Decentralized capacity allocation of a single-facility with fuzzy demand

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Journal of Manufacturing Systems 33 (2014) 7– 15

Contents lists available at ScienceDirect

Journal of Manufacturing Systems

jo ur nal home p age: www.elsev ier .com/ locate / jmansys

echnical Paper

ecentralized capacity allocation of a single-facilityith fuzzy demand

in Huanga, Min Songa,∗, V. Jorge Leonb,c, Xingwei Wanga

College of Information Science and Engineering, Northeastern University; State Key Laboratory of Synthetical Automation for Process IndustriesNortheastern University), Shenyang, Liaoning, 110819, ChinaDepartment of Industrial and Systems Engineering, Texas A&M University, College Station, TX 77843-3131, USADepartment of Engineering Technology and Industrial Distribution, Texas A&M University, College Station, TX 77843-3367, USA

r t i c l e i n f o

rticle history:eceived 9 May 2012eceived in revised form 18 May 2013ccepted 22 September 2013vailable online 2 November 2013

eywords:

a b s t r a c t

Capacity allocation under uncertainty environment is an important decision problem in manufactur-ing. The decentralized capacity allocation of a single-facility among different organizations with fuzzydemand is investigated in this paper. The objective and demand of each organization are assumed to beprivate information that other organizations and the facility cannot access to. In addition, we assumeorganizations have limited view of the capacity and loading of the facility. First, fuzzy optimization mod-els associated with each organization and the facility are set up. Then, based on fuzzy theory, the fuzzy

upply chain managementapacity allocationrivate informationuzzy demand

optimization models are converted into parametric programming models and subsequently an inter-active algorithm is proposed to solve those parametric programming models. The extra benefit of thisalgorithm is that the whole solving process is amenable to decentralized implementation. Finally, exper-imental results illustrate the effectiveness of this work under two levels of information sharing: capacityinformation of the facility unknown to organizations and capacity information of the facility partiallyknown to organizations.

iety o

. Introduction

Capacity allocation is a common occurrence in a supply chainnd arises whenever the demand exceeds the capacity. For exam-le, we need capacity allocation when several product linesompete for the scarce capacity. Capacity allocation decisions areidely recognized as very important strategic decisions in a sup-ly chain [1]. There exists a large body of literature that deals withhe capacity allocation problems [2–4]. Generally speaking, basedn the access to the modeling information and the decision mak-ng authority, supply chain systems can be roughly categorized intoour categories (see Fig. 1). The term ‘centralized’ is used to refer to

multi-member system where there is a single decision maker whoas authority to make decision for all the members in the systemhile the term ‘decentralized’ is used to qualify a multi-member

ystem where no member has authority to make decision for otherembers in the system. The term ‘complete information’ refers to a

ystem where the decision-maker(s) has complete access to model-ng information throughout the system while ‘partial information’

∗ Corresponding author. Tel.: +86 24 83671469; fax: +86 24 83688608.E-mail addresses: bingning1982330@126.com, songmin0330@126.com

M. Song).

278-6125/$ – see front matter © 2013 The Society of Manufacturing Engineers. Publishettp://dx.doi.org/10.1016/j.jmsy.2013.09.004

f Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

refers to a system in which the decision-maker(s) has access to onlylimited information of the system (e.g., private information).

Historically, capacity allocation research has focused on the‘centralized’ supply chain systems [5]. Although these referenceshave contributed centralized decision rules for the capacity alloca-tion problems, in fact, in today’s highly globalized and competitiveenvironments, with the rapid developments in computers andcommunication technology, most real word supply chains are‘decentralized’ systems. During the past years the emphasis ofthe study of the capacity allocation problems has changed fromthe centralized systems to the decentralized systems [6,7]. Thestreams of literature in the field of decentralized capacity allo-cation can roughly be differentiated into the fields of contracttheory on the one hand [8,9], and coordination approaches formathematical programming models on the other hand [10,11].Although these modeling views represent real world problemsmore accurately, a significant shortcoming of all these models isthat they did not take private information factor into considera-tion. With the development of the research, it is widely recognizedthat the most persistent obstacle in the capacity allocation ofthe ‘decentralized’ supply chain systems is the existence of pri-

d by Elsevier Ltd. All rights reserved.

8 M. Huang et al. / Journal of Manufact

P, C P, D

C, C C, D

Centralized Decentralized

Partial

Complete

tczheotptlsseaatHwtsfs

pctafmpfimpr[mCiifkaFouiii

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Fig. 1. Supply chain systems categories.

heir work, and role conflict. An obvious example is a supplyhain where some of its members are highly competitive organi-ations [13]. Recently, private information in supply chain systemsas attracted a lot of interest among researchers [14–16]. How-ver, most of the results are limited to deterministic environmentsr stochastic environments. In fact, with increasing globalization,here are many sources of uncertainty across the entire sup-ly chain, such as demand uncertainty, supply uncertainty, andhe sudden breakdown of production facilities [17]. These prob-ems lead to poor service levels, high inventories, and frequenttock-outs [18]. Thus, it is very necessary to study ‘decentralized’upply chain systems with private information in an uncertainnvironment. Traditionally, uncertain parameters in the capacityllocation problems have been considered as stochastic variablesnd modeled by probability distributions [19]. A probability dis-ribution is usually derived from evidence recorded in the past.owever, the standard probabilistic reasoning methods will failhen historical data is unavailable. In this case, uncertain parame-

ers only can be specified based on the experience and managerialubjective judgment. Fuzzy set theory provides the appropriateramework to describe and treat uncertainty related to impreci-ions.

