Research ArticleA Novel Method for Optimal Solution of Fuzzy ChanceConstraint Single-Period Inventory Model

Anuradha Sahoo and J K Dash

Department of Mathematics ITER Siksha lsquoOrsquo Anusandhan University Bhubaneswar Odisha 751030 India

Correspondence should be addressed to Anuradha Sahoo anuradha25anugmailcom

Received 16 July 2016 Accepted 11 October 2016

Academic Editor Erich Peter Klement

Copyright copy 2016 A Sahoo and J K Dash This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A method is proposed for solving single-period inventory fuzzy probabilistic model (SPIFPM) with fuzzy demand and fuzzystorage space under a chance constraint Our objective is to maximize the total profit for both overstock and understock situationswhere the demand 119895 for each product 119895 in the objective function is considered as a fuzzy random variable (FRV) and with theavailable storage space area which is also a FRV under normal distribution and exponential distribution Initially we usedthe weighted sum method to consider both overstock and understock situations Then the fuzziness of the model is removed byranking function method and the randomness of the model is removed by chance constrained programming problem which is adeterministic nonlinear programming problem (NLPP) model Finally this NLPP is solved by using LINGO software To validateand to demonstrate the results of the proposed model numerical examples are given

1 Introduction

One of the special parts of inventory model is single-periodinventory model (SPIM) The objective of this SPIM is todetermine the optimal order quantity where the order foreach item is only at the beginning of each work period thatis for a day a week a month and so forth to maximize theexpected profit or to minimize the cost The use of SPIM canbe found in fashion products business food business sport-ing style goods newspapers Christmas trees and so forth Inreal life situations to estimate the market demand is a mainfactor for maximizing the profit or for minimizing the costThere are the cases inwhich the probability distribution of thedemand for the new items is completely unknown due to lackof historical data and sometimes due to technical difficultiesor economic reasons or using the linguistic expression bythe expert for demand forecasting This deals with situationswhere the demandof products is uncertain For these demanduncertainties fuzzy set theory has been applied to inventorymodels

Initially Zadeh [1] introduced a feasible approach todeal with this kind of fuzzy problem and established afuzzy set which is characterized by a membership function

which assigns to each object a grade of membership rangingbetween zero and one Later depending upon differentrequirements ofmeasurability it was developed and extendedby several researchers like Buckley [2] Kwakernaak [3] andZimmermann [4] et al Many researchers have done a lot ofwork in this area

Nanda and Kar [5] discussed the concept of convexityand logarithmic convexity for fuzzy mappings and someapplications to optimization are introduced Puri andRalescu[6] defined the concepts of fuzzy random variable and theexpectation of a fuzzy random variable

Petrovic et al [7] presented two fuzzy models for thenewsboy problem in an uncertain environment where theuncertainties appear in demand and in inventory costsMulti-item stochastic and fuzzy-stochastic inventory modelsare formulated under total budgetary and space constraintswhere shortages are allowed but fully backlogged byDas et al[8] Nanda et al [9] presented a chance constrained pro-gramming model where both fuzziness and randomness arepresent in the objective function and constraints

Mahapatra andMaiti [10] developed a multiobjective andsingle-objective inventory models of stochastically deterio-rating items in which demand is a function of inventory

Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2016 Article ID 6132768 10 pageshttpdxdoiorg10115520166132768

2 Advances in Fuzzy Systems

level and selling price of the commodity They derived theresults for both without shortages and partially backloggedshortages Also objectives for profit maximization for eachitem are separately formulated with different goals and com-promise solutions of themultiobjective production inventoryproblems are obtained by goal programming method

Chang et al [11] considered the mixture inventory modelinvolving variable lead time with back orders and lost salesThey used the centroid method of defuzzification to estimatetotal cost in the fuzzy sense and found the optimal solutionfor order quantity and lead time in the fuzzy sense such thatthe total cost has a minimum value

Shao and Ji [12] considered the multiproduct newsboyproblem with fuzzy demands under budget constraint Theydeveloped three types of models under different criteria inwhich the objective functions are to maximize the expectedprofit of newsboy the chance of achieving a target profitand the profit which satisfies some chance constraints withat least some given confidence level A model to help thedecision maker in a newsboy problem to assess the value ofinformation is presented by Lee [13] for the potential benefitsof demand forecasting to decrease the risk of overstocking orshortage

Hayya et al [14] considered the reorder point orderquantity inventory model where the demand and the leadtime are independently and identically distributed randomvariables They used a normal approximation to show howto obtain regression equations for the optimal cost and theoptimal policy parameters in terms of the problem para-meters Dash et al [15] have discussed the deterministicequivalent of fuzzy chance constraint programming problemin different scenarios for the fuzzy random variable

Hu et al [16] presented fuzzy random models for thenewsboy problem in the decentralized and centralized sys-tems facing imperfect quality and the optimal policies for thesystems analyzed and derived

Zhang [17] presented a Lagrangian relaxation approach tosolve the multiproduct newsboy problem as a mixed integernonlinear programming model with both supplier quantitydiscounts and a budget constraint The profit for newsboy-type product is introducing a new capacity index which isbased on statistical hypothesis testing analyzed by Su et al[18]

Kotb and Fergany [19] derived the analytical solutionof the EOQ model of multiple items with both demand-dependent unit cost and leading time using geometric pro-gramming approach by considering continuous functions ofdemand rate and leading time for the varying purchase andleading time for crashing costs respectively They deducedthe optimal order quantity the demand rate and the leadingtime as decision variables and also the optimal total cost Deyand Chakraborty [20] presented a fuzzy random continuousreview system with the annual customer demand distributeduniformly and proposed amethodology tominimize the cost

R Banerjee and S Banerjee [21] considered a proba-bilistic inventory model under probabilistic and impreciseconstraints with uniform lead-time demand and fuzzy costcomponents Nagare and Dutta [22] captured both fuzzy

perception and randomness of customer demand to deter-mine an unambiguous optimal order quantity from a set of 119899fuzzy observations in a news-vendor inventory setting in thepresence of fuzzy random variable demand

Jahantigh et al [23] computed the compromised solutionof a LR fuzzy linear system by the use of ranking fuzzy num-bers Karpagam and Sumathi [24] used the ranking functionto find the fuzzy optimal solution of fully fuzzy linear prog-ramming problems

Dutta and Kumar [25] determined the optimal totalcost and the optimal order quantity for an inventory modelwithout shortages in a fuzzy environment They used signeddistance method to compute economic order quantity Thismodel deals with indeterminacy information for multiprod-uct newsboy problem which was initially transformed intodeterministic model and the solution was obtained by classicinteger programming method proposed by Ding [26]

Zolfagharinia and Isotupa [27] pointed out some flaws inthe simulationmodel and some of the formulae for inventorylevel and ordering quantity Majumder et al [28] formulatedan EPQ model for deteriorating items under partial tradecredit policy with crisp and fuzzy demand They have usedweighted sum method to convert a multiobjective to a singleobjective and the model was solved by Generalized ReducedGradient method

A real-world multiperiod inventory control problemunder budget constraint is investigated and the weightedlinear sum of objectives is applied to generate a single-objective model for the biobjective problem and a harmonysearch algorithm is developed to solve the complex inventoryproblem byMousavi et al [29]The goal is to find the optimalordered quantities of products not only the total inventorycost but also the required storage space as a fuzzy number tostore the products is minimized

Soni and Joshi [30] considered a periodic review inven-tory model in fuzzy-stochastic environment with lead timeand the back order rate as control variables and by applyingthe criterion of credibility they determined the expectedshortages

To capture the uncertainty of demand Kumar andGoswami [31] employed the fuzzy expectation signed dis-tance and possibility or necessitymeasure to the fuzzymodelto transform it into an equivalent deterministic nonlinearprogramming problem where the model is restricted tobudget and allowable shortages

Kazemi et al [32] developed and extended a fuzzy inven-tory model of an existing economic order quantity with backorders in which both demand and lead times are fuzzifiedto decrease the total inventory cost Dash and Sahoo [33]obtained the optimum order quantity and the expected profitby using Buckleyrsquos concept of minimization of fuzzy numbersfor a single-period inventory model

Nagare et al [34] developed an inventory model todetermine the optimal order quantity and weight factor onthe basis of revised forecasts and the results demonstrateeconomic benefits of using revised demand negative impactof constraints and role of demand distribution entropy indetermining the order size and expected profit

Advances in Fuzzy Systems 3

A fuzzy inventory model for fixed deteriorating itemswith shortages under fully backlogged condition is formu-lated and solved by Raula et al [35] They defuzzified themodel by graded unit preference integration method and allrelated inventory parameters were assumed to be hexagonalfuzzy numbers

Sangal et al [36] developed an inventory model fornoninstantaneous decaying items under crisp and fuzzyenvironmentThey have considered stock dependent demandrate and variable deterioration and defuzzified the total costfunction by signed distance method

The important feature of the present work is to determinethe optimum order quantity by considering a SPIFPM withdemand as a fuzzy random variable in the objective functionsubject to a chance constraint where the storage spacedistributed normally and exponentially which is differentfrom the previous studies Initially we used the weighted summethod ranking function to the objective function and sometheorems have been developed to transform the SPIFPM intoa crisp model

The remaining part of the paper is organized as followsSome basic concepts and results of uncertain theory are givenin Section 2 In Section 3 an analytical model is formulatedfor the SPIFPM under a chance constraint In Section 4 wepresent a transformation technique to convert the model toan equivalent deterministic model In Section 5 a numericalexample is given for verification Finally in Section 6 someconclusions are listed

2 Preliminaries

From literature study here we present some basic conceptsregarding the uncertainty theory

Definition 1 (fuzzy set [1]) A fuzzy set over a universal set119883 is defined as a set of ordered pairs = (119909 120583119860(119909)) 119909 isin 119883and is characterized by its membership function 120583119860(119909)where120583119860(119909) 119883 rarr [0 1] for each 119909 isin 119883Definition 2 (triangular fuzzy number [15]) The triangularfuzzy number is a fuzzy number denoted as = (1198601 1198602 1198603)and is interpreted by its membership function as follows

120583 (119909) =

119909 minus 11986011198602 minus 1198601 if 1198601 le 119909 le 11986021198603 minus 1199091198603 minus 1198602 if 1198602 le 119909 le 11986030 otherwise

(1)

Definition 3 (120572-cut of a fuzzy number [15]) The 120572-cut of thefuzzy number is the set 119909 | 120583119860(119909) ge 120572 for 120572 isin (0 1) anddenoted as [120572]

Definition 4 (partial order relation of two fuzzy numbers [9])Let = (1198601 1198602 1198603) and = (1198611 1198612 1198613) be two fuzzynumbers with 120572-cut [120572] = [119860lowast 119860lowast] and [120572] = [119861lowast 119861lowast]respectively then ⪯ if and only if 119860lowast le 119861lowast and ⪰ ifand only if 119860lowast le 119861lowastDefinition 5 (fuzzy random variable [3 6]) The randomvariables whose parameters like mean variance and so forthare fuzzy numbers are called fuzzy random variables A fuzzyrandom variable with mean 119909 and variance 2119909 is denotedby ( 2)Definition 6 (mean of a continuous fuzzy random variable[6]) Let = 11988611 oplus 11988622 oplus sdot sdot sdot oplus 119886119899119899 be a FRV whichis the linear combination of 119899 number of independent fuzzyrandom variables 1 2 119899 whose means are fuzzynumbers 1198831 1198832 119883119899 respectively Then 119880[120572] =sum119899119895=1 119886119895119883119895[120572]Definition 7 (variance of a continuous fuzzy random variable[6]) Let = 11988611 oplus 11988622 oplus sdot sdot sdot oplus 119886119899119899 be the linearcombination of 119899 number of independent fuzzy random vari-ables 1 2 119899 whose variances are the fuzzy numbers21198831 21198832 2119883119899 respectively Then 2119880[120572] = sum119899119895=1 1198862119895 2119883119895[120572]Definition 8 (ranking function [23 24]) The ranking func-tion is defined on a set of real numbers (119877) which maps eachfuzzy number into the real line where a natural order existsthat is119877 119865(119877) rarr 119877 where119865(119877) is a set of fuzzy numbers Tocalculate the triangular weights ranking function is effectiveLet = (119886 119887 119888) be a triangular fuzzy number We define astandard ranking function as 119877() = (119886 + 2 lowast 119887 + 119888)4Definition 9 (weighted sum method [28]) This method canbe interpreted by some positive weights or priority assignedto the objective criterion by the decision maker Here weexpress the weighted summethod by multiplying the weightsby the users each supplied in the objective and the weightsare chosen in a way such that their sum is one that is

optimize119899sum119894=1

119908119894119891119894 (119909) 119908119894 isin [0 1] 119909 isin 119883 (2)

such that sum119899119894=1 119908119894 = 1 where 119908119894 are the weights of the 119894thobjective functions

Definition 10 (fuzzy normal distribution [15]) A normalfuzzy random variable 119873( 2) is a normally distributedrandom variable with fuzzy mean and fuzzy variance 2as fuzzy parameters Let 119873(119898 1205902) denote the crisp normalrandom variable with mean 119898 and variance 1205902 and let119891(119909119898 1205902) be the density function of the crisp normaldistribution where

119891 (119909119898 1205902) = 1radic2120587120590119890minus(119909minus119898)221205902 120590 gt 0 (3)

4 Advances in Fuzzy Systems

so (1radic2120587120590) intinfinminusinfin

119890minus(119909minus119898)221205902119889119909 = 1 Now the fuzzy proba-bility of the FRV on the interval [119888 119889] is a fuzzy numberwhose 120572-cut is (119888 le le 119889) [120572] = 1radic2120587120590 int

119889

119888119890minus(119909minus119898)221205902119889119909 | 119898

isin [120572] 1205902 isin 2 [120572] = 1radic2120587 int1199112

1199111

119890minus11991122119889119911 | 119898isin [120572] 1205902 isin 2 [120572]

(4)

where 1199111 = (119888 minus 119898)120590 and 1199112 = (119889 minus 119898)120590Definition 11 (fuzzy exponential distribution [15]) Let be aexponentially distributed fuzzy random variable with densityfunction 119891(119909 ) being a fuzzy set and its 120572-cut is

119891 (119909 ) [120572] = 119890minus119909 | 120582 isin (120572) (5)

Now the fuzzy probability of on the interval [119888 119889] is a fuzzynumber and denoted by (119888 le le 119889) where 119875lowast(120572) =minint119889

119888120582119890minus120582119909119889119909 | 120582 isin [120572] and 119875lowast(120572) = maxint119889

119888120582119890minus120582119909119889119909 |120582 isin [120572]

Definition 12 (Fuzzy Chance Constrained Programmingproblem (FCCP) [26]) A general chance constraint pro-gramming problem CCP is of the following form

(CCP) Optimize119896sum119895=1

119888119895119909119895Subject to 119875( 119896sum

119895=1

119886119894119895119909119895 le 119887119894) ⪰ 120574119894119896sum119895=1

119887119897119895119909119895 ge ℎ119897119909119895 ge 0119887119897119895 ℎ119897 isin 119877119894 = 1 2 119898119897 = 1 2 1198990 le 120574119894 le 1

(6)

where at least one of 119886119894119895 119888119895 and 119887119894 is treated as randomvariable

So FCCP is a CCP given as above form where at least oneof 119886119894119895 119888119895 and 119887119894 is treated as FRV

3 Mathematical Formulation

Notations used in the proposed model are presented inNotations section

31 Assumptions We imposed the following assumptions todevelop the model

(i) For all 119895 119895 are independent fuzzy random variableswhose mean 119863 are triangular fuzzy numbers

(ii) For all 119895 119895 are independent fuzzy random variableswhose mean 119882 and variance 2119882 are triangular fuzzynumbers

(iii) The time period is precise for all 119895 that is singleperiod

(iv) The decision makerrsquos order quantity 119876119895 is optimal forall products at the beginning of the period

(v) At the end of the season the leftover items aresalvaged

(vi) The lost sales penalty cost is zero

32 Single-Period Inventory Fuzzy Probabilistic Model(SPIFPM) The profit function119885119895 associated with each orderquantity 119876119895 and fuzzy demand 119895 is given as follows

119885119895 (119876119895 119895) = 119901119895min 119876119895 119895 + 119904119895max 119876119895 minus 119895 0minus 119888119895119876119895 119895 = 1 2 119896 (7)

Here two situations are explored to discuss the profit func-tion that is overstock situation (119895 le 119876119895) and understocksituation (119895 ge 119876119895) Then the above equation boils down tothe following form

119885119895 (119876119895 119895)=

119895 (119876119895 119895) = (119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895 119895 le 119876119895119895 (119876119895 119895) = (119901119895 minus 119888119895)119876119895 119895 ge 119876119895

(8)

where 119885119895 and 119885119895 are profit functions for each product inoverstock and understock situations respectively Due tofuzziness of demand 119895 we get either a fuzzy overstock profit119895(119876119895 119895) or a fuzzy understock profit 119895(119876119895 119895) Hencethe total expected profit 119895(119876119895 119895) corresponding to bothoverstock and understock situations is given as follows

119895 (119876119895 119895) = 119895 (119876119895 119895) cup 119895 (119876119895 119895) (9)

Advances in Fuzzy Systems 5

Thus a multi-item SPIFPM subject to a storage space chanceconstraint with a confidence level 120574 can be stated as follows

(SPIFPM) Maximize (1198761 1198762 119876119896)= 119896sum119895=1

119895 (119876119895 119895)

subject to ( sum119895=1

119886119895119876119895 le ) ⪰ 119876119895 ge 0for 119895 = 1 2 119896

(10)

In the above model we consider demand 119895 as a FRVpresent in the objective function and the total storage space are independent FRVs present in the constraint distributednormally and exponentially

4 Transformation Technique

41 Crisp Equivalent of the Fuzzy Objective Function InSPIM the defined resultant profit function is fuzzy since thecustomer demand for each product is imprecise Now our

objective is to convert the fuzzy model to crisp model Usingthe weighted function to the profit function we have

119895 (119876119895 119895) = 120596119895 (119876119895 119895) + (1 minus 120596) 119895 (119876119895 119895) 997904rArr119895 (119876119895 119895) = 120596[[

119896sum119895=1

(119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]] (11)

Here 119895 are FRVs whose mean 119898119863119895 are triangular fuzzynumbers To make it crisp apply the ranking function as119877(119898119863119895) = (119898119863119895 +2119898119863119895 +119898119863119895)4 So the above profit functionbecomes

119895 (119876119895 119895) = 120596[[[119896sum119895=1

(119901119895 minus 119904119895) (119898119863119895 + 2119898119863119895 + 119898119863119895)4

+ (119904119895 minus 119888119895)119876119895]]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]] (12)

Therefore the crisp equivalent of the objective function is asfollows

Maximize 120596[[[119896sum119895=1

(119901119895 minus 119904119895) (119898119863119895 + 2119898119863119895 + 119898119863119895)4 + (119904119895 minus 119888119895)119876119895]]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]] (13)

42 Crisp Equivalent of the Fuzzy Chance Constraint Thefollowing results are proved to convert the fuzzy chanceconstraint to a deterministic form

Theorem 13 (Case 1 ( is normally distributed fuzzy randomvariable)) If is a normally distributed FRV 119886119895 isin 119877and with predetermined confidence level 120574 the inequality(sum119896119895=1 119886119895119876119895 le ) ⪰ is equivalent to

119896sum119895=1

119886119895119876119895 le 120583119882lowast (120572) + 120590119882lowast (120572) 119865minus1 (1 minus 120574lowast (120572)) (14)

for 120572 isin [0 1] and 119865 is the cumulative distribution function of119873(0 1) distributionProof Let us consider the constraint of the model as follows

( 119896sum119895=1

119886119895119876119895 le ) ⪰ 119886119895 isin 119877 (15)

where is a normally distributed FRV whose mean andvariance are fuzzy numbers denoted by 119882 and 2119882 andtheir 120572-cuts are 119882[120572] = [120583119882lowast(120572) 120583lowast119882(120572)] and 2119882[120572] =[1205902119882lowast(120572) 1205902119882lowast(120572)] respectively The 120572-cut of is [120572] =[120574lowast(120572) 120574lowast(120572)]

Let sum119896119895=1 119886119895119876119895 = 119880 Here 119880 is a real number since 119886119895119895 = 1 2 119896 are real numbers Then the above inequalitycan be written as 119875(119880 le ) ⪰ Here is a FRV whosemean and variance are the fuzzy numbers 119882 and 2119882 whichcan be computed from the crisp random variable119882 that is119882 isin [120572] That means 120583119882 isin 119882[120572] and 1205902119882 isin 2119882[120572] whichsatisfy (1radic2120587120590119882) intinfinminusinfin 119890minus(119882minus120583119882)221205902119882119889119882 = 1

Now the 120572-cut of the probability (sum119896119895=1 119886119895119876119895 le ) is ( 119896sum119895=1

119886119895119876119895 le ) [120572] = 119875 (119880 le 119882) | 120583119882isin 119882 [120572] 1205902119882 isin 2119882 [120572] = 1 minus 119875 (119882 le 119880) | 120583119882

6 Advances in Fuzzy Systems

isin 119882 [120572] 1205902119882 isin 2119882 [120572] = 1minus 1radic2120587120590119882 int

119880

minusinfin119890minus(119882minus120583119882)221205902119882119889119882 | 120583119882

isin 119882 [120572] 1205902119882 isin 2119882 [120572] (16)

Let 119881 = (119882 minus 120583119882)120590119882 Then

( 119896sum119895=1

119886119895119876119895 le ) [120572] = 1minus 1radic2120587 int

(119880minus120583119882)120590119882

minusinfin119890minus11988122119889119881 | 120583119882 isin 119882 [120572] 1205902119908

isin 2119908 [120572] = 1 minus 119865(119880 minus 120583119882120590119882 ) | 120583119882isin 119882 [120572] 1205902119882 isin 2119882 [120572]

(17)

where 119865 is the cumulative distribution function of 119873(0 1)distribution which is an increasing function So

min1 minus 119865(119880 minus 120583119882120590119882 ) | 120583119882 isin 119882 [120572] 1205902119882isin 2119882 [120572] = 1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) )

max1 minus 119865(119880 minus 120583119882120590119882 ) | 120583119882 isin 119882 [120572] 1205902119882isin 2119882 [120572] = 1 minus 119865(119880 minus 120583lowast119882 (120572)120590lowast119882 (120572) )

(18)

So

( 119896sum119895=1

119886119895119876119895 le ) [120572]= [1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) 1 minus 119865(119880 minus 120583lowast119882 (120572)120590lowast119882 (120572) )]

(19)

Using the Inequality law due toNanda et al [9] the constraint(sum119896119895=1 119886119895119876119895 le ) ⪰ becomes

1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) ge 120574lowast (120572) 997904rArr119896sum119895=1

119886119895119876119895 le 120583119882lowast (120572) + 120590119882lowast (120572) 119865minus1 (1 minus 120574lowast (120572)) (20)

Hence the proof is complete

Therefore finally the SPIFPM is transformed to anequivalent single-period inventory crisp model (SPICM) asfollows

(SPICM) Maximize 120596[[[119896sum119895=1

(119901119895 minus 119904119895) (119898119863119895 + 2119898119863119895 + 119898119863119895)4 + (119904119895 minus 119888119895)119876119895]]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]]Subject to

119896sum119895=1

119886119895119876119895 le 120583119882lowast (120572) + 120590119882lowast (120572) 119865minus1 (1 minus 120574lowast (120572))119876119895 ge 0for 120572 isin [0 1] 119895 = 1 2 119896

(21)

