Ramezanian P Fattahi and M HeydarildquoA robust optimization model for multi-product two-stagecapacitated production planning under uncertaintyrdquo AppliedMathematical Modelling vol 37 no 20-21 pp 8957ndash8971 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

012

01

008

006

004

002

0

120

130

150

160

180

190

200

220

230

240

260

270

280

300

310

320

(Values in thousands)

Rela

tive p

roba

bilit

y

Objective fct pessimistic

Objective fct pessimistic

(a)

(Values in thousands)

Rela

tive p

roba

bilit

y

016

014

012

01

008

006

004

002

0

minus25

minus23

minus21

minus19

minus17

minus15

minus13

minus11

minus8

minus6

minus5 3 0 2 3 5

Loss fct pessimistic

Loss fct pessimistic

(b)

Figure 3 Profit and loss function distribution in a pessimistic scenario

018

016

014

012

01

008

006

004

002

0

100

150

170

220

240

270

300

330

370

390

430

450

470

530

550

580

Objective fct normal

Objective fct normal

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

016

014

012

01

008

006

004

002

0

minus40

minus35

minus30

minus27

minus23

minus17

minus13

minus10

minus5 0 3 7

13

17

22

27

Loss fct normal

Loss fct normal

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

Figure 4 Profit and loss function distribution in a normal scenario

Table 3 Production plan with the demand quantities in a pes-simistic scenario

Product 1 Product 2 Product 3 Product 4Demand nextseason 17969 16549 10169 14945

Start of production 3 2 1 1Expected deliverydate 6 6 5 4

Produced InFacility 1 0 1 1 1Facility 2 1 0 0 0

the facilities (Table 4) In this case the expected averageprofit is about 450000 and the loss function is negative in975 of all possible realizations of the stochastic parameters(Figure 4)

53 Optimistic Scenario Since the optimistic scenario pre-dicts a boom economic scenario the production reaches its

Table 4 Production plan with the demand quantities in a normalscenario

Product 1 Product 2 Product 3 Product 4Demand next season 32355 19682 14415 26284Start of production 1 4 3 1Expected delivery date 6 8 7 6Produced in

Facility 1 0 1 1 1Facility 2 1 0 0 0

maximum (Table 5) In this case the average profit is about665000 and the loss function is satisfied with 991 of allpossible realizations of the stochastic parameters (Figure 5)

6 Conclusion and Outlook

In this paper we have presented a risk-constrained profit-expected maximization model for a textile industry scenario

Mathematical Problems in Engineering 9

045

04

035

03

025

02

015

01

005

0

225

250

275

325

350

400

425

475

500

525

575

600

625

675

625

750

Objective fct optimistic

Objective fct optimistic

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

04

035

03

025

02

015

01

005

0

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

minus2 2 7

11

16

20

Loss fct optimistic

Loss fct optimistic

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

Figure 5 Profit and loss function distribution in an optimistic scenario

Table 5 Production plan with the demand quantities in an opti-mistic scenario

Product 1 Product 2 Product 3 Product 4Demand next season 37933 29312 19410 31600Start of production 1 4 3 1Expected delivery date 6 9 7 6Produced in

Facility 1 0 1 1 1Facility 2 1 0 0 0

The forecast demand uncertainty has been explicitly con-sidered in the model by means of the robust optimizationand conditional value at risk theory by introducing andrestricting a loss function The robust optimization modeloriginally proposed by [32] was adopted as the benchmarkformulation of uncertainty considered in this paper A casestudy has been used to demonstrate the viability of theproposed mathematical model Results indicate that theproduction planning model using CVaR as the risk measureperforms robustly and provides flexibility in decision analysisbetween different scenarios The next step will be the furtherdevelopment of the profit maximization model to considerother types of uncertainties for example concerning theavailability of the production plants transportation meansand demand due dates

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Datamonitor Global Apparel Retail 2010 httpwwwdata-monitorcom

[2] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of Banking and Finance vol26 no 7 pp 1443ndash1471 2002

[3] S Zhu and M Fukushima ldquoWorst-case conditional value-at-risk with application to robust portfolio managementrdquo Opera-tions Research vol 57 no 5 pp 1155ndash1168 2009