Considering the above discussion, this paper considers theroblem associated with the decentralized allocation of the finiteapacity of a single facility among different business organiza-ions with fuzzy demand. In the decentralized allocation, decisionuthorities and the system information are dispersed amid theacility and organizations. Due to the existence of private infor-

ation and fuzzy demand, the capacity allocation problem in thisaper is so difficult to solve for the global optimal solution. There-ore, we propose a heuristics to treat the fuzzy demand and privatenformation separately. For fuzzy demand, we convert the decision

aking problems with fuzzy demand constraints into parametricrogramming models by the “ level cut set” method [20]. An algo-ithm based on Cooperative Interaction via Coupling Agent (CICA)21] is proposed to solve the resulting parametric programming

odels where private information still exists. The main idea of theICA-based algorithm is as follows: the facility and organizations

nteract by passing partial information between them. By receiv-ng system-wide information vectors from iteration-to-iterationrom the facility, the coupled organizations gradually gain globalnowledge and the lack of global information is compensated by

partial information exchange through collaborative interactions.inally, this work is tested by extensive experiments for two levelsf information sharing, namely capacity information of the facilitynknown to organizations and capacity information of the facil-

ty partially known to organizations. Exhaustive results analysisllustrates the efficiency of this work despite the loss of optimal-ty.

The rest of this paper is organized as follows. The capacity allo-ation problem is formulated in Section 2. In Section 3 the solution

uring Systems 33 (2014) 7– 15

2. Problem formulation

In this paper, capacity allocation refers to the problem of allo-cating the finite capacity of a single-facility to satisfy the demandassociated with multiple organizations within a given planninghorizon. We assume that the facility and all organizations are affil-iated with the same company, but they are independent as is oftenobserved in practice. In this setting, these organizations wouldlike to participate in the capacity planning of the company, but inorder to maximize their individual profits, they are understandablyunwilling to fully disclose their sensitive data with other organiza-tions. In this paper, we assume that:

• Decision authorities are dispersed amid the participating organi-zations,

• Each organization only views its local objective,• Each organization only views its demand,• Organizations have limited view of the facility’s capacity and

loading,• The demand of each organization is fuzzy demand.

The notation used for the development of models is given asfollows.Superscriptsop the notation that means the variable is the private information of the

organization,fp the notation that means the variable is the private information of the

facility,n the number of iterations.

Subscriptsk the index for production horizons, k = 1, 2, . . ., t,i the index for organizations, i = 1, 2, . . ., m.

Decision variablesxik the production quantity of product i at time interval k proposed by

organization i,yik the production quantity of product i at time interval k proposed by the

facility.

Parametersbopik

the benefit of selling unit of product i for organization i at time intervalk,

aik the processing time of unit of product i at time interval k,cfpk

the available service time of the facility at time interval k,

dopi

the fuzzy demand of organization i,∼= the fuzzy equation that implies the equation is met in terms of a

degree of truth.

The above capacity allocation problem is described in Fig. 2.Two types of constraints are considered: demand constraints

and capacity constraints. Demand constraints ensure that organi-zations schedule production on the facility so that their demandsare satisfied at the end of the planning horizon. Capacity con-straints ensure that the production scheduled on the facility doesnot exceed its available capacity at any time period. Such a capac-ity allocation strategy could be found in many industries, such asthe transportation industry with shared carriers, the semiconduc-tor manufacturing industry with shared machines, and the foodproduction industry with shared outlets.

Note: In this paper, two cases are considered for the facility’scapacity constraints in order to study the impact of informationsharing levels on the proposed methodology. In the first case eachorganization has no access to the facility capacity. This case is theso called “without capacity information” (WOCI). In the second caseevery organization has access to the facility capacity for each time

M. Huang et al. / Journal of Manufact

Organization 1 Organization 2 Organization mMax Max Max

11 11 1t1top opxb xb

S.t.

21 21 2t2top opxb xb op op

m1m1 mt mtxb xb

111 1topx dx S.t.

21 22topx dx

S.t.opmmtm1x dx

Facility

1

2

11 11 21 21

12 12 22 22

1t1t 2t2t

fp

fp

fpt

m1m1

m2m2

mt mt

xa xa xa cxa xa xa c

xa xa xa c

f

m

s

∑T

m

s

∑

t

3

l(

3

ama

Fig. 2. The structure of the capacity allocation problem.

The organization i’s problem and the facility’s problem can beormulated as follows:

Organization i’s fuzzy model (OFMi)

axt∑k=1

bopikxik (1)

.t.

t∑k=1

xik ∼= dopi , (2)

m

i=1

aikxik ≤ cfpk

(k = 1, 2, . . ., t). (3)

he facility’s fuzzy model (FFM)

axm∑i=1

t∑k=1

bopikyik (4)

.t.

t∑k=1

yik ∼= dopi (i = 1, 2, . . ., m), (5)

m

i=1

aikyik ≤ cfpk

(k = 1, 2, . . ., t). (6)

The existence of private information and fuzzy demand makeshe problem (1)–(6) difficult to solve.