Theorem 14 (Case 2 ( follows fuzzy exponential distribu-tion)) If is an exponentially distributed FRV with fuzzyparameter 119886119895 isin 119877 then (sum119896119895=1 119886119895119876119895 le ) ⪰ is equivalentto 119890minus120582lowast(120572)119880 ge 120574lowast(120572) for each 120572 isin [0 1] where [120572] =[120582lowast(120572) 120582lowast(120572)]Proof The 120572-cut of the fuzzy probability (sum119896119895=1 119886119895119876119895 le )is

( 119896sum119895=1

119886119895119876119895 le ) [120572] = 119875 (119880 le 119882) | 119882 isin [120572]= intinfin119880120582119890minus120582119882119889119882 | 120582 isin [120572]

= 119890minus120582119880 | 120582lowast (120572) le 120582 le 120582lowast (120572) = [119875lowast (120572) 119875lowast (120572)](say)

(22)

where 119875lowast(120572) and 119875lowast(120572) are the solutions of the optimizationproblems

minimize120582lowast(120572)le120582le120582

lowast(120572)119890minus120582119880

maximize120582lowast(120572)le120582le120582

lowast(120572)119890minus120582119880 (23)

respectively

Advances in Fuzzy Systems 7

Here 119890minus120582119880 attains its minimum at 120582lowast(120572) and maximumat 120582lowast(120572) since 119890minus120582119880 is a decreasing function of 120582 That ismin 119890minus120582119880 = 120582lowast(120572) and max 119890minus120582119880 = 120582lowast(120572)

Hence

( 119896sum119895=1

119886119895119876119895 le ) [120572] = [119890minus120582lowast(120572)119880 119890minus120582lowast(120572)119880] (24)

Therefore the deterministic equivalent of the fuzzy chanceconstraint

( 119896sum119895=1

119886119895119876119895 le ) ⪰ becomes 119890minus120582lowast(120572)119880 ge 120574lowast (120572)

(25)

5 Numerical Example

Let us consider a SPIFPM to illustrate the effectiveness of theabove approach with the following data

(SPIFPM) Maximize [[2sum119895=1

(119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895]]cup [[2sum119895=1

(119901119895 minus 119888119895)119876119895]]Subject to (41198761 oplus 31198762 le ) ⪰ 05

1198761 ge 01198762 ge 0

(26)

where the other data for multi-items is given in Table 1Here 1 2 are independent normally distributed FRVs

with mean 1198631 1198632 being fuzzy numbers Also areindependent normally distributed FRVs with mean 119882 andvariance 2119882 as fuzzy numbers and 05 is also a fuzzy numberAll these fuzzy numbers are assumed to be linear triangularas follows 1198631 = ⟨200300400⟩ 1198632 = ⟨700800900⟩119882 = ⟨200020102020⟩ 2119882 = ⟨300400500⟩ and 05 =⟨040506⟩51 Solution Initially applying the weighted sum method tothe objective function we can write

Maximize (1198761 1198762)= 120596[[

2sum119895=1

(119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895]]+ (1 minus 120596)[[

2sum119895=1

(119901119895 minus 119888119895)119876119895]] (27)

Table 1

Product name 119888119895 119901119895 119904119895 119886119895Product 1 30 34 28 4Product 2 40 43 35 3

Let 120596 = 06 and using the above cost data to the aboveexpression we have

Maximize (1198761 1198762)= 041198761 minus 181198762 + 361 + 482 (28)

Here 1 2 are FRVs whose mean 1198631 1198632 are triangularfuzzy numbers with the 120572-cuts being

1198631 [120572] = [200 + 100120572 400 minus 100120572] 1198632 [120572] = [700 + 100120572 900 minus 100120572] (29)

Applying the ranking function to the above objective functionwe have

Maximize (1198761 1198762)= 041198761 minus 181198762 + 36119877 (1198631)+ 48119877 (1198632)

(30)

Hence the crisp equivalent of the objective function takes thefollowing form

Maximize 041198761 minus 181198762 + 4920 (31)

Now let us consider the constraint (41198761 + 31198762 le ) ⪰ 05We can convert this constraint to its deterministic form as follows normal and exponential distribution

Case 1 ( is normally distributed fuzzy random variable)As the mean 119882 variance 2119882 and 05 are triangular fuzzynumbers their 120572-cuts are

119882 [120572] = [2000 + 10120572 2020 minus 10120572] 2119882 = [300 + 100120572 500 minus 100120572]

05 [120572] = [04 + 120572 06 minus 120572] (32)

Let 41198761 +31198762 = 119880 So the constraint is given by (119880 le ) ⪰05 Using the above theorem result the 120572-cut of (119880 le )becomes

(119880 le ) [120572]= [1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) 1 minus 119865(119880 minus 120583lowast119882 (120572)120590lowast119882 (120572) )] (33)

So the crisp conversion of the constraint is

1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) ge 06 minus 01120572 (34)

8 Advances in Fuzzy Systems

which can be further simplified as

119880 minus 120583119882lowast (120572) le 120590119882lowast (120572) 119865minus1 (06 + 01120572) 997904rArr41198761 + 31198762 le 5 + 120572 + (radic3 + 120572)119865minus1 (04 + 01120572) (35)

Hence the crisp equivalent that is SPICM of the originalSPIFPM takes the following form

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to 41198761 + 31198762

le 5 + 120572+ (radic3 + 120572)119865minus1 (04 + 01120572)

1198761 ge 01198762 ge 0

(36)

Now 119865minus1(04 + 01120572) can be found from the normal distri-bution table For each 120572 there exists a solution of the aboveproblem For example if 120572 = 04 then SPICM reduces to thefollowing nonlinear programming problem

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to (41198761 + 31198762 minus 2004)2

le 7752341198761 ge 01198762 ge 0

(37)

Using any optimization software its local solution can bedetermined Here we have solved this by using LINGO andits solution is1198761 = 5016959 and1198762 = 0 and the correspond-ing objective value is 5120678 Each value of 120572 gives a corres-ponding solution

Case 2 ( follows fuzzy exponential distribution) Let bean exponential FRV with the parameter = ⟨1000 15002000⟩ having 120572-cut as [120572] = [1000+500120572 2000minus500120572] Let41198761+31198762 = 119880 So the constraint is given by (119880 le ) ⪰ 05Using theorem the crisp conversion of the constraint is

exp (minus (2000 minus 500120572) (41198761 + 31198762)) ge 06 minus 01120572 (38)

Hence the crisp equivalent that is SPICM of the originalSPIFPM takes the following form

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to exp (minus (2000 minus 500120572) (41198761 + 31198762))

ge 06 minus 011205721198761 ge 01198762 ge 0

(39)

For each 120572 there exists a solution of the above problem Forexample if 120572 = 04 then SPICM reduces to the followingnonlinear programming problem

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to exp (minus1800 (41198761 + 31198762))

ge 0561198761 ge 01198762 ge 0

(40)

Using any optimization software its local solution can bedetermined Here we have solved this by using LINGO andits solution is 1198761 = 08053035119864 minus 04 and 1198762 = 0 and thecorresponding objective value is 4920 Each value of 120572 givesa corresponding solution

6 Conclusion

This paper presents a solution procedure for solving SPIFPMin which the the storage space in the constraint is a FRVconsidered under two cases that is normal distributionand exponential distribution Also the demand is a FRVwhose mean and variance are fuzzy numbers in the objectivefunction of the given model Initially some theorems areproposed to transform the fuzzy model into an equivalentcrisp model Then the resultant model is solved by using theLINGO software When the storage space in the constraintis normally distributed then the optimal solution is 1198761 =5016959 and 1198762 = 0 and the corresponding objectivevalue is 5120678 and for exponentially distributed randomvariable its solution is 1198761 = 08053035119864 minus 04 and 1198762 = 0and the corresponding objective value is 4920 for one valueof 120572 However more approximate solution can be obtainedby taking more numbers of values of 120572 For the futurescope of the present work triangular fuzzy number can bereplaced by other fuzzy numbers and the demand for eachitems can be considered as other types of nonindependentFRVs instead of normally distributed independent FRVsA multiconstrained optimization model can be developedof our present work and our model can be extended toother mathematical models To analyze the effect of changesin the optimal solution with respect to change in variousparameters sensitivity analysis can be carried out

Notations

119876119895 Order quantity for item 119895119895 Demand as a fuzzy random variable for item 119895119888119895 Purchase cost per unit of item 119895119901119895 Selling price per unit of item 119895119904119895 Salvage value per unit of item 119895119886119895 Area required for each unit of item 119895119895 Fuzzy profit function for product 119895 Fuzzy total profit function for all products Available storage space a fuzzy random variable

Advances in Fuzzy Systems 9

Competing Interests

The authors declare that they have no competing interests

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] J J Buckley Fuzzy Probabilities New Approach and Applica-tions Physica Heidelberg Germany 2003

[3] H Kwakernaak ldquoFuzzy random variablesmdashI definitions andtheoremsrdquo Information Sciences vol 15 no 1 pp 1ndash29 1978

[4] H-J Zimmermann ldquoFuzzymathematical programmingrdquoCom-puters amp Operations Research vol 10 no 4 pp 291ndash298 1983

[5] S Nanda and K Kar ldquoConvex fuzzy mappingsrdquo Fuzzy Sets andSystems vol 48 no 1 pp 129ndash132 1992

[6] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986

[7] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy models forthe newsboy problemrdquo International Journal of Production Eco-nomics vol 45 no 1ndash3 pp 435ndash441 1996

[8] K Das T K Roy and M Maiti ldquoMulti-item stochastic andfuzzy-stochastic inventory models under two restrictionsrdquoComputers and Operations Research vol 31 no 11 pp 1793ndash1806 2004

[9] S Nanda G Panda and J K Dash ldquoA new methodology forcrisp equivalent of fuzzy chance constrained programmingproblemrdquo Fuzzy Optimization and Decision Making vol 7 no1 pp 59ndash74 2008

[10] N K Mahapatra and M Maiti ldquoDecision process for multiob-jective multi-item production-inventory system via interactivefuzzy satisficing techniquerdquo Computers amp Mathematics withApplications vol 49 no 5-6 pp 805ndash821 2005

[11] H-C Chang J-S Yao and L-Y Ouyang ldquoFuzzy mixtureinventory model involving fuzzy random variable lead timedemand and fuzzy total demandrdquo European Journal of Opera-tional Research vol 169 no 1 pp 65ndash80 2006

[12] Z Shao and X Ji ldquoFuzzy multi-product constraint newsboyproblemrdquoAppliedMathematics and Computation vol 180 no 1pp 7ndash15 2006

[13] C-M Lee ldquoA Bayesian approach to determine the value ofinformation in the newsboy problemrdquo International Journal ofProduction Economics vol 112 no 1 pp 391ndash402 2008

[14] J C Hayya T P Harrison and D C Chatfield ldquoA solutionfor the intractable inventory model when both demand andlead time are stochasticrdquo International Journal of ProductionEconomics vol 122 no 2 pp 595ndash605 2009

[15] J K Dash G Panda and S Nanda ldquoChance constrained prog-ramming problem under different fuzzy distributionsrdquo Inter-national Journal of Optimization Theory Methods and Applica-tions vol 1 no 1 pp 58ndash71 2009

[16] J-S Hu H Zheng R-Q Xu Y-P Ji and C-Y Guo ldquoSupplychain coordination for fuzzy random newsboy problem withimperfect qualityrdquo International Journal of Approximate Reason-ing vol 51 no 7 pp 771ndash784 2010

[17] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[18] R-H Su D-Y Yang and W L Pearn ldquoDecision-making ina single-period inventory environment with fuzzy demandrdquo

Expert Systems with Applications vol 38 no 3 pp 1909ndash19162011

[19] K A M Kotb and H A Fergany ldquoMulti-item EOQmodel withboth demand-dependent unit cost and varying leading time viageometric programmingrdquoAppliedMathematics vol 2 no 5 pp551ndash555 2011

[20] O Dey and D Chakraborty ldquoA fuzzy random continuousreview inventory systemrdquo International Journal of ProductionEconomics vol 132 no 1 pp 101ndash106 2011

[21] R Banerjee and S Banerjee ldquoSolution of a probabilistic inven-tory model with chance constraints a general fuzzy program-ming and intuitionistic fuzzy optimization approachrdquo Interna-tional Journal of Pure and Applied Sciences and Technology vol9 no 1 pp 20ndash38 2012

[22] M Nagare and P Dutta ldquoOn solving single-period inventorymodel under hybrid uncertaintyrdquo International Journal of Eco-nomics and Management Sciences vol 6 no 4 pp 290ndash2952012

[23] M A Jahantigh S Khezerloo A A Hosseinzadeh and MKhezerloo ldquoSolutions of fuzzy linear systems using rankingfunctionrdquo International Journal of AppliedOperational Researchvol 2 pp 67ndash75 2012

[24] A Karpagam and P Sumathi ldquoNew approach to solve fuzzy lin-ear programming problems by the ranking functionrdquo BonfringInternational Journal of Data Mining vol 4 no 4 pp 22ndash252014

[25] D Dutta and P Kumar ldquoFuzzy inventory model without short-age using trapezoidal fuzzy number with sensitivity analysisrdquoIOSR Journal of Mathematics vol 4 no 3 pp 32ndash37 2012

[26] S Ding ldquoUncertain multi-product newsboy problem withchance constraintrdquo Applied Mathematics and Computation vol223 pp 139ndash146 2013

[27] H Zolfagharinia and K P S Isotupa ldquoErratum on lsquoOptimalinventory control of empty containers in inland transportationsystemrsquo Int J Prod Econ 133(1) (2011) 451ndash457rdquo InternationalJournal of Production Economics vol 141 no 1 pp 43ndash436 2013

[28] P Majumder U K Bera and M Maiti ldquoAn EPQ model ofdeteriorating items under partial trade credit financing anddemand declining market in crisp and fuzzy environmentrdquoProcedia Computer Science vol 45 pp 780ndash789 2015

[29] S M Mousavi J Sadeghi S T A Niaki N Alikar A Bahre-ininejad and H S C Metselaar ldquoTwo parameter-tuned meta-heuristics for a discounted inventory control problem in a fuzzyenvironmentrdquo Information Sciences vol 276 pp 42ndash62 2014

[30] H N Soni and M Joshi ldquoA periodic review inventory modelwith controllable lead time and backorder rate in fuzzy-stocha-stic environmentrdquo Fuzzy Information and Engineering vol 7 no1 pp 101ndash114 2015

[31] R S Kumar and A Goswami ldquoA fuzzy random EPQ modelfor imperfect quality items with possibility and necessity con-straintsrdquo Applied Soft Computing vol 34 pp 838ndash850 2015

[32] N Kazemi E Shekarian L E C Barron and E U OluguldquoIncorporating human learning into a fuzzy EOQ inventorymodel with backordersrdquo Computers amp Industrial Engineeringvol 87 pp 540ndash542 2015

[33] J K Dash and A Sahoo ldquoOptimal solution for a single periodinventorymodel with fuzzy cost and demand as a fuzzy randomvariablerdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 28 no 3 pp 1195ndash1203 2015

[34] M Nagare P Dutta and N Cheikhrouhou ldquoOptimal orderingpolicy for newsvendor models with bidirectional changes in

10 Advances in Fuzzy Systems

demand using expert judgmentrdquoOPSEARCH vol 53 no 3 pp620ndash647 2016

[35] P Raula S K Indrajitsingha P N Samanta and U K Misra ldquoAFuzzy InventoryModel for constant deteriorating item by usingGMIRmethod in which inventory parameters treated as HFNrdquoOpen Journal of Applied andTheoretical Mathematics (OJATM)vol 2 no 1 pp 13ndash20 2016

[36] I Sangal A Agarwal and S Rani ldquoA fuzzy environment inven-tory model with partial backlogging under learning effectrdquoInternational Journal of Computer Applications vol 137 no 6pp 25ndash32 2016

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Applied Computational Intelligence and Soft Computing

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Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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RoboticsJournal of

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Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

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2 Advances in Fuzzy Systems

level and selling price of the commodity They derived theresults for both without shortages and partially backloggedshortages Also objectives for profit maximization for eachitem are separately formulated with different goals and com-promise solutions of themultiobjective production inventoryproblems are obtained by goal programming method

Chang et al [11] considered the mixture inventory modelinvolving variable lead time with back orders and lost salesThey used the centroid method of defuzzification to estimatetotal cost in the fuzzy sense and found the optimal solutionfor order quantity and lead time in the fuzzy sense such thatthe total cost has a minimum value

Shao and Ji [12] considered the multiproduct newsboyproblem with fuzzy demands under budget constraint Theydeveloped three types of models under different criteria inwhich the objective functions are to maximize the expectedprofit of newsboy the chance of achieving a target profitand the profit which satisfies some chance constraints withat least some given confidence level A model to help thedecision maker in a newsboy problem to assess the value ofinformation is presented by Lee [13] for the potential benefitsof demand forecasting to decrease the risk of overstocking orshortage

Hayya et al [14] considered the reorder point orderquantity inventory model where the demand and the leadtime are independently and identically distributed randomvariables They used a normal approximation to show howto obtain regression equations for the optimal cost and theoptimal policy parameters in terms of the problem para-meters Dash et al [15] have discussed the deterministicequivalent of fuzzy chance constraint programming problemin different scenarios for the fuzzy random variable

Hu et al [16] presented fuzzy random models for thenewsboy problem in the decentralized and centralized sys-tems facing imperfect quality and the optimal policies for thesystems analyzed and derived

Zhang [17] presented a Lagrangian relaxation approach tosolve the multiproduct newsboy problem as a mixed integernonlinear programming model with both supplier quantitydiscounts and a budget constraint The profit for newsboy-type product is introducing a new capacity index which isbased on statistical hypothesis testing analyzed by Su et al[18]

Kotb and Fergany [19] derived the analytical solutionof the EOQ model of multiple items with both demand-dependent unit cost and leading time using geometric pro-gramming approach by considering continuous functions ofdemand rate and leading time for the varying purchase andleading time for crashing costs respectively They deducedthe optimal order quantity the demand rate and the leadingtime as decision variables and also the optimal total cost Deyand Chakraborty [20] presented a fuzzy random continuousreview system with the annual customer demand distributeduniformly and proposed amethodology tominimize the cost

R Banerjee and S Banerjee [21] considered a proba-bilistic inventory model under probabilistic and impreciseconstraints with uniform lead-time demand and fuzzy costcomponents Nagare and Dutta [22] captured both fuzzy

perception and randomness of customer demand to deter-mine an unambiguous optimal order quantity from a set of 119899fuzzy observations in a news-vendor inventory setting in thepresence of fuzzy random variable demand

Jahantigh et al [23] computed the compromised solutionof a LR fuzzy linear system by the use of ranking fuzzy num-bers Karpagam and Sumathi [24] used the ranking functionto find the fuzzy optimal solution of fully fuzzy linear prog-ramming problems

Dutta and Kumar [25] determined the optimal totalcost and the optimal order quantity for an inventory modelwithout shortages in a fuzzy environment They used signeddistance method to compute economic order quantity Thismodel deals with indeterminacy information for multiprod-uct newsboy problem which was initially transformed intodeterministic model and the solution was obtained by classicinteger programming method proposed by Ding [26]

Zolfagharinia and Isotupa [27] pointed out some flaws inthe simulationmodel and some of the formulae for inventorylevel and ordering quantity Majumder et al [28] formulatedan EPQ model for deteriorating items under partial tradecredit policy with crisp and fuzzy demand They have usedweighted sum method to convert a multiobjective to a singleobjective and the model was solved by Generalized ReducedGradient method

A real-world multiperiod inventory control problemunder budget constraint is investigated and the weightedlinear sum of objectives is applied to generate a single-objective model for the biobjective problem and a harmonysearch algorithm is developed to solve the complex inventoryproblem byMousavi et al [29]The goal is to find the optimalordered quantities of products not only the total inventorycost but also the required storage space as a fuzzy number tostore the products is minimized

Soni and Joshi [30] considered a periodic review inven-tory model in fuzzy-stochastic environment with lead timeand the back order rate as control variables and by applyingthe criterion of credibility they determined the expectedshortages

To capture the uncertainty of demand Kumar andGoswami [31] employed the fuzzy expectation signed dis-tance and possibility or necessitymeasure to the fuzzymodelto transform it into an equivalent deterministic nonlinearprogramming problem where the model is restricted tobudget and allowable shortages

Kazemi et al [32] developed and extended a fuzzy inven-tory model of an existing economic order quantity with backorders in which both demand and lead times are fuzzifiedto decrease the total inventory cost Dash and Sahoo [33]obtained the optimum order quantity and the expected profitby using Buckleyrsquos concept of minimization of fuzzy numbersfor a single-period inventory model

Nagare et al [34] developed an inventory model todetermine the optimal order quantity and weight factor onthe basis of revised forecasts and the results demonstrateeconomic benefits of using revised demand negative impactof constraints and role of demand distribution entropy indetermining the order size and expected profit

Advances in Fuzzy Systems 3

A fuzzy inventory model for fixed deteriorating itemswith shortages under fully backlogged condition is formu-lated and solved by Raula et al [35] They defuzzified themodel by graded unit preference integration method and allrelated inventory parameters were assumed to be hexagonalfuzzy numbers

Sangal et al [36] developed an inventory model fornoninstantaneous decaying items under crisp and fuzzyenvironmentThey have considered stock dependent demandrate and variable deterioration and defuzzified the total costfunction by signed distance method

The important feature of the present work is to determinethe optimum order quantity by considering a SPIFPM withdemand as a fuzzy random variable in the objective functionsubject to a chance constraint where the storage spacedistributed normally and exponentially which is differentfrom the previous studies Initially we used the weighted summethod ranking function to the objective function and sometheorems have been developed to transform the SPIFPM intoa crisp model

The remaining part of the paper is organized as followsSome basic concepts and results of uncertain theory are givenin Section 2 In Section 3 an analytical model is formulatedfor the SPIFPM under a chance constraint In Section 4 wepresent a transformation technique to convert the model toan equivalent deterministic model In Section 5 a numericalexample is given for verification Finally in Section 6 someconclusions are listed

2 Preliminaries

From literature study here we present some basic conceptsregarding the uncertainty theory

Definition 1 (fuzzy set [1]) A fuzzy set over a universal set119883 is defined as a set of ordered pairs = (119909 120583119860(119909)) 119909 isin 119883and is characterized by its membership function 120583119860(119909)where120583119860(119909) 119883 rarr [0 1] for each 119909 isin 119883Definition 2 (triangular fuzzy number [15]) The triangularfuzzy number is a fuzzy number denoted as = (1198601 1198602 1198603)and is interpreted by its membership function as follows

120583 (119909) =

119909 minus 11986011198602 minus 1198601 if 1198601 le 119909 le 11986021198603 minus 1199091198603 minus 1198602 if 1198602 le 119909 le 11986030 otherwise

(1)

Definition 3 (120572-cut of a fuzzy number [15]) The 120572-cut of thefuzzy number is the set 119909 | 120583119860(119909) ge 120572 for 120572 isin (0 1) anddenoted as [120572]

