Research Article Multi-Item Multiperiodic Inventory ...downloads.hindawi.com/journals/tswj/2014/136047.pdf · Multi-Item Multiperiodic Inventory Control Problem with Variable Demand

Research ArticleMulti-Item Multiperiodic Inventory ControlProblem with Variable Demand and DiscountsA Particle Swarm Optimization Algorithm

Seyed Mohsen Mousavi1 S T A Niaki2 Ardeshir Bahreininejad1 and Siti Nurmaya Musa1

1 Department of Mechanical Engineering Faculty of Engineering University of Malaya 50603 Kuala Lumpur Malaysia2 Department of Industrial Engineering Sharif University of Technology PO Box 11155-0414 Azadi Avenue Tehran 1458889694 Iran

Correspondence should be addressed to Ardeshir Bahreininejad bahreininejadumedumy

Received 21 April 2014 Accepted 28 May 2014 Published 30 June 2014

Academic Editor Leopoldo Eduardo Cardenas-Barron

Copyright copy 2014 Seyed Mohsen Mousavi et alThis 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

Amulti-itemmultiperiod inventory control model is developed for known-deterministic variable demands under limited availablebudget Assuming the order quantity is more than the shortage quantity in each period the shortage in combination of backorderand lost sale is considered The orders are placed in batch sizes and the decision variables are assumed integer Moreover all unitdiscounts for a number of products and incremental quantity discount for some other items are consideredWhile the objectives aretominimize both the total inventory cost and the required storage space themodel is formulated into a fuzzymulticriteria decisionmaking (FMCDM) framework and is shown to be a mixed integer nonlinear programming type In order to solve the modela multiobjective particle swarm optimization (MOPSO) approach is applied A set of compromise solution including optimumand near optimum ones via MOPSO has been derived for some numerical illustration where the results are compared with thoseobtained using a weighting approach To assess the efficiency of the proposed MOPSO the model is solved using multi-objectivegenetic algorithm (MOGA) as well A large number of numerical examples are generated at the end where graphical and statisticalapproaches show more efficiency of MOPSO compared with MOGA

1 Introduction and Literature ReviewMost real-world problems in industries and commerceare studied as an optimization problem involving a singleobjective The assumption that organizations always seek tomaximize (or minimize) their profit (or their cost) ratherthan making trade-offs among multiple objectives has beencensured for a long time Generally classical inventory mod-els are developed under the basic assumption that the man-agement purchases or produces a single product Howeverin many real-life conditions this assumption does not holdInstead many firms enterprises or vendors are motivated tostore a number of products in their shops for more profitablebusiness affairs Another cause of theirmotivation is to attractthe customers to purchase several items in one showroom orshop

This work proposes a multiperiod inventory model forseasonal and fashion items The multiperiodic inventory

control problems have been investigated in depth in differentresearch Chiang [1] investigated a periodic review inventorymodel in which the period is partly long The importantaspect of his study was to introduce emergency orders toprevent shortages He employed a dynamic programmingapproach to model the problem Mohebbi and Posner [2]investigated an inventory system with periodic review mul-tiple replenishment and multilevel delivery Assuming aPoisson process for the demand shortages were allowed inthis research and the lost sale policy could be employed Leeand Kang [3] developed a model for managing inventory ofa product in multiple periods Their model was first derivedfor one item and then was extended for several productsMousavi et al [4] proposed a multiproduct multiperiodinventory control problem under time value of money andinflation where total storage space and budget were limitedThey solved the problem using twometaheuristic algorithms

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 136047 16 pageshttpdxdoiorg1011552014136047

2 The Scientific World Journal

that is genetic algorithm and simulated annealing Mirza-pour Al-e-hashem and Rekik [5] presented a multiproductmultiperiod inventory routing problem where multiple con-strained vehicles distributed products from multiple suppli-ers to a single plant tomeet the given demand of each productover a finite planning horizon Janakiraman et al [6] analyzedthe multiperiodic newsvendor problem and proposed somenew results

The quantity discount is of increasing attention dueto its practical importance in purchasing and control ofitems Usually one derives the better marginal cost of pur-chaseproduction by taking advantages of the chances of costsavings through bulk purchaseproduction Furthermore insupply chain environments quantity discounts can be consid-ered an inventory coordination mechanism between buyersand suppliers [7] In the literature of quantity discountsBenton [8] considered an inventory system with quantitydiscount with multiple price breaks and alternative purchas-ing and lot-sizing policy Abad [9 10] proposed modelsfor joint price and lot size determination when supplieroffers either incremental (IQD) or all unit (AUD) quantitydiscounts K Maiti andM Maiti [11] developed a model for amulti-item inventory control system of breakable items withAUD and IQD (and a combination of the two policies) andproposed genetic algorithm to solve the model Sana andChaudhuri [12] extended an EOQ model by relaxation ofthe preassumptions related to payments allowing delay ondelivery and discounts They used a mixed integer nonlinearprogramming technique to model the problem Taleizadeh etal [13] considered a genetic algorithm to optimizemultiprod-uct multiconstraint inventory control systems with stochasticreplenishment intervals and discount Recently several workssuch as the ones in [4 14ndash16] have also spotted discounts ininventory control problems Huang and Lin [17] addressed anintegrated model that scheduled multi-item replenishmentwith uncertain demand to determine delivery routes andtruck loads In this study the products are purchased indifferent periods under AUD and IQD policies

Metaheuristic algorithms have been suggested to solvesome of the existing developed inventory problems in theliterature Someof these algorithms are tabu search [18 19 31]genetic algorithms (GA) [20ndash22 32] simulating annealing(SA) [23 33 34] evolutionary algorithm [24 35] thresholdaccepting [30] neural networks [36] ant colony optimization[37] fuzzy simulation [25] and harmony search [26 38ndash41]

Inspired by social behavior of bird flocking or fish school-ing particle swarm optimization (PSO) is also a population-based stochastic optimization metaheuristic developed byKennedy and Eberhart [42] Recently researchers haveemployed this effective technique to find optimal or nearoptimal solutions of their inventory control problems Forexample Taleizadeh et al [43] employed PSO to solvetheir integer nonlinear programming model of a constraintjoint single buyer-single vendor inventory problem withchangeable lead time and (119903 119876) policy in supply chainswith stochastic demand Chen and Dye [44] solved aninventory problem with deteriorating products and variabledemands using a PSO algorithm Further Taleizadeh et al[27] modeled a chancendashconstraint supply chain problem

with uniformly distributed stochastic demand where an AntColony Bee and a PSO algorithm were utilized to solve theproblem

Instead of optimizing a single objective some researcherstried to find Pareto front solutions for their multiple objectiveinventory planning problems that usually consist of multipleconflicting objectives Agrell [45] proposed a multicriteriaframework for inventory control problem in which the solu-tion procedure was an interactive method with preferencesextracted gradually in decision analysis process to determinebatch size and security stock Roy and Maiti [28] presenteda multiobjective inventory model of deteriorating items withstock-dependent demand under limited imprecise storagearea and total cost budget Tsou [46] developed a multiob-jective reorder point and order size system and proposeda multiobjective PSO (MOPSO) to generate Pareto frontsolutions He employed TOPSIS to sort the nondominatedsolutions The objectives therein were to maximize the profitand to minimize the wastage cost where the profit goalwastage cost and storage areawere fuzzy in natureOne of thesuccessful applications of PSO toMOOPs is the seminal workof Coello Coello and Lechuga [47] Yaghin et al [29] firstaddressed an inventory-marketing system to determine theproduction lot size marketing expenditure and selling pricesin which the model was formulated as a fuzzy nonlinearmultiobjective program Then they converted the model toa classical single-objective one by a fuzzy goal programmingmethod where an efficient solution procedure using PSOwas provided to solve the resulting nonlinear problem Intheir study MOPSO is not only a viable alternative tosolve MOOPs but also the only one compared with thenondominated sorting genetic algorithm-II (NSGA-II) [48]the Pareto archive evolutionary strategy (PAES) [49] and themicrogenetic algorithm [50] for multiobjective optimizationproblems [51] Table 1 shows the literature review of theworksreviewed in this work where DOE is an abbreviation of termldquodesign of experimentsrdquo

In this research the contribution of the problem is con-sidering a new biobjective multi-item multiperiodic inven-tory control model where some items are purchased underAUD and the other items are bought from IQDThe demandsvary in different periods the budget is limited the ordersare placed in batch sizes and shortages in combination ofbackorder and lost sale are considered The goal is to find theoptimum inventory levels of the items in each period suchthat the total inventory cost and the total required warehousespace are minimized simultaneously Since it is not easy forthe managers to allocate the crisp values to the weights ofthe objectives in a decisionmaking process considering theseweights as fuzzy numbers will be taken as an advantage

In order to be more understanding of the problem wetry to explain the model with taking an example in the realworld We consider a company which produces some kindsof fashion clothes including trousers t-shirt and shirt in acertain period The customers (wholesales) of this companywith different demand rates make the orders and receivetheir products in the prespecific boxes each one consistingof a known number of these clothes Moreover due tosome unforeseen matters such as production limitation the

The Scientific World Journal 3

Table 1 Literature review of the related works

References Multiproduct Multiperiod Fuzzy multiobjective Discount policy Solving methodology Shortages DOE[1] mdash mdash mdash B and B mdash mdash[2] mdash mdash mdash mdash Level-crossing mdash[3] mdash mdash IQD Numerical methods mdash mdash[4] mdash IQD GA mdash mdash[5] mdash mdash CPLEX mdash mdash[6] mdash mdash mdash mdash mdash[7] mdash mdash mdash IQD and AUD Simulation mdash mdash[8] mdash mdash mdash AUD Simulation mdash mdash[9] mdash mdash mdash IQD Numerical methods mdash mdash[10] mdash mdash mdash IQD Numerical methods mdash mdash[11] mdash mdash IQD and AUD GA mdash mdash[12] mdash mdash AUD Numerical methods mdash mdash[13] mdash mdash IQD and AUD GA mdash[14] mdash mdash mdash IQD Yager ranking mdash mdash[15] mdash mdash mdash IQD Excel macro mdash mdash[16] mdash mdash mdash TOPSIS and GA mdash[17] mdash mdash mdash ACO mdash mdash[18] mdash mdash mdash mdash Tabu search and Lagrangian mdash mdash[19] mdash mdash mdash Tabu search mdash mdash[20] mdash mdash mdash IQD Goal programming and GA mdash[21] mdash mdash IQD GA and fuzzy simulation mdash[22] mdash mdash mdash GA mdash[23] mdash mdash mdash SA and GA mdash[24] mdash mdash mdash TOPSIS and GA mdash[25] mdash mdash IQD fuzzy simulation mdash[26] mdash mdash mdash Harmony search mdash[27] mdash mdash mdash Bee colony and PSO mdash[28] mdash mdash Fuzzy programming algorithm mdash mdash[29] mdash mdash Fuzzy method mdash mdashThis research PSO and GA Taguchi

companies are not responsive to all of the demands in aperiod and hence some customers must wait until the nextperiod to receive their orders Furthermore it is assumed thecompany is going to extend the production part and thereforethe owner has a plan to build and optimize a new storagesubject to the available space

The remainder of the paper is organized as follows InSection 2 the problem along with its assumptions is definedIn Section 3 the defined problem of Section 2 is modeledTo do this the parameters and the variables of the problemare first introduced A MOPSO algorithm is presented inSection 4 to solve the model Section 5 contains a numericalexample for a problem with 5 items and 3 periods forwhich a multiobjective genetic algorithm (MOGA) is alsoapplied as benchmark for comparisons Finally conclusionand recommendations for future research comes in Section 6

2 Problem DefinitionAssumptions and Notations

Consider a biobjective multi-item multiperiod inventorycontrol problem in which an AUD policy is used for someitems and an IQD policy for some other itemsThe inventorycontrol problem of this research is similar to the seasonal

items problem where the planning horizon starts in a period(or season) and finish in a certain period (or season) Thetotal available budget in the planning horizon is limited andfixed Due to existing ordering limitations or productionconstraints the order quantities of all items in differentperiods cannot be more than their predetermined upperbounds The demands of the products are constant anddistinct and in case of shortage a fraction is consideredbackorder and a fraction lost sale The costs associated withthe inventory control system are holding backorder lost saleand purchasing costs Moreover due to current managerialdecision adaptations on production policies (ie settingup a new manufacturing line extending the warehouse orbuilding a new storage area) minimizing the total storagespace is required as well as minimizing the total inventorycosts Therefore the goal is to identify the inventory levels ofthe items in each period such that the two objective functionstotal inventory costs and total storage space are minimized

In order to simplify the modeling the following assump-tions are set to the problem at hand

(1) The demand rate of an item is independent of theothers and is constant in a period However it can bedifferent in different periods


(2) At most one order can be placed in a period Thisorder can include or exclude an item

(3) The items are delivered in a special container Thusthe order quantitiesmust be amultiple of a fixed-sizedbatch

(4) The vendor uses an AUD policy for some items andan IQD policy for others

(5) A fraction of the shortages is considered backorderand a fraction lost sale

(6) The initial inventory level of all items is zero(7) The budget is limited(8) Theplanning horizon is finite and known In the plan-

ning horizon there are119873 periods of equal duration(9) The order quantity on an item in a period is greater

than or equal to its shortage quantity in the previousperiod (ie 119876

119894119895+1ge 119887119894119895defined below)

In order to model the problem at hand in what comes nextwe first define the variables and the parameters Then theproblem is formulated in Section 3

For 119894 = 1 2 119898 and 119895 = 1 2 119873 minus 1 and 119896 =1 2 119870 the variables and the parameters of the model aredefined as follows

119873 number of replenishment cycles during the plan-ning horizon119898 number of items119870 number of price break points119878119894 required storage space per unit of the 119894th product119879119895 total time elapsed up to and including the 119895th

replenishment cycle1198791015840119894119895 119895th period in which the inventory level of item 119894

is zero (a decision variable)119861119894 batch size of the 119894th product119881119894119895 number of the packets for the 119894th product order

in period j (a decision variable)119863119894119895 demand of the 119894th product in period j

119876119894119895 purchase quantity of item 119894 in period 119895 (a decision

variable where 119876119894119895= 119861119894119881119894119895)

119860119894 ordering cost per replenishment of product 119894 (If

an order is placed for one or more items in period 119895this cost appears in that period)119887119894119895 shortage quantity of the 119894th product in period 119895 (a

decision variable)119883119894119895 the beginning positive inventory level of the

119894th product in period 119895 (in 119895 = 1 the beginningpositive inventory level of all items is zero) (a decisionvariable)119868119894119895 inventory position of the 119894th product in period j

(it is119883119894119895+1+ 119876119894119895+1

if 119868119894119895ge 0 otherwise equals 119887

119894119895)

119868119894(119905) the inventory level of the 119894th item at time t119867119894 unit inventory holding cost for item i

119902119894119896 kth discount point for the 119894th product (119902

1198941= 0)

119898119894119896 discount rate of item 119894 in kth price break point

(1198981198941= 0)

119875119894 purchasing cost per unit of the 119894th product119875119894119896 purchasing cost per unit of the 119894th product at the

kth price break point119880119894119895119896

a binary variable set 1 if item 119894 is purchased atprice break point 119896 in period j and 0 otherwise119882119894119895 a binary variable set 1 if a purchase of item 119894 is

made in period j and 0 otherwise119871119894119895 a binary variable set 1 if a shortage for item 119894

occurs in period j and 0 otherwise120573119894 percentage of unsatisfied demands of the 119894th

product that is back ordered120587119894119895 backorder cost per unit demandof the 119894th product

in period j119894119895 shortage cost per unit of the 119894th product in period

119895 that is lost1198851 total inventory cost

1198852 total storage space

119879119861 total available budget1198721 an upper bound for order quantity of the 119894th item

in period j1198722 an upper bound for order quantities of all items

in each period (the truck capacity)TMF objective function (the weighted combinationof the total inventory cost and the total storage space)1199081 a weight associated with the total inventory cost

(0 le 1199081le 1)

1199082 a weight associated with the total storage space

(0 le 1199082le 1)

3 Problem Formulation

A graphical representation of the inventory control problemat hand with 5 periods for item 119894 is given in Figure 1 to obtainthe inventory costs At the beginning of the primary period(1198790) it is assumed the starting inventory level of item 119894 is

zero and that the order quantity has been received and isavailable In the following periods shortages can either occuror not If shortage occurs the corresponding binary variableis 1 otherwise it is zero In the latter case the inventory levelsat the beginning of each period may be positive

31 The Objective Functions The first objective function ofthe problem the total inventory cost is obtained as

1198851= Total Inventory Cost

= Total Ordering Cost + Total Holding Cost

+ Total Shortage Cost + Total Purchasing Cost

(1)

where each part is derived as follows


Qi1 Qi2

Qi3

Qi4

Di1

bi1bi3

Xi3

T1T0 T4T3T2 T5T9984001 T9984003120573i bi1

(1 minus 120573i)bi1

Figure 1 Some possible situations for the inventory of item 119894 in 5 periods

The ordering cost of an item in a period occurs when anorder is placed for it in that period Using a binary variable119882119894119895 where it is 1 if an order for the 119894th product in period 119895

is placed and zero otherwise and knowing that orders can beplaced in periods 1 to119873minus1 the total ordering cost is obtainedas