[4] A Bruckner and S Muller Supply Chain Management in derBekleidungsindustrie Forschungsstelle der Bekleidungsindus-trie Cologne Germany 2003

[5] J Fissahn Marktorientierte Beschaffung in der Bekleidungsin-dustrie Munster University Munster Germany edition 2001

[6] A Schneider Internationalisierungsstrategien in der deutschenTextil-und Bekleidungsindustrie-eine empirische Untersuchung[MS dissertation] RWTH Aachen University Aachen Ger-many 2003

[7] G Gereffi and O Memedovic The Global Apparel ValueChainmdashWhat Prospects for Upgrading by Developing CountriesUNIDO Vienna Austria 2003

[8] MGromling and JMatthesGlobalisierung und Strukturwandelder Deutschen Textil-und Bekleidungsindustrie Dt Inst-VerlagColougne Germany 2003

[9] I Langenhorst Shop-logistik in der bekleidungswirtschaft eineanalyse der anforderungen herstellerinitiierter shop-systeme andie logistikprozesse der bekleidungsindustrie [PhD thesis]Munster University 1999

[10] H-C Pfohl M Gomm and X Shen China Textil-undBekleidungs-Supply Chain Zwischen Deutschland und ChinaFree Beratung Korschenbroich Germany in H Wolf-Kluthausen Jahrbuch der Logistik 2007

[11] M Teucke and B Scholz-Reiter ldquoImproving order allocationin fashion supply chains using radio frequency identification(RFID) technologiesrdquo in Fashion Supply Chain ManagementUsing Radio Frequency Identification (RFID) Technologies WK Wong and Z X Guo Eds vol 152 of Woodhead PublishingSeries in Textiles Woodhead Publishing Cambridge UK 2014

[12] S Thomassey ldquoSales forecasts in clothing industry the keysuccess factor of the supply chain managementrdquo InternationalJournal of Production Economics vol 128 no 2 pp 470ndash4832010

[13] D Ahlert and G Dieckheuer Marktorientierte Beschaffung inder Bekleidungsindustrie FATM Munster Germany 2001

[14] J T Mentzer C C Bienstock and K B Kahn ldquoBenchmarkingsales forecastingmanagementrdquo Business Horizons vol 42 no 3pp 48ndash56 1999

10 Mathematical Problems in Engineering

[15] J Quick M Rinis C Schmidt and B Walber SupplyTexmdashErfolgreiches Supply-Management in KMU Der Textil- UndBekleidungsindustrie Forschungsinstitut fur Rationalisierung(FIR) eV an der Universitat Aachen Aachen Germany 2010

[16] Z-L Sun T-M Choi K-F Au and Y Yu ldquoSales forecastingusing extreme learning machine with applications in fashionretailingrdquo Decision Support Systems vol 46 no 1 pp 411ndash4192008

[17] S Thomassey M Happiette and J-M Castelain ldquoA globalforecasting support system adapted to textile distributionrdquoInternational Journal of Production Economics vol 96 no 1 pp81ndash95 2005

[18] J Mostard R Teunter and R de Koster ldquoForecasting demandfor single-period products a case study in the apparel industryrdquoEuropean Journal of Operational Research vol 211 no 1 pp 139ndash147 2011

[19] K Hoyndorff S Hulsmann D Spee and M ten HompelFashion Logistics Grundlagen Uber Prozesse und IT Entlang derSupply Chain Munich Huss-Verlag GmbH 2010

[20] J Richardson ldquoVertical integration and rapid response infashion apparelrdquoOrganization Science vol 7 no 4 pp 400ndash4121996

[21] Z-L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[22] J Mula D Peidro M Dıaz-Madronero and E Vicens ldquoMathe-matical programming models for supply chain production andtransport planningrdquo European Journal of Operational Researchvol 204 no 3 pp 377ndash390 2010

[23] S A Torabi and E Hassini ldquoAn interactive possibilistic pro-gramming approach for multiple objective supply chain masterplanningrdquo Fuzzy Sets and Systems vol 159 no 2 pp 193ndash2142008