Definition 4 (partial order relation of two fuzzy numbers [9])Let = (1198601 1198602 1198603) and = (1198611 1198612 1198613) be two fuzzynumbers with 120572-cut [120572] = [119860lowast 119860lowast] and [120572] = [119861lowast 119861lowast]respectively then ⪯ if and only if 119860lowast le 119861lowast and ⪰ ifand only if 119860lowast le 119861lowastDefinition 5 (fuzzy random variable [3 6]) The randomvariables whose parameters like mean variance and so forthare fuzzy numbers are called fuzzy random variables A fuzzyrandom variable with mean 119909 and variance 2119909 is denotedby ( 2)Definition 6 (mean of a continuous fuzzy random variable[6]) Let = 11988611 oplus 11988622 oplus sdot sdot sdot oplus 119886119899119899 be a FRV whichis the linear combination of 119899 number of independent fuzzyrandom variables 1 2 119899 whose means are fuzzynumbers 1198831 1198832 119883119899 respectively Then 119880[120572] =sum119899119895=1 119886119895119883119895[120572]Definition 7 (variance of a continuous fuzzy random variable[6]) Let = 11988611 oplus 11988622 oplus sdot sdot sdot oplus 119886119899119899 be the linearcombination of 119899 number of independent fuzzy random vari-ables 1 2 119899 whose variances are the fuzzy numbers21198831 21198832 2119883119899 respectively Then 2119880[120572] = sum119899119895=1 1198862119895 2119883119895[120572]Definition 8 (ranking function [23 24]) The ranking func-tion is defined on a set of real numbers (119877) which maps eachfuzzy number into the real line where a natural order existsthat is119877 119865(119877) rarr 119877 where119865(119877) is a set of fuzzy numbers Tocalculate the triangular weights ranking function is effectiveLet = (119886 119887 119888) be a triangular fuzzy number We define astandard ranking function as 119877() = (119886 + 2 lowast 119887 + 119888)4Definition 9 (weighted sum method [28]) This method canbe interpreted by some positive weights or priority assignedto the objective criterion by the decision maker Here weexpress the weighted summethod by multiplying the weightsby the users each supplied in the objective and the weightsare chosen in a way such that their sum is one that is

optimize119899sum119894=1

119908119894119891119894 (119909) 119908119894 isin [0 1] 119909 isin 119883 (2)

such that sum119899119894=1 119908119894 = 1 where 119908119894 are the weights of the 119894thobjective functions

Definition 10 (fuzzy normal distribution [15]) A normalfuzzy random variable 119873( 2) is a normally distributedrandom variable with fuzzy mean and fuzzy variance 2as fuzzy parameters Let 119873(119898 1205902) denote the crisp normalrandom variable with mean 119898 and variance 1205902 and let119891(119909119898 1205902) be the density function of the crisp normaldistribution where

119891 (119909119898 1205902) = 1radic2120587120590119890minus(119909minus119898)221205902 120590 gt 0 (3)

4 Advances in Fuzzy Systems

so (1radic2120587120590) intinfinminusinfin

119890minus(119909minus119898)221205902119889119909 = 1 Now the fuzzy proba-bility of the FRV on the interval [119888 119889] is a fuzzy numberwhose 120572-cut is (119888 le le 119889) [120572] = 1radic2120587120590 int

119889

119888119890minus(119909minus119898)221205902119889119909 | 119898

isin [120572] 1205902 isin 2 [120572] = 1radic2120587 int1199112

1199111

119890minus11991122119889119911 | 119898isin [120572] 1205902 isin 2 [120572]

(4)

where 1199111 = (119888 minus 119898)120590 and 1199112 = (119889 minus 119898)120590Definition 11 (fuzzy exponential distribution [15]) Let be aexponentially distributed fuzzy random variable with densityfunction 119891(119909 ) being a fuzzy set and its 120572-cut is

119891 (119909 ) [120572] = 119890minus119909 | 120582 isin (120572) (5)

Now the fuzzy probability of on the interval [119888 119889] is a fuzzynumber and denoted by (119888 le le 119889) where 119875lowast(120572) =minint119889

119888120582119890minus120582119909119889119909 | 120582 isin [120572] and 119875lowast(120572) = maxint119889

119888120582119890minus120582119909119889119909 |120582 isin [120572]

Definition 12 (Fuzzy Chance Constrained Programmingproblem (FCCP) [26]) A general chance constraint pro-gramming problem CCP is of the following form

(CCP) Optimize119896sum119895=1

119888119895119909119895Subject to 119875( 119896sum

119895=1

119886119894119895119909119895 le 119887119894) ⪰ 120574119894119896sum119895=1

119887119897119895119909119895 ge ℎ119897119909119895 ge 0119887119897119895 ℎ119897 isin 119877119894 = 1 2 119898119897 = 1 2 1198990 le 120574119894 le 1

(6)

where at least one of 119886119894119895 119888119895 and 119887119894 is treated as randomvariable

So FCCP is a CCP given as above form where at least oneof 119886119894119895 119888119895 and 119887119894 is treated as FRV

3 Mathematical Formulation

Notations used in the proposed model are presented inNotations section

31 Assumptions We imposed the following assumptions todevelop the model

(i) For all 119895 119895 are independent fuzzy random variableswhose mean 119863 are triangular fuzzy numbers

(ii) For all 119895 119895 are independent fuzzy random variableswhose mean 119882 and variance 2119882 are triangular fuzzynumbers

(iii) The time period is precise for all 119895 that is singleperiod

(iv) The decision makerrsquos order quantity 119876119895 is optimal forall products at the beginning of the period

(v) At the end of the season the leftover items aresalvaged

(vi) The lost sales penalty cost is zero

32 Single-Period Inventory Fuzzy Probabilistic Model(SPIFPM) The profit function119885119895 associated with each orderquantity 119876119895 and fuzzy demand 119895 is given as follows

119885119895 (119876119895 119895) = 119901119895min 119876119895 119895 + 119904119895max 119876119895 minus 119895 0minus 119888119895119876119895 119895 = 1 2 119896 (7)

Here two situations are explored to discuss the profit func-tion that is overstock situation (119895 le 119876119895) and understocksituation (119895 ge 119876119895) Then the above equation boils down tothe following form

119885119895 (119876119895 119895)=

119895 (119876119895 119895) = (119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895 119895 le 119876119895119895 (119876119895 119895) = (119901119895 minus 119888119895)119876119895 119895 ge 119876119895

(8)

where 119885119895 and 119885119895 are profit functions for each product inoverstock and understock situations respectively Due tofuzziness of demand 119895 we get either a fuzzy overstock profit119895(119876119895 119895) or a fuzzy understock profit 119895(119876119895 119895) Hencethe total expected profit 119895(119876119895 119895) corresponding to bothoverstock and understock situations is given as follows

119895 (119876119895 119895) = 119895 (119876119895 119895) cup 119895 (119876119895 119895) (9)

Advances in Fuzzy Systems 5

Thus a multi-item SPIFPM subject to a storage space chanceconstraint with a confidence level 120574 can be stated as follows

(SPIFPM) Maximize (1198761 1198762 119876119896)= 119896sum119895=1

119895 (119876119895 119895)

subject to ( sum119895=1

119886119895119876119895 le ) ⪰ 119876119895 ge 0for 119895 = 1 2 119896

(10)

In the above model we consider demand 119895 as a FRVpresent in the objective function and the total storage space are independent FRVs present in the constraint distributednormally and exponentially

4 Transformation Technique

41 Crisp Equivalent of the Fuzzy Objective Function InSPIM the defined resultant profit function is fuzzy since thecustomer demand for each product is imprecise Now our

objective is to convert the fuzzy model to crisp model Usingthe weighted function to the profit function we have

119895 (119876119895 119895) = 120596119895 (119876119895 119895) + (1 minus 120596) 119895 (119876119895 119895) 997904rArr119895 (119876119895 119895) = 120596[[

119896sum119895=1

(119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]] (11)

Here 119895 are FRVs whose mean 119898119863119895 are triangular fuzzynumbers To make it crisp apply the ranking function as119877(119898119863119895) = (119898119863119895 +2119898119863119895 +119898119863119895)4 So the above profit functionbecomes

119895 (119876119895 119895) = 120596[[[119896sum119895=1

(119901119895 minus 119904119895) (119898119863119895 + 2119898119863119895 + 119898119863119895)4

+ (119904119895 minus 119888119895)119876119895]]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]] (12)

Therefore the crisp equivalent of the objective function is asfollows

Maximize 120596[[[119896sum119895=1

(119901119895 minus 119904119895) (119898119863119895 + 2119898119863119895 + 119898119863119895)4 + (119904119895 minus 119888119895)119876119895]]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]] (13)

42 Crisp Equivalent of the Fuzzy Chance Constraint Thefollowing results are proved to convert the fuzzy chanceconstraint to a deterministic form

Theorem 13 (Case 1 ( is normally distributed fuzzy randomvariable)) If is a normally distributed FRV 119886119895 isin 119877and with predetermined confidence level 120574 the inequality(sum119896119895=1 119886119895119876119895 le ) ⪰ is equivalent to

119896sum119895=1

119886119895119876119895 le 120583119882lowast (120572) + 120590119882lowast (120572) 119865minus1 (1 minus 120574lowast (120572)) (14)

for 120572 isin [0 1] and 119865 is the cumulative distribution function of119873(0 1) distributionProof Let us consider the constraint of the model as follows

( 119896sum119895=1

119886119895119876119895 le ) ⪰ 119886119895 isin 119877 (15)

where is a normally distributed FRV whose mean andvariance are fuzzy numbers denoted by 119882 and 2119882 andtheir 120572-cuts are 119882[120572] = [120583119882lowast(120572) 120583lowast119882(120572)] and 2119882[120572] =[1205902119882lowast(120572) 1205902119882lowast(120572)] respectively The 120572-cut of is [120572] =[120574lowast(120572) 120574lowast(120572)]

Let sum119896119895=1 119886119895119876119895 = 119880 Here 119880 is a real number since 119886119895119895 = 1 2 119896 are real numbers Then the above inequalitycan be written as 119875(119880 le ) ⪰ Here is a FRV whosemean and variance are the fuzzy numbers 119882 and 2119882 whichcan be computed from the crisp random variable119882 that is119882 isin [120572] That means 120583119882 isin 119882[120572] and 1205902119882 isin 2119882[120572] whichsatisfy (1radic2120587120590119882) intinfinminusinfin 119890minus(119882minus120583119882)221205902119882119889119882 = 1

Now the 120572-cut of the probability (sum119896119895=1 119886119895119876119895 le ) is ( 119896sum119895=1

119886119895119876119895 le ) [120572] = 119875 (119880 le 119882) | 120583119882isin 119882 [120572] 1205902119882 isin 2119882 [120572] = 1 minus 119875 (119882 le 119880) | 120583119882

6 Advances in Fuzzy Systems

isin 119882 [120572] 1205902119882 isin 2119882 [120572] = 1minus 1radic2120587120590119882 int

119880

minusinfin119890minus(119882minus120583119882)221205902119882119889119882 | 120583119882

isin 119882 [120572] 1205902119882 isin 2119882 [120572] (16)

Let 119881 = (119882 minus 120583119882)120590119882 Then

( 119896sum119895=1

119886119895119876119895 le ) [120572] = 1minus 1radic2120587 int

(119880minus120583119882)120590119882

minusinfin119890minus11988122119889119881 | 120583119882 isin 119882 [120572] 1205902119908

isin 2119908 [120572] = 1 minus 119865(119880 minus 120583119882120590119882 ) | 120583119882isin 119882 [120572] 1205902119882 isin 2119882 [120572]

(17)

where 119865 is the cumulative distribution function of 119873(0 1)distribution which is an increasing function So

min1 minus 119865(119880 minus 120583119882120590119882 ) | 120583119882 isin 119882 [120572] 1205902119882isin 2119882 [120572] = 1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) )

max1 minus 119865(119880 minus 120583119882120590119882 ) | 120583119882 isin 119882 [120572] 1205902119882isin 2119882 [120572] = 1 minus 119865(119880 minus 120583lowast119882 (120572)120590lowast119882 (120572) )

(18)

So

( 119896sum119895=1

119886119895119876119895 le ) [120572]= [1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) 1 minus 119865(119880 minus 120583lowast119882 (120572)120590lowast119882 (120572) )]

(19)

Using the Inequality law due toNanda et al [9] the constraint(sum119896119895=1 119886119895119876119895 le ) ⪰ becomes

1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) ge 120574lowast (120572) 997904rArr119896sum119895=1

119886119895119876119895 le 120583119882lowast (120572) + 120590119882lowast (120572) 119865minus1 (1 minus 120574lowast (120572)) (20)

Hence the proof is complete

Therefore finally the SPIFPM is transformed to anequivalent single-period inventory crisp model (SPICM) asfollows

(SPICM) Maximize 120596[[[119896sum119895=1

(119901119895 minus 119904119895) (119898119863119895 + 2119898119863119895 + 119898119863119895)4 + (119904119895 minus 119888119895)119876119895]]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]]Subject to

119896sum119895=1

119886119895119876119895 le 120583119882lowast (120572) + 120590119882lowast (120572) 119865minus1 (1 minus 120574lowast (120572))119876119895 ge 0for 120572 isin [0 1] 119895 = 1 2 119896

(21)

Theorem 14 (Case 2 ( follows fuzzy exponential distribu-tion)) If is an exponentially distributed FRV with fuzzyparameter 119886119895 isin 119877 then (sum119896119895=1 119886119895119876119895 le ) ⪰ is equivalentto 119890minus120582lowast(120572)119880 ge 120574lowast(120572) for each 120572 isin [0 1] where [120572] =[120582lowast(120572) 120582lowast(120572)]Proof The 120572-cut of the fuzzy probability (sum119896119895=1 119886119895119876119895 le )is

( 119896sum119895=1

119886119895119876119895 le ) [120572] = 119875 (119880 le 119882) | 119882 isin [120572]= intinfin119880120582119890minus120582119882119889119882 | 120582 isin [120572]

= 119890minus120582119880 | 120582lowast (120572) le 120582 le 120582lowast (120572) = [119875lowast (120572) 119875lowast (120572)](say)

(22)

where 119875lowast(120572) and 119875lowast(120572) are the solutions of the optimizationproblems

minimize120582lowast(120572)le120582le120582

lowast(120572)119890minus120582119880

maximize120582lowast(120572)le120582le120582

lowast(120572)119890minus120582119880 (23)

respectively

Advances in Fuzzy Systems 7

Here 119890minus120582119880 attains its minimum at 120582lowast(120572) and maximumat 120582lowast(120572) since 119890minus120582119880 is a decreasing function of 120582 That ismin 119890minus120582119880 = 120582lowast(120572) and max 119890minus120582119880 = 120582lowast(120572)

Hence

( 119896sum119895=1

119886119895119876119895 le ) [120572] = [119890minus120582lowast(120572)119880 119890minus120582lowast(120572)119880] (24)

Therefore the deterministic equivalent of the fuzzy chanceconstraint

( 119896sum119895=1

119886119895119876119895 le ) ⪰ becomes 119890minus120582lowast(120572)119880 ge 120574lowast (120572)

(25)

5 Numerical Example

Let us consider a SPIFPM to illustrate the effectiveness of theabove approach with the following data

(SPIFPM) Maximize [[2sum119895=1

(119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895]]cup [[2sum119895=1

(119901119895 minus 119888119895)119876119895]]Subject to (41198761 oplus 31198762 le ) ⪰ 05

1198761 ge 01198762 ge 0

(26)

where the other data for multi-items is given in Table 1Here 1 2 are independent normally distributed FRVs

with mean 1198631 1198632 being fuzzy numbers Also areindependent normally distributed FRVs with mean 119882 andvariance 2119882 as fuzzy numbers and 05 is also a fuzzy numberAll these fuzzy numbers are assumed to be linear triangularas follows 1198631 = ⟨200300400⟩ 1198632 = ⟨700800900⟩119882 = ⟨200020102020⟩ 2119882 = ⟨300400500⟩ and 05 =⟨040506⟩51 Solution Initially applying the weighted sum method tothe objective function we can write

Maximize (1198761 1198762)= 120596[[

2sum119895=1

(119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895]]+ (1 minus 120596)[[

2sum119895=1

(119901119895 minus 119888119895)119876119895]] (27)

Table 1

Product name 119888119895 119901119895 119904119895 119886119895Product 1 30 34 28 4Product 2 40 43 35 3

Let 120596 = 06 and using the above cost data to the aboveexpression we have

Maximize (1198761 1198762)= 041198761 minus 181198762 + 361 + 482 (28)

Here 1 2 are FRVs whose mean 1198631 1198632 are triangularfuzzy numbers with the 120572-cuts being

1198631 [120572] = [200 + 100120572 400 minus 100120572] 1198632 [120572] = [700 + 100120572 900 minus 100120572] (29)

Applying the ranking function to the above objective functionwe have

Maximize (1198761 1198762)= 041198761 minus 181198762 + 36119877 (1198631)+ 48119877 (1198632)

(30)

Hence the crisp equivalent of the objective function takes thefollowing form

Maximize 041198761 minus 181198762 + 4920 (31)

Now let us consider the constraint (41198761 + 31198762 le ) ⪰ 05We can convert this constraint to its deterministic form as follows normal and exponential distribution

Case 1 ( is normally distributed fuzzy random variable)As the mean 119882 variance 2119882 and 05 are triangular fuzzynumbers their 120572-cuts are

119882 [120572] = [2000 + 10120572 2020 minus 10120572] 2119882 = [300 + 100120572 500 minus 100120572]

05 [120572] = [04 + 120572 06 minus 120572] (32)

Let 41198761 +31198762 = 119880 So the constraint is given by (119880 le ) ⪰05 Using the above theorem result the 120572-cut of (119880 le )becomes

(119880 le ) [120572]= [1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) 1 minus 119865(119880 minus 120583lowast119882 (120572)120590lowast119882 (120572) )] (33)

So the crisp conversion of the constraint is

1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) ge 06 minus 01120572 (34)

8 Advances in Fuzzy Systems

which can be further simplified as

119880 minus 120583119882lowast (120572) le 120590119882lowast (120572) 119865minus1 (06 + 01120572) 997904rArr41198761 + 31198762 le 5 + 120572 + (radic3 + 120572)119865minus1 (04 + 01120572) (35)

Hence the crisp equivalent that is SPICM of the originalSPIFPM takes the following form

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to 41198761 + 31198762

le 5 + 120572+ (radic3 + 120572)119865minus1 (04 + 01120572)

1198761 ge 01198762 ge 0

(36)

Now 119865minus1(04 + 01120572) can be found from the normal distri-bution table For each 120572 there exists a solution of the aboveproblem For example if 120572 = 04 then SPICM reduces to thefollowing nonlinear programming problem

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to (41198761 + 31198762 minus 2004)2

le 7752341198761 ge 01198762 ge 0

(37)

Using any optimization software its local solution can bedetermined Here we have solved this by using LINGO andits solution is1198761 = 5016959 and1198762 = 0 and the correspond-ing objective value is 5120678 Each value of 120572 gives a corres-ponding solution

Case 2 ( follows fuzzy exponential distribution) Let bean exponential FRV with the parameter = ⟨1000 15002000⟩ having 120572-cut as [120572] = [1000+500120572 2000minus500120572] Let41198761+31198762 = 119880 So the constraint is given by (119880 le ) ⪰ 05Using theorem the crisp conversion of the constraint is

exp (minus (2000 minus 500120572) (41198761 + 31198762)) ge 06 minus 01120572 (38)

Hence the crisp equivalent that is SPICM of the originalSPIFPM takes the following form

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to exp (minus (2000 minus 500120572) (41198761 + 31198762))

ge 06 minus 011205721198761 ge 01198762 ge 0

(39)

For each 120572 there exists a solution of the above problem Forexample if 120572 = 04 then SPICM reduces to the followingnonlinear programming problem

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to exp (minus1800 (41198761 + 31198762))

ge 0561198761 ge 01198762 ge 0

(40)

Using any optimization software its local solution can bedetermined Here we have solved this by using LINGO andits solution is 1198761 = 08053035119864 minus 04 and 1198762 = 0 and thecorresponding objective value is 4920 Each value of 120572 givesa corresponding solution

6 Conclusion

This paper presents a solution procedure for solving SPIFPMin which the the storage space in the constraint is a FRVconsidered under two cases that is normal distributionand exponential distribution Also the demand is a FRVwhose mean and variance are fuzzy numbers in the objectivefunction of the given model Initially some theorems areproposed to transform the fuzzy model into an equivalentcrisp model Then the resultant model is solved by using theLINGO software When the storage space in the constraintis normally distributed then the optimal solution is 1198761 =5016959 and 1198762 = 0 and the corresponding objectivevalue is 5120678 and for exponentially distributed randomvariable its solution is 1198761 = 08053035119864 minus 04 and 1198762 = 0and the corresponding objective value is 4920 for one valueof 120572 However more approximate solution can be obtainedby taking more numbers of values of 120572 For the futurescope of the present work triangular fuzzy number can bereplaced by other fuzzy numbers and the demand for eachitems can be considered as other types of nonindependentFRVs instead of normally distributed independent FRVsA multiconstrained optimization model can be developedof our present work and our model can be extended toother mathematical models To analyze the effect of changesin the optimal solution with respect to change in variousparameters sensitivity analysis can be carried out

Notations

119876119895 Order quantity for item 119895119895 Demand as a fuzzy random variable for item 119895119888119895 Purchase cost per unit of item 119895119901119895 Selling price per unit of item 119895119904119895 Salvage value per unit of item 119895119886119895 Area required for each unit of item 119895119895 Fuzzy profit function for product 119895 Fuzzy total profit function for all products Available storage space a fuzzy random variable

Advances in Fuzzy Systems 9

Competing Interests

The authors declare that they have no competing interests

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] J J Buckley Fuzzy Probabilities New Approach and Applica-tions Physica Heidelberg Germany 2003

[3] H Kwakernaak ldquoFuzzy random variablesmdashI definitions andtheoremsrdquo Information Sciences vol 15 no 1 pp 1ndash29 1978

[4] H-J Zimmermann ldquoFuzzymathematical programmingrdquoCom-puters amp Operations Research vol 10 no 4 pp 291ndash298 1983

[5] S Nanda and K Kar ldquoConvex fuzzy mappingsrdquo Fuzzy Sets andSystems vol 48 no 1 pp 129ndash132 1992

[6] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986

[7] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy models forthe newsboy problemrdquo International Journal of Production Eco-nomics vol 45 no 1ndash3 pp 435ndash441 1996

[8] K Das T K Roy and M Maiti ldquoMulti-item stochastic andfuzzy-stochastic inventory models under two restrictionsrdquoComputers and Operations Research vol 31 no 11 pp 1793ndash1806 2004

[9] S Nanda G Panda and J K Dash ldquoA new methodology forcrisp equivalent of fuzzy chance constrained programmingproblemrdquo Fuzzy Optimization and Decision Making vol 7 no1 pp 59ndash74 2008

[10] N K Mahapatra and M Maiti ldquoDecision process for multiob-jective multi-item production-inventory system via interactivefuzzy satisficing techniquerdquo Computers amp Mathematics withApplications vol 49 no 5-6 pp 805ndash821 2005

[11] H-C Chang J-S Yao and L-Y Ouyang ldquoFuzzy mixtureinventory model involving fuzzy random variable lead timedemand and fuzzy total demandrdquo European Journal of Opera-tional Research vol 169 no 1 pp 65ndash80 2006

[12] Z Shao and X Ji ldquoFuzzy multi-product constraint newsboyproblemrdquoAppliedMathematics and Computation vol 180 no 1pp 7ndash15 2006

[13] C-M Lee ldquoA Bayesian approach to determine the value ofinformation in the newsboy problemrdquo International Journal ofProduction Economics vol 112 no 1 pp 391ndash402 2008

[14] J C Hayya T P Harrison and D C Chatfield ldquoA solutionfor the intractable inventory model when both demand andlead time are stochasticrdquo International Journal of ProductionEconomics vol 122 no 2 pp 595ndash605 2009

[15] J K Dash G Panda and S Nanda ldquoChance constrained prog-ramming problem under different fuzzy distributionsrdquo Inter-national Journal of Optimization Theory Methods and Applica-tions vol 1 no 1 pp 58ndash71 2009

[16] J-S Hu H Zheng R-Q Xu Y-P Ji and C-Y Guo ldquoSupplychain coordination for fuzzy random newsboy problem withimperfect qualityrdquo International Journal of Approximate Reason-ing vol 51 no 7 pp 771ndash784 2010

[17] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[18] R-H Su D-Y Yang and W L Pearn ldquoDecision-making ina single-period inventory environment with fuzzy demandrdquo