Total Ordering Cost =119898

sum119894=1

119873minus1

sum119895=1

119860119894119882119894119895 (2)

Since it is assumed a shortage may occur for a productin a period or not the holding cost derivation is notas straightforward as the ordering cost derivation Takingadvantage of a binary variable 119871

119894119895 where it is 1 if a shortage

for item 119894 in period 119895 occurs and otherwise zero and usingFigure 1 the holding cost for item 119894 in the time interval119879

119895minus1le

119905 le 119879119895(1 minus 119871

119894119895) + 1198791015840119894119895119871119894119895is obtained as

119867119894int119879119895(1minus119871 119894119895)+119879

1015840119894119895119871 119894119895

119879119895minus1

119868119894 (119905) 119889119905 (3)

where 119868119894(119905) is the inventory level of the 119894th item at time t

In (3) if a shortage for item 119894 occurs 119871119894119895becomes 1 and the

term 119879119895(1 minus 119871

119894119895) + 1198791015840119894119895119871119894119895becomes 1198791015840

119894119895 Otherwise 119871

119894119895= 0

and119879119895(1minus119871119894119895)+1198791015840119894119895119871119894119895= 119879119895 In Figure 1 the trapezoidal area

above the horizontal timeline in each period whenmultipliedby the unit inventory holding cost of an item 119867

119894 represents

the holding cost of the item in that period In other wordsince

119868119894119895+1= 119868119894119895+ 119876119894119895minus 119863119894119895 (4)

and if 119868119894119895+1ge 0 then 119868

119894119895+1= 119883119894119895+1

otherwise 119868119894119895+1= 119887119894119895 (3)

becomes

119867119894int119879119895(1minus119871 119894119895)+119879

1015840119894119895119871 119894119895

119879119895minus1

119868119894 (119905) 119889119905

=119883119894119895+ 119876119894119895minus 119863119894119895

2(119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

(5)

Therefore the total holding cost is obtained in

Total Holding Cost

=

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895+ 119883119894119895+1

2)

times (119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

(6)

The total shortage cost consists of two parts the totalbackorder cost and the total lost sale cost In Figure 1 thetrapezoidal area underneath the horizontal timeline in eachperiod (shown for the primary period) when multiplied bythe backorder cost per unit demand of the 119894th product inperiod 119895 120587

119894119895 is equal to the backorder cost of the item in that

period Therefore the total backorder cost will be

Total Backorder Cost

=

119898

sum119894=1

119873minus1

sum119895=1

(120587119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) 120573119894)

(7)

Furthermore since (1 minus 120573119894) represents the percentage

demands of the 119894th product that is lost sale the total lost salebecomes

Total Lost Sale Cost

=

119898

sum119894=1

119873minus1

sum119895=1

(119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) (1 minus 120573

119894))

(8)

in which 119879119895minus 1198791015840119894119895= 119887119894119895119863119894119895

The total purchase cost also consists of twoAUD and IQDcosts The purchasing offered by AUD policy is modeled by

119875119894=

1198751198941 0 lt 119876

119894119895le 1199021198942

1198751198942 1199021198942lt 119876119894119895le 1199021198943

119875119894119870 119902119894119870lt 119876119894119895

(9)

Hence the purchasing cost of this policy is obtained as

AUD Purchasing Cost

=

119898

sum119894=1

119873minus1

sum119895=1

119870

sum119896=1

119875119894119896119876119894119895119880119894119895119896

(10)


PijK

PijKminus1

Pij2

Pij1

qijKqijKminus1qij2qij1

Figure 2 AUD policy for purchasing the products in different periods

PijKPijKminus1Pij2Pij1


Pi1qi2 + Pi2(qi3 minus qi2)(1 minus mi2) PiK(Qij minus qiK)(1 minus miK)

Pi1qi2 + Pi2(Qij minus qi2)(1 minus mi2)

Pi1Qij

+ middot middot middot +

Figure 3 IQD policy for purchasing the products in different periods

A graphical representation of the AUD policy employedto purchase the products in different periods is shown inFigure 2 In this Figure the relation between the price breakpoints and the purchasing costs is demonstrated clearlyMoreover 119880

119894119895119896is a binary variable set 1 if the 119894th item is

purchased with price break 119896 in period 119895 and 0 otherwiseIn the IQD policy the purchasing cost per unit of the 119894th

product depends on its order quantity Therefore for eachprice break point we have

1198751198941119876119894119895 0 lt 119876

119894119895le 1199021198942

11987511989411199021198942+ 1198751198942(119876119894119895minus 1199021198942) (1 minus 119898

1198942) 119902

1198942lt 119876119894119895le 1199021198943

11987511989411199021198942+ 1198751198942(1199021198943minus 1199021198942) (1 minus 119898

1198942)

+ sdot sdot sdot + 119875119894119870(119876119894119895minus 119902119894119870) (1 minus 119898

119894119870)

119902119894119870lt 119876119894119895

(11)

Hence the total purchasing cost under the IQD policy isobtained as

IQD Purchasing Cost

=

119898

sum119894=1

119873minus1

sum119895=1

(119876119894119895minus 119902119894119870) 119875119894119880119894119895119870(1 minus 119898

119894119870)

+

119870minus1

sum119896=1

(119902119894119896+1minus 119902119894119896) 119875119894(1 minus 119898

119894119896)

(12)

Figure 3 graphically depicts the IQD policy for eachproduct in different periods

Thus the first objective function of the problem at handbecomes

1198851=

119898

sum119894=1

119873minus1

sum119895=1

119860119894119882119894119895+

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895+ 119883119894119895+1

2)

times (119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

+

119898

sum119894=1

119873minus1

sum119895=1

(120587119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) 120573119894)

+

119898

sum119894=1

119873minus1

sum119895=1

(119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) (1 minus 120573

119894))

+

119898

sum119894=1

119873minus1

sum119895=1

119870

sum119896=1

119876119894119895119875119894119896119880119894119895119896

+

119898

sum119894=1

119873minus1

sum119895=1

(119876119894119895minus 119902119894119870) 119875119894119880119894119895119870(1 minus 119898

119894119870)

+

119870minus1

sum119896=1

(119902119894119896+1minus 119902119894119896) 119875119894(1 minus 119898

119894119896)

(13)

The second objective of the problem is to minimize thetotal required storage space Since in each period orderquantities 119876

119894119895enter the storage and the beginning inventory


of a period is the remainder inventory of the previous period119883119894119895 the second objective function of the problem is modeled

by

1198852=

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895) 119878119894 (14)

Finally the fitness function is defined as the weightedcombination of the total inventory cost and the requiredstorage space as

TMF = 11990811198851+ 11990821198852 (15)

32 The Constraints In real-world inventory planning prob-lems due to existing constraints on either supplying orproducing goods (eg budget labor production carryingequipment and the like) objectives are not met simplyThis section presents formulations for some real-world con-straints

The first limitation is given in (4) where it relates thebeginning inventory of the items in a period to the beginninginventory of the items in the previous period plus the orderquantity of the previous period minus the demand of theprevious period

The second limitation is due to delivering the items inpackets of batches Since119876

119894119895represents the purchase quantity

of item 119894 in period 119895 denoting the batch size by 119861119894and the

number of packets by 119881119894119895 we have

119876119894119895= 119861119894119881119894119895 (16)

Furthermore since 119876119894119895can only be purchased based on one

price break point the following constraint must hold

119870

sum119896=1

119880119894119895119896= 1 (17)

The prerequisite of using this strategy is that the lowest 119902119894119896in

the AUD table must be zero (ie 1199021198941= 0)

Since the total available budget is TB the unit purchasingcost of the product is 119875

119894 and the order quantity is 119876

119894119895 the

budget constraint will be

119898

sum119894=1

119873minus1

sum119895=1

119876119894119895119875119894le TB (18)

In real-world environments the order quantity of an itemin a period may be limited Defining119872

1an upper bound for

this quantity for 119894 = 1 2 119898 and 119895 = 1 2 119873 minus 1 wehave

119876119894119895le 1198721 (19)

Moreover due to transportation contract and the truckcapacity the number of product orders and the total orderquantities in a period are limited as well Hence for 119895 =1 2 119873 minus 1 we have

119898

sum119894=1

119876119894119895119882119894119895le 1198722 (20)

where if an order occurs for item 119894 in period 119895 119882119894119895= 1

otherwise119882119894119895= 0 Further119872

2is an upper boundon the total

number of orders and the total order quantities in a periodAs a result the complete mathematical model of the

problem is

Min TMF = 11990811198851+ 11990821198852 (21)

subject to

1198851=

119898

sum119894=1

119873minus1

sum119895=1

119860119894119882119894119895+

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895+ 119883119894119895+1

2)

times (119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

+

119898

sum119894=1

119873minus1

sum119895=1

(120587119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) 120573119894)

+

119898

sum119894=1

119873minus1

sum119895=1

(119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) (1 minus 120573

119894))

+

119898

sum119894=1

119873minus1

sum119895=1

119870

sum119896=1

119876119894119895119875119894119896119880119894119895119896

+

119898

sum119894=1

119873minus1

sum119895=1

(119876119894119895minus 119902119894119870) 119875119894119880119894119895119870(1 minus 119898

119894119870)

+

119870minus1

sum119896=1

(119902119894119896+1minus 119902119894119896) 119875119894(1 minus 119898

119894119896)

1198852=

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895) 119878119894

119868119894119895+1= 119868119894119895+ 119876119894119895minus 119863119894119895

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119868119894119895+1= 119883119894119895+1 119868119894119895+1ge 0

119887119894119895 119868

119894119895+1lt 0

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119876119894119895= 119861119894119881119894119895

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119870

sum119896=1

119880119894119895119896= 1

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119898

sum119894=1

119873minus1

sum119895=1

119876119894119895119875119894le TB

119876119894119895le 1198721


(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119882119894119895isin 0 1 (119895 = 1 2 119873 minus 1)

119880119894119895119896isin 0 1

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1) (119896 = 1 2 119870)

119898

sum119894=1

119876119894119895119882119894119895le 1198722 (119895 = 1 2 119873 minus 1)

119876119894119895+1ge 119887119894119895

(22)

Inmost inventory-planningmodels that have been devel-oped so far researchers have imposed some unrealisticassumptions such that the objective function of the modelbecomes concave and the model can easily be solved bysome mathematical approaches like the Lagrangian or thederivative methods However since the model in (22) whichis obtained based on assumptions that are more realisticis an integer nonlinear programming mixed with binaryvariables reaching an analytical solution (if any) to theproblem is difficult In addition efficient treatment of integernonlinear optimization is one of the most difficult problemsin practical optimization [52] As a result in the next sectiona metaheuristic algorithm is proposed to solve the model in(22)

4 The Proposed Multiobjective ParticleSwarm Optimization Algorithm

Many researchers have successfully usedmetaheuristicmeth-ods to solve complicated optimization problems in differentfields of scientific and engineering disciplines among themthe particle swarmoptimization (PSO) algorithm is one of themost efficient methods That is why this approach is taken inthis research to solve the model in (22) The structure of theproposedMOPSO that is based on the PSO algorithm for themultiobjective inventory planning problem at hand is givenas follows

41 Generating and Initializing the Particles Positions andVelocities PSO is initialized by a group of random particles(solutions) called generation and then searches for optima byupdating generationsThe initial population is constructed byrandomly generated 119877 particles (similar to the chromosomesof a genetic algorithm) In a 119889-dimensional search spacelet 119894119896= 1199091198941198961 1199091198941198962 119909119894

119896119889 and V119894

119896= V1198941198961 V1198941198962 V119894

119896119889

be respectively the position and the velocity of particle 119894 attime 119896 Then (23) is applied to generate initial particles inwhich 119909min and 119909max are the lower and the upper bounds onthe design variable values and RAND is a random numberbetween 0 and 1 Consider

119909119894

0= 119909min + RAND (119909max minus 119909min)

V1198940= 119909min + RAND (119909max minus 119909min)

(23)

An important note for the generating and initializing step ofthe PSO is that solutionsmust be feasible andmust satisfy theconstraints As a result if a solution vector does not satisfy aconstraint the related vector solution will be penalized by abig penalty on its fitness

42 Selecting the Best Position andVelocity For every particledenote the best solution (fitness) that has been achieved so faras

997888997888997888997888997888rarr119901best119894119896= 119901best119894

1198961 119901best119894

1198962 119901best119894

119896119889 (24)

997888997888997888997888997888rarr119892best119894119896= 119892best119894

1198961 119892best119894

1198962 119892best119894

119896119889 (25)

where997888997888997888997888997888rarr119901best119894119896in (25) is the best position already found by

particle 119894 until time 119896 and997888997888997888997888997888rarr119892best119894119896in (24) is the best position

already found by a neighbor until time 119896

43 Velocity and Position Update The new velocities andpositions of the particles for the next fitness evaluation arecalculated using [53 54]

V119894119896+1119889

= 119908 sdot V119894119896119889+ 1198621sdot 1199031sdot (119901best119894

119896119889minus 119909119894

119896119889)

+ 1198622sdot 1199032sdot (119892best119894

119896119889minus 119909119894

119896119889)

119909119894

119896+1119889= 119909119894

119896119889+ V119894119896+1119889

(26)

where 1199031and 1199032are random numbers between 0 and 1 coef-

ficients 1198621and 119862

2are given acceleration constants towards

997888997888997888997888rarr119901best and

997888997888997888997888rarr119892best respectively and 119908 is the inertia weight

Introducing a linearly decreasing inertia weight into theoriginal PSO significantly improves its performance throughthe parameter study of inertia weight [55 56] Moreover thelinear distribution of the inertia weight is expressed as follows[55]

119908 = 119908max minus119908max minus 119908miniter max

iteration (27)

where iter max is the maximum number of iterations anditeration is the current number of iteration Equation (27)presents how the inertia weight is updated considering 119908maxand119908min are the initial and the final weights respectivelyTheparameters 119908max = 09 and 119908min = 04 that were previouslyinvestigated by Naka et al [55] and Shi and Eberhart [56] areused in this research as well

44 Stopping Criterion Achieving a predetermined solutionsteady-state mean and standard deviations of a solution inseveral consecutive generations stopping the algorithm at acertain computer CPU time or stopping when a maximumnumber of iterations is reached are usual stopping rules thathave been used so far in different research works In currentresearch the PSO algorithm stops when the maximumnumber of iterations is reached

Pseudocode 1 shows the pseudocode of the proposedMOPSO algorithm Moreover since the problem and hence


for 119894 = 1 to Popinitialize position (119894)initialize velocity (119894)if position (119894) and velocity (119894) be a feasible candidate solution

penalty = 0else penalty = a positive numberend if

end for119908 = [04 09]

do while Iter lt= Genfor 119895 = 1 to Pop

Calculate new velocity of the particleCalculate new position of the particle119901best(iter) = min(119901best(119894))

end for119892best(iter) = min(119892best)119908 = 119908max minus ((119908max minus 119908min)iter max) times itermodifying the velocity and position of the particle

end while

Pseudocode 1 The pseudocode of MOPSO algorithm

BeginSet 119875119862 119875119898 Pop and Gen

119878 rarr 0

initialize Population (119878)evaluate Population (119878)while (non terminating condition)repeat119878 rarr 119878 + 1

select Population (119878) from Population with roulette wheel (119878 minus 1)uniform crossoverone point mutationevaluate Population (119878)end repeatprint optimum resultend

Pseudocode 2 The pseudocode of MOGA algorithm

the model is new and there is no other available algorithmto compare the results a multiobjective genetic algorithm(MOGA) is developed in this research for validation andbenchmarking MOGA was coded using roulette wheel inselection operator population size of 40 uniform crossoverwith probability of 064 one-point random mutation withprobability 02 and a maximum number of 500 iterationsPseudocode 2 shows the pseudocode of the proposedMOGAalgorithm The computer programs of the MOPSO andMOGAalgorithmswere developed inMATLAB software andare executed on a computer with 250GHz of core 2 CPUand 300GB of RAM Furthermore all the graphical andstatistical analyses are performed in MINITAB 15

In the next section some numerical examples are given toillustrate the application of the proposed MOPSO algorithmin real-world environments and to evaluate and compare itsperformances with the ones obtained by a MOGAmethod

5 Numerical Illustrations

The decision variables in the inventory model (22) are 119876119894119895

119883119894119895119881119894119895 and 119887

119894119895 We note that the determination of the order

quantity of the items in different periods that is 119876119894119895 results

in the determination of the other decision variables as wellHence we first randomly generate 119876

119894119895 that is modeled by

the particlesrsquo position and velocity Equation (28) shows apictorial representation of the matrix 119876 for a problem with4 items in 4 periods where rows and columns correspond tothe items and the periods respectively

The structure of a particle

11987644=[[[

[

124 116 50 0

205 190 58 0

114 68 107 0

43 87 210 0

]]]

]

(28)


Table 2 Different problems and their optimal TMF values obtained by the two algorithms