. Solution approach

For the solution to problem (1)–(6), we have to answer the fol-owing two key questions: (1) How to deal with fuzzy equations?2) How to deal with private information?

.1. How to deal with fuzzy equations?

Fuzzy programming is one of the most popular decision makingpproaches based on fuzzy set theory. Many fuzzy programmingodels have been proposed to incorporate fuzziness of objective

nd constraint functions [22,23]. In the last two decades, many

uring Systems 33 (2014) 7– 15 9

studies regarding fuzzy programming have been presented. Severalworks provide thorough reviews of fuzzy programming [20].

In this paper, we assume that fuzzy demands are triangular fuzzynumbers, i.e., dop

i= (dop

i, ˛opi, ˇopi

), where dopi, dopi

− ˛opi, dopi

+ ˇopi

are the most possible, the most pessimistic and the most optimisticvalue of dop

i, respectively. Motivated by the work mentioned above,

fuzzy equations can be expressed as a membership function �i(=)depending on dop

ias follows:

�i(=) = �dopi

(�i) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩1 − (dop

i− �i)

˛opi

, �i ∈ [dopi

− ˛opi, dopi

]

1 − (�i − dopi

)

ˇopi

, �i ∈ [dopi, dopi

+ ˇopi

]

0, else

(7)

where �i denotes the sum of levels of production i at all periods. For-mulation (7) indicates that �i should not differ from the demand toomuch. The larger the difference, the smaller the degree of truth thatthe equation holds, and the less likely that market demands can bemet. In this sense, �i(=) reflects the possibility level at which themarket demands are met. For example, if

∑tk=1xik = dop

i, the differ-

ence is zero, and the equation is strictly satisfied, and the marketdemand is most possibly met, i.e., the possibility level equals one.On the contrary, if

∑tk=1xik ≤ dop

i− ˛op

i, the difference is greater

than the tolerance, and the equation holds with a degree of truthbeing zero, and the market demand is unlikely met. Of course,when the right-hand of the equation becomes dop

i+ ˇop

i, the mar-

ket demand will be likely met. However, at the same time it willalso bring higher inventory levels and holding costs, and may benot always practical. Therefore, we define that the degree of truththat the equation holds with is also zero.

With this interpretation, given the decision maker’s request thatthe production planning is to meet the market demand for prod-uct i with possibility level �i, fuzzy equations can be formulated asfollows:

�i(=)≥�i (i = 1, 2, . . ., m). (8)

Then the organization problem and the facility problem can beconverted into the following equivalent parametric programmingmodels.

Organization i’s crisp model (OCMi)

maxt∑k=1

bopikxik (9)

s.t.

t∑k=1

xik + (1 − �i)˛opi

≥dopi, (10)

t∑k=1

xik − (1 − �i)ˇopi

≤ dopi, (11)

m∑i=1

aikxik ≤ cfpk

(k = 1, 2, . . ., t). (12)

The facility’s crisp model (FCM)

maxm∑i=1

t∑k=1

bopikyik (13)

s.t.

t∑k=1

yik + (1 − �i)˛opi

≥dopi

(i = 1, 2, . . ., m), (14)

10 M. Huang et al. / Journal of Manufact

Facility Problem(FP)

Organization 1(OP1)

Organization i(OP)i

Organization m(OP m)

,(x ),nn nik ik ik ,(y ),nn n

ik ik ik

∑∑

cet

3

tta

mct

pBttppt

•

•

soccfcic

Fig. 3. CICA model for a capacity allocation problem with fuzzy demand.

t

k=1

yik − (1 − �i)ˇopi

≤ dopi

(i = 1, 2, . . ., m), (15)

m

i=1

aikyik ≤ cfpk

(k = 1, 2, . . ., t). (16)

Note that, if complete information were available, the problemould be solved by standard linear programming techniques. How-ver, the existence of private information still makes the solving ofhe above problem challenging.

.2. How to deal with private information?

CICA algorithm allows the existence of multiple coupling agentshat operate over arbitrary subsets of coupling constraints. In addi-ion, the method requires only limited local information sharingmong subproblems, i.e., coupled autonomous organizations.

This section describes the proposed CICA-based solutionethodology where the facility and organizations correspond the

oupling agent and coupled autonomous organizations, respec-ively.

Fig. 3 illustrates how the capacity allocation problem is decom-osed and how the information exchanges among the subsystems.ased on the information distribution described in Section 2,he facility and organizations interact by passing the informa-ion triplets between them. The information triplet consists of theroduction quantity scheduled in each period and two marginalenalties associated with scheduling under or over the given quan-ity. Specifically at the nth iteration:

(xnik, �n

ik, �nik

)is passed from organization i to the facility, where�nik

and �nik

are the marginal penalties specified by organization iif deviations from xn

ikoccur. These penalties reflect both losses in

the organization’s objective value and demand constraints viola-tions.(yn−1ik, �n−1

ik, �n−1

ik)is passed from the facility to organization i,

where �n−1ik

and �n−1ik

are the marginal penalties specified bythe facility to organization i if the scheduled quantity deviatesfrom yn−1

ik. These penalties reflect both losses in overall system

compromise and capacity constraints violations.