Expert Systems with Applications vol 38 no 3 pp 1909ndash19162011

[19] K A M Kotb and H A Fergany ldquoMulti-item EOQmodel withboth demand-dependent unit cost and varying leading time viageometric programmingrdquoAppliedMathematics vol 2 no 5 pp551ndash555 2011

[20] O Dey and D Chakraborty ldquoA fuzzy random continuousreview inventory systemrdquo International Journal of ProductionEconomics vol 132 no 1 pp 101ndash106 2011

[21] R Banerjee and S Banerjee ldquoSolution of a probabilistic inven-tory model with chance constraints a general fuzzy program-ming and intuitionistic fuzzy optimization approachrdquo Interna-tional Journal of Pure and Applied Sciences and Technology vol9 no 1 pp 20ndash38 2012

[22] M Nagare and P Dutta ldquoOn solving single-period inventorymodel under hybrid uncertaintyrdquo International Journal of Eco-nomics and Management Sciences vol 6 no 4 pp 290ndash2952012

[23] M A Jahantigh S Khezerloo A A Hosseinzadeh and MKhezerloo ldquoSolutions of fuzzy linear systems using rankingfunctionrdquo International Journal of AppliedOperational Researchvol 2 pp 67ndash75 2012

[24] A Karpagam and P Sumathi ldquoNew approach to solve fuzzy lin-ear programming problems by the ranking functionrdquo BonfringInternational Journal of Data Mining vol 4 no 4 pp 22ndash252014

[25] D Dutta and P Kumar ldquoFuzzy inventory model without short-age using trapezoidal fuzzy number with sensitivity analysisrdquoIOSR Journal of Mathematics vol 4 no 3 pp 32ndash37 2012

[26] S Ding ldquoUncertain multi-product newsboy problem withchance constraintrdquo Applied Mathematics and Computation vol223 pp 139ndash146 2013

[27] H Zolfagharinia and K P S Isotupa ldquoErratum on lsquoOptimalinventory control of empty containers in inland transportationsystemrsquo Int J Prod Econ 133(1) (2011) 451ndash457rdquo InternationalJournal of Production Economics vol 141 no 1 pp 43ndash436 2013

[28] P Majumder U K Bera and M Maiti ldquoAn EPQ model ofdeteriorating items under partial trade credit financing anddemand declining market in crisp and fuzzy environmentrdquoProcedia Computer Science vol 45 pp 780ndash789 2015

[29] S M Mousavi J Sadeghi S T A Niaki N Alikar A Bahre-ininejad and H S C Metselaar ldquoTwo parameter-tuned meta-heuristics for a discounted inventory control problem in a fuzzyenvironmentrdquo Information Sciences vol 276 pp 42ndash62 2014

[30] H N Soni and M Joshi ldquoA periodic review inventory modelwith controllable lead time and backorder rate in fuzzy-stocha-stic environmentrdquo Fuzzy Information and Engineering vol 7 no1 pp 101ndash114 2015

[31] R S Kumar and A Goswami ldquoA fuzzy random EPQ modelfor imperfect quality items with possibility and necessity con-straintsrdquo Applied Soft Computing vol 34 pp 838ndash850 2015

[32] N Kazemi E Shekarian L E C Barron and E U OluguldquoIncorporating human learning into a fuzzy EOQ inventorymodel with backordersrdquo Computers amp Industrial Engineeringvol 87 pp 540ndash542 2015

[33] J K Dash and A Sahoo ldquoOptimal solution for a single periodinventorymodel with fuzzy cost and demand as a fuzzy randomvariablerdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 28 no 3 pp 1195ndash1203 2015

[34] M Nagare P Dutta and N Cheikhrouhou ldquoOptimal orderingpolicy for newsvendor models with bidirectional changes in

10 Advances in Fuzzy Systems

demand using expert judgmentrdquoOPSEARCH vol 53 no 3 pp620ndash647 2016

[35] P Raula S K Indrajitsingha P N Samanta and U K Misra ldquoAFuzzy InventoryModel for constant deteriorating item by usingGMIRmethod in which inventory parameters treated as HFNrdquoOpen Journal of Applied andTheoretical Mathematics (OJATM)vol 2 no 1 pp 13ndash20 2016

[36] I Sangal A Agarwal and S Rani ldquoA fuzzy environment inven-tory model with partial backlogging under learning effectrdquoInternational Journal of Computer Applications vol 137 no 6pp 25ndash32 2016

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Advances in Fuzzy Systems 3

A fuzzy inventory model for fixed deteriorating itemswith shortages under fully backlogged condition is formu-lated and solved by Raula et al [35] They defuzzified themodel by graded unit preference integration method and allrelated inventory parameters were assumed to be hexagonalfuzzy numbers

Sangal et al [36] developed an inventory model fornoninstantaneous decaying items under crisp and fuzzyenvironmentThey have considered stock dependent demandrate and variable deterioration and defuzzified the total costfunction by signed distance method

The important feature of the present work is to determinethe optimum order quantity by considering a SPIFPM withdemand as a fuzzy random variable in the objective functionsubject to a chance constraint where the storage spacedistributed normally and exponentially which is differentfrom the previous studies Initially we used the weighted summethod ranking function to the objective function and sometheorems have been developed to transform the SPIFPM intoa crisp model

The remaining part of the paper is organized as followsSome basic concepts and results of uncertain theory are givenin Section 2 In Section 3 an analytical model is formulatedfor the SPIFPM under a chance constraint In Section 4 wepresent a transformation technique to convert the model toan equivalent deterministic model In Section 5 a numericalexample is given for verification Finally in Section 6 someconclusions are listed

2 Preliminaries

From literature study here we present some basic conceptsregarding the uncertainty theory

Definition 1 (fuzzy set [1]) A fuzzy set over a universal set119883 is defined as a set of ordered pairs = (119909 120583119860(119909)) 119909 isin 119883and is characterized by its membership function 120583119860(119909)where120583119860(119909) 119883 rarr [0 1] for each 119909 isin 119883Definition 2 (triangular fuzzy number [15]) The triangularfuzzy number is a fuzzy number denoted as = (1198601 1198602 1198603)and is interpreted by its membership function as follows

120583 (119909) =

119909 minus 11986011198602 minus 1198601 if 1198601 le 119909 le 11986021198603 minus 1199091198603 minus 1198602 if 1198602 le 119909 le 11986030 otherwise

(1)

Definition 3 (120572-cut of a fuzzy number [15]) The 120572-cut of thefuzzy number is the set 119909 | 120583119860(119909) ge 120572 for 120572 isin (0 1) anddenoted as [120572]

Definition 4 (partial order relation of two fuzzy numbers [9])Let = (1198601 1198602 1198603) and = (1198611 1198612 1198613) be two fuzzynumbers with 120572-cut [120572] = [119860lowast 119860lowast] and [120572] = [119861lowast 119861lowast]respectively then ⪯ if and only if 119860lowast le 119861lowast and ⪰ ifand only if 119860lowast le 119861lowastDefinition 5 (fuzzy random variable [3 6]) The randomvariables whose parameters like mean variance and so forthare fuzzy numbers are called fuzzy random variables A fuzzyrandom variable with mean 119909 and variance 2119909 is denotedby ( 2)Definition 6 (mean of a continuous fuzzy random variable[6]) Let = 11988611 oplus 11988622 oplus sdot sdot sdot oplus 119886119899119899 be a FRV whichis the linear combination of 119899 number of independent fuzzyrandom variables 1 2 119899 whose means are fuzzynumbers 1198831 1198832 119883119899 respectively Then 119880[120572] =sum119899119895=1 119886119895119883119895[120572]Definition 7 (variance of a continuous fuzzy random variable[6]) Let = 11988611 oplus 11988622 oplus sdot sdot sdot oplus 119886119899119899 be the linearcombination of 119899 number of independent fuzzy random vari-ables 1 2 119899 whose variances are the fuzzy numbers21198831 21198832 2119883119899 respectively Then 2119880[120572] = sum119899119895=1 1198862119895 2119883119895[120572]Definition 8 (ranking function [23 24]) The ranking func-tion is defined on a set of real numbers (119877) which maps eachfuzzy number into the real line where a natural order existsthat is119877 119865(119877) rarr 119877 where119865(119877) is a set of fuzzy numbers Tocalculate the triangular weights ranking function is effectiveLet = (119886 119887 119888) be a triangular fuzzy number We define astandard ranking function as 119877() = (119886 + 2 lowast 119887 + 119888)4Definition 9 (weighted sum method [28]) This method canbe interpreted by some positive weights or priority assignedto the objective criterion by the decision maker Here weexpress the weighted summethod by multiplying the weightsby the users each supplied in the objective and the weightsare chosen in a way such that their sum is one that is

optimize119899sum119894=1

119908119894119891119894 (119909) 119908119894 isin [0 1] 119909 isin 119883 (2)

such that sum119899119894=1 119908119894 = 1 where 119908119894 are the weights of the 119894thobjective functions

Definition 10 (fuzzy normal distribution [15]) A normalfuzzy random variable 119873( 2) is a normally distributedrandom variable with fuzzy mean and fuzzy variance 2as fuzzy parameters Let 119873(119898 1205902) denote the crisp normalrandom variable with mean 119898 and variance 1205902 and let119891(119909119898 1205902) be the density function of the crisp normaldistribution where

119891 (119909119898 1205902) = 1radic2120587120590119890minus(119909minus119898)221205902 120590 gt 0 (3)

4 Advances in Fuzzy Systems

so (1radic2120587120590) intinfinminusinfin

119890minus(119909minus119898)221205902119889119909 = 1 Now the fuzzy proba-bility of the FRV on the interval [119888 119889] is a fuzzy numberwhose 120572-cut is (119888 le le 119889) [120572] = 1radic2120587120590 int

119889

119888119890minus(119909minus119898)221205902119889119909 | 119898

isin [120572] 1205902 isin 2 [120572] = 1radic2120587 int1199112

1199111

119890minus11991122119889119911 | 119898isin [120572] 1205902 isin 2 [120572]

(4)

where 1199111 = (119888 minus 119898)120590 and 1199112 = (119889 minus 119898)120590Definition 11 (fuzzy exponential distribution [15]) Let be aexponentially distributed fuzzy random variable with densityfunction 119891(119909 ) being a fuzzy set and its 120572-cut is

119891 (119909 ) [120572] = 119890minus119909 | 120582 isin (120572) (5)

Now the fuzzy probability of on the interval [119888 119889] is a fuzzynumber and denoted by (119888 le le 119889) where 119875lowast(120572) =minint119889

119888120582119890minus120582119909119889119909 | 120582 isin [120572] and 119875lowast(120572) = maxint119889

119888120582119890minus120582119909119889119909 |120582 isin [120572]

Definition 12 (Fuzzy Chance Constrained Programmingproblem (FCCP) [26]) A general chance constraint pro-gramming problem CCP is of the following form

(CCP) Optimize119896sum119895=1

119888119895119909119895Subject to 119875( 119896sum

119895=1

119886119894119895119909119895 le 119887119894) ⪰ 120574119894119896sum119895=1

119887119897119895119909119895 ge ℎ119897119909119895 ge 0119887119897119895 ℎ119897 isin 119877119894 = 1 2 119898119897 = 1 2 1198990 le 120574119894 le 1

(6)

where at least one of 119886119894119895 119888119895 and 119887119894 is treated as randomvariable

So FCCP is a CCP given as above form where at least oneof 119886119894119895 119888119895 and 119887119894 is treated as FRV

3 Mathematical Formulation

Notations used in the proposed model are presented inNotations section

31 Assumptions We imposed the following assumptions todevelop the model

(i) For all 119895 119895 are independent fuzzy random variableswhose mean 119863 are triangular fuzzy numbers

(ii) For all 119895 119895 are independent fuzzy random variableswhose mean 119882 and variance 2119882 are triangular fuzzynumbers

(iii) The time period is precise for all 119895 that is singleperiod

(iv) The decision makerrsquos order quantity 119876119895 is optimal forall products at the beginning of the period

(v) At the end of the season the leftover items aresalvaged

(vi) The lost sales penalty cost is zero

32 Single-Period Inventory Fuzzy Probabilistic Model(SPIFPM) The profit function119885119895 associated with each orderquantity 119876119895 and fuzzy demand 119895 is given as follows

119885119895 (119876119895 119895) = 119901119895min 119876119895 119895 + 119904119895max 119876119895 minus 119895 0minus 119888119895119876119895 119895 = 1 2 119896 (7)

Here two situations are explored to discuss the profit func-tion that is overstock situation (119895 le 119876119895) and understocksituation (119895 ge 119876119895) Then the above equation boils down tothe following form

119885119895 (119876119895 119895)=

119895 (119876119895 119895) = (119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895 119895 le 119876119895119895 (119876119895 119895) = (119901119895 minus 119888119895)119876119895 119895 ge 119876119895

(8)

where 119885119895 and 119885119895 are profit functions for each product inoverstock and understock situations respectively Due tofuzziness of demand 119895 we get either a fuzzy overstock profit119895(119876119895 119895) or a fuzzy understock profit 119895(119876119895 119895) Hencethe total expected profit 119895(119876119895 119895) corresponding to bothoverstock and understock situations is given as follows

119895 (119876119895 119895) = 119895 (119876119895 119895) cup 119895 (119876119895 119895) (9)

Advances in Fuzzy Systems 5

Thus a multi-item SPIFPM subject to a storage space chanceconstraint with a confidence level 120574 can be stated as follows

(SPIFPM) Maximize (1198761 1198762 119876119896)= 119896sum119895=1

119895 (119876119895 119895)

subject to ( sum119895=1

119886119895119876119895 le ) ⪰ 119876119895 ge 0for 119895 = 1 2 119896

(10)

In the above model we consider demand 119895 as a FRVpresent in the objective function and the total storage space are independent FRVs present in the constraint distributednormally and exponentially

4 Transformation Technique

41 Crisp Equivalent of the Fuzzy Objective Function InSPIM the defined resultant profit function is fuzzy since thecustomer demand for each product is imprecise Now our

objective is to convert the fuzzy model to crisp model Usingthe weighted function to the profit function we have

119895 (119876119895 119895) = 120596119895 (119876119895 119895) + (1 minus 120596) 119895 (119876119895 119895) 997904rArr119895 (119876119895 119895) = 120596[[

119896sum119895=1

(119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]] (11)

Here 119895 are FRVs whose mean 119898119863119895 are triangular fuzzynumbers To make it crisp apply the ranking function as119877(119898119863119895) = (119898119863119895 +2119898119863119895 +119898119863119895)4 So the above profit functionbecomes

119895 (119876119895 119895) = 120596[[[119896sum119895=1

(119901119895 minus 119904119895) (119898119863119895 + 2119898119863119895 + 119898119863119895)4

+ (119904119895 minus 119888119895)119876119895]]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]] (12)

Therefore the crisp equivalent of the objective function is asfollows

Maximize 120596[[[119896sum119895=1

(119901119895 minus 119904119895) (119898119863119895 + 2119898119863119895 + 119898119863119895)4 + (119904119895 minus 119888119895)119876119895]]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]] (13)

42 Crisp Equivalent of the Fuzzy Chance Constraint Thefollowing results are proved to convert the fuzzy chanceconstraint to a deterministic form

Theorem 13 (Case 1 ( is normally distributed fuzzy randomvariable)) If is a normally distributed FRV 119886119895 isin 119877and with predetermined confidence level 120574 the inequality(sum119896119895=1 119886119895119876119895 le ) ⪰ is equivalent to

119896sum119895=1

119886119895119876119895 le 120583119882lowast (120572) + 120590119882lowast (120572) 119865minus1 (1 minus 120574lowast (120572)) (14)

for 120572 isin [0 1] and 119865 is the cumulative distribution function of119873(0 1) distributionProof Let us consider the constraint of the model as follows

( 119896sum119895=1

119886119895119876119895 le ) ⪰ 119886119895 isin 119877 (15)

where is a normally distributed FRV whose mean andvariance are fuzzy numbers denoted by 119882 and 2119882 andtheir 120572-cuts are 119882[120572] = [120583119882lowast(120572) 120583lowast119882(120572)] and 2119882[120572] =[1205902119882lowast(120572) 1205902119882lowast(120572)] respectively The 120572-cut of is [120572] =[120574lowast(120572) 120574lowast(120572)]

Let sum119896119895=1 119886119895119876119895 = 119880 Here 119880 is a real number since 119886119895119895 = 1 2 119896 are real numbers Then the above inequalitycan be written as 119875(119880 le ) ⪰ Here is a FRV whosemean and variance are the fuzzy numbers 119882 and 2119882 whichcan be computed from the crisp random variable119882 that is119882 isin [120572] That means 120583119882 isin 119882[120572] and 1205902119882 isin 2119882[120572] whichsatisfy (1radic2120587120590119882) intinfinminusinfin 119890minus(119882minus120583119882)221205902119882119889119882 = 1

Now the 120572-cut of the probability (sum119896119895=1 119886119895119876119895 le ) is ( 119896sum119895=1

119886119895119876119895 le ) [120572] = 119875 (119880 le 119882) | 120583119882isin 119882 [120572] 1205902119882 isin 2119882 [120572] = 1 minus 119875 (119882 le 119880) | 120583119882

6 Advances in Fuzzy Systems

isin 119882 [120572] 1205902119882 isin 2119882 [120572] = 1minus 1radic2120587120590119882 int

119880

minusinfin119890minus(119882minus120583119882)221205902119882119889119882 | 120583119882

isin 119882 [120572] 1205902119882 isin 2119882 [120572] (16)

Let 119881 = (119882 minus 120583119882)120590119882 Then

( 119896sum119895=1

119886119895119876119895 le ) [120572] = 1minus 1radic2120587 int

(119880minus120583119882)120590119882

minusinfin119890minus11988122119889119881 | 120583119882 isin 119882 [120572] 1205902119908

isin 2119908 [120572] = 1 minus 119865(119880 minus 120583119882120590119882 ) | 120583119882isin 119882 [120572] 1205902119882 isin 2119882 [120572]

(17)

where 119865 is the cumulative distribution function of 119873(0 1)distribution which is an increasing function So

min1 minus 119865(119880 minus 120583119882120590119882 ) | 120583119882 isin 119882 [120572] 1205902119882isin 2119882 [120572] = 1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) )

max1 minus 119865(119880 minus 120583119882120590119882 ) | 120583119882 isin 119882 [120572] 1205902119882isin 2119882 [120572] = 1 minus 119865(119880 minus 120583lowast119882 (120572)120590lowast119882 (120572) )

(18)

So

( 119896sum119895=1

119886119895119876119895 le ) [120572]= [1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) 1 minus 119865(119880 minus 120583lowast119882 (120572)120590lowast119882 (120572) )]

(19)

Using the Inequality law due toNanda et al [9] the constraint(sum119896119895=1 119886119895119876119895 le ) ⪰ becomes

1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) ge 120574lowast (120572) 997904rArr119896sum119895=1

119886119895119876119895 le 120583119882lowast (120572) + 120590119882lowast (120572) 119865minus1 (1 minus 120574lowast (120572)) (20)

Hence the proof is complete

Therefore finally the SPIFPM is transformed to anequivalent single-period inventory crisp model (SPICM) asfollows

(SPICM) Maximize 120596[[[119896sum119895=1

(119901119895 minus 119904119895) (119898119863119895 + 2119898119863119895 + 119898119863119895)4 + (119904119895 minus 119888119895)119876119895]]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]]Subject to

119896sum119895=1

119886119895119876119895 le 120583119882lowast (120572) + 120590119882lowast (120572) 119865minus1 (1 minus 120574lowast (120572))119876119895 ge 0for 120572 isin [0 1] 119895 = 1 2 119896

(21)

Theorem 14 (Case 2 ( follows fuzzy exponential distribu-tion)) If is an exponentially distributed FRV with fuzzyparameter 119886119895 isin 119877 then (sum119896119895=1 119886119895119876119895 le ) ⪰ is equivalentto 119890minus120582lowast(120572)119880 ge 120574lowast(120572) for each 120572 isin [0 1] where [120572] =[120582lowast(120572) 120582lowast(120572)]Proof The 120572-cut of the fuzzy probability (sum119896119895=1 119886119895119876119895 le )is

( 119896sum119895=1

119886119895119876119895 le ) [120572] = 119875 (119880 le 119882) | 119882 isin [120572]= intinfin119880120582119890minus120582119882119889119882 | 120582 isin [120572]

= 119890minus120582119880 | 120582lowast (120572) le 120582 le 120582lowast (120572) = [119875lowast (120572) 119875lowast (120572)](say)

(22)

where 119875lowast(120572) and 119875lowast(120572) are the solutions of the optimizationproblems

minimize120582lowast(120572)le120582le120582

lowast(120572)119890minus120582119880

maximize120582lowast(120572)le120582le120582

lowast(120572)119890minus120582119880 (23)

respectively

Advances in Fuzzy Systems 7

Here 119890minus120582119880 attains its minimum at 120582lowast(120572) and maximumat 120582lowast(120572) since 119890minus120582119880 is a decreasing function of 120582 That ismin 119890minus120582119880 = 120582lowast(120572) and max 119890minus120582119880 = 120582lowast(120572)

Hence

( 119896sum119895=1

119886119895119876119895 le ) [120572] = [119890minus120582lowast(120572)119880 119890minus120582lowast(120572)119880] (24)

Therefore the deterministic equivalent of the fuzzy chanceconstraint

( 119896sum119895=1

119886119895119876119895 le ) ⪰ becomes 119890minus120582lowast(120572)119880 ge 120574lowast (120572)

(25)

5 Numerical Example

Let us consider a SPIFPM to illustrate the effectiveness of theabove approach with the following data

(SPIFPM) Maximize [[2sum119895=1

(119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895]]cup [[2sum119895=1

(119901119895 minus 119888119895)119876119895]]Subject to (41198761 oplus 31198762 le ) ⪰ 05

1198761 ge 01198762 ge 0

(26)

where the other data for multi-items is given in Table 1Here 1 2 are independent normally distributed FRVs

with mean 1198631 1198632 being fuzzy numbers Also areindependent normally distributed FRVs with mean 119882 andvariance 2119882 as fuzzy numbers and 05 is also a fuzzy numberAll these fuzzy numbers are assumed to be linear triangularas follows 1198631 = ⟨200300400⟩ 1198632 = ⟨700800900⟩119882 = ⟨200020102020⟩ 2119882 = ⟨300400500⟩ and 05 =⟨040506⟩51 Solution Initially applying the weighted sum method tothe objective function we can write

Maximize (1198761 1198762)= 120596[[

2sum119895=1

(119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895]]+ (1 minus 120596)[[

2sum119895=1

(119901119895 minus 119888119895)119876119895]] (27)

Table 1

Product name 119888119895 119901119895 119904119895 119886119895Product 1 30 34 28 4Product 2 40 43 35 3

Let 120596 = 06 and using the above cost data to the aboveexpression we have

Maximize (1198761 1198762)= 041198761 minus 181198762 + 361 + 482 (28)

Here 1 2 are FRVs whose mean 1198631 1198632 are triangularfuzzy numbers with the 120572-cuts being

1198631 [120572] = [200 + 100120572 400 minus 100120572] 1198632 [120572] = [700 + 100120572 900 minus 100120572] (29)

Applying the ranking function to the above objective functionwe have

Maximize (1198761 1198762)= 041198761 minus 181198762 + 36119877 (1198631)+ 48119877 (1198632)

(30)

Hence the crisp equivalent of the objective function takes thefollowing form

Maximize 041198761 minus 181198762 + 4920 (31)

Now let us consider the constraint (41198761 + 31198762 le ) ⪰ 05We can convert this constraint to its deterministic form as follows normal and exponential distribution

Case 1 ( is normally distributed fuzzy random variable)As the mean 119882 variance 2119882 and 05 are triangular fuzzynumbers their 120572-cuts are