[24] B Scholz-Reiter J Hartmann T Makuschewitz and E MFrazzon ldquoA generic approach for the graph-based integratedproduction and intermodal transport scheduling with capacityrestrictionsrdquo in Procedia CIRP vol 7 pp 109ndash114 2013

[25] A D Toni andAMeneghetti ldquoProduction planning process fora network of firms in the textile-apparel industryrdquo InternationalJournal of Production Economics vol 65 no 1 pp 17ndash32 2000

[26] E M Frazzon A G N Novaes B Scholz-Reiter J Hartmannand T Makuschewitz ldquoAn approach for the rescheduling ofintegrated production and transport systemsrdquo in Proceedingsof the 4th World Conference Production amp Operations Manage-ment Serving the World Production amp Operations ManagementSociety p 10 Amsterdam The Netherlands 2012

[27] H Ishikura ldquoStudy on the production planning of apparel prod-ucts determining optimal production times and quantitiesrdquoComputers and Industrial Engineering vol 27 no 1ndash4 pp 19ndash22 1994

[28] Y W Tan J B Guan and H R Karimi ldquoThe Impact ofthe subsidy policy on total factor productivity an empiricalanalysis of Chinarsquos cotton productionrdquo Mathematical Problemsin Engineering vol 2013 Article ID 248537 8 pages 2013

[29] H R Karimi N A Duffie and S Dashkovskiy ldquoLocalcapacity Hinfin control for production networks of autonomouswork systems with time-varying delaysrdquo IEEE Transactions onAutomation Science and Engineering vol 7 no 4 pp 849ndash8572010

[30] S Dashkovskiy H R Karimi and M Kosmykov ldquoStabilityanalysis of logistics networks with time-delaysrdquo Journal ofProduction Planning amp Control vol 24 no 7 pp 567ndash574 2013

[31] A Mehrsai H R Karimi and B Scholz-Reiter ldquoApplication oflearning pallets for real-time scheduling by the use of radialbasis function networkrdquo Neurocomputing vol 101 pp 82ndash932013

[32] S C H Leung S O S Tsang W L Ng and Y Wu ldquoArobust optimization model for multi-site production planningproblem in an uncertain environmentrdquo European Journal ofOperational Research vol 181 no 1 pp 224ndash238 2007

[33] A-G Quaranta and A Zaffaroni ldquoRobust optimization ofconditional value at risk and portfolio selectionrdquo Journal ofBanking and Finance vol 32 no 10 pp 2046ndash2056 2008

[34] R A Jabr ldquoRobust self-scheduling under price uncertaintyusing conditional Value-at-Riskrdquo IEEE Transactions on PowerSystems vol 20 no 4 pp 1852ndash1858 2005

[35] D Peidro J Mula R Poler and J-L Verdegay ldquoFuzzy opti-mization for supply chain planning under supply demand andprocess uncertaintiesrdquo Fuzzy Sets and Systems vol 160 no 18pp 2640ndash2657 2009

[36] F Pan and R Nagi ldquoRobust supply chain design underuncertain demand in agile manufacturingrdquo Computers andOperations Research vol 37 no 4 pp 668ndash683 2010

[37] P M Verderame and C A Floudas ldquoOperational planning oflarge-scale industrial batch plants under demand due date andamount uncertainty II Conditional value-at-risk frameworkrdquoIndustrial and Engineering Chemistry Research vol 49 no 1 pp260ndash275 2010

[38] P M Verderame and C A Floudas ldquoMultisite planning underdemand and transportation time uncertainty Robust optimiza-tion and conditional value-at-risk frameworksrdquo Industrial andEngineering Chemistry Research vol 50 no 9 pp 4959ndash49822011

[39] S M J Mirzapour Al-E-Hashem H Malekly and M BAryanezhad ldquoA multi-objective robust optimization modelfor multi-product multi-site aggregate production planningin a supply chain under uncertaintyrdquo International Journal ofProduction Economics vol 134 no 1 pp 28ndash42 2011