Problem 119898 119873 1198721

TB Objective values MOPSO MOGAMOPSO MOGA 119862

11198622

Pop Gen 119875119862

119875119898

Pop Gen1 1 3 2000 30000 18510 18510 2 25 30 200 06 01 40 5002 2 3 3000 85000 44622 44943 15 25 20 200 05 008 30 2003 3 3 5000 170000 85830 86089 15 2 30 500 06 008 50 3004 2 4 3500 130000 92758 93423 2 2 30 100 07 02 50 2005 3 4 5000 240000 147480 146450 15 25 40 100 06 02 50 2006 4 3 6000 260000 132910 133400 25 15 20 200 06 008 40 5007 5 3 9000 370000 153840 154550 2 25 40 200 06 02 50 2008 6 3 8800 360000 214620 215510 15 25 30 500 05 01 30 2009 7 3 10500 400000 265020 266020 2 2 30 200 07 01 40 30010 8 5 12000 470000 322850 328250 25 25 30 500 06 008 50 20011 8 8 15000 550000 431245 442100 2 2 20 200 07 02 30 30012 9 5 15000 530000 438790 449835 15 25 40 100 07 02 50 30013 9 8 18000 600000 495470 513725 2 2 40 500 07 008 50 50014 9 9 20000 630000 553276 571250 15 25 20 100 05 01 40 50015 10 5 20000 620000 579030 593167 25 25 20 200 05 02 30 20016 10 8 25000 700000 642870 665890 15 25 40 200 07 008 30 20017 10 10 34000 780000 710035 731280 15 15 30 200 05 008 50 50018 10 15 45000 900000 823210 852400 2 25 20 500 06 01 40 50019 11 10 40000 850000 827659 859080 15 25 30 500 06 01 30 50020 11 15 45000 870000 867840 897500 25 2 40 100 07 02 50 20021 12 10 48000 870000 902720 948956 15 25 30 200 05 02 50 30022 12 12 53000 900000 932760 974380 15 2 30 100 06 02 40 20023 12 15 58000 950000 965470 1023950 2 25 20 100 05 02 30 20024 13 10 52000 890000 973200 1043569 15 15 20 500 07 008 40 30025 13 13 55000 930000 985439 1089210 15 25 40 500 06 01 30 30026 13 15 62000 980000 1056810 1104325 2 25 40 500 05 01 40 50027 15 8 57000 900000 1059835 1110360 2 2 20 200 06 008 50 50028 15 10 63000 900000 1095430 1176509 25 25 30 100 06 02 30 20029 15 12 68000 950000 1198720 1332900 25 15 30 500 05 01 40 20030 15 15 75000 1000000 1256980 1447905 15 2 20 500 06 008 50 20031 16 12 70000 940000 1298750 1473400 2 15 40 100 07 01 50 50032 16 15 80000 1050000 1454328 1772349 15 25 40 500 07 02 40 20033 17 15 87000 1100000 1543890 1809850 2 2 40 200 07 01 50 50034 17 17 93000 1140000 1630215 1865780 25 25 30 500 06 01 40 30035 18 10 80000 1000000 1678345 1890437 15 15 30 100 07 008 50 30036 18 15 98000 1150000 1768950 1924670 2 25 20 500 05 008 40 30037 18 18 103000 1230000 1832450 1987320 15 2 30 100 07 01 40 20038 20 10 100000 1200000 1876895 1998230 25 25 20 500 06 01 50 50039 20 15 108000 1260000 1904564 2035689 15 2 30 500 07 02 50 50040 20 20 115000 1330000 1987350 2154670 15 15 30 100 07 008 30 500Mean mdash mdash mdash mdash 897145 971541 mdash mdash mdash mdash mdash mdash mdash mdashSt Dev mdash mdash mdash mdash 581013 652778 mdash mdash mdash mdash mdash mdash mdash mdash

Table 2 shows partial data for 40 different problemswith different sizes along with their near optimal solutionsobtained by MOPSO and MOGA In these problems thenumber of items varies between 1 and 20 and the numberof periods takes values between 3 and 15 In addition thetotal available budgets and the upper bounds for the orderquantities (119872

1) are given in Table 1 for each problem

In order to illustrate how the results are obtained con-sider a typical problem with 5 items and 3 periods (theseventh row in Table 2) for which the complete input data is

given in Table 3 The parameters of the MOPSO and MOGAalgorithms are set by Taguchi method where119862

1 1198622the num-

ber of populations (Pop) and number of generations (Gen)are the parameters of MOPSO and crossover probability andtheir level values are shown in Table 4 Furthermore therest of MOPSOrsquos parameters are set as 119908min = 04 119908max =09 and the time periods 119879

119895= 3 for 119895 = 0 1 2 3 The

above parameter settings are obtained performing intensiveruns Furthermore the amount of 119881

119894119895will be obtained

automatically after gaining the order quantity 119876119894119895


Table 3 The general data for a problem with 5 items and 3 periods

Product 1198631198941

1198631198942

1205871198941

1205871198942

1198941

1198942

119861119894

119867119894

119860119894

120573119894

119878119894

1 1200 800 20 18 9 10 3 5 20 05 42 1300 900 20 18 9 10 7 5 15 05 63 1500 1200 11 14 8 12 5 6 25 08 74 2100 2000 11 14 8 12 8 6 18 08 55 1800 1600 12 15 9 11 7 7 19 06 6

0

1

Variable

120583

Wa Wb Wc

Figure 4 The triangular fuzzy numbers

Table 4 The parameters of the two algorithms and their levels

Algorithms Factors Levels [1 2 3]

MOPSO

1198621(119860) [15 2 25]

1198622(119861) [15 2 25]

Pop(119862) [20 30 40]

Gen(119863) [100 200 500]

MOGA

119875119862(119860) [05 06 07]

119875119898(119861) [008 01 02]

Pop(119862) [30 40 50]

Gen(119863) [200 300 500]

Theweights associated with the objectives are as triangu-lar fuzzy number 119908 = [119908

119886 119908119887 119908119888] shown in Figure 4 where

membership function of variable 119909 is given by

120583 (119909) =

0 119909 lt 119908119886

119909 minus 119908119886

119908119887minus 119908119886

119908119886lt 119909 lt 119908

119887

119908119888minus 119909

119908119888minus 119908119887

119908119887lt 119909 lt 119908

119888

0 119908119888lt 119909

(29)

Now in order to get crisp interval by 120572-cut operationinterval 119908

120572can be obtained as follows (forall120572 isin [0 1])119908(120572)119886minus 119908119886

119908119887minus 119908119886

= 120572119908119888minus 119908(120572)119888

119908119888minus 119908119887

= 120572 (30)

We have

119908(120572)

119886= (119908119887minus 119908119886) 120572 + 119908

119886

119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572

(31)

Therefore

119908120572= [119908(120572)

119886 119908(120572)

119888]

= [(119908119887minus 119908119886) 120572 + 119908

119886 119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572]

(32)

where 1199081= [03 05 07] 119908

2= [02 03 06] and 120572 = 05

Table 5 The Taguchi 1198719design along with objective values of the

algorithms

Run number 119860 119861 119862 119863 MOPSO MOGA1 1 1 1 1 154040 1549802 1 2 2 2 154367 1547603 1 3 3 3 154220 1550754 2 1 2 3 153944 1548755 2 2 3 1 153985 1552306 2 3 1 2 154568 1551027 3 1 3 2 154215 1547808 3 2 1 3 154320 1547509 3 3 2 1 154100 155111

Table 6 The optimal levels of the algorithmsrsquo parameters forproblem 7 of Table 2

Algorithms Factors Optimal levels

MOPSO

1198621

21198622

25Pop 40Gen 200

MOGA

119875119862

06119875119898

02Pop 50Gen 200

To perform Taguchi approach in this paper a 1198719design is

utilized based on which the results for problem 7 describedin Table 2 are shown in Table 5 as an example The optimalvalues of the levels of the algorithmsrsquo parameters shown inTable 5 are represented by Table 6 Figure 5 depicts the meanSN ratio plot each level of the factors ofMOPSO andMOGAfor problem 7 in Table 2

Tables 7 and 8 show the best result obtained by MOPSOand MOGA for the problem with 5 items and 3 periods(problem 7) respectively including the amounts of decisionvariables and the optimal objective values In these tables


Mea

n of

SN

ratio

s

321 321

321 321

Main effects plot of MOPSO mean for SN ratios

Signal-to-noise smaller is better

(A) (B)

(D)(C)

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

(a)

Mea

n of

SN

ratio

s

321 321

321 321

(A) (B)

(D)(C)

Main effects plot of MOGA mean for SN ratios


minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

(b)

Figure 5 The mean SN ratio plot for parameter levels of MOPSO and MOGA in problem 7 of Table 2


Table 7 The best result of the MOPSO algorithm

Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1215 159 15 0 405 53 0 626 1538402 1162 252 0 0 166 36 138 7863 1555 190 55 0 311 38 0 9554 1360 864 0 0 170 108 740 18765 1435 420 0 0 205 60 365 1545

Table 8 The best result of the MOGA algorithm

Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1221 168 21 0 407 56 0 611 1545502 959 392 0 0 137 56 341 8493 1220 390 0 0 244 78 280 10904 1168 960 0 0 146 120 932 19725 168 2254 0 0 24 322 1632 978

Table 9 The ANOVA analysis of the performances

Source DF SS MS 119865 119875 valueFactor 1 111119864 + 11 111119864 + 11 028 06Error 78 311119864 + 13 399119864 + 11 mdash mdashTotal 79 312119864 + 13 mdash mdash mdash

TMF is the best value of the biobjective inventory planningproblem which is given in the last two columns of Table 2Similarly the best TMF for the other problems is obtainedand is summarized in Table 2

To compare the performances of the MOPSO andMOGA several statistical and graphical approaches areemployed A one-way ANOVA analysis of the means ofthe algorithms in confidence 095 is used to compare andevaluate the objective values of the generated 40 problemsTable 9 shows the ANOVA analysis of the results of thetwo algorithms that demonstrates no significant differencebetween both algorithms Moreover the mean and stan-dard deviation (Std Dev) of the objective values of the 30generated problems shows that the MOPSO has the betterperformance in terms of the objective values in comparisonwith the MOGA In addition a pictorial presentation ofthe performances of the two algorithms shown by Figure 6displays that the MOPSO is more efficient than the MOGAalgorithm in the large number of the problems

Figure 7 depicts the boxplot and the individual value plotand Figure 8 shows the residual plots for the algorithms

A comparison of the results in Table 2 shows that theMOPSO algorithm performs better than theMOGA in termsof the fitness functions values

6 Conclusion and Recommendations forFuture Research

In this paper a biobjective multi-item multiperiod inventoryplanning problem with total available budget under all unitdiscount for some items and incremental quantity discount

Obj

ectiv

e val

ue

39373533312927252321191715131197531

22

20

18

16

14

12

10

8

6

4

2

0

VariableMOPSOMOGA

times105

Figure 6 The pictorial representation of the performances of thealgorithms

for other items was considered The orders were assumedto be placed in batch sizes and the order quantities at theend period were zeros Shortages were allowed and containbackorder and lost sale It was assumed that the beginninginventory level in primary period was zeros and the orderquantity in each period was more than the shortage quantityin the previous period Due to adopting decisions related to acertain department of production planning (extending ware-house or building a new manufacturing line) the managerdecided to build a new warehouse for the ordered items Theobjectives were to minimize both the total inventory costs


MOGAMOPSO

20

15

10

5

0

Boxplot of MOPSO and MOGAD

ata

times105

(a)

Dat

a

MOGAMOPSO

20

15

10

5

0

Individual value plot of MOPSO and MOGAtimes105

(b)

Figure 7 The boxplot and the individual value plot of the performances of the algorithms

Residual

()

200000010000000

99999

90

50

10

101

Fitted value

Resid

ual

960000940000920000900000880000

1000000

500000

0

Residual

Freq

uenc

y

10000005000000

16

12

8

4

0

Normal probability plot of the residuals Residuals versus the fitted values

Histogram of the residuals

Residual Plots for MOPSO and MOGA

minus1000000minus2000000

minus500000

minus1000000


Figure 8 The residual plots of the algorithms

and the total required storage space for which a weightedcombination was defined as the objective function The aimof the study was to determine the optimal order quantity andthe shortage quantity of each product in each period suchthat the objective function is minimized and the constraintsholdThedevelopedmodel of the problemwas shown to be aninteger nonlinear programming mixed with binary variablesTo solve the model both a multiobjective particle swarmoptimization and a multiobjective genetic algorithm wereapplied The results showed that for the 10 specific problems

the MOPSO performs better than the MOGA in terms of thefitness function values

Some recommendations for future works are to expandthe model to cover a supply chain environment to considerfuzzy or stochastic demands andor to take into account theinflation and the time value of the money

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


Acknowledgments

The authors would like to acknowledge the University ofMalaya for providing the necessary facilities and resources forthis research This research was fully funded by the Ministryof Higher EducationMalaysia with the high impact research(HIR) Grant no HIR-MOHE-16001-D000001

References

[1] C Chiang ldquoOptimal replenishment for a periodic reviewinventory system with two supply modesrdquo European Journal ofOperational Research vol 149 no 1 pp 229ndash244 2003

[2] E Mohebbi and M J M Posner ldquoMultiple replenishmentorders in a continuous-review inventory system with lost salesrdquoOperations Research Letters vol 30 no 2 pp 117ndash129 2002

[3] A H I Lee and H-Y Kang ldquoAmixed 0-1 integer programmingfor inventory model a case study of TFT-LCD manufacturingcompany in Taiwanrdquo Kybernetes vol 37 no 1 pp 66ndash82 2008

[4] S M Mousavi V Hajipour S T A Niaki and N AlikarldquoOptimizing multi-item multi-period inventory control systemwith discounted cash flow and inflation two calibrated meta-heuristic algorithmsrdquo Applied Mathematical Modelling vol 37no 4 pp 2241ndash2256 2013

[5] S M J Mirzapour Al-e-hashem and Y Rekik ldquoMulti-productmulti-period Inventory Routing problem with a transshipmentoption a green approachrdquo International Journal of ProductionEconomics 2013

[6] G Janakiraman S J Park S Seshadri and QWu ldquoNew resultson the newsvendor model and the multi-period inventorymodel with backorderingrdquo Operations Research Letters vol 41no 4 pp 373ndash376 2013

[7] H J Shin and W C Benton ldquoQuantity discount-based inven-tory coordination effectiveness and critical environmentalfactorsrdquo Production and Operations Management vol 13 no 1pp 63ndash76 2004

[8] W C Benton ldquoMultiple price breaks and alternative purchaselot-sizing procedures in material requirements planning sys-temsrdquo International Journal of Production Research vol 23 no5 pp 1025ndash1047 1985

[9] P L Abad ldquoJoint price and lot size determinationwhen supplieroffers incremental quantity discountrdquo Journal of the OperationalResearch Society vol 39 no 6 pp 603ndash607 1988

[10] P L Abad ldquoDetermining optimal selling price and lot size whenthe supplier offers all unit quantity discountrdquo Decision Sciencevol 19 no 6 pp 632ndash634 1988

[11] A K Maiti and M Maiti ldquoDiscounted multi-item inventorymodel via genetic algorithmwith roulettewheel selection arith-metic crossover and uniform mutation in constraints boundeddomainsrdquo International Journal of Computer Mathematics vol85 no 9 pp 1341ndash1353 2008

[12] S S Sana and K S Chaudhuri ldquoA deterministic EOQ modelwith delays in payments and price-discount offersrdquo EuropeanJournal of Operational Research vol 184 no 2 pp 509ndash5332008

[13] A A Taleizadeh S T A Niaki M-B Aryanezhad and A FTafti ldquoA genetic algorithm to optimize multiproduct multicon-straint inventory control systems with stochastic replenishmentintervals and discountrdquo International Journal of AdvancedManufacturing Technology vol 51 no 1ndash4 pp 311ndash323 2010

[14] S-P Chen and Y-H Ho ldquoOptimal inventory policy for thefuzzy newsboy problem with quantity discountsrdquo InformationSciences vol 228 pp 75ndash89 2013

[15] M Khan M Y Jaber and A-R Ahmad ldquoAn integrated supplychain model with errors in quality inspection and learning inproductionrdquo Omega vol 42 no 1 pp 16ndash24 2014

[16] A A Taleizadeh S T A Niaki M-B Aryanezhad and NShafii ldquoA hybrid method of fuzzy simulation and geneticalgorithm to optimize constrained inventory control systemswith stochastic replenishments and fuzzy demandrdquo InformationSciences vol 220 pp 425ndash441 2013

[17] S-H Huang and P-C Lin ldquoAmodified ant colony optimizationalgorithm for multi-item inventory routing problems withdemand uncertaintyrdquo Transportation Research E Logistics andTransportation Review vol 46 no 5 pp 598ndash611 2010

[18] K Li B Chen A I Sivakumar and Y Wu ldquoAn inventory-routing problemwith the objective of travel timeminimizationrdquoEuropean Journal of Operational Research vol 236 no 3 pp936ndash945 2014