119882 [120572] = [2000 + 10120572 2020 minus 10120572] 2119882 = [300 + 100120572 500 minus 100120572]

05 [120572] = [04 + 120572 06 minus 120572] (32)

Let 41198761 +31198762 = 119880 So the constraint is given by (119880 le ) ⪰05 Using the above theorem result the 120572-cut of (119880 le )becomes

(119880 le ) [120572]= [1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) 1 minus 119865(119880 minus 120583lowast119882 (120572)120590lowast119882 (120572) )] (33)

So the crisp conversion of the constraint is

1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) ge 06 minus 01120572 (34)

8 Advances in Fuzzy Systems

which can be further simplified as

119880 minus 120583119882lowast (120572) le 120590119882lowast (120572) 119865minus1 (06 + 01120572) 997904rArr41198761 + 31198762 le 5 + 120572 + (radic3 + 120572)119865minus1 (04 + 01120572) (35)

Hence the crisp equivalent that is SPICM of the originalSPIFPM takes the following form

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to 41198761 + 31198762

le 5 + 120572+ (radic3 + 120572)119865minus1 (04 + 01120572)

1198761 ge 01198762 ge 0

(36)

Now 119865minus1(04 + 01120572) can be found from the normal distri-bution table For each 120572 there exists a solution of the aboveproblem For example if 120572 = 04 then SPICM reduces to thefollowing nonlinear programming problem

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to (41198761 + 31198762 minus 2004)2

le 7752341198761 ge 01198762 ge 0

(37)

Using any optimization software its local solution can bedetermined Here we have solved this by using LINGO andits solution is1198761 = 5016959 and1198762 = 0 and the correspond-ing objective value is 5120678 Each value of 120572 gives a corres-ponding solution

Case 2 ( follows fuzzy exponential distribution) Let bean exponential FRV with the parameter = ⟨1000 15002000⟩ having 120572-cut as [120572] = [1000+500120572 2000minus500120572] Let41198761+31198762 = 119880 So the constraint is given by (119880 le ) ⪰ 05Using theorem the crisp conversion of the constraint is

exp (minus (2000 minus 500120572) (41198761 + 31198762)) ge 06 minus 01120572 (38)

Hence the crisp equivalent that is SPICM of the originalSPIFPM takes the following form

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to exp (minus (2000 minus 500120572) (41198761 + 31198762))

ge 06 minus 011205721198761 ge 01198762 ge 0

(39)

For each 120572 there exists a solution of the above problem Forexample if 120572 = 04 then SPICM reduces to the followingnonlinear programming problem

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to exp (minus1800 (41198761 + 31198762))

ge 0561198761 ge 01198762 ge 0

(40)

Using any optimization software its local solution can bedetermined Here we have solved this by using LINGO andits solution is 1198761 = 08053035119864 minus 04 and 1198762 = 0 and thecorresponding objective value is 4920 Each value of 120572 givesa corresponding solution

6 Conclusion

This paper presents a solution procedure for solving SPIFPMin which the the storage space in the constraint is a FRVconsidered under two cases that is normal distributionand exponential distribution Also the demand is a FRVwhose mean and variance are fuzzy numbers in the objectivefunction of the given model Initially some theorems areproposed to transform the fuzzy model into an equivalentcrisp model Then the resultant model is solved by using theLINGO software When the storage space in the constraintis normally distributed then the optimal solution is 1198761 =5016959 and 1198762 = 0 and the corresponding objectivevalue is 5120678 and for exponentially distributed randomvariable its solution is 1198761 = 08053035119864 minus 04 and 1198762 = 0and the corresponding objective value is 4920 for one valueof 120572 However more approximate solution can be obtainedby taking more numbers of values of 120572 For the futurescope of the present work triangular fuzzy number can bereplaced by other fuzzy numbers and the demand for eachitems can be considered as other types of nonindependentFRVs instead of normally distributed independent FRVsA multiconstrained optimization model can be developedof our present work and our model can be extended toother mathematical models To analyze the effect of changesin the optimal solution with respect to change in variousparameters sensitivity analysis can be carried out

Notations

119876119895 Order quantity for item 119895119895 Demand as a fuzzy random variable for item 119895119888119895 Purchase cost per unit of item 119895119901119895 Selling price per unit of item 119895119904119895 Salvage value per unit of item 119895119886119895 Area required for each unit of item 119895119895 Fuzzy profit function for product 119895 Fuzzy total profit function for all products Available storage space a fuzzy random variable

Advances in Fuzzy Systems 9

Competing Interests

The authors declare that they have no competing interests

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] J J Buckley Fuzzy Probabilities New Approach and Applica-tions Physica Heidelberg Germany 2003

[3] H Kwakernaak ldquoFuzzy random variablesmdashI definitions andtheoremsrdquo Information Sciences vol 15 no 1 pp 1ndash29 1978

[4] H-J Zimmermann ldquoFuzzymathematical programmingrdquoCom-puters amp Operations Research vol 10 no 4 pp 291ndash298 1983

[5] S Nanda and K Kar ldquoConvex fuzzy mappingsrdquo Fuzzy Sets andSystems vol 48 no 1 pp 129ndash132 1992

[6] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986

[7] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy models forthe newsboy problemrdquo International Journal of Production Eco-nomics vol 45 no 1ndash3 pp 435ndash441 1996

[8] K Das T K Roy and M Maiti ldquoMulti-item stochastic andfuzzy-stochastic inventory models under two restrictionsrdquoComputers and Operations Research vol 31 no 11 pp 1793ndash1806 2004

[9] S Nanda G Panda and J K Dash ldquoA new methodology forcrisp equivalent of fuzzy chance constrained programmingproblemrdquo Fuzzy Optimization and Decision Making vol 7 no1 pp 59ndash74 2008

[10] N K Mahapatra and M Maiti ldquoDecision process for multiob-jective multi-item production-inventory system via interactivefuzzy satisficing techniquerdquo Computers amp Mathematics withApplications vol 49 no 5-6 pp 805ndash821 2005

[11] H-C Chang J-S Yao and L-Y Ouyang ldquoFuzzy mixtureinventory model involving fuzzy random variable lead timedemand and fuzzy total demandrdquo European Journal of Opera-tional Research vol 169 no 1 pp 65ndash80 2006

[12] Z Shao and X Ji ldquoFuzzy multi-product constraint newsboyproblemrdquoAppliedMathematics and Computation vol 180 no 1pp 7ndash15 2006

[13] C-M Lee ldquoA Bayesian approach to determine the value ofinformation in the newsboy problemrdquo International Journal ofProduction Economics vol 112 no 1 pp 391ndash402 2008

[14] J C Hayya T P Harrison and D C Chatfield ldquoA solutionfor the intractable inventory model when both demand andlead time are stochasticrdquo International Journal of ProductionEconomics vol 122 no 2 pp 595ndash605 2009

[15] J K Dash G Panda and S Nanda ldquoChance constrained prog-ramming problem under different fuzzy distributionsrdquo Inter-national Journal of Optimization Theory Methods and Applica-tions vol 1 no 1 pp 58ndash71 2009

[16] J-S Hu H Zheng R-Q Xu Y-P Ji and C-Y Guo ldquoSupplychain coordination for fuzzy random newsboy problem withimperfect qualityrdquo International Journal of Approximate Reason-ing vol 51 no 7 pp 771ndash784 2010

[17] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[18] R-H Su D-Y Yang and W L Pearn ldquoDecision-making ina single-period inventory environment with fuzzy demandrdquo

Expert Systems with Applications vol 38 no 3 pp 1909ndash19162011

[19] K A M Kotb and H A Fergany ldquoMulti-item EOQmodel withboth demand-dependent unit cost and varying leading time viageometric programmingrdquoAppliedMathematics vol 2 no 5 pp551ndash555 2011

[20] O Dey and D Chakraborty ldquoA fuzzy random continuousreview inventory systemrdquo International Journal of ProductionEconomics vol 132 no 1 pp 101ndash106 2011

[21] R Banerjee and S Banerjee ldquoSolution of a probabilistic inven-tory model with chance constraints a general fuzzy program-ming and intuitionistic fuzzy optimization approachrdquo Interna-tional Journal of Pure and Applied Sciences and Technology vol9 no 1 pp 20ndash38 2012

[22] M Nagare and P Dutta ldquoOn solving single-period inventorymodel under hybrid uncertaintyrdquo International Journal of Eco-nomics and Management Sciences vol 6 no 4 pp 290ndash2952012

[23] M A Jahantigh S Khezerloo A A Hosseinzadeh and MKhezerloo ldquoSolutions of fuzzy linear systems using rankingfunctionrdquo International Journal of AppliedOperational Researchvol 2 pp 67ndash75 2012

[24] A Karpagam and P Sumathi ldquoNew approach to solve fuzzy lin-ear programming problems by the ranking functionrdquo BonfringInternational Journal of Data Mining vol 4 no 4 pp 22ndash252014

[25] D Dutta and P Kumar ldquoFuzzy inventory model without short-age using trapezoidal fuzzy number with sensitivity analysisrdquoIOSR Journal of Mathematics vol 4 no 3 pp 32ndash37 2012

[26] S Ding ldquoUncertain multi-product newsboy problem withchance constraintrdquo Applied Mathematics and Computation vol223 pp 139ndash146 2013

[27] H Zolfagharinia and K P S Isotupa ldquoErratum on lsquoOptimalinventory control of empty containers in inland transportationsystemrsquo Int J Prod Econ 133(1) (2011) 451ndash457rdquo InternationalJournal of Production Economics vol 141 no 1 pp 43ndash436 2013

[28] P Majumder U K Bera and M Maiti ldquoAn EPQ model ofdeteriorating items under partial trade credit financing anddemand declining market in crisp and fuzzy environmentrdquoProcedia Computer Science vol 45 pp 780ndash789 2015

[29] S M Mousavi J Sadeghi S T A Niaki N Alikar A Bahre-ininejad and H S C Metselaar ldquoTwo parameter-tuned meta-heuristics for a discounted inventory control problem in a fuzzyenvironmentrdquo Information Sciences vol 276 pp 42ndash62 2014

[30] H N Soni and M Joshi ldquoA periodic review inventory modelwith controllable lead time and backorder rate in fuzzy-stocha-stic environmentrdquo Fuzzy Information and Engineering vol 7 no1 pp 101ndash114 2015

[31] R S Kumar and A Goswami ldquoA fuzzy random EPQ modelfor imperfect quality items with possibility and necessity con-straintsrdquo Applied Soft Computing vol 34 pp 838ndash850 2015

[32] N Kazemi E Shekarian L E C Barron and E U OluguldquoIncorporating human learning into a fuzzy EOQ inventorymodel with backordersrdquo Computers amp Industrial Engineeringvol 87 pp 540ndash542 2015

[33] J K Dash and A Sahoo ldquoOptimal solution for a single periodinventorymodel with fuzzy cost and demand as a fuzzy randomvariablerdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 28 no 3 pp 1195ndash1203 2015

[34] M Nagare P Dutta and N Cheikhrouhou ldquoOptimal orderingpolicy for newsvendor models with bidirectional changes in

10 Advances in Fuzzy Systems

demand using expert judgmentrdquoOPSEARCH vol 53 no 3 pp620ndash647 2016

[35] P Raula S K Indrajitsingha P N Samanta and U K Misra ldquoAFuzzy InventoryModel for constant deteriorating item by usingGMIRmethod in which inventory parameters treated as HFNrdquoOpen Journal of Applied andTheoretical Mathematics (OJATM)vol 2 no 1 pp 13ndash20 2016

[36] I Sangal A Agarwal and S Rani ldquoA fuzzy environment inven-tory model with partial backlogging under learning effectrdquoInternational Journal of Computer Applications vol 137 no 6pp 25ndash32 2016

Submit your manuscripts athttpwwwhindawicom

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Human-ComputerInteraction

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4 Advances in Fuzzy Systems

so (1radic2120587120590) intinfinminusinfin

119890minus(119909minus119898)221205902119889119909 = 1 Now the fuzzy proba-bility of the FRV on the interval [119888 119889] is a fuzzy numberwhose 120572-cut is (119888 le le 119889) [120572] = 1radic2120587120590 int

119889

119888119890minus(119909minus119898)221205902119889119909 | 119898

isin [120572] 1205902 isin 2 [120572] = 1radic2120587 int1199112

1199111

119890minus11991122119889119911 | 119898isin [120572] 1205902 isin 2 [120572]

(4)

where 1199111 = (119888 minus 119898)120590 and 1199112 = (119889 minus 119898)120590Definition 11 (fuzzy exponential distribution [15]) Let be aexponentially distributed fuzzy random variable with densityfunction 119891(119909 ) being a fuzzy set and its 120572-cut is

119891 (119909 ) [120572] = 119890minus119909 | 120582 isin (120572) (5)

Now the fuzzy probability of on the interval [119888 119889] is a fuzzynumber and denoted by (119888 le le 119889) where 119875lowast(120572) =minint119889

119888120582119890minus120582119909119889119909 | 120582 isin [120572] and 119875lowast(120572) = maxint119889

119888120582119890minus120582119909119889119909 |120582 isin [120572]

Definition 12 (Fuzzy Chance Constrained Programmingproblem (FCCP) [26]) A general chance constraint pro-gramming problem CCP is of the following form

(CCP) Optimize119896sum119895=1

119888119895119909119895Subject to 119875( 119896sum

119895=1

119886119894119895119909119895 le 119887119894) ⪰ 120574119894119896sum119895=1

119887119897119895119909119895 ge ℎ119897119909119895 ge 0119887119897119895 ℎ119897 isin 119877119894 = 1 2 119898119897 = 1 2 1198990 le 120574119894 le 1

(6)

where at least one of 119886119894119895 119888119895 and 119887119894 is treated as randomvariable

So FCCP is a CCP given as above form where at least oneof 119886119894119895 119888119895 and 119887119894 is treated as FRV

3 Mathematical Formulation

Notations used in the proposed model are presented inNotations section

31 Assumptions We imposed the following assumptions todevelop the model

(i) For all 119895 119895 are independent fuzzy random variableswhose mean 119863 are triangular fuzzy numbers

(ii) For all 119895 119895 are independent fuzzy random variableswhose mean 119882 and variance 2119882 are triangular fuzzynumbers

(iii) The time period is precise for all 119895 that is singleperiod

(iv) The decision makerrsquos order quantity 119876119895 is optimal forall products at the beginning of the period

(v) At the end of the season the leftover items aresalvaged

(vi) The lost sales penalty cost is zero

32 Single-Period Inventory Fuzzy Probabilistic Model(SPIFPM) The profit function119885119895 associated with each orderquantity 119876119895 and fuzzy demand 119895 is given as follows

119885119895 (119876119895 119895) = 119901119895min 119876119895 119895 + 119904119895max 119876119895 minus 119895 0minus 119888119895119876119895 119895 = 1 2 119896 (7)

Here two situations are explored to discuss the profit func-tion that is overstock situation (119895 le 119876119895) and understocksituation (119895 ge 119876119895) Then the above equation boils down tothe following form

119885119895 (119876119895 119895)=

119895 (119876119895 119895) = (119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895 119895 le 119876119895119895 (119876119895 119895) = (119901119895 minus 119888119895)119876119895 119895 ge 119876119895

(8)

where 119885119895 and 119885119895 are profit functions for each product inoverstock and understock situations respectively Due tofuzziness of demand 119895 we get either a fuzzy overstock profit119895(119876119895 119895) or a fuzzy understock profit 119895(119876119895 119895) Hencethe total expected profit 119895(119876119895 119895) corresponding to bothoverstock and understock situations is given as follows

119895 (119876119895 119895) = 119895 (119876119895 119895) cup 119895 (119876119895 119895) (9)

Advances in Fuzzy Systems 5

Thus a multi-item SPIFPM subject to a storage space chanceconstraint with a confidence level 120574 can be stated as follows

(SPIFPM) Maximize (1198761 1198762 119876119896)= 119896sum119895=1

119895 (119876119895 119895)

subject to ( sum119895=1

119886119895119876119895 le ) ⪰ 119876119895 ge 0for 119895 = 1 2 119896

(10)

In the above model we consider demand 119895 as a FRVpresent in the objective function and the total storage space are independent FRVs present in the constraint distributednormally and exponentially

4 Transformation Technique

41 Crisp Equivalent of the Fuzzy Objective Function InSPIM the defined resultant profit function is fuzzy since thecustomer demand for each product is imprecise Now our

objective is to convert the fuzzy model to crisp model Usingthe weighted function to the profit function we have

119895 (119876119895 119895) = 120596119895 (119876119895 119895) + (1 minus 120596) 119895 (119876119895 119895) 997904rArr119895 (119876119895 119895) = 120596[[

119896sum119895=1

(119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]] (11)

Here 119895 are FRVs whose mean 119898119863119895 are triangular fuzzynumbers To make it crisp apply the ranking function as119877(119898119863119895) = (119898119863119895 +2119898119863119895 +119898119863119895)4 So the above profit functionbecomes

119895 (119876119895 119895) = 120596[[[119896sum119895=1

(119901119895 minus 119904119895) (119898119863119895 + 2119898119863119895 + 119898119863119895)4

+ (119904119895 minus 119888119895)119876119895]]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]] (12)

Therefore the crisp equivalent of the objective function is asfollows

Maximize 120596[[[119896sum119895=1

(119901119895 minus 119904119895) (119898119863119895 + 2119898119863119895 + 119898119863119895)4 + (119904119895 minus 119888119895)119876119895]]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]] (13)

42 Crisp Equivalent of the Fuzzy Chance Constraint Thefollowing results are proved to convert the fuzzy chanceconstraint to a deterministic form

Theorem 13 (Case 1 ( is normally distributed fuzzy randomvariable)) If is a normally distributed FRV 119886119895 isin 119877and with predetermined confidence level 120574 the inequality(sum119896119895=1 119886119895119876119895 le ) ⪰ is equivalent to

119896sum119895=1

119886119895119876119895 le 120583119882lowast (120572) + 120590119882lowast (120572) 119865minus1 (1 minus 120574lowast (120572)) (14)

for 120572 isin [0 1] and 119865 is the cumulative distribution function of119873(0 1) distributionProof Let us consider the constraint of the model as follows

( 119896sum119895=1

119886119895119876119895 le ) ⪰ 119886119895 isin 119877 (15)

where is a normally distributed FRV whose mean andvariance are fuzzy numbers denoted by 119882 and 2119882 andtheir 120572-cuts are 119882[120572] = [120583119882lowast(120572) 120583lowast119882(120572)] and 2119882[120572] =[1205902119882lowast(120572) 1205902119882lowast(120572)] respectively The 120572-cut of is [120572] =[120574lowast(120572) 120574lowast(120572)]

Let sum119896119895=1 119886119895119876119895 = 119880 Here 119880 is a real number since 119886119895119895 = 1 2 119896 are real numbers Then the above inequalitycan be written as 119875(119880 le ) ⪰ Here is a FRV whosemean and variance are the fuzzy numbers 119882 and 2119882 whichcan be computed from the crisp random variable119882 that is119882 isin [120572] That means 120583119882 isin 119882[120572] and 1205902119882 isin 2119882[120572] whichsatisfy (1radic2120587120590119882) intinfinminusinfin 119890minus(119882minus120583119882)221205902119882119889119882 = 1

Now the 120572-cut of the probability (sum119896119895=1 119886119895119876119895 le ) is ( 119896sum119895=1

119886119895119876119895 le ) [120572] = 119875 (119880 le 119882) | 120583119882isin 119882 [120572] 1205902119882 isin 2119882 [120572] = 1 minus 119875 (119882 le 119880) | 120583119882

6 Advances in Fuzzy Systems

isin 119882 [120572] 1205902119882 isin 2119882 [120572] = 1minus 1radic2120587120590119882 int

119880

minusinfin119890minus(119882minus120583119882)221205902119882119889119882 | 120583119882

isin 119882 [120572] 1205902119882 isin 2119882 [120572] (16)

Let 119881 = (119882 minus 120583119882)120590119882 Then

( 119896sum119895=1

119886119895119876119895 le ) [120572] = 1minus 1radic2120587 int

(119880minus120583119882)120590119882

minusinfin119890minus11988122119889119881 | 120583119882 isin 119882 [120572] 1205902119908

isin 2119908 [120572] = 1 minus 119865(119880 minus 120583119882120590119882 ) | 120583119882isin 119882 [120572] 1205902119882 isin 2119882 [120572]

(17)

where 119865 is the cumulative distribution function of 119873(0 1)distribution which is an increasing function So

min1 minus 119865(119880 minus 120583119882120590119882 ) | 120583119882 isin 119882 [120572] 1205902119882isin 2119882 [120572] = 1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) )

max1 minus 119865(119880 minus 120583119882120590119882 ) | 120583119882 isin 119882 [120572] 1205902119882isin 2119882 [120572] = 1 minus 119865(119880 minus 120583lowast119882 (120572)120590lowast119882 (120572) )

(18)

So

( 119896sum119895=1

119886119895119876119895 le ) [120572]= [1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) 1 minus 119865(119880 minus 120583lowast119882 (120572)120590lowast119882 (120572) )]

(19)

Using the Inequality law due toNanda et al [9] the constraint(sum119896119895=1 119886119895119876119895 le ) ⪰ becomes

1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) ge 120574lowast (120572) 997904rArr119896sum119895=1

119886119895119876119895 le 120583119882lowast (120572) + 120590119882lowast (120572) 119865minus1 (1 minus 120574lowast (120572)) (20)

Hence the proof is complete

Therefore finally the SPIFPM is transformed to anequivalent single-period inventory crisp model (SPICM) asfollows

(SPICM) Maximize 120596[[[119896sum119895=1

(119901119895 minus 119904119895) (119898119863119895 + 2119898119863119895 + 119898119863119895)4 + (119904119895 minus 119888119895)119876119895]]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]]Subject to

119896sum119895=1

119886119895119876119895 le 120583119882lowast (120572) + 120590119882lowast (120572) 119865minus1 (1 minus 120574lowast (120572))119876119895 ge 0for 120572 isin [0 1] 119895 = 1 2 119896

(21)

Theorem 14 (Case 2 ( follows fuzzy exponential distribu-tion)) If is an exponentially distributed FRV with fuzzyparameter 119886119895 isin 119877 then (sum119896119895=1 119886119895119876119895 le ) ⪰ is equivalentto 119890minus120582lowast(120572)119880 ge 120574lowast(120572) for each 120572 isin [0 1] where [120572] =[120582lowast(120572) 120582lowast(120572)]Proof The 120572-cut of the fuzzy probability (sum119896119895=1 119886119895119876119895 le )is

( 119896sum119895=1

119886119895119876119895 le ) [120572] = 119875 (119880 le 119882) | 119882 isin [120572]= intinfin119880120582119890minus120582119882119889119882 | 120582 isin [120572]

= 119890minus120582119880 | 120582lowast (120572) le 120582 le 120582lowast (120572) = [119875lowast (120572) 119875lowast (120572)](say)

(22)

where 119875lowast(120572) and 119875lowast(120572) are the solutions of the optimizationproblems

minimize120582lowast(120572)le120582le120582

lowast(120572)119890minus120582119880

maximize120582lowast(120572)le120582le120582

lowast(120572)119890minus120582119880 (23)

respectively

Advances in Fuzzy Systems 7

Here 119890minus120582119880 attains its minimum at 120582lowast(120572) and maximumat 120582lowast(120572) since 119890minus120582119880 is a decreasing function of 120582 That ismin 119890minus120582119880 = 120582lowast(120572) and max 119890minus120582119880 = 120582lowast(120572)

Hence

( 119896sum119895=1

119886119895119876119895 le ) [120572] = [119890minus120582lowast(120572)119880 119890minus120582lowast(120572)119880] (24)

Therefore the deterministic equivalent of the fuzzy chanceconstraint

( 119896sum119895=1

119886119895119876119895 le ) ⪰ becomes 119890minus120582lowast(120572)119880 ge 120574lowast (120572)