Given each organization’s triplet, the facility problem (FP) con-ists of minimizing the total costs caused from deviating from eachrganization’s specified quantities, subject to the facility capacityonstraints. Given the facility triplet, organization i’s problem (OPi)onsists of optimizing its local objective and deviations from the

uring Systems 33 (2014) 7– 15

information is compensated by a partial information exchangethrough collaborative interactions. A detailed description of thedecision problems and derivations, and the calculation of thesetriplets are given in the following subsections.

3.2.1. The organization i’s problem (OPi)In the decentralized environment, passing complete informa-

tion about capacity constraints is not allowed since it is assumed tobe the private information of the facility. Thus, the facility passesonly surrogate information about capacity constraints to organiza-tion i through a facility information vector that consists of a facility’ssolution (yn−1

ik) and penalty weights (�n−1

ikand �n−1

ik).

Theorem 1. Given the facility information vector (yn−1ik

, �n−1ik

, �n−1ik

),the problem associated with organization i can be approximated asfollows:

(OPi) maxt∑k=1

bopikxik +

t∑k=1

(�n−1ik

max(0, yn−1ik

− xik)

+ �n−1ik

max(0, xik − yn−1ik

)) (17)

s.t.

t∑k=1

xik + (1 − �i)˛opi

≥dopi, (18)

t∑k=1

xik − (1 − �i)ˇopi

≤ dopi. (19)

Proof. If complete information about capacity constraints wereavailable from the shared facility, then organization i could explic-itly incorporate capacity constraints in its own local objectivefunction as follows:

maxt∑k=1

bopikxik +

t∑k=1

�k

(cfpk

−m∑i=1

aikxik

)(20)

s.t.

t∑k=1

xik + (1 − �i)˛opi

≥dopi, (21)

t∑k=1

xik − (1 − �i)ˇopi

≤ dopi, (22)

where �k is the Lagrangian multiplier of the capacity constraint attime interval k. Define Xi = [xi1, xi2, . . ., xit]T, Yi = [yi1, yi2, . . ., yit]T, X =[XT1 , XT2 , . . ., XTm]

T, Y = [YT1 , YT2 , . . ., YTm]

T, and H(X) =

∑tk=1�k(c

fpk

−∑mi=1aikxik). By the first order (Taylor series) approximation of H(X)

at X = Y, we achieve

H(X) = H(Y) + ∇TH(Y)(X − Y). (23)

Eq. (23) is equivalent to the following equation:

H(X) = H(Y) − ∇TH(Y) max(0, Y − X) + ∇TH(Y) max(0, X − Y),(24)

where max(0, Y − X) = (max(0, yik − xik))mt×1, i = 1, 2, . . ., m, k =1, 2, . . ., t. Then formula (24) can be further rewritten as follows:

H(X) = H(Y) +m∑i=1

t∑k=1

(�it max(0, yik − xik) + �it max(0, xik − yik)),

(25)

ufact

w

�

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G

M. Huang et al. / Journal of Man

here

ik = −∂(H(X)∂xik

∣∣∣∣xik=yik−

= �kaik, (26)

ik = ∂(H(X)∂xik

∣∣∣∣xik=yik+

= −�kaik. (27)

ue to the fact that the constant item H(Y) does not affect the opti-ization process, we only return the appropriate penalty items in

25) to the corresponding organization i. This completes the prooff Theorem 1.

.2.2. The facility problem (FP)In this paper, organizations neither disclose their local objective

unctions nor their demand satisfaction constraints to the facility.ather, the organization information vector is composed of tripletlements formed by its solution (xn

ik) and penalty weights (�n

ikand

nik

).

heorem 2. Given all organizations information vectorsxnik, �n

ik, �nik

), i = 1, 2, . . ., m, k = 1, 2, . . ., t, the problem associatedith the facility can be approximated as follows:

FP) maxm∑i=1

t∑k=1

(�nik max(0, xnik − yik) + �nik max(0, yik − xnik))

(28)

.t.

m∑i=1

aikyik ≤ cfpk

(k = 1, 2, . . ., t). (29)

roof. Given complete information, the facility’s objective is tochieve the global goal subject to coupling constraints of the systemi.e., capacity constraints) as follows:

axm∑i=1

(t∑k=1

bopikyik + �i

(t∑k=1

yik + (1 − �i)˛i − dopi

)

+ �i

(dopi

−t∑k=1

yik + (1 − �i)ˇi

))(30)

.t.

m∑i=1

aikyik ≤ cfpk

(k = 1, 2, . . ., t), (31)

here �i and � i are the Lagrangian multipliers of the demandatisfaction constraints of organization i. Let Gi(Yi) =