[19] L Qin L Miao Q Ruan and Y Zhang ldquoA local search methodfor periodic inventory routing problemrdquo Expert Systems withApplications vol 41 no 2 pp 765ndash778 2014

[20] A A Taleizadeh S T A Niaki and V Hosseini ldquoThe multi-product multi-constraint newsboy problem with incrementaldiscount and batch orderrdquo Asian Journal of Applied Science vol1 no 2 pp 110ndash122 2008

[21] A A Taleizadeh S T A Niaki and V Hoseini ldquoOptimizing themulti-product multi-constraint bi-objective newsboy problemwith discount by a hybrid method of goal programming andgenetic algorithmrdquo Engineering Optimization vol 41 no 5 pp437ndash457 2009

[22] S H R Pasandideh S T A Niaki and M Hemmati farldquoOptimization of vendor managed inventory of multiproductEPQ model with multiple constraints using genetic algorithmrdquoInternational Journal of Advanced Manufacturing Technologyvol 71 no 1ndash4 pp 365ndash376 2014

[23] A A Taleizadeh M-B Aryanezhad and S T A NiakildquoOptimizing multi-product multi-constraint inventory controlsystems with stochastic replenishmentsrdquo Journal of AppliedSciences vol 8 no 7 pp 1228ndash1234 2008

[24] A A Taleizadeh S T A Niaki and M-B AryanezhadldquoA hybrid method of Pareto TOPSIS and genetic algorithmto optimize multi-product multi-constraint inventory controlsystems with random fuzzy replenishmentsrdquoMathematical andComputer Modelling vol 49 no 5-6 pp 1044ndash1057 2009

[25] A A Taleizadeh S T A Niaki and M-B AryaneznadldquoMulti-product multi-constraint inventory control systemswith stochastic replenishment and discount under fuzzy pur-chasing price and holding costsrdquo The American Journal ofApplied Sciences vol 6 no 1 pp 1ndash12 2009

[26] A A Taleizadeh S T A Niaki and F Barzinpour ldquoMultiple-buyer multiple-vendor multi-product multi-constraint supplychain problem with stochastic demand and variable lead-time a harmony search algorithmrdquo Applied Mathematics andComputation vol 217 no 22 pp 9234ndash9253 2011

[27] A A Taleizadeh S T A Niaki and H-M Wee ldquoJoint singlevendor-single buyer supply chain problem with stochasticdemand and fuzzy lead-timerdquo Knowledge-Based Systems vol48 pp 1ndash9 2013


[28] T K Roy and M Maiti ldquoMulti-objective inventory models ofdeteriorating items with some constraints in a fuzzy environ-mentrdquo Computers and Operations Research vol 25 no 12 pp1085ndash1095 1998

[29] R G Yaghin S M T Fatemi Ghomi and S A Torabi ldquoApossibilistic multiple objective pricing and lot-sizing modelwith multiple demand classesrdquo Fuzzy Sets and Systems vol 231pp 26ndash44 2013

[30] G Dueck ldquoThreshold accepting a general purpose optimiza-tion algorithm appearing superior to simulated annealingrdquoJournal of Computational Physics vol 90 no 1 pp 161ndash175 1990

[31] S J Joo and J Y Bong ldquoConstruction of exact D-optimaldesigns by Tabu searchrdquo Computational Statistics and DataAnalysis vol 21 no 2 pp 181ndash191 1996

[32] J Sadeghi S Sadeghi and S T A Niaki ldquoOptimizing ahybrid vendor-managed inventory and transportation problemwith fuzzy demand an improved particle swarm optimizationalgorithmrdquo Information Sciences vol 272 pp 126ndash144 2014

[33] E H L Aarts and J H M Korst Simulated Annealing andBoltzmann Machine A stochastic Approach to Computing JohnWiley amp Sons Chichester UK 1st edition 1989

[34] A Varyani and A Jalilvand-Nejad ldquoDetermining the optimumproduction quantity in three-echelon production system withstochastic demandrdquo International Journal of Advanced Manu-facturing Technology vol 72 no 1ndash4 pp 119ndash133 2014

[35] M Laumanns L Thiele K Deb and E Zitzler ldquoCombiningconvergence and diversity in evolutionary multiobjective opti-mizationrdquoEvolutionary Computation vol 10 no 3 pp 263ndash2822002

[36] A R Gaiduk Y A Vershinin and M J West ldquoNeuralnetworks and optimization problemsrdquo in Proceedings of theIEEE International Conference onControl Applications vol 1 pp37ndash41 September 2002

[37] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCambridge Mass USA 2004

[38] Z W Geem J H Kim and G V Loganathan ldquoA new heuristicoptimization algorithm harmony searchrdquo Simulation vol 76no 2 pp 60ndash68 2001

[39] K S Lee and Z W Geem ldquoA new structural optimizationmethod based on the harmony search algorithmrdquo Computersand Structures vol 82 no 9-10 pp 781ndash798 2004

[40] A A Taleizadeh HMoghadasi S T A Niaki andA EftekharildquoAn economic order quantity under joint replenishment policyto supply expensive imported raw materials with payment inadvancerdquo Journal of Applied Sciences vol 8 no 23 pp 4263ndash4273 2008

[41] F Fu ldquoIntegrated scheduling and batch ordering for construc-tion projectrdquoApplied Mathematical Modelling vol 38 no 2 pp784ndash797 2014

[42] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Wash USA 1995

[43] A A Taleizadeh S T A Niaki N Shafii R G Meibodi andA Jabbarzadeh ldquoA particle swarm optimization approach forconstraint joint single buyer-single vendor inventory problemwith changeable lead time and (rQ) policy in supply chainrdquoInternational Journal of Advanced Manufacturing Technologyvol 51 no 9ndash12 pp 1209ndash1223 2010

[44] Y-R Chen and C-Y Dye ldquoApplication of particle swarmoptimisation for solving deteriorating inventory model with

fluctuating demand and controllable deterioration raterdquo Inter-national Journal of Systems Science vol 44 no 6 pp 1026ndash10392013

[45] P J Agrell ldquoA multicriteria framework for inventory controlrdquoInternational Journal of Production Economics vol 41 no 1ndash3pp 59ndash70 1995

[46] C-S Tsou ldquoMulti-objective inventory planning using MOPSOand TOPSISrdquo Expert Systems with Applications vol 35 no 1-2pp 136ndash142 2008

[47] C A Coello Coello andM S Lechuga ldquoMOPSO a proposal formultiple objective particle swarm optimizationrdquo in Proceedingsof the IEEE Congress on Computational Intelligence vol 2 pp1051ndash1056 Honolulu Hawaii USA May 2002

[48] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[49] J D Knowles and D W Corne ldquoApproximating the non-dominated front using the Pareto archived evolution strategyrdquoEvolutionary Computation vol 8 no 2 pp 149ndash172 2000

[50] C A Coello Coello and G T Pulido ldquoMultiobjective opti-mization using a micro-genetic algorithmrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference (GECCO01) pp 274ndash282 San Francisco Calif USA 2001

[51] C A Coello Coello G T Pulido andM S Lechuga ldquoHandlingmultiple objectives with particle swarm optimizationrdquo IEEETransactions on Evolutionary Computation vol 8 no 3 pp256ndash279 2004

[52] L El-Sharkawi Modern Heuristic Optimization TechniquesWiley InterScience New Jersey NJ USA 1st edition 2008

[53] H Shayeghi H A Shayanfar S Jalilzadeh and A Safari ldquoAPSO based unified power flow controller for damping of powersystem oscillationsrdquo Energy Conversion and Management vol50 no 10 pp 2583ndash2592 2009

[54] J Kennedy R Eberhart and Y Shi Swarm Intelligence MorganKaufmann San Francisco Calif USA 2001

[55] S Naka T Genji T Yura and Y Fukuyama ldquoPractical distribu-tion state estimation using hybrid particle swarm optimizationrdquoin Proceedings of the IEEE Power Engineering Society WinterMeeting vol 2 pp 815ndash820 Columbus Ohio USA February2001

[56] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation Washington DC USA July 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


that is genetic algorithm and simulated annealing Mirza-pour Al-e-hashem and Rekik [5] presented a multiproductmultiperiod inventory routing problem where multiple con-strained vehicles distributed products from multiple suppli-ers to a single plant tomeet the given demand of each productover a finite planning horizon Janakiraman et al [6] analyzedthe multiperiodic newsvendor problem and proposed somenew results

The quantity discount is of increasing attention dueto its practical importance in purchasing and control ofitems Usually one derives the better marginal cost of pur-chaseproduction by taking advantages of the chances of costsavings through bulk purchaseproduction Furthermore insupply chain environments quantity discounts can be consid-ered an inventory coordination mechanism between buyersand suppliers [7] In the literature of quantity discountsBenton [8] considered an inventory system with quantitydiscount with multiple price breaks and alternative purchas-ing and lot-sizing policy Abad [9 10] proposed modelsfor joint price and lot size determination when supplieroffers either incremental (IQD) or all unit (AUD) quantitydiscounts K Maiti andM Maiti [11] developed a model for amulti-item inventory control system of breakable items withAUD and IQD (and a combination of the two policies) andproposed genetic algorithm to solve the model Sana andChaudhuri [12] extended an EOQ model by relaxation ofthe preassumptions related to payments allowing delay ondelivery and discounts They used a mixed integer nonlinearprogramming technique to model the problem Taleizadeh etal [13] considered a genetic algorithm to optimizemultiprod-uct multiconstraint inventory control systems with stochasticreplenishment intervals and discount Recently several workssuch as the ones in [4 14ndash16] have also spotted discounts ininventory control problems Huang and Lin [17] addressed anintegrated model that scheduled multi-item replenishmentwith uncertain demand to determine delivery routes andtruck loads In this study the products are purchased indifferent periods under AUD and IQD policies

Metaheuristic algorithms have been suggested to solvesome of the existing developed inventory problems in theliterature Someof these algorithms are tabu search [18 19 31]genetic algorithms (GA) [20ndash22 32] simulating annealing(SA) [23 33 34] evolutionary algorithm [24 35] thresholdaccepting [30] neural networks [36] ant colony optimization[37] fuzzy simulation [25] and harmony search [26 38ndash41]

Inspired by social behavior of bird flocking or fish school-ing particle swarm optimization (PSO) is also a population-based stochastic optimization metaheuristic developed byKennedy and Eberhart [42] Recently researchers haveemployed this effective technique to find optimal or nearoptimal solutions of their inventory control problems Forexample Taleizadeh et al [43] employed PSO to solvetheir integer nonlinear programming model of a constraintjoint single buyer-single vendor inventory problem withchangeable lead time and (119903 119876) policy in supply chainswith stochastic demand Chen and Dye [44] solved aninventory problem with deteriorating products and variabledemands using a PSO algorithm Further Taleizadeh et al[27] modeled a chancendashconstraint supply chain problem

with uniformly distributed stochastic demand where an AntColony Bee and a PSO algorithm were utilized to solve theproblem

Instead of optimizing a single objective some researcherstried to find Pareto front solutions for their multiple objectiveinventory planning problems that usually consist of multipleconflicting objectives Agrell [45] proposed a multicriteriaframework for inventory control problem in which the solu-tion procedure was an interactive method with preferencesextracted gradually in decision analysis process to determinebatch size and security stock Roy and Maiti [28] presenteda multiobjective inventory model of deteriorating items withstock-dependent demand under limited imprecise storagearea and total cost budget Tsou [46] developed a multiob-jective reorder point and order size system and proposeda multiobjective PSO (MOPSO) to generate Pareto frontsolutions He employed TOPSIS to sort the nondominatedsolutions The objectives therein were to maximize the profitand to minimize the wastage cost where the profit goalwastage cost and storage areawere fuzzy in natureOne of thesuccessful applications of PSO toMOOPs is the seminal workof Coello Coello and Lechuga [47] Yaghin et al [29] firstaddressed an inventory-marketing system to determine theproduction lot size marketing expenditure and selling pricesin which the model was formulated as a fuzzy nonlinearmultiobjective program Then they converted the model toa classical single-objective one by a fuzzy goal programmingmethod where an efficient solution procedure using PSOwas provided to solve the resulting nonlinear problem Intheir study MOPSO is not only a viable alternative tosolve MOOPs but also the only one compared with thenondominated sorting genetic algorithm-II (NSGA-II) [48]the Pareto archive evolutionary strategy (PAES) [49] and themicrogenetic algorithm [50] for multiobjective optimizationproblems [51] Table 1 shows the literature review of theworksreviewed in this work where DOE is an abbreviation of termldquodesign of experimentsrdquo

In this research the contribution of the problem is con-sidering a new biobjective multi-item multiperiodic inven-tory control model where some items are purchased underAUD and the other items are bought from IQDThe demandsvary in different periods the budget is limited the ordersare placed in batch sizes and shortages in combination ofbackorder and lost sale are considered The goal is to find theoptimum inventory levels of the items in each period suchthat the total inventory cost and the total required warehousespace are minimized simultaneously Since it is not easy forthe managers to allocate the crisp values to the weights ofthe objectives in a decisionmaking process considering theseweights as fuzzy numbers will be taken as an advantage

In order to be more understanding of the problem wetry to explain the model with taking an example in the realworld We consider a company which produces some kindsof fashion clothes including trousers t-shirt and shirt in acertain period The customers (wholesales) of this companywith different demand rates make the orders and receivetheir products in the prespecific boxes each one consistingof a known number of these clothes Moreover due tosome unforeseen matters such as production limitation the



















119894119895+1ge 119887119894119895defined below)








variable where 119876119894119895= 119861119894119881119894119895)





(it is119883119894119895+1+ 119876119894119895+1


119894119895)



1198941= 0)


(1198981198941= 0)













(0 le 1199081le 1)


(0 le 1199082le 1)








(1)



Qi1 Qi2

Qi3

Qi4

Di1

bi1bi3

Xi3

T1T0 T4T3T2 T5T9984001 T9984003120573i bi1






sum119894=1

119873minus1

sum119895=1

119860119894119882119894119895 (2)




119895minus1le

119905 le 119879119895(1 minus 119871

119894119895) + 1198791015840119894119895119871119894119895is obtained as

119867119894int119879119895(1minus119871 119894119895)+119879

1015840119894119895119871 119894119895

119879119895minus1

119868119894 (119905) 119889119905 (3)



term 119879119895(1 minus 119871

119894119895) + 1198791015840119894119895119871119894119895becomes 1198791015840

119894119895 Otherwise 119871

119894119895= 0



119894 represents


119868119894119895+1= 119868119894119895+ 119876119894119895minus 119863119894119895 (4)

and if 119868119894119895+1ge 0 then 119868

119894119895+1= 119883119894119895+1

otherwise 119868119894119895+1= 119887119894119895 (3)

becomes

119867119894int119879119895(1minus119871 119894119895)+119879

1015840119894119895119871 119894119895

119879119895minus1

119868119894 (119905) 119889119905

=119883119894119895+ 119876119894119895minus 119863119894119895

2(119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

(5)


Total Holding Cost

=

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895+ 119883119894119895+1

2)

times (119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

(6)





=

119898

sum119894=1

119873minus1

sum119895=1

(120587119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) 120573119894)

(7)




=

119898

sum119894=1

119873minus1

sum119895=1

(119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) (1 minus 120573

119894))

(8)

in which 119879119895minus 1198791015840119894119895= 119887119894119895119863119894119895


119875119894=

1198751198941 0 lt 119876

119894119895le 1199021198942

1198751198942 1199021198942lt 119876119894119895le 1199021198943

119875119894119870 119902119894119870lt 119876119894119895

(9)


AUD Purchasing Cost

=

119898

sum119894=1

119873minus1

sum119895=1

119870

sum119896=1

119875119894119896119876119894119895119880119894119895119896

(10)


PijK

PijKminus1

Pij2

Pij1







Pi1Qij







1198751198941119876119894119895 0 lt 119876

119894119895le 1199021198942

11987511989411199021198942+ 1198751198942(119876119894119895minus 1199021198942) (1 minus 119898

1198942) 119902

1198942lt 119876119894119895le 1199021198943

11987511989411199021198942+ 1198751198942(1199021198943minus 1199021198942) (1 minus 119898

1198942)


119894119870)

119902119894119870lt 119876119894119895

(11)


IQD Purchasing Cost

=

119898

sum119894=1

119873minus1

sum119895=1

(119876119894119895minus 119902119894119870) 119875119894119880119894119895119870(1 minus 119898

119894119870)

+

119870minus1

sum119896=1

(119902119894119896+1minus 119902119894119896) 119875119894(1 minus 119898

119894119896)

(12)



1198851=

119898

sum119894=1

119873minus1

sum119895=1

119860119894119882119894119895+

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895+ 119883119894119895+1

2)

times (119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

+

119898

sum119894=1

119873minus1

sum119895=1

(120587119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) 120573119894)

+

119898

sum119894=1

119873minus1

sum119895=1

(119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) (1 minus 120573

119894))

+

119898

sum119894=1

119873minus1

sum119895=1

119870

sum119896=1

119876119894119895119875119894119896119880119894119895119896

+

119898

sum119894=1

119873minus1

sum119895=1

(119876119894119895minus 119902119894119870) 119875119894119880119894119895119870(1 minus 119898