(25)

5 Numerical Example

Let us consider a SPIFPM to illustrate the effectiveness of theabove approach with the following data

(SPIFPM) Maximize [[2sum119895=1

(119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895]]cup [[2sum119895=1

(119901119895 minus 119888119895)119876119895]]Subject to (41198761 oplus 31198762 le ) ⪰ 05

1198761 ge 01198762 ge 0

(26)

where the other data for multi-items is given in Table 1Here 1 2 are independent normally distributed FRVs

with mean 1198631 1198632 being fuzzy numbers Also areindependent normally distributed FRVs with mean 119882 andvariance 2119882 as fuzzy numbers and 05 is also a fuzzy numberAll these fuzzy numbers are assumed to be linear triangularas follows 1198631 = ⟨200300400⟩ 1198632 = ⟨700800900⟩119882 = ⟨200020102020⟩ 2119882 = ⟨300400500⟩ and 05 =⟨040506⟩51 Solution Initially applying the weighted sum method tothe objective function we can write

Maximize (1198761 1198762)= 120596[[

2sum119895=1

(119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895]]+ (1 minus 120596)[[

2sum119895=1

(119901119895 minus 119888119895)119876119895]] (27)

Table 1

Product name 119888119895 119901119895 119904119895 119886119895Product 1 30 34 28 4Product 2 40 43 35 3

Let 120596 = 06 and using the above cost data to the aboveexpression we have

Maximize (1198761 1198762)= 041198761 minus 181198762 + 361 + 482 (28)

Here 1 2 are FRVs whose mean 1198631 1198632 are triangularfuzzy numbers with the 120572-cuts being

1198631 [120572] = [200 + 100120572 400 minus 100120572] 1198632 [120572] = [700 + 100120572 900 minus 100120572] (29)

Applying the ranking function to the above objective functionwe have

Maximize (1198761 1198762)= 041198761 minus 181198762 + 36119877 (1198631)+ 48119877 (1198632)

(30)

Hence the crisp equivalent of the objective function takes thefollowing form

Maximize 041198761 minus 181198762 + 4920 (31)

Now let us consider the constraint (41198761 + 31198762 le ) ⪰ 05We can convert this constraint to its deterministic form as follows normal and exponential distribution

Case 1 ( is normally distributed fuzzy random variable)As the mean 119882 variance 2119882 and 05 are triangular fuzzynumbers their 120572-cuts are

119882 [120572] = [2000 + 10120572 2020 minus 10120572] 2119882 = [300 + 100120572 500 minus 100120572]

05 [120572] = [04 + 120572 06 minus 120572] (32)

Let 41198761 +31198762 = 119880 So the constraint is given by (119880 le ) ⪰05 Using the above theorem result the 120572-cut of (119880 le )becomes

(119880 le ) [120572]= [1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) 1 minus 119865(119880 minus 120583lowast119882 (120572)120590lowast119882 (120572) )] (33)

So the crisp conversion of the constraint is

1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) ge 06 minus 01120572 (34)

8 Advances in Fuzzy Systems

which can be further simplified as

119880 minus 120583119882lowast (120572) le 120590119882lowast (120572) 119865minus1 (06 + 01120572) 997904rArr41198761 + 31198762 le 5 + 120572 + (radic3 + 120572)119865minus1 (04 + 01120572) (35)

Hence the crisp equivalent that is SPICM of the originalSPIFPM takes the following form

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to 41198761 + 31198762

le 5 + 120572+ (radic3 + 120572)119865minus1 (04 + 01120572)

1198761 ge 01198762 ge 0

(36)

Now 119865minus1(04 + 01120572) can be found from the normal distri-bution table For each 120572 there exists a solution of the aboveproblem For example if 120572 = 04 then SPICM reduces to thefollowing nonlinear programming problem

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to (41198761 + 31198762 minus 2004)2

le 7752341198761 ge 01198762 ge 0

(37)

Using any optimization software its local solution can bedetermined Here we have solved this by using LINGO andits solution is1198761 = 5016959 and1198762 = 0 and the correspond-ing objective value is 5120678 Each value of 120572 gives a corres-ponding solution

Case 2 ( follows fuzzy exponential distribution) Let bean exponential FRV with the parameter = ⟨1000 15002000⟩ having 120572-cut as [120572] = [1000+500120572 2000minus500120572] Let41198761+31198762 = 119880 So the constraint is given by (119880 le ) ⪰ 05Using theorem the crisp conversion of the constraint is

exp (minus (2000 minus 500120572) (41198761 + 31198762)) ge 06 minus 01120572 (38)

Hence the crisp equivalent that is SPICM of the originalSPIFPM takes the following form

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to exp (minus (2000 minus 500120572) (41198761 + 31198762))

ge 06 minus 011205721198761 ge 01198762 ge 0

(39)

For each 120572 there exists a solution of the above problem Forexample if 120572 = 04 then SPICM reduces to the followingnonlinear programming problem

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to exp (minus1800 (41198761 + 31198762))

ge 0561198761 ge 01198762 ge 0

(40)

Using any optimization software its local solution can bedetermined Here we have solved this by using LINGO andits solution is 1198761 = 08053035119864 minus 04 and 1198762 = 0 and thecorresponding objective value is 4920 Each value of 120572 givesa corresponding solution

6 Conclusion

This paper presents a solution procedure for solving SPIFPMin which the the storage space in the constraint is a FRVconsidered under two cases that is normal distributionand exponential distribution Also the demand is a FRVwhose mean and variance are fuzzy numbers in the objectivefunction of the given model Initially some theorems areproposed to transform the fuzzy model into an equivalentcrisp model Then the resultant model is solved by using theLINGO software When the storage space in the constraintis normally distributed then the optimal solution is 1198761 =5016959 and 1198762 = 0 and the corresponding objectivevalue is 5120678 and for exponentially distributed randomvariable its solution is 1198761 = 08053035119864 minus 04 and 1198762 = 0and the corresponding objective value is 4920 for one valueof 120572 However more approximate solution can be obtainedby taking more numbers of values of 120572 For the futurescope of the present work triangular fuzzy number can bereplaced by other fuzzy numbers and the demand for eachitems can be considered as other types of nonindependentFRVs instead of normally distributed independent FRVsA multiconstrained optimization model can be developedof our present work and our model can be extended toother mathematical models To analyze the effect of changesin the optimal solution with respect to change in variousparameters sensitivity analysis can be carried out

Notations

119876119895 Order quantity for item 119895119895 Demand as a fuzzy random variable for item 119895119888119895 Purchase cost per unit of item 119895119901119895 Selling price per unit of item 119895119904119895 Salvage value per unit of item 119895119886119895 Area required for each unit of item 119895119895 Fuzzy profit function for product 119895 Fuzzy total profit function for all products Available storage space a fuzzy random variable

Advances in Fuzzy Systems 9

Competing Interests

The authors declare that they have no competing interests

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] J J Buckley Fuzzy Probabilities New Approach and Applica-tions Physica Heidelberg Germany 2003

[3] H Kwakernaak ldquoFuzzy random variablesmdashI definitions andtheoremsrdquo Information Sciences vol 15 no 1 pp 1ndash29 1978

[4] H-J Zimmermann ldquoFuzzymathematical programmingrdquoCom-puters amp Operations Research vol 10 no 4 pp 291ndash298 1983

[5] S Nanda and K Kar ldquoConvex fuzzy mappingsrdquo Fuzzy Sets andSystems vol 48 no 1 pp 129ndash132 1992

[6] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986

[7] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy models forthe newsboy problemrdquo International Journal of Production Eco-nomics vol 45 no 1ndash3 pp 435ndash441 1996

[8] K Das T K Roy and M Maiti ldquoMulti-item stochastic andfuzzy-stochastic inventory models under two restrictionsrdquoComputers and Operations Research vol 31 no 11 pp 1793ndash1806 2004

[9] S Nanda G Panda and J K Dash ldquoA new methodology forcrisp equivalent of fuzzy chance constrained programmingproblemrdquo Fuzzy Optimization and Decision Making vol 7 no1 pp 59ndash74 2008

[10] N K Mahapatra and M Maiti ldquoDecision process for multiob-jective multi-item production-inventory system via interactivefuzzy satisficing techniquerdquo Computers amp Mathematics withApplications vol 49 no 5-6 pp 805ndash821 2005

[11] H-C Chang J-S Yao and L-Y Ouyang ldquoFuzzy mixtureinventory model involving fuzzy random variable lead timedemand and fuzzy total demandrdquo European Journal of Opera-tional Research vol 169 no 1 pp 65ndash80 2006

[12] Z Shao and X Ji ldquoFuzzy multi-product constraint newsboyproblemrdquoAppliedMathematics and Computation vol 180 no 1pp 7ndash15 2006

[13] C-M Lee ldquoA Bayesian approach to determine the value ofinformation in the newsboy problemrdquo International Journal ofProduction Economics vol 112 no 1 pp 391ndash402 2008

[14] J C Hayya T P Harrison and D C Chatfield ldquoA solutionfor the intractable inventory model when both demand andlead time are stochasticrdquo International Journal of ProductionEconomics vol 122 no 2 pp 595ndash605 2009

[15] J K Dash G Panda and S Nanda ldquoChance constrained prog-ramming problem under different fuzzy distributionsrdquo Inter-national Journal of Optimization Theory Methods and Applica-tions vol 1 no 1 pp 58ndash71 2009

[16] J-S Hu H Zheng R-Q Xu Y-P Ji and C-Y Guo ldquoSupplychain coordination for fuzzy random newsboy problem withimperfect qualityrdquo International Journal of Approximate Reason-ing vol 51 no 7 pp 771ndash784 2010

[17] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[18] R-H Su D-Y Yang and W L Pearn ldquoDecision-making ina single-period inventory environment with fuzzy demandrdquo

Expert Systems with Applications vol 38 no 3 pp 1909ndash19162011

[19] K A M Kotb and H A Fergany ldquoMulti-item EOQmodel withboth demand-dependent unit cost and varying leading time viageometric programmingrdquoAppliedMathematics vol 2 no 5 pp551ndash555 2011

[20] O Dey and D Chakraborty ldquoA fuzzy random continuousreview inventory systemrdquo International Journal of ProductionEconomics vol 132 no 1 pp 101ndash106 2011

[21] R Banerjee and S Banerjee ldquoSolution of a probabilistic inven-tory model with chance constraints a general fuzzy program-ming and intuitionistic fuzzy optimization approachrdquo Interna-tional Journal of Pure and Applied Sciences and Technology vol9 no 1 pp 20ndash38 2012

[22] M Nagare and P Dutta ldquoOn solving single-period inventorymodel under hybrid uncertaintyrdquo International Journal of Eco-nomics and Management Sciences vol 6 no 4 pp 290ndash2952012

[23] M A Jahantigh S Khezerloo A A Hosseinzadeh and MKhezerloo ldquoSolutions of fuzzy linear systems using rankingfunctionrdquo International Journal of AppliedOperational Researchvol 2 pp 67ndash75 2012

[24] A Karpagam and P Sumathi ldquoNew approach to solve fuzzy lin-ear programming problems by the ranking functionrdquo BonfringInternational Journal of Data Mining vol 4 no 4 pp 22ndash252014

[25] D Dutta and P Kumar ldquoFuzzy inventory model without short-age using trapezoidal fuzzy number with sensitivity analysisrdquoIOSR Journal of Mathematics vol 4 no 3 pp 32ndash37 2012

[26] S Ding ldquoUncertain multi-product newsboy problem withchance constraintrdquo Applied Mathematics and Computation vol223 pp 139ndash146 2013

[27] H Zolfagharinia and K P S Isotupa ldquoErratum on lsquoOptimalinventory control of empty containers in inland transportationsystemrsquo Int J Prod Econ 133(1) (2011) 451ndash457rdquo InternationalJournal of Production Economics vol 141 no 1 pp 43ndash436 2013

[28] P Majumder U K Bera and M Maiti ldquoAn EPQ model ofdeteriorating items under partial trade credit financing anddemand declining market in crisp and fuzzy environmentrdquoProcedia Computer Science vol 45 pp 780ndash789 2015

[29] S M Mousavi J Sadeghi S T A Niaki N Alikar A Bahre-ininejad and H S C Metselaar ldquoTwo parameter-tuned meta-heuristics for a discounted inventory control problem in a fuzzyenvironmentrdquo Information Sciences vol 276 pp 42ndash62 2014

[30] H N Soni and M Joshi ldquoA periodic review inventory modelwith controllable lead time and backorder rate in fuzzy-stocha-stic environmentrdquo Fuzzy Information and Engineering vol 7 no1 pp 101ndash114 2015

[31] R S Kumar and A Goswami ldquoA fuzzy random EPQ modelfor imperfect quality items with possibility and necessity con-straintsrdquo Applied Soft Computing vol 34 pp 838ndash850 2015

[32] N Kazemi E Shekarian L E C Barron and E U OluguldquoIncorporating human learning into a fuzzy EOQ inventorymodel with backordersrdquo Computers amp Industrial Engineeringvol 87 pp 540ndash542 2015

[33] J K Dash and A Sahoo ldquoOptimal solution for a single periodinventorymodel with fuzzy cost and demand as a fuzzy randomvariablerdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 28 no 3 pp 1195ndash1203 2015

[34] M Nagare P Dutta and N Cheikhrouhou ldquoOptimal orderingpolicy for newsvendor models with bidirectional changes in

10 Advances in Fuzzy Systems

demand using expert judgmentrdquoOPSEARCH vol 53 no 3 pp620ndash647 2016

[35] P Raula S K Indrajitsingha P N Samanta and U K Misra ldquoAFuzzy InventoryModel for constant deteriorating item by usingGMIRmethod in which inventory parameters treated as HFNrdquoOpen Journal of Applied andTheoretical Mathematics (OJATM)vol 2 no 1 pp 13ndash20 2016

[36] I Sangal A Agarwal and S Rani ldquoA fuzzy environment inven-tory model with partial backlogging under learning effectrdquoInternational Journal of Computer Applications vol 137 no 6pp 25ndash32 2016

Submit your manuscripts athttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Fuzzy Systems 5

Thus a multi-item SPIFPM subject to a storage space chanceconstraint with a confidence level 120574 can be stated as follows

(SPIFPM) Maximize (1198761 1198762 119876119896)= 119896sum119895=1

119895 (119876119895 119895)

subject to ( sum119895=1

119886119895119876119895 le ) ⪰ 119876119895 ge 0for 119895 = 1 2 119896

(10)

In the above model we consider demand 119895 as a FRVpresent in the objective function and the total storage space are independent FRVs present in the constraint distributednormally and exponentially

4 Transformation Technique

41 Crisp Equivalent of the Fuzzy Objective Function InSPIM the defined resultant profit function is fuzzy since thecustomer demand for each product is imprecise Now our

objective is to convert the fuzzy model to crisp model Usingthe weighted function to the profit function we have

119895 (119876119895 119895) = 120596119895 (119876119895 119895) + (1 minus 120596) 119895 (119876119895 119895) 997904rArr119895 (119876119895 119895) = 120596[[

119896sum119895=1

(119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]] (11)

Here 119895 are FRVs whose mean 119898119863119895 are triangular fuzzynumbers To make it crisp apply the ranking function as119877(119898119863119895) = (119898119863119895 +2119898119863119895 +119898119863119895)4 So the above profit functionbecomes

119895 (119876119895 119895) = 120596[[[119896sum119895=1

(119901119895 minus 119904119895) (119898119863119895 + 2119898119863119895 + 119898119863119895)4

+ (119904119895 minus 119888119895)119876119895]]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]] (12)

Therefore the crisp equivalent of the objective function is asfollows

Maximize 120596[[[119896sum119895=1

(119901119895 minus 119904119895) (119898119863119895 + 2119898119863119895 + 119898119863119895)4 + (119904119895 minus 119888119895)119876119895]]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]] (13)

42 Crisp Equivalent of the Fuzzy Chance Constraint Thefollowing results are proved to convert the fuzzy chanceconstraint to a deterministic form

Theorem 13 (Case 1 ( is normally distributed fuzzy randomvariable)) If is a normally distributed FRV 119886119895 isin 119877and with predetermined confidence level 120574 the inequality(sum119896119895=1 119886119895119876119895 le ) ⪰ is equivalent to

119896sum119895=1

119886119895119876119895 le 120583119882lowast (120572) + 120590119882lowast (120572) 119865minus1 (1 minus 120574lowast (120572)) (14)

for 120572 isin [0 1] and 119865 is the cumulative distribution function of119873(0 1) distributionProof Let us consider the constraint of the model as follows

( 119896sum119895=1

119886119895119876119895 le ) ⪰ 119886119895 isin 119877 (15)

where is a normally distributed FRV whose mean andvariance are fuzzy numbers denoted by 119882 and 2119882 andtheir 120572-cuts are 119882[120572] = [120583119882lowast(120572) 120583lowast119882(120572)] and 2119882[120572] =[1205902119882lowast(120572) 1205902119882lowast(120572)] respectively The 120572-cut of is [120572] =[120574lowast(120572) 120574lowast(120572)]

Let sum119896119895=1 119886119895119876119895 = 119880 Here 119880 is a real number since 119886119895119895 = 1 2 119896 are real numbers Then the above inequalitycan be written as 119875(119880 le ) ⪰ Here is a FRV whosemean and variance are the fuzzy numbers 119882 and 2119882 whichcan be computed from the crisp random variable119882 that is119882 isin [120572] That means 120583119882 isin 119882[120572] and 1205902119882 isin 2119882[120572] whichsatisfy (1radic2120587120590119882) intinfinminusinfin 119890minus(119882minus120583119882)221205902119882119889119882 = 1

Now the 120572-cut of the probability (sum119896119895=1 119886119895119876119895 le ) is ( 119896sum119895=1

119886119895119876119895 le ) [120572] = 119875 (119880 le 119882) | 120583119882isin 119882 [120572] 1205902119882 isin 2119882 [120572] = 1 minus 119875 (119882 le 119880) | 120583119882

6 Advances in Fuzzy Systems

isin 119882 [120572] 1205902119882 isin 2119882 [120572] = 1minus 1radic2120587120590119882 int

119880

minusinfin119890minus(119882minus120583119882)221205902119882119889119882 | 120583119882

isin 119882 [120572] 1205902119882 isin 2119882 [120572] (16)

Let 119881 = (119882 minus 120583119882)120590119882 Then

( 119896sum119895=1

119886119895119876119895 le ) [120572] = 1minus 1radic2120587 int

(119880minus120583119882)120590119882

minusinfin119890minus11988122119889119881 | 120583119882 isin 119882 [120572] 1205902119908

isin 2119908 [120572] = 1 minus 119865(119880 minus 120583119882120590119882 ) | 120583119882isin 119882 [120572] 1205902119882 isin 2119882 [120572]

(17)

where 119865 is the cumulative distribution function of 119873(0 1)distribution which is an increasing function So

min1 minus 119865(119880 minus 120583119882120590119882 ) | 120583119882 isin 119882 [120572] 1205902119882isin 2119882 [120572] = 1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) )

max1 minus 119865(119880 minus 120583119882120590119882 ) | 120583119882 isin 119882 [120572] 1205902119882isin 2119882 [120572] = 1 minus 119865(119880 minus 120583lowast119882 (120572)120590lowast119882 (120572) )

(18)

So

( 119896sum119895=1

119886119895119876119895 le ) [120572]= [1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) 1 minus 119865(119880 minus 120583lowast119882 (120572)120590lowast119882 (120572) )]

(19)

Using the Inequality law due toNanda et al [9] the constraint(sum119896119895=1 119886119895119876119895 le ) ⪰ becomes

1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) ge 120574lowast (120572) 997904rArr119896sum119895=1

119886119895119876119895 le 120583119882lowast (120572) + 120590119882lowast (120572) 119865minus1 (1 minus 120574lowast (120572)) (20)

Hence the proof is complete

Therefore finally the SPIFPM is transformed to anequivalent single-period inventory crisp model (SPICM) asfollows

(SPICM) Maximize 120596[[[119896sum119895=1

(119901119895 minus 119904119895) (119898119863119895 + 2119898119863119895 + 119898119863119895)4 + (119904119895 minus 119888119895)119876119895]]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]]Subject to

119896sum119895=1

119886119895119876119895 le 120583119882lowast (120572) + 120590119882lowast (120572) 119865minus1 (1 minus 120574lowast (120572))119876119895 ge 0for 120572 isin [0 1] 119895 = 1 2 119896

(21)

Theorem 14 (Case 2 ( follows fuzzy exponential distribu-tion)) If is an exponentially distributed FRV with fuzzyparameter 119886119895 isin 119877 then (sum119896119895=1 119886119895119876119895 le ) ⪰ is equivalentto 119890minus120582lowast(120572)119880 ge 120574lowast(120572) for each 120572 isin [0 1] where [120572] =[120582lowast(120572) 120582lowast(120572)]Proof The 120572-cut of the fuzzy probability (sum119896119895=1 119886119895119876119895 le )is

( 119896sum119895=1

119886119895119876119895 le ) [120572] = 119875 (119880 le 119882) | 119882 isin [120572]= intinfin119880120582119890minus120582119882119889119882 | 120582 isin [120572]

= 119890minus120582119880 | 120582lowast (120572) le 120582 le 120582lowast (120572) = [119875lowast (120572) 119875lowast (120572)](say)

(22)

where 119875lowast(120572) and 119875lowast(120572) are the solutions of the optimizationproblems

minimize120582lowast(120572)le120582le120582

lowast(120572)119890minus120582119880

maximize120582lowast(120572)le120582le120582

lowast(120572)119890minus120582119880 (23)

respectively

Advances in Fuzzy Systems 7

Here 119890minus120582119880 attains its minimum at 120582lowast(120572) and maximumat 120582lowast(120572) since 119890minus120582119880 is a decreasing function of 120582 That ismin 119890minus120582119880 = 120582lowast(120572) and max 119890minus120582119880 = 120582lowast(120572)

Hence

( 119896sum119895=1

119886119895119876119895 le ) [120572] = [119890minus120582lowast(120572)119880 119890minus120582lowast(120572)119880] (24)

Therefore the deterministic equivalent of the fuzzy chanceconstraint

( 119896sum119895=1

119886119895119876119895 le ) ⪰ becomes 119890minus120582lowast(120572)119880 ge 120574lowast (120572)

(25)

5 Numerical Example

Let us consider a SPIFPM to illustrate the effectiveness of theabove approach with the following data

(SPIFPM) Maximize [[2sum119895=1

(119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895]]cup [[2sum119895=1

(119901119895 minus 119888119895)119876119895]]Subject to (41198761 oplus 31198762 le ) ⪰ 05

1198761 ge 01198762 ge 0

(26)

where the other data for multi-items is given in Table 1Here 1 2 are independent normally distributed FRVs

with mean 1198631 1198632 being fuzzy numbers Also areindependent normally distributed FRVs with mean 119882 andvariance 2119882 as fuzzy numbers and 05 is also a fuzzy numberAll these fuzzy numbers are assumed to be linear triangularas follows 1198631 = ⟨200300400⟩ 1198632 = ⟨700800900⟩119882 = ⟨200020102020⟩ 2119882 = ⟨300400500⟩ and 05 =⟨040506⟩51 Solution Initially applying the weighted sum method tothe objective function we can write

Maximize (1198761 1198762)= 120596[[

2sum119895=1

(119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895]]+ (1 minus 120596)[[

2sum119895=1

(119901119895 minus 119888119895)119876119895]] (27)

Table 1

Product name 119888119895 119901119895 119904119895 119886119895Product 1 30 34 28 4Product 2 40 43 35 3

Let 120596 = 06 and using the above cost data to the aboveexpression we have

Maximize (1198761 1198762)= 041198761 minus 181198762 + 361 + 482 (28)