∑tk=1b

opikyik +

i(∑t

k=1yik + (1 − �i)˛i − dopi

) + �i(dopi

−∑t

k=1yik + (1 − �i)ˇopi

).imilar to (23), the first order approximation of Gi(Yi) at Yi = Xi isiven as follows:

i(Yi) = Gi(Xi) + ∇TGi(Yi)(Yi − Xi). (32)

q. (32) is equivalent to the following equation:

i(Yi) = Gi(Xi) − ∇TGi(Yi) max(0, Xi − Yi) + ∇TGi(Yi)× max(0, Yi − Xi), (33)

here max(0, Xi − Yi) = (max(0, xik − yik))t×1, k = 1, 2, . . ., t. Thenq. (33) can be further rewritten as follows:

t

i(Yi) = Gi(Xi) +∑k=1

(�ik max(0, xik − yik)

+�ik max(0, yik − xik)), (34)

uring Systems 33 (2014) 7– 15 11

where

�ik = −∂Gi(Yi)∂yik

∣∣∣∣yik=xik−

= −(bopik

+ �i − �i), (35)

�ik = ∂Gi(Yi)∂yik

∣∣∣∣yik=xik+

= bopik

+ �i − �i. (36)

Similar to the proof of Theorem 1, here we also return the appro-priate penalty items in (34) to the corresponding organization i,since the constant item Gi(Xi) could be dropped without affectingthe optimization process. This completes the proof of Theorem 2.

Given Eqs. (26), (27), (35) and (36), organization problems andthe facility problem can be reformulated by replacing �n−1

ikand

�n−1ik

in formula (17), and by replacing �nik

and �nik

in formula (28)as follows:

(OP ′i) max

t∑k=1

(bopik

+ �n−1ik

)xik (37)

s.t.

t∑k=1

xik + (1 − �i)˛i≥di, (38)

t∑k=1

xik − (1 − �i)ˇi ≤ di. (39)

(FP ′) maxm∑i=1

t∑k=1

�nikyik (40)

s.t.

m∑i=1

aikyik ≤ cfpk

(k = 1, 2, . . ., t). (41)

Note that, both problem OP ′i and problem FP′ are linear program-

ming. Thus the organization’s solution may oscillate between thevertexes of the local feasible region. In addition, the facility’s solu-tion may oscillate between the extreme points of the couplingconstraints. A modified version of the convex combination ruleproposed by [24] is suggested to resolve potential oscillations. Thedetails of this algorithm are explained in the next subsection.

3.3. Updating the Lagrangian multipliers

Given yn−1ik

from the facility, organization i updates the multi-pliers associated with its demand constraints at the nth iteration asfollows:

�ni = max

(0, �n−1

i− tn−1

i

(t∑k=1

yn−1ik

+ (1 − �i)˛opi

− dopi

)), (42)

�ni = max

(0, �n−1

i− tn−1

i

(dopi

−t∑k=1

yn−1ik

+ (1 − �i)ˇopi

)), (43)

tn−1i

= n−1i

∣∣∑t

k=1bopikxnik

−∑t

k=1bopikyn−1ik

∣∣(∑t

k=1yn−1ik

+ (1 − �i)˛opi

− dopi

)2+(dopi

−∑t

k=1yn−1ik

+ (1 − �i)ˇopi

)2,

(44)

1 nufact

w

R

R

tmoi|uecoti

fmp

dw

ti

�

s

a

w

ucodita

.

tep 1.

2 M. Huang et al. / Journal of Ma

n−1i

=

⎧⎨⎩ n−2i, if Rn−1

x ≤ Rn−2x

n−2i

Rn−2x

Rn−1x

, otherwise, (45)

here

n−1x =

∣∣∑t

k=1bopikxnik

−∑t

k=1bopikyn−1ik

∣∣(∑t

k=1yn−1ik

+ (1 − �i)˛opi

− dopi

)2+(dopi

−∑t

k=1yn−1ik

+ (1 − �i)ˇopi

)2,

n−2x =

∣∣∑t

k=1bopikxn−1ik

−∑t

k=1bopikyn−2ik

∣∣(∑t

k=1yn−2ik

+ (1 − �i)˛opi

− dopi

)2+(dopi

−∑t

k=1yn−2ik

+ (1 − �i)ˇopi

)2.

In Eqs. (42) and (43), organization i updates �ni

and �ni

alonghe negative subgradient direction using only the available infor-

ation about the organization’s own demand constraints and localbjective functions. The step size in Eq. (44) uses a different updat-ng rule compared with the classical Lagrangian Relaxation, since∑t

k=1bopikxnik

−∑t

k=1bopikyn−1ik

| is used instead of (zn − z*) to avoidsing a globally feasible solution in calculations. In a decentralizednvironment, it is not easy to find a globally feasible solution toalculate z*, since no one has complete access to constraints or thebjective function for the entire system. If all organizations agreeo accept the facility’s solution proposed at the (n-1)th iteration,.e., xn

ik= yn−1

ik, ∀i, k, then

∑tk=1b

opikxnik

−∑t

k=1bopikyn−1ik

= 0. There-

ore, |∑t

k=1bopikxnik

−∑t

k=1bopikyn−1ik

| can be used as a criterion toeasure whether the facility and organizations achieve a com-

romise solution. As shown in Eq. (45), the step length n−1i

is

ecreased if |∑t

k=1bopikxnik

−∑t

k=1bopikyn−1ik

| is not reduced comparedith the previous iteration.