119894119870)

+

119870minus1

sum119896=1

(119902119894119896+1minus 119902119894119896) 119875119894(1 minus 119898

119894119896)

(13)





by

1198852=

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895) 119878119894 (14)


TMF = 11990811198851+ 11990821198852 (15)







119876119894119895= 119861119894119881119894119895 (16)



119870

sum119896=1

119880119894119895119896= 1 (17)





119894119895 the


119898

sum119894=1

119873minus1

sum119895=1

119876119894119895119875119894le TB (18)


1an upper bound for


119876119894119895le 1198721 (19)


119898

sum119894=1

119876119894119895119882119894119895le 1198722 (20)





problem is

Min TMF = 11990811198851+ 11990821198852 (21)

subject to

1198851=

119898

sum119894=1

119873minus1

sum119895=1

119860119894119882119894119895+

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895+ 119883119894119895+1

2)

times (119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

+

119898

sum119894=1

119873minus1

sum119895=1

(120587119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) 120573119894)

+

119898

sum119894=1

119873minus1

sum119895=1

(119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) (1 minus 120573

119894))

+

119898

sum119894=1

119873minus1

sum119895=1

119870

sum119896=1

119876119894119895119875119894119896119880119894119895119896

+

119898

sum119894=1

119873minus1

sum119895=1

(119876119894119895minus 119902119894119870) 119875119894119880119894119895119870(1 minus 119898

119894119870)

+

119870minus1

sum119896=1

(119902119894119896+1minus 119902119894119896) 119875119894(1 minus 119898

119894119896)

1198852=

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895) 119878119894

119868119894119895+1= 119868119894119895+ 119876119894119895minus 119863119894119895

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119868119894119895+1= 119883119894119895+1 119868119894119895+1ge 0

119887119894119895 119868

119894119895+1lt 0

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119876119894119895= 119861119894119881119894119895

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119870

sum119896=1

119880119894119895119896= 1

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119898

sum119894=1

119873minus1

sum119895=1

119876119894119895119875119894le TB

119876119894119895le 1198721


(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119882119894119895isin 0 1 (119895 = 1 2 119873 minus 1)

119880119894119895119896isin 0 1

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1) (119896 = 1 2 119870)

119898

sum119894=1

119876119894119895119882119894119895le 1198722 (119895 = 1 2 119873 minus 1)

119876119894119895+1ge 119887119894119895

(22)





119896119889 and V119894

119896= V1198941198961 V1198941198962 V119894

119896119889


119909119894



(23)



997888997888997888997888997888rarr119901best119894119896= 119901best119894

1198961 119901best119894

1198962 119901best119894

119896119889 (24)

997888997888997888997888997888rarr119892best119894119896= 119892best119894

1198961 119892best119894

1198962 119892best119894

119896119889 (25)





V119894119896+1119889


119896119889minus 119909119894

119896119889)

+ 1198622sdot 1199032sdot (119892best119894

119896119889minus 119909119894

119896119889)

119909119894

119896+1119889= 119909119894

119896119889+ V119894119896+1119889

(26)




997888997888997888997888rarr119901best and




iteration (27)







end for119908 = [04 09]




end while



119878 rarr 0








119883119894119895119881119894119895 and 119887







11987644=[[[

[

124 116 50 0

205 190 58 0

114 68 107 0

43 87 210 0

]]]

]

(28)



Problem 119898 119873 1198721


11198622

Pop Gen 119875119862

119875119898






1 1198622the num-


119895= 3 for 119895 = 0 1 2 3 The






Product 1198631198941

1198631198942

1205871198941

1205871198942

1198941

1198942

119861119894

119867119894

119860119894

120573119894

119878119894

1 1200 800 20 18 9 10 3 5 20 05 42 1300 900 20 18 9 10 7 5 15 05 63 1500 1200 11 14 8 12 5 6 25 08 74 2100 2000 11 14 8 12 8 6 18 08 55 1800 1600 12 15 9 11 7 7 19 06 6

0

1

Variable

120583

Wa Wb Wc




MOPSO

1198621(119860) [15 2 25]

1198622(119861) [15 2 25]

Pop(119862) [20 30 40]

Gen(119863) [100 200 500]

MOGA

119875119862(119860) [05 06 07]

119875119898(119861) [008 01 02]

Pop(119862) [30 40 50]

Gen(119863) [200 300 500]




120583 (119909) =

0 119909 lt 119908119886

119909 minus 119908119886

119908119887minus 119908119886

119908119886lt 119909 lt 119908

119887

119908119888minus 119909

119908119888minus 119908119887

119908119887lt 119909 lt 119908

119888

0 119908119888lt 119909

(29)



119908119887minus 119908119886

= 120572119908119888minus 119908(120572)119888

119908119888minus 119908119887

= 120572 (30)

We have

119908(120572)

119886= (119908119887minus 119908119886) 120572 + 119908

119886

119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572

(31)

Therefore

119908120572= [119908(120572)

119886 119908(120572)

119888]

= [(119908119887minus 119908119886) 120572 + 119908

119886 119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572]

(32)

where 1199081= [03 05 07] 119908

2= [02 03 06] and 120572 = 05


algorithms




MOPSO

1198621

21198622

25Pop 40Gen 200

MOGA

119875119862

06119875119898

02Pop 50Gen 200





Mea

n of

SN

ratio

s

321 321

321 321



(A) (B)

(D)(C)

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

(a)

Mea

n of

SN

ratio

s

321 321

321 321

(A) (B)

(D)(C)



minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

(b)




Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1215 159 15 0 405 53 0 626 1538402 1162 252 0 0 166 36 138 7863 1555 190 55 0 311 38 0 9554 1360 864 0 0 170 108 740 18765 1435 420 0 0 205 60 365 1545


Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1221 168 21 0 407 56 0 611 1545502 959 392 0 0 137 56 341 8493 1220 390 0 0 244 78 280 10904 1168 960 0 0 146 120 932 19725 168 2254 0 0 24 322 1632 978









Obj

ectiv

e val

ue

39373533312927252321191715131197531

22

20

18

16

14

12

10

8

6

4

2

0

VariableMOPSOMOGA

times105




MOGAMOPSO

20

15

10

5

0


ata

times105

(a)

Dat

a

MOGAMOPSO

20

15

10

5

0


(b)


Residual

()

200000010000000

99999

90

50

10

101

Fitted value

Resid

ual

960000940000920000900000880000

1000000

500000

0

Residual

Freq

uenc

y

10000005000000

16

12

8

4

0





minus500000

minus1000000








Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























119894119895+1ge 119887119894119895defined below)








variable where 119876119894119895= 119861119894119881119894119895)





(it is119883119894119895+1+ 119876119894119895+1


119894119895)



1198941= 0)


(1198981198941= 0)













(0 le 1199081le 1)


(0 le 1199082le 1)








(1)



Qi1 Qi2

Qi3

Qi4

Di1

bi1bi3

Xi3

T1T0 T4T3T2 T5T9984001 T9984003120573i bi1






sum119894=1

119873minus1

sum119895=1

119860119894119882119894119895 (2)




119895minus1le

119905 le 119879119895(1 minus 119871

119894119895) + 1198791015840119894119895119871119894119895is obtained as

119867119894int119879119895(1minus119871 119894119895)+119879

1015840119894119895119871 119894119895

119879119895minus1

119868119894 (119905) 119889119905 (3)



term 119879119895(1 minus 119871

119894119895) + 1198791015840119894119895119871119894119895becomes 1198791015840

119894119895 Otherwise 119871

119894119895= 0



119894 represents


119868119894119895+1= 119868119894119895+ 119876119894119895minus 119863119894119895 (4)

and if 119868119894119895+1ge 0 then 119868

119894119895+1= 119883119894119895+1

otherwise 119868119894119895+1= 119887119894119895 (3)

becomes

119867119894int119879119895(1minus119871 119894119895)+119879

1015840119894119895119871 119894119895

119879119895minus1

119868119894 (119905) 119889119905

=119883119894119895+ 119876119894119895minus 119863119894119895

2(119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

(5)


Total Holding Cost

=

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895+ 119883119894119895+1

2)

times (119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

(6)





=

119898

sum119894=1

119873minus1

sum119895=1

(120587119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) 120573119894)

(7)




=

119898

sum119894=1

119873minus1

sum119895=1

(119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) (1 minus 120573

119894))

(8)

in which 119879119895minus 1198791015840119894119895= 119887119894119895119863119894119895


119875119894=

1198751198941 0 lt 119876

119894119895le 1199021198942

1198751198942 1199021198942lt 119876119894119895le 1199021198943

119875119894119870 119902119894119870lt 119876119894119895

(9)


AUD Purchasing Cost

=

119898

sum119894=1

119873minus1

sum119895=1

119870

sum119896=1

119875119894119896119876119894119895119880119894119895119896

(10)


PijK

PijKminus1

Pij2

Pij1







Pi1Qij







1198751198941119876119894119895 0 lt 119876

119894119895le 1199021198942

11987511989411199021198942+ 1198751198942(119876119894119895minus 1199021198942) (1 minus 119898

1198942) 119902

1198942lt 119876119894119895le 1199021198943

11987511989411199021198942+ 1198751198942(1199021198943minus 1199021198942) (1 minus 119898

1198942)


119894119870)

119902119894119870lt 119876119894119895

(11)


IQD Purchasing Cost

=

119898

sum119894=1

119873minus1

sum119895=1

(119876119894119895minus 119902119894119870) 119875119894119880119894119895119870(1 minus 119898

119894119870)

+

119870minus1

sum119896=1

(119902119894119896+1minus 119902119894119896) 119875119894(1 minus 119898

119894119896)

(12)



1198851=

119898

sum119894=1

119873minus1

sum119895=1

119860119894119882119894119895+

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895+ 119883119894119895+1

2)

times (119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

+

119898

sum119894=1

119873minus1

sum119895=1

(120587119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) 120573119894)

+

119898

sum119894=1

119873minus1

sum119895=1

(119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) (1 minus 120573

119894))

+

119898

sum119894=1

119873minus1

sum119895=1

119870

sum119896=1

119876119894119895119875119894119896119880119894119895119896

+

119898

sum119894=1

119873minus1

sum119895=1

(119876119894119895minus 119902119894119870) 119875119894119880119894119895119870(1 minus 119898

119894119870)

+

119870minus1

sum119896=1

(119902119894119896+1minus 119902119894119896) 119875119894(1 minus 119898

119894119896)

(13)





by

1198852=

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895) 119878119894 (14)


TMF = 11990811198851+ 11990821198852 (15)







119876119894119895= 119861119894119881119894119895 (16)



119870

sum119896=1

119880119894119895119896= 1 (17)





119894119895 the


119898

sum119894=1

119873minus1

sum119895=1

119876119894119895119875119894le TB (18)


1an upper bound for


119876119894119895le 1198721 (19)


119898

sum119894=1

119876119894119895119882119894119895le 1198722 (20)





problem is

Min TMF = 11990811198851+ 11990821198852 (21)

subject to

1198851=

119898

sum119894=1

119873minus1

sum119895=1

119860119894119882119894119895+

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895+ 119883119894119895+1

2)

times (119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

+

119898

sum119894=1

119873minus1

sum119895=1

(120587119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) 120573119894)

+

119898

sum119894=1

119873minus1

sum119895=1

(119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) (1 minus 120573

119894))

+

119898

sum119894=1

119873minus1

sum119895=1

119870

sum119896=1

119876119894119895119875119894119896119880119894119895119896

+

119898

sum119894=1

119873minus1

sum119895=1

(119876119894119895minus 119902119894119870) 119875119894119880119894119895119870(1 minus 119898

119894119870)

+

119870minus1

sum119896=1

(119902119894119896+1minus 119902119894119896) 119875119894(1 minus 119898

119894119896)

1198852=

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895) 119878119894

119868119894119895+1= 119868119894119895+ 119876119894119895minus 119863119894119895

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119868119894119895+1= 119883119894119895+1 119868119894119895+1ge 0

119887119894119895 119868

119894119895+1lt 0

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119876119894119895= 119861119894119881119894119895

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119870

sum119896=1

119880119894119895119896= 1

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119898

sum119894=1

119873minus1

sum119895=1

119876119894119895119875119894le TB

119876119894119895le 1198721


(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119882119894119895isin 0 1 (119895 = 1 2 119873 minus 1)

119880119894119895119896isin 0 1

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1) (119896 = 1 2 119870)

119898

sum119894=1

119876119894119895119882119894119895le 1198722 (119895 = 1 2 119873 minus 1)

119876119894119895+1ge 119887119894119895

(22)





119896119889 and V119894

119896= V1198941198961 V1198941198962 V119894

119896119889


119909119894



(23)



997888997888997888997888997888rarr119901best119894119896= 119901best119894

1198961 119901best119894

1198962 119901best119894

119896119889 (24)

997888997888997888997888997888rarr119892best119894119896= 119892best119894

1198961 119892best119894

1198962 119892best119894

119896119889 (25)





V119894119896+1119889


119896119889minus 119909119894

119896119889)

+ 1198622sdot 1199032sdot (119892best119894

119896119889minus 119909119894

119896119889)

119909119894

119896+1119889= 119909119894

119896119889+ V119894119896+1119889

(26)




997888997888997888997888rarr119901best and




iteration (27)







end for119908 = [04 09]




end while



119878 rarr 0








119883119894119895119881119894119895 and 119887







11987644=[[[

[

124 116 50 0

205 190 58 0

114 68 107 0

43 87 210 0

]]]

]

(28)



Problem 119898 119873 1198721


11198622

Pop Gen 119875119862

119875119898






1 1198622the num-


119895= 3 for 119895 = 0 1 2 3 The






Product 1198631198941

1198631198942

1205871198941

1205871198942

1198941

1198942

119861119894

119867119894

119860119894

120573119894

119878119894

1 1200 800 20 18 9 10 3 5 20 05 42 1300 900 20 18 9 10 7 5 15 05 63 1500 1200 11 14 8 12 5 6 25 08 74 2100 2000 11 14 8 12 8 6 18 08 55 1800 1600 12 15 9 11 7 7 19 06 6

0

1

Variable

120583

Wa Wb Wc




MOPSO

1198621(119860) [15 2 25]

1198622(119861) [15 2 25]

Pop(119862) [20 30 40]

Gen(119863) [100 200 500]

MOGA

119875119862(119860) [05 06 07]

119875119898(119861) [008 01 02]

Pop(119862) [30 40 50]

Gen(119863) [200 300 500]




120583 (119909) =

0 119909 lt 119908119886

119909 minus 119908119886

119908119887minus 119908119886

119908119886lt 119909 lt 119908

119887

119908119888minus 119909

119908119888minus 119908119887

119908119887lt 119909 lt 119908

119888

0 119908119888lt 119909

(29)



119908119887minus 119908119886

= 120572119908119888minus 119908(120572)119888

119908119888minus 119908119887

= 120572 (30)

We have

119908(120572)

119886= (119908119887minus 119908119886) 120572 + 119908

119886

119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572

(31)

Therefore

119908120572= [119908(120572)

119886 119908(120572)

119888]

= [(119908119887minus 119908119886) 120572 + 119908

119886 119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572]

(32)

where 1199081= [03 05 07] 119908

2= [02 03 06] and 120572 = 05


algorithms




MOPSO

1198621

21198622

25Pop 40Gen 200

MOGA

119875119862

06119875119898

02Pop 50Gen 200





Mea

n of

SN

ratio

s

321 321

321 321



(A) (B)

(D)(C)

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

(a)

Mea

n of

SN

ratio

s

321 321

321 321

(A) (B)

(D)(C)



minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

(b)




Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1215 159 15 0 405 53 0 626 1538402 1162 252 0 0 166 36 138 7863 1555 190 55 0 311 38 0 9554 1360 864 0 0 170 108 740 18765 1435 420 0 0 205 60 365 1545


Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1221 168 21 0 407 56 0 611 1545502 959 392 0 0 137 56 341 8493 1220 390 0 0 244 78 280 10904 1168 960 0 0 146 120 932 19725 168 2254 0 0 24 322 1632 978









Obj

ectiv

e val

ue

39373533312927252321191715131197531

22

20

18

16

14

12

10

8

6

4

2

0

VariableMOPSOMOGA

times105




MOGAMOPSO

20

15

10

5

0


ata

times105

(a)

Dat

a

MOGAMOPSO

20

15

10

5

0


(b)


Residual

()

200000010000000

99999

90

50

10

101

Fitted value

Resid

ual

960000940000920000900000880000

1000000

500000

0

Residual

Freq

uenc

y

10000005000000

16

12

8

4

0





minus500000

minus1000000








Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















119894119895+1ge 119887119894119895defined below)








variable where 119876119894119895= 119861119894119881119894119895)





(it is119883119894119895+1+ 119876119894119895+1


119894119895)



1198941= 0)


(1198981198941= 0)













(0 le 1199081le 1)


(0 le 1199082le 1)








(1)