[40] D Rahmani R Ramezanian P Fattahi and M HeydarildquoA robust optimization model for multi-product two-stagecapacitated production planning under uncertaintyrdquo AppliedMathematical Modelling vol 37 no 20-21 pp 8957ndash8971 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

045

04

035

03

025

02

015

01

005

0

225

250

275

325

350

400

425

475

500

525

575

600

625

675

625

750

Objective fct optimistic

Objective fct optimistic

(Values in thousands)

Rela

tive p

roba

bilit

y

(a)

04

035

03

025

02

015

01

005

0

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

minus2 2 7

11

16

20

Loss fct optimistic

Loss fct optimistic

(Values in thousands)

Rela

tive p

roba

bilit

y

(b)

Figure 5 Profit and loss function distribution in an optimistic scenario

Table 5 Production plan with the demand quantities in an opti-mistic scenario

Product 1 Product 2 Product 3 Product 4Demand next season 37933 29312 19410 31600Start of production 1 4 3 1Expected delivery date 6 9 7 6Produced in

Facility 1 0 1 1 1Facility 2 1 0 0 0

The forecast demand uncertainty has been explicitly con-sidered in the model by means of the robust optimizationand conditional value at risk theory by introducing andrestricting a loss function The robust optimization modeloriginally proposed by [32] was adopted as the benchmarkformulation of uncertainty considered in this paper A casestudy has been used to demonstrate the viability of theproposed mathematical model Results indicate that theproduction planning model using CVaR as the risk measureperforms robustly and provides flexibility in decision analysisbetween different scenarios The next step will be the furtherdevelopment of the profit maximization model to considerother types of uncertainties for example concerning theavailability of the production plants transportation meansand demand due dates

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Datamonitor Global Apparel Retail 2010 httpwwwdata-monitorcom

[2] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of Banking and Finance vol26 no 7 pp 1443ndash1471 2002

[3] S Zhu and M Fukushima ldquoWorst-case conditional value-at-risk with application to robust portfolio managementrdquo Opera-tions Research vol 57 no 5 pp 1155ndash1168 2009

[4] A Bruckner and S Muller Supply Chain Management in derBekleidungsindustrie Forschungsstelle der Bekleidungsindus-trie Cologne Germany 2003

[5] J Fissahn Marktorientierte Beschaffung in der Bekleidungsin-dustrie Munster University Munster Germany edition 2001

[6] A Schneider Internationalisierungsstrategien in der deutschenTextil-und Bekleidungsindustrie-eine empirische Untersuchung[MS dissertation] RWTH Aachen University Aachen Ger-many 2003

[7] G Gereffi and O Memedovic The Global Apparel ValueChainmdashWhat Prospects for Upgrading by Developing CountriesUNIDO Vienna Austria 2003

[8] MGromling and JMatthesGlobalisierung und Strukturwandelder Deutschen Textil-und Bekleidungsindustrie Dt Inst-VerlagColougne Germany 2003

[9] I Langenhorst Shop-logistik in der bekleidungswirtschaft eineanalyse der anforderungen herstellerinitiierter shop-systeme andie logistikprozesse der bekleidungsindustrie [PhD thesis]Munster University 1999

[10] H-C Pfohl M Gomm and X Shen China Textil-undBekleidungs-Supply Chain Zwischen Deutschland und ChinaFree Beratung Korschenbroich Germany in H Wolf-Kluthausen Jahrbuch der Logistik 2007

[11] M Teucke and B Scholz-Reiter ldquoImproving order allocationin fashion supply chains using radio frequency identification(RFID) technologiesrdquo in Fashion Supply Chain ManagementUsing Radio Frequency Identification (RFID) Technologies WK Wong and Z X Guo Eds vol 152 of Woodhead PublishingSeries in Textiles Woodhead Publishing Cambridge UK 2014

[12] S Thomassey ldquoSales forecasts in clothing industry the keysuccess factor of the supply chain managementrdquo InternationalJournal of Production Economics vol 128 no 2 pp 470ndash4832010

[13] D Ahlert and G Dieckheuer Marktorientierte Beschaffung inder Bekleidungsindustrie FATM Munster Germany 2001