Here 1 2 are FRVs whose mean 1198631 1198632 are triangularfuzzy numbers with the 120572-cuts being

1198631 [120572] = [200 + 100120572 400 minus 100120572] 1198632 [120572] = [700 + 100120572 900 minus 100120572] (29)

Applying the ranking function to the above objective functionwe have

Maximize (1198761 1198762)= 041198761 minus 181198762 + 36119877 (1198631)+ 48119877 (1198632)

(30)

Hence the crisp equivalent of the objective function takes thefollowing form

Maximize 041198761 minus 181198762 + 4920 (31)

Now let us consider the constraint (41198761 + 31198762 le ) ⪰ 05We can convert this constraint to its deterministic form as follows normal and exponential distribution

Case 1 ( is normally distributed fuzzy random variable)As the mean 119882 variance 2119882 and 05 are triangular fuzzynumbers their 120572-cuts are

119882 [120572] = [2000 + 10120572 2020 minus 10120572] 2119882 = [300 + 100120572 500 minus 100120572]

05 [120572] = [04 + 120572 06 minus 120572] (32)

Let 41198761 +31198762 = 119880 So the constraint is given by (119880 le ) ⪰05 Using the above theorem result the 120572-cut of (119880 le )becomes

(119880 le ) [120572]= [1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) 1 minus 119865(119880 minus 120583lowast119882 (120572)120590lowast119882 (120572) )] (33)

So the crisp conversion of the constraint is

1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) ge 06 minus 01120572 (34)

8 Advances in Fuzzy Systems

which can be further simplified as

119880 minus 120583119882lowast (120572) le 120590119882lowast (120572) 119865minus1 (06 + 01120572) 997904rArr41198761 + 31198762 le 5 + 120572 + (radic3 + 120572)119865minus1 (04 + 01120572) (35)

Hence the crisp equivalent that is SPICM of the originalSPIFPM takes the following form

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to 41198761 + 31198762

le 5 + 120572+ (radic3 + 120572)119865minus1 (04 + 01120572)

1198761 ge 01198762 ge 0

(36)

Now 119865minus1(04 + 01120572) can be found from the normal distri-bution table For each 120572 there exists a solution of the aboveproblem For example if 120572 = 04 then SPICM reduces to thefollowing nonlinear programming problem

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to (41198761 + 31198762 minus 2004)2

le 7752341198761 ge 01198762 ge 0

(37)

Using any optimization software its local solution can bedetermined Here we have solved this by using LINGO andits solution is1198761 = 5016959 and1198762 = 0 and the correspond-ing objective value is 5120678 Each value of 120572 gives a corres-ponding solution

Case 2 ( follows fuzzy exponential distribution) Let bean exponential FRV with the parameter = ⟨1000 15002000⟩ having 120572-cut as [120572] = [1000+500120572 2000minus500120572] Let41198761+31198762 = 119880 So the constraint is given by (119880 le ) ⪰ 05Using theorem the crisp conversion of the constraint is

exp (minus (2000 minus 500120572) (41198761 + 31198762)) ge 06 minus 01120572 (38)

Hence the crisp equivalent that is SPICM of the originalSPIFPM takes the following form

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to exp (minus (2000 minus 500120572) (41198761 + 31198762))

ge 06 minus 011205721198761 ge 01198762 ge 0

(39)

For each 120572 there exists a solution of the above problem Forexample if 120572 = 04 then SPICM reduces to the followingnonlinear programming problem

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to exp (minus1800 (41198761 + 31198762))

ge 0561198761 ge 01198762 ge 0

(40)

Using any optimization software its local solution can bedetermined Here we have solved this by using LINGO andits solution is 1198761 = 08053035119864 minus 04 and 1198762 = 0 and thecorresponding objective value is 4920 Each value of 120572 givesa corresponding solution

6 Conclusion

This paper presents a solution procedure for solving SPIFPMin which the the storage space in the constraint is a FRVconsidered under two cases that is normal distributionand exponential distribution Also the demand is a FRVwhose mean and variance are fuzzy numbers in the objectivefunction of the given model Initially some theorems areproposed to transform the fuzzy model into an equivalentcrisp model Then the resultant model is solved by using theLINGO software When the storage space in the constraintis normally distributed then the optimal solution is 1198761 =5016959 and 1198762 = 0 and the corresponding objectivevalue is 5120678 and for exponentially distributed randomvariable its solution is 1198761 = 08053035119864 minus 04 and 1198762 = 0and the corresponding objective value is 4920 for one valueof 120572 However more approximate solution can be obtainedby taking more numbers of values of 120572 For the futurescope of the present work triangular fuzzy number can bereplaced by other fuzzy numbers and the demand for eachitems can be considered as other types of nonindependentFRVs instead of normally distributed independent FRVsA multiconstrained optimization model can be developedof our present work and our model can be extended toother mathematical models To analyze the effect of changesin the optimal solution with respect to change in variousparameters sensitivity analysis can be carried out

Notations

119876119895 Order quantity for item 119895119895 Demand as a fuzzy random variable for item 119895119888119895 Purchase cost per unit of item 119895119901119895 Selling price per unit of item 119895119904119895 Salvage value per unit of item 119895119886119895 Area required for each unit of item 119895119895 Fuzzy profit function for product 119895 Fuzzy total profit function for all products Available storage space a fuzzy random variable

Advances in Fuzzy Systems 9

Competing Interests

The authors declare that they have no competing interests

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] J J Buckley Fuzzy Probabilities New Approach and Applica-tions Physica Heidelberg Germany 2003

[3] H Kwakernaak ldquoFuzzy random variablesmdashI definitions andtheoremsrdquo Information Sciences vol 15 no 1 pp 1ndash29 1978

[4] H-J Zimmermann ldquoFuzzymathematical programmingrdquoCom-puters amp Operations Research vol 10 no 4 pp 291ndash298 1983

[5] S Nanda and K Kar ldquoConvex fuzzy mappingsrdquo Fuzzy Sets andSystems vol 48 no 1 pp 129ndash132 1992

[6] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986

[7] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy models forthe newsboy problemrdquo International Journal of Production Eco-nomics vol 45 no 1ndash3 pp 435ndash441 1996

[8] K Das T K Roy and M Maiti ldquoMulti-item stochastic andfuzzy-stochastic inventory models under two restrictionsrdquoComputers and Operations Research vol 31 no 11 pp 1793ndash1806 2004

[9] S Nanda G Panda and J K Dash ldquoA new methodology forcrisp equivalent of fuzzy chance constrained programmingproblemrdquo Fuzzy Optimization and Decision Making vol 7 no1 pp 59ndash74 2008

[10] N K Mahapatra and M Maiti ldquoDecision process for multiob-jective multi-item production-inventory system via interactivefuzzy satisficing techniquerdquo Computers amp Mathematics withApplications vol 49 no 5-6 pp 805ndash821 2005

[11] H-C Chang J-S Yao and L-Y Ouyang ldquoFuzzy mixtureinventory model involving fuzzy random variable lead timedemand and fuzzy total demandrdquo European Journal of Opera-tional Research vol 169 no 1 pp 65ndash80 2006

[12] Z Shao and X Ji ldquoFuzzy multi-product constraint newsboyproblemrdquoAppliedMathematics and Computation vol 180 no 1pp 7ndash15 2006

[13] C-M Lee ldquoA Bayesian approach to determine the value ofinformation in the newsboy problemrdquo International Journal ofProduction Economics vol 112 no 1 pp 391ndash402 2008

[14] J C Hayya T P Harrison and D C Chatfield ldquoA solutionfor the intractable inventory model when both demand andlead time are stochasticrdquo International Journal of ProductionEconomics vol 122 no 2 pp 595ndash605 2009

[15] J K Dash G Panda and S Nanda ldquoChance constrained prog-ramming problem under different fuzzy distributionsrdquo Inter-national Journal of Optimization Theory Methods and Applica-tions vol 1 no 1 pp 58ndash71 2009

[16] J-S Hu H Zheng R-Q Xu Y-P Ji and C-Y Guo ldquoSupplychain coordination for fuzzy random newsboy problem withimperfect qualityrdquo International Journal of Approximate Reason-ing vol 51 no 7 pp 771ndash784 2010

[17] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[18] R-H Su D-Y Yang and W L Pearn ldquoDecision-making ina single-period inventory environment with fuzzy demandrdquo

Expert Systems with Applications vol 38 no 3 pp 1909ndash19162011

[19] K A M Kotb and H A Fergany ldquoMulti-item EOQmodel withboth demand-dependent unit cost and varying leading time viageometric programmingrdquoAppliedMathematics vol 2 no 5 pp551ndash555 2011

[20] O Dey and D Chakraborty ldquoA fuzzy random continuousreview inventory systemrdquo International Journal of ProductionEconomics vol 132 no 1 pp 101ndash106 2011

[21] R Banerjee and S Banerjee ldquoSolution of a probabilistic inven-tory model with chance constraints a general fuzzy program-ming and intuitionistic fuzzy optimization approachrdquo Interna-tional Journal of Pure and Applied Sciences and Technology vol9 no 1 pp 20ndash38 2012

[22] M Nagare and P Dutta ldquoOn solving single-period inventorymodel under hybrid uncertaintyrdquo International Journal of Eco-nomics and Management Sciences vol 6 no 4 pp 290ndash2952012

[23] M A Jahantigh S Khezerloo A A Hosseinzadeh and MKhezerloo ldquoSolutions of fuzzy linear systems using rankingfunctionrdquo International Journal of AppliedOperational Researchvol 2 pp 67ndash75 2012

[24] A Karpagam and P Sumathi ldquoNew approach to solve fuzzy lin-ear programming problems by the ranking functionrdquo BonfringInternational Journal of Data Mining vol 4 no 4 pp 22ndash252014

[25] D Dutta and P Kumar ldquoFuzzy inventory model without short-age using trapezoidal fuzzy number with sensitivity analysisrdquoIOSR Journal of Mathematics vol 4 no 3 pp 32ndash37 2012

[26] S Ding ldquoUncertain multi-product newsboy problem withchance constraintrdquo Applied Mathematics and Computation vol223 pp 139ndash146 2013

[27] H Zolfagharinia and K P S Isotupa ldquoErratum on lsquoOptimalinventory control of empty containers in inland transportationsystemrsquo Int J Prod Econ 133(1) (2011) 451ndash457rdquo InternationalJournal of Production Economics vol 141 no 1 pp 43ndash436 2013

[28] P Majumder U K Bera and M Maiti ldquoAn EPQ model ofdeteriorating items under partial trade credit financing anddemand declining market in crisp and fuzzy environmentrdquoProcedia Computer Science vol 45 pp 780ndash789 2015

[29] S M Mousavi J Sadeghi S T A Niaki N Alikar A Bahre-ininejad and H S C Metselaar ldquoTwo parameter-tuned meta-heuristics for a discounted inventory control problem in a fuzzyenvironmentrdquo Information Sciences vol 276 pp 42ndash62 2014

[30] H N Soni and M Joshi ldquoA periodic review inventory modelwith controllable lead time and backorder rate in fuzzy-stocha-stic environmentrdquo Fuzzy Information and Engineering vol 7 no1 pp 101ndash114 2015

[31] R S Kumar and A Goswami ldquoA fuzzy random EPQ modelfor imperfect quality items with possibility and necessity con-straintsrdquo Applied Soft Computing vol 34 pp 838ndash850 2015

[32] N Kazemi E Shekarian L E C Barron and E U OluguldquoIncorporating human learning into a fuzzy EOQ inventorymodel with backordersrdquo Computers amp Industrial Engineeringvol 87 pp 540ndash542 2015

[33] J K Dash and A Sahoo ldquoOptimal solution for a single periodinventorymodel with fuzzy cost and demand as a fuzzy randomvariablerdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 28 no 3 pp 1195ndash1203 2015

[34] M Nagare P Dutta and N Cheikhrouhou ldquoOptimal orderingpolicy for newsvendor models with bidirectional changes in

10 Advances in Fuzzy Systems

demand using expert judgmentrdquoOPSEARCH vol 53 no 3 pp620ndash647 2016

[35] P Raula S K Indrajitsingha P N Samanta and U K Misra ldquoAFuzzy InventoryModel for constant deteriorating item by usingGMIRmethod in which inventory parameters treated as HFNrdquoOpen Journal of Applied andTheoretical Mathematics (OJATM)vol 2 no 1 pp 13ndash20 2016

[36] I Sangal A Agarwal and S Rani ldquoA fuzzy environment inven-tory model with partial backlogging under learning effectrdquoInternational Journal of Computer Applications vol 137 no 6pp 25ndash32 2016

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 Advances in Fuzzy Systems

isin 119882 [120572] 1205902119882 isin 2119882 [120572] = 1minus 1radic2120587120590119882 int

119880

minusinfin119890minus(119882minus120583119882)221205902119882119889119882 | 120583119882

isin 119882 [120572] 1205902119882 isin 2119882 [120572] (16)

Let 119881 = (119882 minus 120583119882)120590119882 Then

( 119896sum119895=1

119886119895119876119895 le ) [120572] = 1minus 1radic2120587 int

(119880minus120583119882)120590119882

minusinfin119890minus11988122119889119881 | 120583119882 isin 119882 [120572] 1205902119908

isin 2119908 [120572] = 1 minus 119865(119880 minus 120583119882120590119882 ) | 120583119882isin 119882 [120572] 1205902119882 isin 2119882 [120572]

(17)

where 119865 is the cumulative distribution function of 119873(0 1)distribution which is an increasing function So

min1 minus 119865(119880 minus 120583119882120590119882 ) | 120583119882 isin 119882 [120572] 1205902119882isin 2119882 [120572] = 1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) )

max1 minus 119865(119880 minus 120583119882120590119882 ) | 120583119882 isin 119882 [120572] 1205902119882isin 2119882 [120572] = 1 minus 119865(119880 minus 120583lowast119882 (120572)120590lowast119882 (120572) )

(18)

So

( 119896sum119895=1

119886119895119876119895 le ) [120572]= [1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) 1 minus 119865(119880 minus 120583lowast119882 (120572)120590lowast119882 (120572) )]

(19)

Using the Inequality law due toNanda et al [9] the constraint(sum119896119895=1 119886119895119876119895 le ) ⪰ becomes

1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) ge 120574lowast (120572) 997904rArr119896sum119895=1

119886119895119876119895 le 120583119882lowast (120572) + 120590119882lowast (120572) 119865minus1 (1 minus 120574lowast (120572)) (20)

Hence the proof is complete

Therefore finally the SPIFPM is transformed to anequivalent single-period inventory crisp model (SPICM) asfollows

(SPICM) Maximize 120596[[[119896sum119895=1

(119901119895 minus 119904119895) (119898119863119895 + 2119898119863119895 + 119898119863119895)4 + (119904119895 minus 119888119895)119876119895]]]+ (1 minus 120596)[[

119896sum119895=1

(119901119895 minus 119888119895)119876119895]]Subject to

119896sum119895=1

119886119895119876119895 le 120583119882lowast (120572) + 120590119882lowast (120572) 119865minus1 (1 minus 120574lowast (120572))119876119895 ge 0for 120572 isin [0 1] 119895 = 1 2 119896

(21)

Theorem 14 (Case 2 ( follows fuzzy exponential distribu-tion)) If is an exponentially distributed FRV with fuzzyparameter 119886119895 isin 119877 then (sum119896119895=1 119886119895119876119895 le ) ⪰ is equivalentto 119890minus120582lowast(120572)119880 ge 120574lowast(120572) for each 120572 isin [0 1] where [120572] =[120582lowast(120572) 120582lowast(120572)]Proof The 120572-cut of the fuzzy probability (sum119896119895=1 119886119895119876119895 le )is

( 119896sum119895=1

119886119895119876119895 le ) [120572] = 119875 (119880 le 119882) | 119882 isin [120572]= intinfin119880120582119890minus120582119882119889119882 | 120582 isin [120572]

= 119890minus120582119880 | 120582lowast (120572) le 120582 le 120582lowast (120572) = [119875lowast (120572) 119875lowast (120572)](say)

(22)

where 119875lowast(120572) and 119875lowast(120572) are the solutions of the optimizationproblems

minimize120582lowast(120572)le120582le120582

lowast(120572)119890minus120582119880

maximize120582lowast(120572)le120582le120582

lowast(120572)119890minus120582119880 (23)

respectively

Advances in Fuzzy Systems 7

Here 119890minus120582119880 attains its minimum at 120582lowast(120572) and maximumat 120582lowast(120572) since 119890minus120582119880 is a decreasing function of 120582 That ismin 119890minus120582119880 = 120582lowast(120572) and max 119890minus120582119880 = 120582lowast(120572)

Hence

( 119896sum119895=1

119886119895119876119895 le ) [120572] = [119890minus120582lowast(120572)119880 119890minus120582lowast(120572)119880] (24)

Therefore the deterministic equivalent of the fuzzy chanceconstraint

( 119896sum119895=1

119886119895119876119895 le ) ⪰ becomes 119890minus120582lowast(120572)119880 ge 120574lowast (120572)

(25)

5 Numerical Example

Let us consider a SPIFPM to illustrate the effectiveness of theabove approach with the following data

(SPIFPM) Maximize [[2sum119895=1

(119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895]]cup [[2sum119895=1

(119901119895 minus 119888119895)119876119895]]Subject to (41198761 oplus 31198762 le ) ⪰ 05

1198761 ge 01198762 ge 0

(26)

where the other data for multi-items is given in Table 1Here 1 2 are independent normally distributed FRVs

with mean 1198631 1198632 being fuzzy numbers Also areindependent normally distributed FRVs with mean 119882 andvariance 2119882 as fuzzy numbers and 05 is also a fuzzy numberAll these fuzzy numbers are assumed to be linear triangularas follows 1198631 = ⟨200300400⟩ 1198632 = ⟨700800900⟩119882 = ⟨200020102020⟩ 2119882 = ⟨300400500⟩ and 05 =⟨040506⟩51 Solution Initially applying the weighted sum method tothe objective function we can write

Maximize (1198761 1198762)= 120596[[

2sum119895=1

(119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895]]+ (1 minus 120596)[[

2sum119895=1

(119901119895 minus 119888119895)119876119895]] (27)

Table 1

Product name 119888119895 119901119895 119904119895 119886119895Product 1 30 34 28 4Product 2 40 43 35 3

Let 120596 = 06 and using the above cost data to the aboveexpression we have

Maximize (1198761 1198762)= 041198761 minus 181198762 + 361 + 482 (28)

Here 1 2 are FRVs whose mean 1198631 1198632 are triangularfuzzy numbers with the 120572-cuts being

1198631 [120572] = [200 + 100120572 400 minus 100120572] 1198632 [120572] = [700 + 100120572 900 minus 100120572] (29)

Applying the ranking function to the above objective functionwe have

Maximize (1198761 1198762)= 041198761 minus 181198762 + 36119877 (1198631)+ 48119877 (1198632)

(30)

Hence the crisp equivalent of the objective function takes thefollowing form

Maximize 041198761 minus 181198762 + 4920 (31)

Now let us consider the constraint (41198761 + 31198762 le ) ⪰ 05We can convert this constraint to its deterministic form as follows normal and exponential distribution

Case 1 ( is normally distributed fuzzy random variable)As the mean 119882 variance 2119882 and 05 are triangular fuzzynumbers their 120572-cuts are

119882 [120572] = [2000 + 10120572 2020 minus 10120572] 2119882 = [300 + 100120572 500 minus 100120572]

05 [120572] = [04 + 120572 06 minus 120572] (32)

Let 41198761 +31198762 = 119880 So the constraint is given by (119880 le ) ⪰05 Using the above theorem result the 120572-cut of (119880 le )becomes

(119880 le ) [120572]= [1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) 1 minus 119865(119880 minus 120583lowast119882 (120572)120590lowast119882 (120572) )] (33)

So the crisp conversion of the constraint is

1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) ge 06 minus 01120572 (34)

8 Advances in Fuzzy Systems

which can be further simplified as

119880 minus 120583119882lowast (120572) le 120590119882lowast (120572) 119865minus1 (06 + 01120572) 997904rArr41198761 + 31198762 le 5 + 120572 + (radic3 + 120572)119865minus1 (04 + 01120572) (35)

Hence the crisp equivalent that is SPICM of the originalSPIFPM takes the following form

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to 41198761 + 31198762

le 5 + 120572+ (radic3 + 120572)119865minus1 (04 + 01120572)

1198761 ge 01198762 ge 0

(36)

Now 119865minus1(04 + 01120572) can be found from the normal distri-bution table For each 120572 there exists a solution of the aboveproblem For example if 120572 = 04 then SPICM reduces to thefollowing nonlinear programming problem

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to (41198761 + 31198762 minus 2004)2

le 7752341198761 ge 01198762 ge 0

(37)

Using any optimization software its local solution can bedetermined Here we have solved this by using LINGO andits solution is1198761 = 5016959 and1198762 = 0 and the correspond-ing objective value is 5120678 Each value of 120572 gives a corres-ponding solution

Case 2 ( follows fuzzy exponential distribution) Let bean exponential FRV with the parameter = ⟨1000 15002000⟩ having 120572-cut as [120572] = [1000+500120572 2000minus500120572] Let41198761+31198762 = 119880 So the constraint is given by (119880 le ) ⪰ 05Using theorem the crisp conversion of the constraint is

exp (minus (2000 minus 500120572) (41198761 + 31198762)) ge 06 minus 01120572 (38)

Hence the crisp equivalent that is SPICM of the originalSPIFPM takes the following form

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to exp (minus (2000 minus 500120572) (41198761 + 31198762))

ge 06 minus 011205721198761 ge 01198762 ge 0

(39)

For each 120572 there exists a solution of the above problem Forexample if 120572 = 04 then SPICM reduces to the followingnonlinear programming problem

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to exp (minus1800 (41198761 + 31198762))

ge 0561198761 ge 01198762 ge 0

(40)

Using any optimization software its local solution can bedetermined Here we have solved this by using LINGO andits solution is 1198761 = 08053035119864 minus 04 and 1198762 = 0 and thecorresponding objective value is 4920 Each value of 120572 givesa corresponding solution

6 Conclusion

This paper presents a solution procedure for solving SPIFPMin which the the storage space in the constraint is a FRVconsidered under two cases that is normal distributionand exponential distribution Also the demand is a FRVwhose mean and variance are fuzzy numbers in the objectivefunction of the given model Initially some theorems areproposed to transform the fuzzy model into an equivalentcrisp model Then the resultant model is solved by using theLINGO software When the storage space in the constraintis normally distributed then the optimal solution is 1198761 =5016959 and 1198762 = 0 and the corresponding objectivevalue is 5120678 and for exponentially distributed randomvariable its solution is 1198761 = 08053035119864 minus 04 and 1198762 = 0and the corresponding objective value is 4920 for one valueof 120572 However more approximate solution can be obtainedby taking more numbers of values of 120572 For the futurescope of the present work triangular fuzzy number can bereplaced by other fuzzy numbers and the demand for eachitems can be considered as other types of nonindependentFRVs instead of normally distributed independent FRVsA multiconstrained optimization model can be developedof our present work and our model can be extended toother mathematical models To analyze the effect of changesin the optimal solution with respect to change in variousparameters sensitivity analysis can be carried out

Notations

119876119895 Order quantity for item 119895119895 Demand as a fuzzy random variable for item 119895119888119895 Purchase cost per unit of item 119895119901119895 Selling price per unit of item 119895119904119895 Salvage value per unit of item 119895119886119895 Area required for each unit of item 119895119895 Fuzzy profit function for product 119895 Fuzzy total profit function for all products Available storage space a fuzzy random variable

Advances in Fuzzy Systems 9

Competing Interests

The authors declare that they have no competing interests

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] J J Buckley Fuzzy Probabilities New Approach and Applica-tions Physica Heidelberg Germany 2003