Qi1 Qi2

Qi3

Qi4

Di1

bi1bi3

Xi3

T1T0 T4T3T2 T5T9984001 T9984003120573i bi1






sum119894=1

119873minus1

sum119895=1

119860119894119882119894119895 (2)




119895minus1le

119905 le 119879119895(1 minus 119871

119894119895) + 1198791015840119894119895119871119894119895is obtained as

119867119894int119879119895(1minus119871 119894119895)+119879

1015840119894119895119871 119894119895

119879119895minus1

119868119894 (119905) 119889119905 (3)



term 119879119895(1 minus 119871

119894119895) + 1198791015840119894119895119871119894119895becomes 1198791015840

119894119895 Otherwise 119871

119894119895= 0



119894 represents


119868119894119895+1= 119868119894119895+ 119876119894119895minus 119863119894119895 (4)

and if 119868119894119895+1ge 0 then 119868

119894119895+1= 119883119894119895+1

otherwise 119868119894119895+1= 119887119894119895 (3)

becomes

119867119894int119879119895(1minus119871 119894119895)+119879

1015840119894119895119871 119894119895

119879119895minus1

119868119894 (119905) 119889119905

=119883119894119895+ 119876119894119895minus 119863119894119895

2(119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

(5)


Total Holding Cost

=

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895+ 119883119894119895+1

2)

times (119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

(6)





=

119898

sum119894=1

119873minus1

sum119895=1

(120587119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) 120573119894)

(7)




=

119898

sum119894=1

119873minus1

sum119895=1

(119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) (1 minus 120573

119894))

(8)

in which 119879119895minus 1198791015840119894119895= 119887119894119895119863119894119895


119875119894=

1198751198941 0 lt 119876

119894119895le 1199021198942

1198751198942 1199021198942lt 119876119894119895le 1199021198943

119875119894119870 119902119894119870lt 119876119894119895

(9)


AUD Purchasing Cost

=

119898

sum119894=1

119873minus1

sum119895=1

119870

sum119896=1

119875119894119896119876119894119895119880119894119895119896

(10)


PijK

PijKminus1

Pij2

Pij1







Pi1Qij







1198751198941119876119894119895 0 lt 119876

119894119895le 1199021198942

11987511989411199021198942+ 1198751198942(119876119894119895minus 1199021198942) (1 minus 119898

1198942) 119902

1198942lt 119876119894119895le 1199021198943

11987511989411199021198942+ 1198751198942(1199021198943minus 1199021198942) (1 minus 119898

1198942)


119894119870)

119902119894119870lt 119876119894119895

(11)


IQD Purchasing Cost

=

119898

sum119894=1

119873minus1

sum119895=1

(119876119894119895minus 119902119894119870) 119875119894119880119894119895119870(1 minus 119898

119894119870)

+

119870minus1

sum119896=1

(119902119894119896+1minus 119902119894119896) 119875119894(1 minus 119898

119894119896)

(12)



1198851=

119898

sum119894=1

119873minus1

sum119895=1

119860119894119882119894119895+

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895+ 119883119894119895+1

2)

times (119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

+

119898

sum119894=1

119873minus1

sum119895=1

(120587119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) 120573119894)

+

119898

sum119894=1

119873minus1

sum119895=1

(119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) (1 minus 120573

119894))

+

119898

sum119894=1

119873minus1

sum119895=1

119870

sum119896=1

119876119894119895119875119894119896119880119894119895119896

+

119898

sum119894=1

119873minus1

sum119895=1

(119876119894119895minus 119902119894119870) 119875119894119880119894119895119870(1 minus 119898

119894119870)

+

119870minus1

sum119896=1

(119902119894119896+1minus 119902119894119896) 119875119894(1 minus 119898

119894119896)

(13)





by

1198852=

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895) 119878119894 (14)


TMF = 11990811198851+ 11990821198852 (15)







119876119894119895= 119861119894119881119894119895 (16)



119870

sum119896=1

119880119894119895119896= 1 (17)





119894119895 the


119898

sum119894=1

119873minus1

sum119895=1

119876119894119895119875119894le TB (18)


1an upper bound for


119876119894119895le 1198721 (19)


119898

sum119894=1

119876119894119895119882119894119895le 1198722 (20)





problem is

Min TMF = 11990811198851+ 11990821198852 (21)

subject to

1198851=

119898

sum119894=1

119873minus1

sum119895=1

119860119894119882119894119895+

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895+ 119883119894119895+1

2)

times (119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

+

119898

sum119894=1

119873minus1

sum119895=1

(120587119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) 120573119894)

+

119898

sum119894=1

119873minus1

sum119895=1

(119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) (1 minus 120573

119894))

+

119898

sum119894=1

119873minus1

sum119895=1

119870

sum119896=1

119876119894119895119875119894119896119880119894119895119896

+

119898

sum119894=1

119873minus1

sum119895=1

(119876119894119895minus 119902119894119870) 119875119894119880119894119895119870(1 minus 119898

119894119870)

+

119870minus1

sum119896=1

(119902119894119896+1minus 119902119894119896) 119875119894(1 minus 119898

119894119896)

1198852=

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895) 119878119894

119868119894119895+1= 119868119894119895+ 119876119894119895minus 119863119894119895

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119868119894119895+1= 119883119894119895+1 119868119894119895+1ge 0

119887119894119895 119868

119894119895+1lt 0

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119876119894119895= 119861119894119881119894119895

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119870

sum119896=1

119880119894119895119896= 1

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119898

sum119894=1

119873minus1

sum119895=1

119876119894119895119875119894le TB

119876119894119895le 1198721


(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119882119894119895isin 0 1 (119895 = 1 2 119873 minus 1)

119880119894119895119896isin 0 1

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1) (119896 = 1 2 119870)

119898

sum119894=1

119876119894119895119882119894119895le 1198722 (119895 = 1 2 119873 minus 1)

119876119894119895+1ge 119887119894119895

(22)





119896119889 and V119894

119896= V1198941198961 V1198941198962 V119894

119896119889


119909119894



(23)



997888997888997888997888997888rarr119901best119894119896= 119901best119894

1198961 119901best119894

1198962 119901best119894

119896119889 (24)

997888997888997888997888997888rarr119892best119894119896= 119892best119894

1198961 119892best119894

1198962 119892best119894

119896119889 (25)





V119894119896+1119889


119896119889minus 119909119894

119896119889)

+ 1198622sdot 1199032sdot (119892best119894

119896119889minus 119909119894

119896119889)

119909119894

119896+1119889= 119909119894

119896119889+ V119894119896+1119889

(26)




997888997888997888997888rarr119901best and




iteration (27)







end for119908 = [04 09]




end while



119878 rarr 0








119883119894119895119881119894119895 and 119887







11987644=[[[

[

124 116 50 0

205 190 58 0

114 68 107 0

43 87 210 0

]]]

]

(28)



Problem 119898 119873 1198721


11198622

Pop Gen 119875119862

119875119898






1 1198622the num-


119895= 3 for 119895 = 0 1 2 3 The






Product 1198631198941

1198631198942

1205871198941

1205871198942

1198941

1198942

119861119894

119867119894

119860119894

120573119894

119878119894

1 1200 800 20 18 9 10 3 5 20 05 42 1300 900 20 18 9 10 7 5 15 05 63 1500 1200 11 14 8 12 5 6 25 08 74 2100 2000 11 14 8 12 8 6 18 08 55 1800 1600 12 15 9 11 7 7 19 06 6

0

1

Variable

120583

Wa Wb Wc




MOPSO

1198621(119860) [15 2 25]

1198622(119861) [15 2 25]

Pop(119862) [20 30 40]

Gen(119863) [100 200 500]

MOGA

119875119862(119860) [05 06 07]

119875119898(119861) [008 01 02]

Pop(119862) [30 40 50]

Gen(119863) [200 300 500]




120583 (119909) =

0 119909 lt 119908119886

119909 minus 119908119886

119908119887minus 119908119886

119908119886lt 119909 lt 119908

119887

119908119888minus 119909

119908119888minus 119908119887

119908119887lt 119909 lt 119908

119888

0 119908119888lt 119909

(29)



119908119887minus 119908119886

= 120572119908119888minus 119908(120572)119888

119908119888minus 119908119887

= 120572 (30)

We have

119908(120572)

119886= (119908119887minus 119908119886) 120572 + 119908

119886

119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572

(31)

Therefore

119908120572= [119908(120572)

119886 119908(120572)

119888]

= [(119908119887minus 119908119886) 120572 + 119908

119886 119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572]

(32)

where 1199081= [03 05 07] 119908

2= [02 03 06] and 120572 = 05


algorithms




MOPSO

1198621

21198622

25Pop 40Gen 200

MOGA

119875119862

06119875119898

02Pop 50Gen 200





Mea

n of

SN

ratio

s

321 321

321 321



(A) (B)

(D)(C)

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

(a)

Mea

n of

SN

ratio

s

321 321

321 321

(A) (B)

(D)(C)



minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

(b)




Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1215 159 15 0 405 53 0 626 1538402 1162 252 0 0 166 36 138 7863 1555 190 55 0 311 38 0 9554 1360 864 0 0 170 108 740 18765 1435 420 0 0 205 60 365 1545


Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1221 168 21 0 407 56 0 611 1545502 959 392 0 0 137 56 341 8493 1220 390 0 0 244 78 280 10904 1168 960 0 0 146 120 932 19725 168 2254 0 0 24 322 1632 978









Obj

ectiv

e val

ue

39373533312927252321191715131197531

22

20

18

16

14

12

10

8

6

4

2

0

VariableMOPSOMOGA

times105




MOGAMOPSO

20

15

10

5

0


ata

times105

(a)

Dat

a

MOGAMOPSO

20

15

10

5

0


(b)


Residual

()

200000010000000

99999

90

50

10

101

Fitted value

Resid

ual

960000940000920000900000880000

1000000

500000

0

Residual

Freq

uenc

y

10000005000000

16

12

8

4

0





minus500000

minus1000000








Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Qi1 Qi2

Qi3

Qi4

Di1

bi1bi3

Xi3

T1T0 T4T3T2 T5T9984001 T9984003120573i bi1






sum119894=1

119873minus1

sum119895=1

119860119894119882119894119895 (2)




119895minus1le

119905 le 119879119895(1 minus 119871

119894119895) + 1198791015840119894119895119871119894119895is obtained as

119867119894int119879119895(1minus119871 119894119895)+119879

1015840119894119895119871 119894119895

119879119895minus1

119868119894 (119905) 119889119905 (3)



term 119879119895(1 minus 119871

119894119895) + 1198791015840119894119895119871119894119895becomes 1198791015840

119894119895 Otherwise 119871

119894119895= 0



119894 represents


119868119894119895+1= 119868119894119895+ 119876119894119895minus 119863119894119895 (4)

and if 119868119894119895+1ge 0 then 119868

119894119895+1= 119883119894119895+1

otherwise 119868119894119895+1= 119887119894119895 (3)

becomes

119867119894int119879119895(1minus119871 119894119895)+119879

1015840119894119895119871 119894119895

119879119895minus1

119868119894 (119905) 119889119905

=119883119894119895+ 119876119894119895minus 119863119894119895

2(119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

(5)


Total Holding Cost

=

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895+ 119883119894119895+1

2)

times (119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

(6)





=

119898

sum119894=1

119873minus1

sum119895=1

(120587119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) 120573119894)

(7)




=

119898

sum119894=1

119873minus1

sum119895=1

(119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) (1 minus 120573

119894))

(8)

in which 119879119895minus 1198791015840119894119895= 119887119894119895119863119894119895


119875119894=

1198751198941 0 lt 119876

119894119895le 1199021198942

1198751198942 1199021198942lt 119876119894119895le 1199021198943

119875119894119870 119902119894119870lt 119876119894119895

(9)


AUD Purchasing Cost

=

119898

sum119894=1

119873minus1

sum119895=1

119870

sum119896=1

119875119894119896119876119894119895119880119894119895119896

(10)


PijK

PijKminus1

Pij2

Pij1







Pi1Qij







1198751198941119876119894119895 0 lt 119876

119894119895le 1199021198942

11987511989411199021198942+ 1198751198942(119876119894119895minus 1199021198942) (1 minus 119898

1198942) 119902

1198942lt 119876119894119895le 1199021198943

11987511989411199021198942+ 1198751198942(1199021198943minus 1199021198942) (1 minus 119898

1198942)


119894119870)

119902119894119870lt 119876119894119895

(11)


IQD Purchasing Cost

=

119898

sum119894=1

119873minus1

sum119895=1

(119876119894119895minus 119902119894119870) 119875119894119880119894119895119870(1 minus 119898

119894119870)

+

119870minus1

sum119896=1

(119902119894119896+1minus 119902119894119896) 119875119894(1 minus 119898

119894119896)

(12)



1198851=

119898

sum119894=1

119873minus1

sum119895=1

119860119894119882119894119895+

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895+ 119883119894119895+1

2)

times (119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

+

119898

sum119894=1

119873minus1

sum119895=1

(120587119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) 120573119894)

+

119898

sum119894=1

119873minus1

sum119895=1

(119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) (1 minus 120573

119894))

+

119898

sum119894=1

119873minus1

sum119895=1

119870

sum119896=1

119876119894119895119875119894119896119880119894119895119896

+

119898

sum119894=1

119873minus1

sum119895=1

(119876119894119895minus 119902119894119870) 119875119894119880119894119895119870(1 minus 119898

119894119870)

+

119870minus1

sum119896=1

(119902119894119896+1minus 119902119894119896) 119875119894(1 minus 119898

119894119896)

(13)





by

1198852=

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895) 119878119894 (14)


TMF = 11990811198851+ 11990821198852 (15)







119876119894119895= 119861119894119881119894119895 (16)



119870

sum119896=1

119880119894119895119896= 1 (17)





119894119895 the


119898

sum119894=1

119873minus1

sum119895=1

119876119894119895119875119894le TB (18)


1an upper bound for


119876119894119895le 1198721 (19)


119898

sum119894=1

119876119894119895119882119894119895le 1198722 (20)





problem is

Min TMF = 11990811198851+ 11990821198852 (21)

subject to

1198851=

119898

sum119894=1

119873minus1

sum119895=1

119860119894119882119894119895+

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895+ 119883119894119895+1

2)

times (119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

+

119898

sum119894=1

119873minus1

sum119895=1

(120587119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) 120573119894)

+

119898

sum119894=1

119873minus1

sum119895=1

(119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) (1 minus 120573

119894))

+

119898

sum119894=1

119873minus1

sum119895=1

119870

sum119896=1

119876119894119895119875119894119896119880119894119895119896

+

119898

sum119894=1

119873minus1

sum119895=1

(119876119894119895minus 119902119894119870) 119875119894119880119894119895119870(1 minus 119898

119894119870)

+

119870minus1

sum119896=1

(119902119894119896+1minus 119902119894119896) 119875119894(1 minus 119898

119894119896)

1198852=

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895) 119878119894

119868119894119895+1= 119868119894119895+ 119876119894119895minus 119863119894119895

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119868119894119895+1= 119883119894119895+1 119868119894119895+1ge 0

119887119894119895 119868

119894119895+1lt 0

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119876119894119895= 119861119894119881119894119895

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119870

sum119896=1

119880119894119895119896= 1

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119898

sum119894=1

119873minus1

sum119895=1

119876119894119895119875119894le TB

119876119894119895le 1198721


(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119882119894119895isin 0 1 (119895 = 1 2 119873 minus 1)

119880119894119895119896isin 0 1

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1) (119896 = 1 2 119870)

119898

sum119894=1

119876119894119895119882119894119895le 1198722 (119895 = 1 2 119873 minus 1)

119876119894119895+1ge 119887119894119895

(22)





119896119889 and V119894

119896= V1198941198961 V1198941198962 V119894

119896119889


119909119894



(23)



997888997888997888997888997888rarr119901best119894119896= 119901best119894

1198961 119901best119894

1198962 119901best119894

119896119889 (24)

997888997888997888997888997888rarr119892best119894119896= 119892best119894

1198961 119892best119894

1198962 119892best119894

119896119889 (25)





V119894119896+1119889


119896119889minus 119909119894

119896119889)

+ 1198622sdot 1199032sdot (119892best119894

119896119889minus 119909119894

119896119889)

119909119894

119896+1119889= 119909119894

119896119889+ V119894119896+1119889

(26)




997888997888997888997888rarr119901best and




iteration (27)







end for119908 = [04 09]




end while



119878 rarr 0








119883119894119895119881119894119895 and 119887







11987644=[[[

[

124 116 50 0

205 190 58 0

114 68 107 0

43 87 210 0

]]]

]

(28)



Problem 119898 119873 1198721


11198622

Pop Gen 119875119862

119875119898






1 1198622the num-


119895= 3 for 119895 = 0 1 2 3 The






Product 1198631198941

1198631198942

1205871198941

1205871198942

1198941

1198942

119861119894

119867119894

119860119894

120573119894

119878119894

1 1200 800 20 18 9 10 3 5 20 05 42 1300 900 20 18 9 10 7 5 15 05 63 1500 1200 11 14 8 12 5 6 25 08 74 2100 2000 11 14 8 12 8 6 18 08 55 1800 1600 12 15 9 11 7 7 19 06 6