Conversely, given xnik

from the organizations, the facility updateshe Lagrangian multiplier of the capacity constraint at the kth timenterval for the nth iteration as follows:

nk = max

(0, �n−1

k− sn

(cfpk

−m∑i=1

aikxnik

)), (46)

n = n|ZnCA|∑tk=1(cfp

k−∑m

i=1aikxnik

)2, (47)

nd

n =

⎧⎪⎨⎪⎩ n−1, if Rny ≤ Rn−1

y

n−1Rn−1y

Rny, otherwise

, (48)

here

Rny = |ZnCA|∑tk=1

(cfpk

−∑m

i=1aikxnik

)2,

Rn−1y = |Zn−1

CA |∑tk=1

(cfpk

−∑m

i=1aikxn−1ik

)2.

The facility updates �nk

along the negative subgradient directionsing only the available information about the facility’s capacityonstraints. In order to avoid using global information, ZnCA, thebjective value of the FP model, will be used instead of (zn − z*) in

ty agrees with the solution proposed by organizations, and thushe rule does not change �n−1

k. When ZnCA /= 0, the system solution

nd local solution are different and the rule changes the current

uring Systems 33 (2014) 7– 15

Lagrangian multiplier proportionate to |ZnCA|. In addition, the step

length is reduced whenever |ZnCA| has failed to improve comparedwith the previous interaction as shown in Eq. (48).

It is well known that the subgradient method does not work wellon linear programming [25]. The local solution always oscillates thevertex of the local feasible region because the objective function ofthe Lagrangian subproblem is also a linear function. Meanwhile, theglobal optimal solution may lie inside the local feasible region. Toresolve the oscillation, this paper exploits the convex combinationrule proposed by [24]. Let Xni be the primal solution proposed bythe organization i and Yn be the primal solution proposed by thefacility at the nth interaction. Using the convex combination rule,the primal solutions are defined as follows:

Xni =n∑j=1

Xji

n(i = 1, 2, . . ., m), Yn =

n∑j=1

Yj

n, (49)

where Xji

is the solution to OP ′i model and Yj is the solution to FP′

model at the jthiteration, j = 1, 2, . . ., n. Thus formula (49) impliesthat the primal solution is recovered by giving equal weights toorganizations’ solutions and the facility’s solutions generated sofar.

3.4. CICA-based algorithm

This section presents the proposed CICA-based algorithm fordecentralized capacity allocation problems with fuzzy demand.Initialization :

Set the number of maximum iteration N.Set �0

ik= �0

ik= y0

ik= 0, ∀i, k, 0

i= 0 = 2, ∀i.

Set n=1.Step 1. Organization i’s problem. For all i = 1, 2, . . ., m.

Step 1.1 Solve the model OP ′i .

Step 1.2 Store the organization’s primal solution Xni , ∀ i as shown in formula (49)Step 1.3 Update step length n−1

ias shown in Eq. (45).

Step 1.4 Update step size tn−1i

as shown in Eq. (44).Step 1.5 Update �n

iand �n

ias shown in Eqs. (42) and (43).

Step 1.6 Calculate �nik, �nik

as shown in Eqs. (35) and (36).Step 1.7 Pass xn

ik, �n

ik, and �n

ik, ∀ k to the facility.

Step 2. The facility’s problem.Step 2.1 Solve the model FP’.Step 2.2 Store the facility’s primal solution Yn as shown in formula (49).Step 2.3 Update step length n as shown in Eq. (48).Step 2.4 Update step size sn as shown in Eq. (47).Step 2.5 Update �n

kas shown in Eq. (46).

Step 2.6 Calculate �nik, �n

ikas shown in Eqs. (26) and (27).

Step 2.7 Pass ynik, �n

ik, and �n

ik, ∀ k to organization i, i = 1, 2, . . ., m.

Step 3. If n + 1 > N or a convergence is achieved, stop. Otherwise n = n + 1 and go to SThe algorithm stops if one of the following conditions holds:

(i) Convergence or compromise is achieved, i.e., X = Y. The com-promise solution X = Y is a globally feasible but not necessarilyoptimal solution.

(ii) All step parameters are close to zero, i.e., i ≤ ε, ∀ i and ≤ ε,where ε is a small positive real constant.

(iii) N is reached.

3.5. Performance measures

To evaluate experimentally the performance of our algorithm,performance measures for solution quality and feasibility are devel-oped in this section.

The closeness of X and Y to the centralized solution will bemeasured. The closeness of individual solution X and Y to the

PDx = |Zx − Z∗|Z∗ , PDy = |Zy − Z∗|

Z∗ ,

M. Huang et al. / Journal of Manufacturing Systems 33 (2014) 7– 15 13

Table 1Performance measures of different algorithms.

Objective value PDx PDy CGx CGy

Centralized algorithm 23,680 0 N/A N/A N/ALagrangian Relaxation 23,961 0.0119 N/A N/A N/ACICA-WOCI

Organization’s solution 25,379 0.0718 N/A 0.0525Facility’s solution 24,045 N/A 0.0154 0.0555

0.0059N/A

wzr

b

C

wCatTP

4

4

ot

24

wa

tctεimpcz

m

aTbi

small. Compared to CICA-WPCI, CICA-WOCI takes relatively moreiterations to converge, with a larger gap between the CICA-WOCIand the Opt value. This phenomenon coincides with our intuitionthat the performance of CICA can be improved by sending more

2.6

2.8

3

3.2

3.4

x 104

Ob

ject

ive

valu

e

COCAaLR

CICA-WPCIOrganization’s solution 23,819

Facility’s solution 23,894

here Zx, Zy, and Z∗ represent the objective value of an organi-ation, the objective value of the facility and the optimal value,espectively.