[14] J T Mentzer C C Bienstock and K B Kahn ldquoBenchmarkingsales forecastingmanagementrdquo Business Horizons vol 42 no 3pp 48ndash56 1999

10 Mathematical Problems in Engineering

[15] J Quick M Rinis C Schmidt and B Walber SupplyTexmdashErfolgreiches Supply-Management in KMU Der Textil- UndBekleidungsindustrie Forschungsinstitut fur Rationalisierung(FIR) eV an der Universitat Aachen Aachen Germany 2010

[16] Z-L Sun T-M Choi K-F Au and Y Yu ldquoSales forecastingusing extreme learning machine with applications in fashionretailingrdquo Decision Support Systems vol 46 no 1 pp 411ndash4192008

[17] S Thomassey M Happiette and J-M Castelain ldquoA globalforecasting support system adapted to textile distributionrdquoInternational Journal of Production Economics vol 96 no 1 pp81ndash95 2005

[18] J Mostard R Teunter and R de Koster ldquoForecasting demandfor single-period products a case study in the apparel industryrdquoEuropean Journal of Operational Research vol 211 no 1 pp 139ndash147 2011

[19] K Hoyndorff S Hulsmann D Spee and M ten HompelFashion Logistics Grundlagen Uber Prozesse und IT Entlang derSupply Chain Munich Huss-Verlag GmbH 2010

[20] J Richardson ldquoVertical integration and rapid response infashion apparelrdquoOrganization Science vol 7 no 4 pp 400ndash4121996

[21] Z-L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[22] J Mula D Peidro M Dıaz-Madronero and E Vicens ldquoMathe-matical programming models for supply chain production andtransport planningrdquo European Journal of Operational Researchvol 204 no 3 pp 377ndash390 2010

[23] S A Torabi and E Hassini ldquoAn interactive possibilistic pro-gramming approach for multiple objective supply chain masterplanningrdquo Fuzzy Sets and Systems vol 159 no 2 pp 193ndash2142008

[24] B Scholz-Reiter J Hartmann T Makuschewitz and E MFrazzon ldquoA generic approach for the graph-based integratedproduction and intermodal transport scheduling with capacityrestrictionsrdquo in Procedia CIRP vol 7 pp 109ndash114 2013

[25] A D Toni andAMeneghetti ldquoProduction planning process fora network of firms in the textile-apparel industryrdquo InternationalJournal of Production Economics vol 65 no 1 pp 17ndash32 2000

[26] E M Frazzon A G N Novaes B Scholz-Reiter J Hartmannand T Makuschewitz ldquoAn approach for the rescheduling ofintegrated production and transport systemsrdquo in Proceedingsof the 4th World Conference Production amp Operations Manage-ment Serving the World Production amp Operations ManagementSociety p 10 Amsterdam The Netherlands 2012

[27] H Ishikura ldquoStudy on the production planning of apparel prod-ucts determining optimal production times and quantitiesrdquoComputers and Industrial Engineering vol 27 no 1ndash4 pp 19ndash22 1994

[28] Y W Tan J B Guan and H R Karimi ldquoThe Impact ofthe subsidy policy on total factor productivity an empiricalanalysis of Chinarsquos cotton productionrdquo Mathematical Problemsin Engineering vol 2013 Article ID 248537 8 pages 2013

[29] H R Karimi N A Duffie and S Dashkovskiy ldquoLocalcapacity Hinfin control for production networks of autonomouswork systems with time-varying delaysrdquo IEEE Transactions onAutomation Science and Engineering vol 7 no 4 pp 849ndash8572010

[30] S Dashkovskiy H R Karimi and M Kosmykov ldquoStabilityanalysis of logistics networks with time-delaysrdquo Journal ofProduction Planning amp Control vol 24 no 7 pp 567ndash574 2013

[31] A Mehrsai H R Karimi and B Scholz-Reiter ldquoApplication oflearning pallets for real-time scheduling by the use of radialbasis function networkrdquo Neurocomputing vol 101 pp 82ndash932013

[32] S C H Leung S O S Tsang W L Ng and Y Wu ldquoArobust optimization model for multi-site production planningproblem in an uncertain environmentrdquo European Journal ofOperational Research vol 181 no 1 pp 224ndash238 2007