0

1

Variable

120583

Wa Wb Wc




MOPSO

1198621(119860) [15 2 25]

1198622(119861) [15 2 25]

Pop(119862) [20 30 40]

Gen(119863) [100 200 500]

MOGA

119875119862(119860) [05 06 07]

119875119898(119861) [008 01 02]

Pop(119862) [30 40 50]

Gen(119863) [200 300 500]




120583 (119909) =

0 119909 lt 119908119886

119909 minus 119908119886

119908119887minus 119908119886

119908119886lt 119909 lt 119908

119887

119908119888minus 119909

119908119888minus 119908119887

119908119887lt 119909 lt 119908

119888

0 119908119888lt 119909

(29)



119908119887minus 119908119886

= 120572119908119888minus 119908(120572)119888

119908119888minus 119908119887

= 120572 (30)

We have

119908(120572)

119886= (119908119887minus 119908119886) 120572 + 119908

119886

119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572

(31)

Therefore

119908120572= [119908(120572)

119886 119908(120572)

119888]

= [(119908119887minus 119908119886) 120572 + 119908

119886 119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572]

(32)

where 1199081= [03 05 07] 119908

2= [02 03 06] and 120572 = 05


algorithms




MOPSO

1198621

21198622

25Pop 40Gen 200

MOGA

119875119862

06119875119898

02Pop 50Gen 200





Mea

n of

SN

ratio

s

321 321

321 321



(A) (B)

(D)(C)

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

(a)

Mea

n of

SN

ratio

s

321 321

321 321

(A) (B)

(D)(C)



minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

(b)




Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1215 159 15 0 405 53 0 626 1538402 1162 252 0 0 166 36 138 7863 1555 190 55 0 311 38 0 9554 1360 864 0 0 170 108 740 18765 1435 420 0 0 205 60 365 1545


Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1221 168 21 0 407 56 0 611 1545502 959 392 0 0 137 56 341 8493 1220 390 0 0 244 78 280 10904 1168 960 0 0 146 120 932 19725 168 2254 0 0 24 322 1632 978









Obj

ectiv

e val

ue

39373533312927252321191715131197531

22

20

18

16

14

12

10

8

6

4

2

0

VariableMOPSOMOGA

times105




MOGAMOPSO

20

15

10

5

0


ata

times105

(a)

Dat

a

MOGAMOPSO

20

15

10

5

0


(b)


Residual

()

200000010000000

99999

90

50

10

101

Fitted value

Resid

ual

960000940000920000900000880000

1000000

500000

0

Residual

Freq

uenc

y

10000005000000

16

12

8

4

0





minus500000

minus1000000








Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










PijK

PijKminus1

Pij2

Pij1







Pi1Qij







1198751198941119876119894119895 0 lt 119876

119894119895le 1199021198942

11987511989411199021198942+ 1198751198942(119876119894119895minus 1199021198942) (1 minus 119898

1198942) 119902

1198942lt 119876119894119895le 1199021198943

11987511989411199021198942+ 1198751198942(1199021198943minus 1199021198942) (1 minus 119898

1198942)


119894119870)

119902119894119870lt 119876119894119895

(11)


IQD Purchasing Cost

=

119898

sum119894=1

119873minus1

sum119895=1

(119876119894119895minus 119902119894119870) 119875119894119880119894119895119870(1 minus 119898

119894119870)

+

119870minus1

sum119896=1

(119902119894119896+1minus 119902119894119896) 119875119894(1 minus 119898

119894119896)

(12)



1198851=

119898

sum119894=1

119873minus1

sum119895=1

119860119894119882119894119895+

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895+ 119883119894119895+1

2)

times (119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

+

119898

sum119894=1

119873minus1

sum119895=1

(120587119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) 120573119894)

+

119898

sum119894=1

119873minus1

sum119895=1

(119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) (1 minus 120573

119894))

+

119898

sum119894=1

119873minus1

sum119895=1

119870

sum119896=1

119876119894119895119875119894119896119880119894119895119896

+

119898

sum119894=1

119873minus1

sum119895=1

(119876119894119895minus 119902119894119870) 119875119894119880119894119895119870(1 minus 119898

119894119870)

+

119870minus1

sum119896=1

(119902119894119896+1minus 119902119894119896) 119875119894(1 minus 119898

119894119896)

(13)





by

1198852=

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895) 119878119894 (14)


TMF = 11990811198851+ 11990821198852 (15)







119876119894119895= 119861119894119881119894119895 (16)



119870

sum119896=1

119880119894119895119896= 1 (17)





119894119895 the


119898

sum119894=1

119873minus1

sum119895=1

119876119894119895119875119894le TB (18)


1an upper bound for


119876119894119895le 1198721 (19)


119898

sum119894=1

119876119894119895119882119894119895le 1198722 (20)





problem is

Min TMF = 11990811198851+ 11990821198852 (21)

subject to

1198851=

119898

sum119894=1

119873minus1

sum119895=1

119860119894119882119894119895+

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895+ 119883119894119895+1

2)

times (119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

+

119898

sum119894=1

119873minus1

sum119895=1

(120587119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) 120573119894)

+

119898

sum119894=1

119873minus1

sum119895=1

(119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) (1 minus 120573

119894))

+

119898

sum119894=1

119873minus1

sum119895=1

119870

sum119896=1

119876119894119895119875119894119896119880119894119895119896

+

119898

sum119894=1

119873minus1

sum119895=1

(119876119894119895minus 119902119894119870) 119875119894119880119894119895119870(1 minus 119898

119894119870)

+

119870minus1

sum119896=1

(119902119894119896+1minus 119902119894119896) 119875119894(1 minus 119898

119894119896)

1198852=

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895) 119878119894

119868119894119895+1= 119868119894119895+ 119876119894119895minus 119863119894119895

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119868119894119895+1= 119883119894119895+1 119868119894119895+1ge 0

119887119894119895 119868

119894119895+1lt 0

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119876119894119895= 119861119894119881119894119895

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119870

sum119896=1

119880119894119895119896= 1

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119898

sum119894=1

119873minus1

sum119895=1

119876119894119895119875119894le TB

119876119894119895le 1198721


(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119882119894119895isin 0 1 (119895 = 1 2 119873 minus 1)

119880119894119895119896isin 0 1

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1) (119896 = 1 2 119870)

119898

sum119894=1

119876119894119895119882119894119895le 1198722 (119895 = 1 2 119873 minus 1)

119876119894119895+1ge 119887119894119895

(22)





119896119889 and V119894

119896= V1198941198961 V1198941198962 V119894

119896119889


119909119894



(23)



997888997888997888997888997888rarr119901best119894119896= 119901best119894

1198961 119901best119894

1198962 119901best119894

119896119889 (24)

997888997888997888997888997888rarr119892best119894119896= 119892best119894

1198961 119892best119894

1198962 119892best119894

119896119889 (25)





V119894119896+1119889


119896119889minus 119909119894

119896119889)

+ 1198622sdot 1199032sdot (119892best119894

119896119889minus 119909119894

119896119889)

119909119894

119896+1119889= 119909119894

119896119889+ V119894119896+1119889

(26)




997888997888997888997888rarr119901best and




iteration (27)







end for119908 = [04 09]




end while



119878 rarr 0








119883119894119895119881119894119895 and 119887







11987644=[[[

[

124 116 50 0

205 190 58 0

114 68 107 0

43 87 210 0

]]]

]

(28)



Problem 119898 119873 1198721


11198622

Pop Gen 119875119862

119875119898






1 1198622the num-


119895= 3 for 119895 = 0 1 2 3 The






Product 1198631198941

1198631198942

1205871198941

1205871198942

1198941

1198942

119861119894

119867119894

119860119894

120573119894

119878119894

1 1200 800 20 18 9 10 3 5 20 05 42 1300 900 20 18 9 10 7 5 15 05 63 1500 1200 11 14 8 12 5 6 25 08 74 2100 2000 11 14 8 12 8 6 18 08 55 1800 1600 12 15 9 11 7 7 19 06 6

0

1

Variable

120583

Wa Wb Wc




MOPSO

1198621(119860) [15 2 25]

1198622(119861) [15 2 25]

Pop(119862) [20 30 40]

Gen(119863) [100 200 500]

MOGA

119875119862(119860) [05 06 07]

119875119898(119861) [008 01 02]

Pop(119862) [30 40 50]

Gen(119863) [200 300 500]




120583 (119909) =

0 119909 lt 119908119886

119909 minus 119908119886

119908119887minus 119908119886

119908119886lt 119909 lt 119908

119887

119908119888minus 119909

119908119888minus 119908119887

119908119887lt 119909 lt 119908

119888

0 119908119888lt 119909

(29)



119908119887minus 119908119886

= 120572119908119888minus 119908(120572)119888

119908119888minus 119908119887

= 120572 (30)

We have

119908(120572)

119886= (119908119887minus 119908119886) 120572 + 119908

119886

119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572

(31)

Therefore

119908120572= [119908(120572)

119886 119908(120572)

119888]

= [(119908119887minus 119908119886) 120572 + 119908

119886 119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572]

(32)

where 1199081= [03 05 07] 119908

2= [02 03 06] and 120572 = 05


algorithms




MOPSO

1198621

21198622

25Pop 40Gen 200

MOGA

119875119862

06119875119898

02Pop 50Gen 200





Mea

n of

SN

ratio

s

321 321

321 321



(A) (B)

(D)(C)

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

(a)

Mea

n of

SN

ratio

s

321 321

321 321

(A) (B)

(D)(C)



minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

(b)




Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1215 159 15 0 405 53 0 626 1538402 1162 252 0 0 166 36 138 7863 1555 190 55 0 311 38 0 9554 1360 864 0 0 170 108 740 18765 1435 420 0 0 205 60 365 1545


Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1221 168 21 0 407 56 0 611 1545502 959 392 0 0 137 56 341 8493 1220 390 0 0 244 78 280 10904 1168 960 0 0 146 120 932 19725 168 2254 0 0 24 322 1632 978









Obj

ectiv

e val

ue

39373533312927252321191715131197531

22

20

18

16

14

12

10

8

6

4

2

0

VariableMOPSOMOGA

times105




MOGAMOPSO

20

15

10

5

0


ata

times105

(a)

Dat

a

MOGAMOPSO

20

15

10

5

0


(b)


Residual

()

200000010000000

99999

90

50

10

101

Fitted value

Resid

ual

960000940000920000900000880000

1000000

500000

0

Residual

Freq

uenc

y

10000005000000

16

12

8

4

0





minus500000

minus1000000








Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











by

1198852=

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895) 119878119894 (14)


TMF = 11990811198851+ 11990821198852 (15)







119876119894119895= 119861119894119881119894119895 (16)



119870

sum119896=1

119880119894119895119896= 1 (17)





119894119895 the


119898

sum119894=1

119873minus1

sum119895=1

119876119894119895119875119894le TB (18)


1an upper bound for


119876119894119895le 1198721 (19)


119898

sum119894=1

119876119894119895119882119894119895le 1198722 (20)





problem is

Min TMF = 11990811198851+ 11990821198852 (21)

subject to

1198851=

119898

sum119894=1

119873minus1

sum119895=1

119860119894119882119894119895+

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895+ 119883119894119895+1

2)

times (119879119895(1 minus 119871

119894119895) + 1198791015840

119894119895119871119894119895minus 119879119895minus1)119867119894

+

119898

sum119894=1

119873minus1

sum119895=1

(120587119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) 120573119894)

+

119898

sum119894=1

119873minus1

sum119895=1

(119894119895119887119894119895

2(119879119895minus 1198791015840

119894119895) (1 minus 120573

119894))

+

119898

sum119894=1

119873minus1

sum119895=1

119870

sum119896=1

119876119894119895119875119894119896119880119894119895119896

+

119898

sum119894=1

119873minus1

sum119895=1

(119876119894119895minus 119902119894119870) 119875119894119880119894119895119870(1 minus 119898

119894119870)

+

119870minus1

sum119896=1

(119902119894119896+1minus 119902119894119896) 119875119894(1 minus 119898

119894119896)

1198852=

119898

sum119894=1

119873minus1

sum119895=1

(119883119894119895+ 119876119894119895) 119878119894

119868119894119895+1= 119868119894119895+ 119876119894119895minus 119863119894119895

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119868119894119895+1= 119883119894119895+1 119868119894119895+1ge 0

119887119894119895 119868

119894119895+1lt 0

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119876119894119895= 119861119894119881119894119895

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119870

sum119896=1

119880119894119895119896= 1

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119898

sum119894=1

119873minus1

sum119895=1

119876119894119895119875119894le TB

119876119894119895le 1198721


(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119882119894119895isin 0 1 (119895 = 1 2 119873 minus 1)

119880119894119895119896isin 0 1

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1) (119896 = 1 2 119870)

119898

sum119894=1

119876119894119895119882119894119895le 1198722 (119895 = 1 2 119873 minus 1)

119876119894119895+1ge 119887119894119895

(22)





119896119889 and V119894

119896= V1198941198961 V1198941198962 V119894

119896119889


119909119894



(23)



997888997888997888997888997888rarr119901best119894119896= 119901best119894

1198961 119901best119894

1198962 119901best119894

119896119889 (24)

997888997888997888997888997888rarr119892best119894119896= 119892best119894

1198961 119892best119894

1198962 119892best119894

119896119889 (25)





V119894119896+1119889


119896119889minus 119909119894

119896119889)

+ 1198622sdot 1199032sdot (119892best119894

119896119889minus 119909119894

119896119889)

119909119894

119896+1119889= 119909119894

119896119889+ V119894119896+1119889

(26)




997888997888997888997888rarr119901best and




iteration (27)







end for119908 = [04 09]




end while



119878 rarr 0








119883119894119895119881119894119895 and 119887







11987644=[[[

[

124 116 50 0

205 190 58 0

114 68 107 0

43 87 210 0

]]]

]

(28)



Problem 119898 119873 1198721


11198622

Pop Gen 119875119862

119875119898






1 1198622the num-


119895= 3 for 119895 = 0 1 2 3 The






Product 1198631198941

1198631198942

1205871198941

1205871198942

1198941

1198942

119861119894

119867119894

119860119894

120573119894

119878119894

1 1200 800 20 18 9 10 3 5 20 05 42 1300 900 20 18 9 10 7 5 15 05 63 1500 1200 11 14 8 12 5 6 25 08 74 2100 2000 11 14 8 12 8 6 18 08 55 1800 1600 12 15 9 11 7 7 19 06 6

0

1

Variable

120583

Wa Wb Wc




MOPSO

1198621(119860) [15 2 25]

1198622(119861) [15 2 25]

Pop(119862) [20 30 40]

Gen(119863) [100 200 500]

MOGA

119875119862(119860) [05 06 07]

119875119898(119861) [008 01 02]

Pop(119862) [30 40 50]

Gen(119863) [200 300 500]




120583 (119909) =

0 119909 lt 119908119886

119909 minus 119908119886

119908119887minus 119908119886

119908119886lt 119909 lt 119908

119887

119908119888minus 119909

119908119888minus 119908119887

119908119887lt 119909 lt 119908

119888

0 119908119888lt 119909

(29)



119908119887minus 119908119886

= 120572119908119888minus 119908(120572)119888

119908119888minus 119908119887

= 120572 (30)

We have

119908(120572)

119886= (119908119887minus 119908119886) 120572 + 119908

119886

119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572

(31)

Therefore

119908120572= [119908(120572)

119886 119908(120572)

119888]

= [(119908119887minus 119908119886) 120572 + 119908

119886 119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572]

(32)

where 1199081= [03 05 07] 119908

2= [02 03 06] and 120572 = 05


algorithms




MOPSO

1198621

21198622

25Pop 40Gen 200

MOGA

119875119862

06119875119898

02Pop 50Gen 200





Mea

n of

SN

ratio

s

321 321

321 321



(A) (B)

(D)(C)

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

(a)

Mea

n of

SN

ratio

s

321 321

321 321

(A) (B)

(D)(C)



minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

(b)




Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1215 159 15 0 405 53 0 626 1538402 1162 252 0 0 166 36 138 7863 1555 190 55 0 311 38 0 9554 1360 864 0 0 170 108 740 18765 1435 420 0 0 205 60 365 1545


Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1221 168 21 0 407 56 0 611 1545502 959 392 0 0 137 56 341 8493 1220 390 0 0 244 78 280 10904 1168 960 0 0 146 120 932 19725 168 2254 0 0 24 322 1632 978









Obj

ectiv

e val

ue

39373533312927252321191715131197531

22

20

18

16

14

12

10

8

6

4

2

0

VariableMOPSOMOGA

times105




MOGAMOPSO

20

15

10

5

0


ata

times105

(a)

Dat

a

MOGAMOPSO

20

15

10

5

0


(b)


Residual

()

200000010000000

99999

90

50

10

101

Fitted value

Resid

ual

960000940000920000900000880000

1000000

500000

0

Residual

Freq

uenc

y

10000005000000

16

12

8

4

0





minus500000

minus1000000








Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1)

119882119894119895isin 0 1 (119895 = 1 2 119873 minus 1)