The second type of closeness is the Compromise Gap (CG)etween X and Y . The CG is defined as follows:

Gx = |Zx − Zy|Zx

, CGy = |Zx − Zy|Zy

,

here CGx represents the percentage deviation of Y from X andGy is the percentage deviation of X from Y . A small CG does notlways mean the solutions are close to the optimal solution sincehe solutions may deviate greatly from the optimal objective value.herefore, the quality of solutions must be evaluated by consideringD and CG simultaneously.

. Experimental results

.1. Example

Consider a capacity allocation problem with one facility, tworganizations (m = 2) and four time intervals (t = 4). An example ofhe centralized problem is formulated as follows:

max 20x11 + 30x12 + 10x13 + 50x14 + 10x21 + 20x22 + 40x23 + 30x

s.t. x11 + x12 + x13 + x14 ∼= d1

x21 + x22 + x23 + x24 ∼= d22x11 + 3x21 ≤ 6004x12 + 4x22 ≤ 6003x13 + 5x23 ≤ 6005x14 + 2x24 ≤ 600

here the fuzzy demands d1 = (400, 20, 15), d2 = (450, 10, 20)nd the possibility levels �1 = 0.8 and �2 = 0.9.

The global solution can be calculated easily since it can beransformed into a simple linear programming. Table 1 shows theentralized solution, the Lagrangian Relaxation-based solution andhe CICA-based solution for the example problem with N = 100 and

= 0.000011. As mentioned in Section 2, two versions of CICA aremplemented. The first version is the CICA without capacity infor-

ation (CICA-WOCI), while the second version is the CICA withartial capacity information (CICA-WPCI). In the latter case, theapacity of the facility at any time interval is known to the organi-ation i as follows:

axt∑

(bopik

+ �n−1ik

)xik

1 Similar to other Lagrangian Relaxation based methodologies, the CICA-basedlgorithm does not guarantee the global feasibility and the optimal convergence.herefore, typically simple heuristic methods are applied to restore the global feasi-ility. However, feasibility restoration heuristics are not implemented in this paper

n order to compare the basic forms of the methodologies under consideration.

N/A 0.00310.0090 0.0031

s.t.

t∑k=1

xik + (1 − �i)˛opi

≥dopi,

t∑k=1

xik − (1 − �i)ˇopi

≤ dopi, aikxik ≤ ck∀k.

However, the organizations do not know the specific allocationresult among them, i.e., organization i does not know xjk, j /= i.Compared to CICA-WOCI, CICA-WPCI enables the organizations toreceive more information about the system. The final results ofLagrangian Relaxation and two versions of CICA are shown inTable 1. The algorithms are programmed by MATLAB 7.0. Note thatCICA-WPCI gives a much closer solution to the optimal solution thanCICA-WOCI does. It is also interesting to compare the LagrangianRelaxation solution with the organization’s solution because bothsolutions satisfy the demand constraints and may violate the capac-ity constraints. The comparison results show that the LagrangianRelaxation solution dominates the organization’s solution in thecase CICA-WOCI and the organization’s solution outperforms theLagrangian Relaxation solution in the case CICA-WPCI.

Figs. 4 and 5 depict sequences of objective values of theLagrangian Relaxation (LR) solution, the CICA-WPCI (CO: organiza-tion; CA: facility) solution, the CICA-WOCI (CO: organization; CA:facility), and the optimal solution (Opt) to the example problem. Weobserve that CICA-WPCI converges even faster than the well-knownLR and the gap between the CICA-WPCI and the Opt value is very

5 10 15 20 25 30 35 40 45 502.2

2.4

Iteration

Fig. 4. LR and CICA-WPCI.

14 M. Huang et al. / Journal of Manufacturing Systems 33 (2014) 7– 15

Table 2Factors considered.

Factor Description Level 1 (small variance) Level 2 (large variance)

A Number of time intervals 5 10B Benefit coefficients U(10, 30) U(10, 50)C Processing times U(2, 5) U(2, 10)D d1, d2 U(300, 500) U(300, 600)E Capacity ratio 1.0 1.1F ˛, U(10, 20) U(10, 40)

Note: U(a, b) denotes discrete uniform distribution with minimum a and maximum b.

Table 3The test results for CICA and Lagrangian Relaxation.

PDx PDy CGx CGy Number of iterations

CICA-WOCIMin. 0.0000 0.0001 0.0004 0.0004 23.0000Avg. 0.1095 0.0857 0.1211 0.1267 73.7062Max. 0.6804 0.4363 0.6061 0.6138 100.0000Significant factors E, B C, E C E *

CICA-WPCIMin. 0.0000 0.0000 0.0001 0.0001 26.0000Avg. 0.0208 0.0844 0.0975 0.0866 88.3718Max. 0.3424 0.4413 0.4447 0.3730 100.0000Significant factors E, A E, C E, C C, E *

Lagrangian RelaxationMin. 0.0000 N/A N/A N/A 20.0000Avg. 0.0796 N/A N/A N/A 23.0875Max. 0.5160 N/A N/A N/A 100.0000

N

iaii

4

tstpb

Significant factors A, E, B, D N/A

ote: * means that we could not transform the data to maintain independency.

nformation about the system to organizations. The example resultslso suggest this work as a candidate decentralized algorithm work-ng well under this problem context. In Section 4.2, our algorithms tested on a large set of randomly generated problems.