[33] A-G Quaranta and A Zaffaroni ldquoRobust optimization ofconditional value at risk and portfolio selectionrdquo Journal ofBanking and Finance vol 32 no 10 pp 2046ndash2056 2008

[34] R A Jabr ldquoRobust self-scheduling under price uncertaintyusing conditional Value-at-Riskrdquo IEEE Transactions on PowerSystems vol 20 no 4 pp 1852ndash1858 2005

[35] D Peidro J Mula R Poler and J-L Verdegay ldquoFuzzy opti-mization for supply chain planning under supply demand andprocess uncertaintiesrdquo Fuzzy Sets and Systems vol 160 no 18pp 2640ndash2657 2009

[36] F Pan and R Nagi ldquoRobust supply chain design underuncertain demand in agile manufacturingrdquo Computers andOperations Research vol 37 no 4 pp 668ndash683 2010

[37] P M Verderame and C A Floudas ldquoOperational planning oflarge-scale industrial batch plants under demand due date andamount uncertainty II Conditional value-at-risk frameworkrdquoIndustrial and Engineering Chemistry Research vol 49 no 1 pp260ndash275 2010

[38] P M Verderame and C A Floudas ldquoMultisite planning underdemand and transportation time uncertainty Robust optimiza-tion and conditional value-at-risk frameworksrdquo Industrial andEngineering Chemistry Research vol 50 no 9 pp 4959ndash49822011

[39] S M J Mirzapour Al-E-Hashem H Malekly and M BAryanezhad ldquoA multi-objective robust optimization modelfor multi-product multi-site aggregate production planningin a supply chain under uncertaintyrdquo International Journal ofProduction Economics vol 134 no 1 pp 28ndash42 2011

[40] D Rahmani R Ramezanian P Fattahi and M HeydarildquoA robust optimization model for multi-product two-stagecapacitated production planning under uncertaintyrdquo AppliedMathematical Modelling vol 37 no 20-21 pp 8957ndash8971 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[15] J Quick M Rinis C Schmidt and B Walber SupplyTexmdashErfolgreiches Supply-Management in KMU Der Textil- UndBekleidungsindustrie Forschungsinstitut fur Rationalisierung(FIR) eV an der Universitat Aachen Aachen Germany 2010

[16] Z-L Sun T-M Choi K-F Au and Y Yu ldquoSales forecastingusing extreme learning machine with applications in fashionretailingrdquo Decision Support Systems vol 46 no 1 pp 411ndash4192008

[17] S Thomassey M Happiette and J-M Castelain ldquoA globalforecasting support system adapted to textile distributionrdquoInternational Journal of Production Economics vol 96 no 1 pp81ndash95 2005

[18] J Mostard R Teunter and R de Koster ldquoForecasting demandfor single-period products a case study in the apparel industryrdquoEuropean Journal of Operational Research vol 211 no 1 pp 139ndash147 2011

[19] K Hoyndorff S Hulsmann D Spee and M ten HompelFashion Logistics Grundlagen Uber Prozesse und IT Entlang derSupply Chain Munich Huss-Verlag GmbH 2010

[20] J Richardson ldquoVertical integration and rapid response infashion apparelrdquoOrganization Science vol 7 no 4 pp 400ndash4121996

[21] Z-L Chen ldquoIntegrated production and outbound distributionscheduling review and extensionsrdquo Operations Research vol58 no 1 pp 130ndash148 2010

[22] J Mula D Peidro M Dıaz-Madronero and E Vicens ldquoMathe-matical programming models for supply chain production andtransport planningrdquo European Journal of Operational Researchvol 204 no 3 pp 377ndash390 2010

[23] S A Torabi and E Hassini ldquoAn interactive possibilistic pro-gramming approach for multiple objective supply chain masterplanningrdquo Fuzzy Sets and Systems vol 159 no 2 pp 193ndash2142008

[24] B Scholz-Reiter J Hartmann T Makuschewitz and E MFrazzon ldquoA generic approach for the graph-based integratedproduction and intermodal transport scheduling with capacityrestrictionsrdquo in Procedia CIRP vol 7 pp 109ndash114 2013