119880119894119895119896isin 0 1

(119894 = 1 2 119898) (119895 = 1 2 119873 minus 1) (119896 = 1 2 119870)

119898

sum119894=1

119876119894119895119882119894119895le 1198722 (119895 = 1 2 119873 minus 1)

119876119894119895+1ge 119887119894119895

(22)





119896119889 and V119894

119896= V1198941198961 V1198941198962 V119894

119896119889


119909119894



(23)



997888997888997888997888997888rarr119901best119894119896= 119901best119894

1198961 119901best119894

1198962 119901best119894

119896119889 (24)

997888997888997888997888997888rarr119892best119894119896= 119892best119894

1198961 119892best119894

1198962 119892best119894

119896119889 (25)





V119894119896+1119889


119896119889minus 119909119894

119896119889)

+ 1198622sdot 1199032sdot (119892best119894

119896119889minus 119909119894

119896119889)

119909119894

119896+1119889= 119909119894

119896119889+ V119894119896+1119889

(26)




997888997888997888997888rarr119901best and




iteration (27)







end for119908 = [04 09]




end while



119878 rarr 0








119883119894119895119881119894119895 and 119887







11987644=[[[

[

124 116 50 0

205 190 58 0

114 68 107 0

43 87 210 0

]]]

]

(28)



Problem 119898 119873 1198721


11198622

Pop Gen 119875119862

119875119898






1 1198622the num-


119895= 3 for 119895 = 0 1 2 3 The






Product 1198631198941

1198631198942

1205871198941

1205871198942

1198941

1198942

119861119894

119867119894

119860119894

120573119894

119878119894

1 1200 800 20 18 9 10 3 5 20 05 42 1300 900 20 18 9 10 7 5 15 05 63 1500 1200 11 14 8 12 5 6 25 08 74 2100 2000 11 14 8 12 8 6 18 08 55 1800 1600 12 15 9 11 7 7 19 06 6

0

1

Variable

120583

Wa Wb Wc




MOPSO

1198621(119860) [15 2 25]

1198622(119861) [15 2 25]

Pop(119862) [20 30 40]

Gen(119863) [100 200 500]

MOGA

119875119862(119860) [05 06 07]

119875119898(119861) [008 01 02]

Pop(119862) [30 40 50]

Gen(119863) [200 300 500]




120583 (119909) =

0 119909 lt 119908119886

119909 minus 119908119886

119908119887minus 119908119886

119908119886lt 119909 lt 119908

119887

119908119888minus 119909

119908119888minus 119908119887

119908119887lt 119909 lt 119908

119888

0 119908119888lt 119909

(29)



119908119887minus 119908119886

= 120572119908119888minus 119908(120572)119888

119908119888minus 119908119887

= 120572 (30)

We have

119908(120572)

119886= (119908119887minus 119908119886) 120572 + 119908

119886

119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572

(31)

Therefore

119908120572= [119908(120572)

119886 119908(120572)

119888]

= [(119908119887minus 119908119886) 120572 + 119908

119886 119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572]

(32)

where 1199081= [03 05 07] 119908

2= [02 03 06] and 120572 = 05


algorithms




MOPSO

1198621

21198622

25Pop 40Gen 200

MOGA

119875119862

06119875119898

02Pop 50Gen 200





Mea

n of

SN

ratio

s

321 321

321 321



(A) (B)

(D)(C)

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

(a)

Mea

n of

SN

ratio

s

321 321

321 321

(A) (B)

(D)(C)



minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

(b)




Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1215 159 15 0 405 53 0 626 1538402 1162 252 0 0 166 36 138 7863 1555 190 55 0 311 38 0 9554 1360 864 0 0 170 108 740 18765 1435 420 0 0 205 60 365 1545


Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1221 168 21 0 407 56 0 611 1545502 959 392 0 0 137 56 341 8493 1220 390 0 0 244 78 280 10904 1168 960 0 0 146 120 932 19725 168 2254 0 0 24 322 1632 978









Obj

ectiv

e val

ue

39373533312927252321191715131197531

22

20

18

16

14

12

10

8

6

4

2

0

VariableMOPSOMOGA

times105




MOGAMOPSO

20

15

10

5

0


ata

times105

(a)

Dat

a

MOGAMOPSO

20

15

10

5

0


(b)


Residual

()

200000010000000

99999

90

50

10

101

Fitted value

Resid

ual

960000940000920000900000880000

1000000

500000

0

Residual

Freq

uenc

y

10000005000000

16

12

8

4

0





minus500000

minus1000000








Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












end for119908 = [04 09]




end while



119878 rarr 0








119883119894119895119881119894119895 and 119887







11987644=[[[

[

124 116 50 0

205 190 58 0

114 68 107 0

43 87 210 0

]]]

]

(28)



Problem 119898 119873 1198721


11198622

Pop Gen 119875119862

119875119898






1 1198622the num-


119895= 3 for 119895 = 0 1 2 3 The






Product 1198631198941

1198631198942

1205871198941

1205871198942

1198941

1198942

119861119894

119867119894

119860119894

120573119894

119878119894

1 1200 800 20 18 9 10 3 5 20 05 42 1300 900 20 18 9 10 7 5 15 05 63 1500 1200 11 14 8 12 5 6 25 08 74 2100 2000 11 14 8 12 8 6 18 08 55 1800 1600 12 15 9 11 7 7 19 06 6

0

1

Variable

120583

Wa Wb Wc




MOPSO

1198621(119860) [15 2 25]

1198622(119861) [15 2 25]

Pop(119862) [20 30 40]

Gen(119863) [100 200 500]

MOGA

119875119862(119860) [05 06 07]

119875119898(119861) [008 01 02]

Pop(119862) [30 40 50]

Gen(119863) [200 300 500]




120583 (119909) =

0 119909 lt 119908119886

119909 minus 119908119886

119908119887minus 119908119886

119908119886lt 119909 lt 119908

119887

119908119888minus 119909

119908119888minus 119908119887

119908119887lt 119909 lt 119908

119888

0 119908119888lt 119909

(29)



119908119887minus 119908119886

= 120572119908119888minus 119908(120572)119888

119908119888minus 119908119887

= 120572 (30)

We have

119908(120572)

119886= (119908119887minus 119908119886) 120572 + 119908

119886

119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572

(31)

Therefore

119908120572= [119908(120572)

119886 119908(120572)

119888]

= [(119908119887minus 119908119886) 120572 + 119908

119886 119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572]

(32)

where 1199081= [03 05 07] 119908

2= [02 03 06] and 120572 = 05


algorithms




MOPSO

1198621

21198622

25Pop 40Gen 200

MOGA

119875119862

06119875119898

02Pop 50Gen 200





Mea

n of

SN

ratio

s

321 321

321 321



(A) (B)

(D)(C)

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

(a)

Mea

n of

SN

ratio

s

321 321

321 321

(A) (B)

(D)(C)



minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

(b)




Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1215 159 15 0 405 53 0 626 1538402 1162 252 0 0 166 36 138 7863 1555 190 55 0 311 38 0 9554 1360 864 0 0 170 108 740 18765 1435 420 0 0 205 60 365 1545


Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1221 168 21 0 407 56 0 611 1545502 959 392 0 0 137 56 341 8493 1220 390 0 0 244 78 280 10904 1168 960 0 0 146 120 932 19725 168 2254 0 0 24 322 1632 978









Obj

ectiv

e val

ue

39373533312927252321191715131197531

22

20

18

16

14

12

10

8

6

4

2

0

VariableMOPSOMOGA

times105




MOGAMOPSO

20

15

10

5

0


ata

times105

(a)

Dat

a

MOGAMOPSO

20

15

10

5

0


(b)


Residual

()

200000010000000

99999

90

50

10

101

Fitted value

Resid

ual

960000940000920000900000880000

1000000

500000

0

Residual

Freq

uenc

y

10000005000000

16

12

8

4

0





minus500000

minus1000000








Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Problem 119898 119873 1198721


11198622

Pop Gen 119875119862

119875119898






1 1198622the num-


119895= 3 for 119895 = 0 1 2 3 The






Product 1198631198941

1198631198942

1205871198941

1205871198942

1198941

1198942

119861119894

119867119894

119860119894

120573119894

119878119894

1 1200 800 20 18 9 10 3 5 20 05 42 1300 900 20 18 9 10 7 5 15 05 63 1500 1200 11 14 8 12 5 6 25 08 74 2100 2000 11 14 8 12 8 6 18 08 55 1800 1600 12 15 9 11 7 7 19 06 6

0

1

Variable

120583

Wa Wb Wc




MOPSO

1198621(119860) [15 2 25]

1198622(119861) [15 2 25]

Pop(119862) [20 30 40]

Gen(119863) [100 200 500]

MOGA

119875119862(119860) [05 06 07]

119875119898(119861) [008 01 02]

Pop(119862) [30 40 50]

Gen(119863) [200 300 500]




120583 (119909) =

0 119909 lt 119908119886

119909 minus 119908119886

119908119887minus 119908119886

119908119886lt 119909 lt 119908

119887

119908119888minus 119909

119908119888minus 119908119887

119908119887lt 119909 lt 119908

119888

0 119908119888lt 119909

(29)



119908119887minus 119908119886

= 120572119908119888minus 119908(120572)119888

119908119888minus 119908119887

= 120572 (30)

We have

119908(120572)

119886= (119908119887minus 119908119886) 120572 + 119908

119886

119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572

(31)

Therefore

119908120572= [119908(120572)

119886 119908(120572)

119888]

= [(119908119887minus 119908119886) 120572 + 119908

119886 119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572]

(32)

where 1199081= [03 05 07] 119908

2= [02 03 06] and 120572 = 05


algorithms




MOPSO

1198621

21198622

25Pop 40Gen 200

MOGA

119875119862

06119875119898

02Pop 50Gen 200





Mea

n of

SN

ratio

s

321 321

321 321



(A) (B)

(D)(C)

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

(a)

Mea

n of

SN

ratio

s

321 321

321 321

(A) (B)

(D)(C)



minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

(b)




Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1215 159 15 0 405 53 0 626 1538402 1162 252 0 0 166 36 138 7863 1555 190 55 0 311 38 0 9554 1360 864 0 0 170 108 740 18765 1435 420 0 0 205 60 365 1545


Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1221 168 21 0 407 56 0 611 1545502 959 392 0 0 137 56 341 8493 1220 390 0 0 244 78 280 10904 1168 960 0 0 146 120 932 19725 168 2254 0 0 24 322 1632 978









Obj

ectiv

e val

ue

39373533312927252321191715131197531

22

20

18

16

14

12

10

8

6

4

2

0

VariableMOPSOMOGA

times105




MOGAMOPSO

20

15

10

5

0


ata

times105

(a)

Dat

a

MOGAMOPSO

20

15

10

5

0


(b)


Residual

()

200000010000000

99999

90

50

10

101

Fitted value

Resid

ual

960000940000920000900000880000

1000000

500000

0

Residual

Freq

uenc

y

10000005000000

16

12

8

4

0





minus500000

minus1000000








Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Product 1198631198941

1198631198942

1205871198941

1205871198942

1198941

1198942

119861119894

119867119894

119860119894

120573119894

119878119894

1 1200 800 20 18 9 10 3 5 20 05 42 1300 900 20 18 9 10 7 5 15 05 63 1500 1200 11 14 8 12 5 6 25 08 74 2100 2000 11 14 8 12 8 6 18 08 55 1800 1600 12 15 9 11 7 7 19 06 6

0

1

Variable

120583

Wa Wb Wc




MOPSO

1198621(119860) [15 2 25]

1198622(119861) [15 2 25]

Pop(119862) [20 30 40]

Gen(119863) [100 200 500]

MOGA

119875119862(119860) [05 06 07]

119875119898(119861) [008 01 02]

Pop(119862) [30 40 50]

Gen(119863) [200 300 500]




120583 (119909) =

0 119909 lt 119908119886

119909 minus 119908119886

119908119887minus 119908119886

119908119886lt 119909 lt 119908

119887

119908119888minus 119909

119908119888minus 119908119887

119908119887lt 119909 lt 119908

119888

0 119908119888lt 119909

(29)



119908119887minus 119908119886

= 120572119908119888minus 119908(120572)119888

119908119888minus 119908119887

= 120572 (30)

We have

119908(120572)

119886= (119908119887minus 119908119886) 120572 + 119908

119886

119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572

(31)

Therefore

119908120572= [119908(120572)

119886 119908(120572)

119888]

= [(119908119887minus 119908119886) 120572 + 119908

119886 119908(120572)

119888= 119908119888minus (119908119888minus 119908119887) 120572]

(32)

where 1199081= [03 05 07] 119908

2= [02 03 06] and 120572 = 05


algorithms




MOPSO

1198621

21198622

25Pop 40Gen 200

MOGA

119875119862

06119875119898

02Pop 50Gen 200





Mea

n of

SN

ratio

s

321 321

321 321



(A) (B)

(D)(C)

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

(a)

Mea

n of

SN

ratio

s

321 321

321 321

(A) (B)

(D)(C)



minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

(b)




Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1215 159 15 0 405 53 0 626 1538402 1162 252 0 0 166 36 138 7863 1555 190 55 0 311 38 0 9554 1360 864 0 0 170 108 740 18765 1435 420 0 0 205 60 365 1545


Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1221 168 21 0 407 56 0 611 1545502 959 392 0 0 137 56 341 8493 1220 390 0 0 244 78 280 10904 1168 960 0 0 146 120 932 19725 168 2254 0 0 24 322 1632 978









Obj

ectiv

e val

ue

39373533312927252321191715131197531

22

20

18

16

14

12

10

8

6

4

2

0

VariableMOPSOMOGA

times105




MOGAMOPSO

20

15

10

5

0


ata

times105

(a)

Dat

a

MOGAMOPSO

20

15

10

5

0


(b)


Residual

()

200000010000000

99999

90

50

10

101

Fitted value

Resid

ual

960000940000920000900000880000

1000000

500000

0

Residual

Freq

uenc

y

10000005000000

16

12

8

4

0





minus500000

minus1000000








Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Mea

n of

SN

ratio

s

321 321

321 321



(A) (B)

(D)(C)

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

Mea

n of

SN

ratio

s

minus103755

minus103760

minus103765

minus103770

minus103775

(a)

Mea

n of

SN

ratio

s

321 321

321 321

(A) (B)

(D)(C)



minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

Mea

n of

SN

ratio

s

minus103800

minus103803

minus103806

minus103809

minus103812

(b)




Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1215 159 15 0 405 53 0 626 1538402 1162 252 0 0 166 36 138 7863 1555 190 55 0 311 38 0 9554 1360 864 0 0 170 108 740 18765 1435 420 0 0 205 60 365 1545


Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1221 168 21 0 407 56 0 611 1545502 959 392 0 0 137 56 341 8493 1220 390 0 0 244 78 280 10904 1168 960 0 0 146 120 932 19725 168 2254 0 0 24 322 1632 978









Obj

ectiv

e val

ue

39373533312927252321191715131197531

22

20

18

16

14

12

10

8

6

4

2

0

VariableMOPSOMOGA

times105




MOGAMOPSO

20

15

10

5

0


ata

times105

(a)

Dat

a

MOGAMOPSO

20

15

10

5

0


(b)


Residual

()

200000010000000

99999

90

50

10

101

Fitted value

Resid

ual

960000940000920000900000880000

1000000

500000

0

Residual

Freq

uenc

y

10000005000000

16

12

8

4

0





minus500000

minus1000000








Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1215 159 15 0 405 53 0 626 1538402 1162 252 0 0 166 36 138 7863 1555 190 55 0 311 38 0 9554 1360 864 0 0 170 108 740 18765 1435 420 0 0 205 60 365 1545


Product 1198761198941

1198761198942

1198831198942

1198831198943

1198811198941

1198811198942

1198871198941

1198871198942

TMF1 1221 168 21 0 407 56 0 611 1545502 959 392 0 0 137 56 341 8493 1220 390 0 0 244 78 280 10904 1168 960 0 0 146 120 932 19725 168 2254 0 0 24 322 1632 978









Obj

ectiv

e val

ue

39373533312927252321191715131197531

22

20

18

16

14

12

10

8

6

4

2

0

VariableMOPSOMOGA

times105




MOGAMOPSO

20

15

10

5

0


ata

times105

(a)

Dat

a

MOGAMOPSO

20

15

10

5

0


(b)


Residual

()

200000010000000

99999

90

50

10

101

Fitted value

Resid

ual

960000940000920000900000880000

1000000

500000

0

Residual

Freq

uenc

y

10000005000000

16

12

8

4

0





minus500000

minus1000000








Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










MOGAMOPSO

20

15

10

5

0


ata

times105

(a)

Dat

a

MOGAMOPSO

20

15

10

5

0


(b)


Residual

()

200000010000000

99999

90

50

10

101

Fitted value

Resid

ual

960000940000920000900000880000

1000000

500000

0

Residual

Freq

uenc

y

10000005000000

16

12

8

4

0





minus500000

minus1000000








Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Acknowledgments


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Multi-Item Multiperiodic Inventory ...downloads.hindawi.com/journals/tswj/2014/136047.pdf · Multi-Item Multiperiodic Inventory Control Problem with Variable Demand