.2. Experiments

The experiments consider a capacity allocation problem withwo organizations (m = 2). Table 2 shows the six factors and corre-ponding test levels that are considered in the experiments. In order

ik, ∀i, k

e a random solution that satisfies the demand constraints. Then,

5 10 15 20 25 30 35 40 45 502.2

2.4

2.6

2.8

3

3.2

3.4

x 104

Iteration

Ob

ject

ive

valu

e

COCAOptLR

Fig. 5. LR and CICA-WOCI.

N/A N/A A

the capacity at the time interval k is determined by the followingequation:

cfpk

= M

(2∑i=1

aikx′ik

), k = 1, 2, . . ., t. (50)

If M ≥ 1.0, the centralized problem will always be feasible since theproblem has at least one feasible solution x′

ik, ∀i, k. If M < 1.0, the

problems may be infeasible.The number of problem types is 26 and ten random problems

are generated for each problem type. Thus, the total number oftest problems is 640. Statistical analysis is performed by DESIGNEXPERT and SPSS 13.0. The algorithms are programmed by MATLAB7.0.

Table 3 shows the results of CICA-WOCI, CICA-WPCI andLagrangian Relaxation with N = 100 and ε = 0.00001. In each case,the table shows the minimum, the average and the maximum ofthe performance measures of the organization’s solution and thefacility’s solution, the minimum, the average and the maximum ofthe iterations, and the statistically significant factors for each per-formance measure. The significant factors are selected under thenull hypothesis that there is no effect from considering the factor.

CICA-WOCI and CICA-WPCI are tested under the null hypothe-sis: �WPCI = �WOCI and the alternative hypothesis: �WPCI < �WOCI,where �WPCI and �WOCI are the means of CICA-WPCI and CICA-WOCI,respectively. The result shows that the null hypothesis is rejectedwith a p-value less than 0.001 for all performance measures exceptthat for PDy, where the p-value is 0.318 although CICA-WPCI needsmore iterations to reach a solution than CICA-WOCI does.

Our implementation of the Lagrangian Relaxation with Choiand Kim’s algorithm is applied to the same set of problems. The

and the alternative hypothesis: �WPCI < �LR, where �LR is the meanof Lagrangian Relaxation. The p-value for PDx is less than 0.001.Therefore, CICA-WPCI is significantly better than LR for PDx. In

ufact

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5

tommpiiittec

A

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M. Huang et al. / Journal of Man

ddition, Lagrangian Relaxation takes less iteration to convergehan CICA-WPCI does.

The statistically significant factors with a 1% significance levelre shown in Table 3. From Table 3, we conclude that the capac-ty ratio factor E and the processing times factor C are the top twoignificant factors for both CICA-WOCI and CICA-WPCI. This can bexplained by the way we generate random problems in experi-ents. In generating random problems, a random solution x′

ik, ∀i, k,

s generated. The capacity at the time interval k is determined byultiplying the capacity ratio M to the required capacity to produce

he random solution, that is M(∑2

i=1aikx′ik

). On the one hand, if M isncreased, the capacity is also increased. Also, the increased capac-ty implies that the increased solution space, that is PDx and PDy,s decreased. On the other hand, for CICA-WPCI, some constraintsre aikxik ≤ ck, which are equivalent to aikxik ≤ M(

∑2i=1aikx

′ik

), k =, 2, . . ., t. Thus the second level of processing times factor C canlso increase the capacity and expand the solution space. Theseesults are true only for CICA-WPCI since capacity information isnavailable to organizations in the case CICA-WOCI.

. Conclusion

Considering the private information and fuzzy demand simul-aneously, this paper investigates the capacity allocation problemf a single-facility. First, the decision-making problems are for-ulated as fuzzy optimization models. Based on “ level cut set”ethod, these fuzzy optimization models are then converted into

arametric programming models that cater for different scenar-os and different preferences of the decision maker. At last, annteractive CICA-based algorithm that is amenable to decentralizedmplementation is proposed. The experimental results show thathe proposed approach works well, especially when the capacity ofhe facility is partially known to organizations. Further research willvaluate the performance of the proposed approach with multiplelasses of uncertainties when more than one facilities exist.

cknowledgments

This work was supported by the National Natural Scienceoundation of China under Grant Nos. 71071028, 70931001 and1021061, the National Science Foundation for Distinguishedoung Scholars of China under Grant No. 61225012; the Special-

zed Research Fund of the Doctoral Program of Higher Education forhe Priority Development Areas under Grant No. 20120042130003;pecialized Research Fund for the Doctoral Program of Higherducation under Grant No. 20110042110024; the Fundamentalesearch Funds for the Central Universities under Grant Nos.110204003 and N120104001.

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