[25] A D Toni andAMeneghetti ldquoProduction planning process fora network of firms in the textile-apparel industryrdquo InternationalJournal of Production Economics vol 65 no 1 pp 17ndash32 2000

[26] E M Frazzon A G N Novaes B Scholz-Reiter J Hartmannand T Makuschewitz ldquoAn approach for the rescheduling ofintegrated production and transport systemsrdquo in Proceedingsof the 4th World Conference Production amp Operations Manage-ment Serving the World Production amp Operations ManagementSociety p 10 Amsterdam The Netherlands 2012

[27] H Ishikura ldquoStudy on the production planning of apparel prod-ucts determining optimal production times and quantitiesrdquoComputers and Industrial Engineering vol 27 no 1ndash4 pp 19ndash22 1994

[28] Y W Tan J B Guan and H R Karimi ldquoThe Impact ofthe subsidy policy on total factor productivity an empiricalanalysis of Chinarsquos cotton productionrdquo Mathematical Problemsin Engineering vol 2013 Article ID 248537 8 pages 2013

[29] H R Karimi N A Duffie and S Dashkovskiy ldquoLocalcapacity Hinfin control for production networks of autonomouswork systems with time-varying delaysrdquo IEEE Transactions onAutomation Science and Engineering vol 7 no 4 pp 849ndash8572010

[30] S Dashkovskiy H R Karimi and M Kosmykov ldquoStabilityanalysis of logistics networks with time-delaysrdquo Journal ofProduction Planning amp Control vol 24 no 7 pp 567ndash574 2013

[31] A Mehrsai H R Karimi and B Scholz-Reiter ldquoApplication oflearning pallets for real-time scheduling by the use of radialbasis function networkrdquo Neurocomputing vol 101 pp 82ndash932013

[32] S C H Leung S O S Tsang W L Ng and Y Wu ldquoArobust optimization model for multi-site production planningproblem in an uncertain environmentrdquo European Journal ofOperational Research vol 181 no 1 pp 224ndash238 2007

[33] A-G Quaranta and A Zaffaroni ldquoRobust optimization ofconditional value at risk and portfolio selectionrdquo Journal ofBanking and Finance vol 32 no 10 pp 2046ndash2056 2008

[34] R A Jabr ldquoRobust self-scheduling under price uncertaintyusing conditional Value-at-Riskrdquo IEEE Transactions on PowerSystems vol 20 no 4 pp 1852ndash1858 2005

[35] D Peidro J Mula R Poler and J-L Verdegay ldquoFuzzy opti-mization for supply chain planning under supply demand andprocess uncertaintiesrdquo Fuzzy Sets and Systems vol 160 no 18pp 2640ndash2657 2009

[36] F Pan and R Nagi ldquoRobust supply chain design underuncertain demand in agile manufacturingrdquo Computers andOperations Research vol 37 no 4 pp 668ndash683 2010

[37] P M Verderame and C A Floudas ldquoOperational planning oflarge-scale industrial batch plants under demand due date andamount uncertainty II Conditional value-at-risk frameworkrdquoIndustrial and Engineering Chemistry Research vol 49 no 1 pp260ndash275 2010

[38] P M Verderame and C A Floudas ldquoMultisite planning underdemand and transportation time uncertainty Robust optimiza-tion and conditional value-at-risk frameworksrdquo Industrial andEngineering Chemistry Research vol 50 no 9 pp 4959ndash49822011

[39] S M J Mirzapour Al-E-Hashem H Malekly and M BAryanezhad ldquoA multi-objective robust optimization modelfor multi-product multi-site aggregate production planningin a supply chain under uncertaintyrdquo International Journal ofProduction Economics vol 134 no 1 pp 28ndash42 2011

[40] D Rahmani R Ramezanian P Fattahi and M HeydarildquoA robust optimization model for multi-product two-stagecapacitated production planning under uncertaintyrdquo AppliedMathematical Modelling vol 37 no 20-21 pp 8957ndash8971 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of