[3] H Kwakernaak ldquoFuzzy random variablesmdashI definitions andtheoremsrdquo Information Sciences vol 15 no 1 pp 1ndash29 1978

[4] H-J Zimmermann ldquoFuzzymathematical programmingrdquoCom-puters amp Operations Research vol 10 no 4 pp 291ndash298 1983

[5] S Nanda and K Kar ldquoConvex fuzzy mappingsrdquo Fuzzy Sets andSystems vol 48 no 1 pp 129ndash132 1992

[6] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986

[7] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy models forthe newsboy problemrdquo International Journal of Production Eco-nomics vol 45 no 1ndash3 pp 435ndash441 1996

[8] K Das T K Roy and M Maiti ldquoMulti-item stochastic andfuzzy-stochastic inventory models under two restrictionsrdquoComputers and Operations Research vol 31 no 11 pp 1793ndash1806 2004

[9] S Nanda G Panda and J K Dash ldquoA new methodology forcrisp equivalent of fuzzy chance constrained programmingproblemrdquo Fuzzy Optimization and Decision Making vol 7 no1 pp 59ndash74 2008

[10] N K Mahapatra and M Maiti ldquoDecision process for multiob-jective multi-item production-inventory system via interactivefuzzy satisficing techniquerdquo Computers amp Mathematics withApplications vol 49 no 5-6 pp 805ndash821 2005

[11] H-C Chang J-S Yao and L-Y Ouyang ldquoFuzzy mixtureinventory model involving fuzzy random variable lead timedemand and fuzzy total demandrdquo European Journal of Opera-tional Research vol 169 no 1 pp 65ndash80 2006

[12] Z Shao and X Ji ldquoFuzzy multi-product constraint newsboyproblemrdquoAppliedMathematics and Computation vol 180 no 1pp 7ndash15 2006

[13] C-M Lee ldquoA Bayesian approach to determine the value ofinformation in the newsboy problemrdquo International Journal ofProduction Economics vol 112 no 1 pp 391ndash402 2008

[14] J C Hayya T P Harrison and D C Chatfield ldquoA solutionfor the intractable inventory model when both demand andlead time are stochasticrdquo International Journal of ProductionEconomics vol 122 no 2 pp 595ndash605 2009

[15] J K Dash G Panda and S Nanda ldquoChance constrained prog-ramming problem under different fuzzy distributionsrdquo Inter-national Journal of Optimization Theory Methods and Applica-tions vol 1 no 1 pp 58ndash71 2009

[16] J-S Hu H Zheng R-Q Xu Y-P Ji and C-Y Guo ldquoSupplychain coordination for fuzzy random newsboy problem withimperfect qualityrdquo International Journal of Approximate Reason-ing vol 51 no 7 pp 771ndash784 2010

[17] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[18] R-H Su D-Y Yang and W L Pearn ldquoDecision-making ina single-period inventory environment with fuzzy demandrdquo

Expert Systems with Applications vol 38 no 3 pp 1909ndash19162011

[19] K A M Kotb and H A Fergany ldquoMulti-item EOQmodel withboth demand-dependent unit cost and varying leading time viageometric programmingrdquoAppliedMathematics vol 2 no 5 pp551ndash555 2011

[20] O Dey and D Chakraborty ldquoA fuzzy random continuousreview inventory systemrdquo International Journal of ProductionEconomics vol 132 no 1 pp 101ndash106 2011

[21] R Banerjee and S Banerjee ldquoSolution of a probabilistic inven-tory model with chance constraints a general fuzzy program-ming and intuitionistic fuzzy optimization approachrdquo Interna-tional Journal of Pure and Applied Sciences and Technology vol9 no 1 pp 20ndash38 2012

[22] M Nagare and P Dutta ldquoOn solving single-period inventorymodel under hybrid uncertaintyrdquo International Journal of Eco-nomics and Management Sciences vol 6 no 4 pp 290ndash2952012

[23] M A Jahantigh S Khezerloo A A Hosseinzadeh and MKhezerloo ldquoSolutions of fuzzy linear systems using rankingfunctionrdquo International Journal of AppliedOperational Researchvol 2 pp 67ndash75 2012

[24] A Karpagam and P Sumathi ldquoNew approach to solve fuzzy lin-ear programming problems by the ranking functionrdquo BonfringInternational Journal of Data Mining vol 4 no 4 pp 22ndash252014

[25] D Dutta and P Kumar ldquoFuzzy inventory model without short-age using trapezoidal fuzzy number with sensitivity analysisrdquoIOSR Journal of Mathematics vol 4 no 3 pp 32ndash37 2012

[26] S Ding ldquoUncertain multi-product newsboy problem withchance constraintrdquo Applied Mathematics and Computation vol223 pp 139ndash146 2013

[27] H Zolfagharinia and K P S Isotupa ldquoErratum on lsquoOptimalinventory control of empty containers in inland transportationsystemrsquo Int J Prod Econ 133(1) (2011) 451ndash457rdquo InternationalJournal of Production Economics vol 141 no 1 pp 43ndash436 2013

[28] P Majumder U K Bera and M Maiti ldquoAn EPQ model ofdeteriorating items under partial trade credit financing anddemand declining market in crisp and fuzzy environmentrdquoProcedia Computer Science vol 45 pp 780ndash789 2015

[29] S M Mousavi J Sadeghi S T A Niaki N Alikar A Bahre-ininejad and H S C Metselaar ldquoTwo parameter-tuned meta-heuristics for a discounted inventory control problem in a fuzzyenvironmentrdquo Information Sciences vol 276 pp 42ndash62 2014

[30] H N Soni and M Joshi ldquoA periodic review inventory modelwith controllable lead time and backorder rate in fuzzy-stocha-stic environmentrdquo Fuzzy Information and Engineering vol 7 no1 pp 101ndash114 2015

[31] R S Kumar and A Goswami ldquoA fuzzy random EPQ modelfor imperfect quality items with possibility and necessity con-straintsrdquo Applied Soft Computing vol 34 pp 838ndash850 2015

[32] N Kazemi E Shekarian L E C Barron and E U OluguldquoIncorporating human learning into a fuzzy EOQ inventorymodel with backordersrdquo Computers amp Industrial Engineeringvol 87 pp 540ndash542 2015

[33] J K Dash and A Sahoo ldquoOptimal solution for a single periodinventorymodel with fuzzy cost and demand as a fuzzy randomvariablerdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 28 no 3 pp 1195ndash1203 2015

[34] M Nagare P Dutta and N Cheikhrouhou ldquoOptimal orderingpolicy for newsvendor models with bidirectional changes in

10 Advances in Fuzzy Systems

demand using expert judgmentrdquoOPSEARCH vol 53 no 3 pp620ndash647 2016

[35] P Raula S K Indrajitsingha P N Samanta and U K Misra ldquoAFuzzy InventoryModel for constant deteriorating item by usingGMIRmethod in which inventory parameters treated as HFNrdquoOpen Journal of Applied andTheoretical Mathematics (OJATM)vol 2 no 1 pp 13ndash20 2016

[36] I Sangal A Agarwal and S Rani ldquoA fuzzy environment inven-tory model with partial backlogging under learning effectrdquoInternational Journal of Computer Applications vol 137 no 6pp 25ndash32 2016

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Fuzzy Systems 7

Here 119890minus120582119880 attains its minimum at 120582lowast(120572) and maximumat 120582lowast(120572) since 119890minus120582119880 is a decreasing function of 120582 That ismin 119890minus120582119880 = 120582lowast(120572) and max 119890minus120582119880 = 120582lowast(120572)

Hence

( 119896sum119895=1

119886119895119876119895 le ) [120572] = [119890minus120582lowast(120572)119880 119890minus120582lowast(120572)119880] (24)

Therefore the deterministic equivalent of the fuzzy chanceconstraint

( 119896sum119895=1

119886119895119876119895 le ) ⪰ becomes 119890minus120582lowast(120572)119880 ge 120574lowast (120572)

(25)

5 Numerical Example

Let us consider a SPIFPM to illustrate the effectiveness of theabove approach with the following data

(SPIFPM) Maximize [[2sum119895=1

(119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895]]cup [[2sum119895=1

(119901119895 minus 119888119895)119876119895]]Subject to (41198761 oplus 31198762 le ) ⪰ 05

1198761 ge 01198762 ge 0

(26)

where the other data for multi-items is given in Table 1Here 1 2 are independent normally distributed FRVs

with mean 1198631 1198632 being fuzzy numbers Also areindependent normally distributed FRVs with mean 119882 andvariance 2119882 as fuzzy numbers and 05 is also a fuzzy numberAll these fuzzy numbers are assumed to be linear triangularas follows 1198631 = ⟨200300400⟩ 1198632 = ⟨700800900⟩119882 = ⟨200020102020⟩ 2119882 = ⟨300400500⟩ and 05 =⟨040506⟩51 Solution Initially applying the weighted sum method tothe objective function we can write

Maximize (1198761 1198762)= 120596[[

2sum119895=1

(119901119895 minus 119904119895) 119895 + (119904119895 minus 119888119895)119876119895]]+ (1 minus 120596)[[

2sum119895=1

(119901119895 minus 119888119895)119876119895]] (27)

Table 1

Product name 119888119895 119901119895 119904119895 119886119895Product 1 30 34 28 4Product 2 40 43 35 3

Let 120596 = 06 and using the above cost data to the aboveexpression we have

Maximize (1198761 1198762)= 041198761 minus 181198762 + 361 + 482 (28)

Here 1 2 are FRVs whose mean 1198631 1198632 are triangularfuzzy numbers with the 120572-cuts being

1198631 [120572] = [200 + 100120572 400 minus 100120572] 1198632 [120572] = [700 + 100120572 900 minus 100120572] (29)

Applying the ranking function to the above objective functionwe have

Maximize (1198761 1198762)= 041198761 minus 181198762 + 36119877 (1198631)+ 48119877 (1198632)

(30)

Hence the crisp equivalent of the objective function takes thefollowing form

Maximize 041198761 minus 181198762 + 4920 (31)

Now let us consider the constraint (41198761 + 31198762 le ) ⪰ 05We can convert this constraint to its deterministic form as follows normal and exponential distribution

Case 1 ( is normally distributed fuzzy random variable)As the mean 119882 variance 2119882 and 05 are triangular fuzzynumbers their 120572-cuts are

119882 [120572] = [2000 + 10120572 2020 minus 10120572] 2119882 = [300 + 100120572 500 minus 100120572]

05 [120572] = [04 + 120572 06 minus 120572] (32)

Let 41198761 +31198762 = 119880 So the constraint is given by (119880 le ) ⪰05 Using the above theorem result the 120572-cut of (119880 le )becomes

(119880 le ) [120572]= [1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) 1 minus 119865(119880 minus 120583lowast119882 (120572)120590lowast119882 (120572) )] (33)

So the crisp conversion of the constraint is

1 minus 119865(119880 minus 120583119882lowast (120572)120590119882lowast (120572) ) ge 06 minus 01120572 (34)

8 Advances in Fuzzy Systems

which can be further simplified as

119880 minus 120583119882lowast (120572) le 120590119882lowast (120572) 119865minus1 (06 + 01120572) 997904rArr41198761 + 31198762 le 5 + 120572 + (radic3 + 120572)119865minus1 (04 + 01120572) (35)

Hence the crisp equivalent that is SPICM of the originalSPIFPM takes the following form

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to 41198761 + 31198762

le 5 + 120572+ (radic3 + 120572)119865minus1 (04 + 01120572)

1198761 ge 01198762 ge 0

(36)

Now 119865minus1(04 + 01120572) can be found from the normal distri-bution table For each 120572 there exists a solution of the aboveproblem For example if 120572 = 04 then SPICM reduces to thefollowing nonlinear programming problem

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to (41198761 + 31198762 minus 2004)2

le 7752341198761 ge 01198762 ge 0

(37)

Using any optimization software its local solution can bedetermined Here we have solved this by using LINGO andits solution is1198761 = 5016959 and1198762 = 0 and the correspond-ing objective value is 5120678 Each value of 120572 gives a corres-ponding solution

Case 2 ( follows fuzzy exponential distribution) Let bean exponential FRV with the parameter = ⟨1000 15002000⟩ having 120572-cut as [120572] = [1000+500120572 2000minus500120572] Let41198761+31198762 = 119880 So the constraint is given by (119880 le ) ⪰ 05Using theorem the crisp conversion of the constraint is

exp (minus (2000 minus 500120572) (41198761 + 31198762)) ge 06 minus 01120572 (38)

Hence the crisp equivalent that is SPICM of the originalSPIFPM takes the following form

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to exp (minus (2000 minus 500120572) (41198761 + 31198762))

ge 06 minus 011205721198761 ge 01198762 ge 0

(39)

For each 120572 there exists a solution of the above problem Forexample if 120572 = 04 then SPICM reduces to the followingnonlinear programming problem

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to exp (minus1800 (41198761 + 31198762))

ge 0561198761 ge 01198762 ge 0

(40)

Using any optimization software its local solution can bedetermined Here we have solved this by using LINGO andits solution is 1198761 = 08053035119864 minus 04 and 1198762 = 0 and thecorresponding objective value is 4920 Each value of 120572 givesa corresponding solution

6 Conclusion

This paper presents a solution procedure for solving SPIFPMin which the the storage space in the constraint is a FRVconsidered under two cases that is normal distributionand exponential distribution Also the demand is a FRVwhose mean and variance are fuzzy numbers in the objectivefunction of the given model Initially some theorems areproposed to transform the fuzzy model into an equivalentcrisp model Then the resultant model is solved by using theLINGO software When the storage space in the constraintis normally distributed then the optimal solution is 1198761 =5016959 and 1198762 = 0 and the corresponding objectivevalue is 5120678 and for exponentially distributed randomvariable its solution is 1198761 = 08053035119864 minus 04 and 1198762 = 0and the corresponding objective value is 4920 for one valueof 120572 However more approximate solution can be obtainedby taking more numbers of values of 120572 For the futurescope of the present work triangular fuzzy number can bereplaced by other fuzzy numbers and the demand for eachitems can be considered as other types of nonindependentFRVs instead of normally distributed independent FRVsA multiconstrained optimization model can be developedof our present work and our model can be extended toother mathematical models To analyze the effect of changesin the optimal solution with respect to change in variousparameters sensitivity analysis can be carried out

Notations

119876119895 Order quantity for item 119895119895 Demand as a fuzzy random variable for item 119895119888119895 Purchase cost per unit of item 119895119901119895 Selling price per unit of item 119895119904119895 Salvage value per unit of item 119895119886119895 Area required for each unit of item 119895119895 Fuzzy profit function for product 119895 Fuzzy total profit function for all products Available storage space a fuzzy random variable

Advances in Fuzzy Systems 9

Competing Interests

The authors declare that they have no competing interests

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] J J Buckley Fuzzy Probabilities New Approach and Applica-tions Physica Heidelberg Germany 2003

[3] H Kwakernaak ldquoFuzzy random variablesmdashI definitions andtheoremsrdquo Information Sciences vol 15 no 1 pp 1ndash29 1978

[4] H-J Zimmermann ldquoFuzzymathematical programmingrdquoCom-puters amp Operations Research vol 10 no 4 pp 291ndash298 1983

[5] S Nanda and K Kar ldquoConvex fuzzy mappingsrdquo Fuzzy Sets andSystems vol 48 no 1 pp 129ndash132 1992

[6] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986

[7] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy models forthe newsboy problemrdquo International Journal of Production Eco-nomics vol 45 no 1ndash3 pp 435ndash441 1996

[8] K Das T K Roy and M Maiti ldquoMulti-item stochastic andfuzzy-stochastic inventory models under two restrictionsrdquoComputers and Operations Research vol 31 no 11 pp 1793ndash1806 2004

[9] S Nanda G Panda and J K Dash ldquoA new methodology forcrisp equivalent of fuzzy chance constrained programmingproblemrdquo Fuzzy Optimization and Decision Making vol 7 no1 pp 59ndash74 2008

[10] N K Mahapatra and M Maiti ldquoDecision process for multiob-jective multi-item production-inventory system via interactivefuzzy satisficing techniquerdquo Computers amp Mathematics withApplications vol 49 no 5-6 pp 805ndash821 2005

[11] H-C Chang J-S Yao and L-Y Ouyang ldquoFuzzy mixtureinventory model involving fuzzy random variable lead timedemand and fuzzy total demandrdquo European Journal of Opera-tional Research vol 169 no 1 pp 65ndash80 2006

[12] Z Shao and X Ji ldquoFuzzy multi-product constraint newsboyproblemrdquoAppliedMathematics and Computation vol 180 no 1pp 7ndash15 2006

[13] C-M Lee ldquoA Bayesian approach to determine the value ofinformation in the newsboy problemrdquo International Journal ofProduction Economics vol 112 no 1 pp 391ndash402 2008

[14] J C Hayya T P Harrison and D C Chatfield ldquoA solutionfor the intractable inventory model when both demand andlead time are stochasticrdquo International Journal of ProductionEconomics vol 122 no 2 pp 595ndash605 2009

[15] J K Dash G Panda and S Nanda ldquoChance constrained prog-ramming problem under different fuzzy distributionsrdquo Inter-national Journal of Optimization Theory Methods and Applica-tions vol 1 no 1 pp 58ndash71 2009

[16] J-S Hu H Zheng R-Q Xu Y-P Ji and C-Y Guo ldquoSupplychain coordination for fuzzy random newsboy problem withimperfect qualityrdquo International Journal of Approximate Reason-ing vol 51 no 7 pp 771ndash784 2010

[17] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[18] R-H Su D-Y Yang and W L Pearn ldquoDecision-making ina single-period inventory environment with fuzzy demandrdquo

Expert Systems with Applications vol 38 no 3 pp 1909ndash19162011

[19] K A M Kotb and H A Fergany ldquoMulti-item EOQmodel withboth demand-dependent unit cost and varying leading time viageometric programmingrdquoAppliedMathematics vol 2 no 5 pp551ndash555 2011

[20] O Dey and D Chakraborty ldquoA fuzzy random continuousreview inventory systemrdquo International Journal of ProductionEconomics vol 132 no 1 pp 101ndash106 2011

[21] R Banerjee and S Banerjee ldquoSolution of a probabilistic inven-tory model with chance constraints a general fuzzy program-ming and intuitionistic fuzzy optimization approachrdquo Interna-tional Journal of Pure and Applied Sciences and Technology vol9 no 1 pp 20ndash38 2012

[22] M Nagare and P Dutta ldquoOn solving single-period inventorymodel under hybrid uncertaintyrdquo International Journal of Eco-nomics and Management Sciences vol 6 no 4 pp 290ndash2952012

[23] M A Jahantigh S Khezerloo A A Hosseinzadeh and MKhezerloo ldquoSolutions of fuzzy linear systems using rankingfunctionrdquo International Journal of AppliedOperational Researchvol 2 pp 67ndash75 2012

[24] A Karpagam and P Sumathi ldquoNew approach to solve fuzzy lin-ear programming problems by the ranking functionrdquo BonfringInternational Journal of Data Mining vol 4 no 4 pp 22ndash252014

[25] D Dutta and P Kumar ldquoFuzzy inventory model without short-age using trapezoidal fuzzy number with sensitivity analysisrdquoIOSR Journal of Mathematics vol 4 no 3 pp 32ndash37 2012

[26] S Ding ldquoUncertain multi-product newsboy problem withchance constraintrdquo Applied Mathematics and Computation vol223 pp 139ndash146 2013

[27] H Zolfagharinia and K P S Isotupa ldquoErratum on lsquoOptimalinventory control of empty containers in inland transportationsystemrsquo Int J Prod Econ 133(1) (2011) 451ndash457rdquo InternationalJournal of Production Economics vol 141 no 1 pp 43ndash436 2013

[28] P Majumder U K Bera and M Maiti ldquoAn EPQ model ofdeteriorating items under partial trade credit financing anddemand declining market in crisp and fuzzy environmentrdquoProcedia Computer Science vol 45 pp 780ndash789 2015

[29] S M Mousavi J Sadeghi S T A Niaki N Alikar A Bahre-ininejad and H S C Metselaar ldquoTwo parameter-tuned meta-heuristics for a discounted inventory control problem in a fuzzyenvironmentrdquo Information Sciences vol 276 pp 42ndash62 2014

[30] H N Soni and M Joshi ldquoA periodic review inventory modelwith controllable lead time and backorder rate in fuzzy-stocha-stic environmentrdquo Fuzzy Information and Engineering vol 7 no1 pp 101ndash114 2015

[31] R S Kumar and A Goswami ldquoA fuzzy random EPQ modelfor imperfect quality items with possibility and necessity con-straintsrdquo Applied Soft Computing vol 34 pp 838ndash850 2015

[32] N Kazemi E Shekarian L E C Barron and E U OluguldquoIncorporating human learning into a fuzzy EOQ inventorymodel with backordersrdquo Computers amp Industrial Engineeringvol 87 pp 540ndash542 2015

[33] J K Dash and A Sahoo ldquoOptimal solution for a single periodinventorymodel with fuzzy cost and demand as a fuzzy randomvariablerdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 28 no 3 pp 1195ndash1203 2015

[34] M Nagare P Dutta and N Cheikhrouhou ldquoOptimal orderingpolicy for newsvendor models with bidirectional changes in

10 Advances in Fuzzy Systems

demand using expert judgmentrdquoOPSEARCH vol 53 no 3 pp620ndash647 2016

[35] P Raula S K Indrajitsingha P N Samanta and U K Misra ldquoAFuzzy InventoryModel for constant deteriorating item by usingGMIRmethod in which inventory parameters treated as HFNrdquoOpen Journal of Applied andTheoretical Mathematics (OJATM)vol 2 no 1 pp 13ndash20 2016

[36] I Sangal A Agarwal and S Rani ldquoA fuzzy environment inven-tory model with partial backlogging under learning effectrdquoInternational Journal of Computer Applications vol 137 no 6pp 25ndash32 2016

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

8 Advances in Fuzzy Systems

which can be further simplified as

119880 minus 120583119882lowast (120572) le 120590119882lowast (120572) 119865minus1 (06 + 01120572) 997904rArr41198761 + 31198762 le 5 + 120572 + (radic3 + 120572)119865minus1 (04 + 01120572) (35)

Hence the crisp equivalent that is SPICM of the originalSPIFPM takes the following form

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to 41198761 + 31198762

le 5 + 120572+ (radic3 + 120572)119865minus1 (04 + 01120572)

1198761 ge 01198762 ge 0

(36)

Now 119865minus1(04 + 01120572) can be found from the normal distri-bution table For each 120572 there exists a solution of the aboveproblem For example if 120572 = 04 then SPICM reduces to thefollowing nonlinear programming problem

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to (41198761 + 31198762 minus 2004)2

le 7752341198761 ge 01198762 ge 0

(37)

Using any optimization software its local solution can bedetermined Here we have solved this by using LINGO andits solution is1198761 = 5016959 and1198762 = 0 and the correspond-ing objective value is 5120678 Each value of 120572 gives a corres-ponding solution

Case 2 ( follows fuzzy exponential distribution) Let bean exponential FRV with the parameter = ⟨1000 15002000⟩ having 120572-cut as [120572] = [1000+500120572 2000minus500120572] Let41198761+31198762 = 119880 So the constraint is given by (119880 le ) ⪰ 05Using theorem the crisp conversion of the constraint is

exp (minus (2000 minus 500120572) (41198761 + 31198762)) ge 06 minus 01120572 (38)

Hence the crisp equivalent that is SPICM of the originalSPIFPM takes the following form

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to exp (minus (2000 minus 500120572) (41198761 + 31198762))

ge 06 minus 011205721198761 ge 01198762 ge 0

(39)

For each 120572 there exists a solution of the above problem Forexample if 120572 = 04 then SPICM reduces to the followingnonlinear programming problem

(SPICM) Maximize 041198761 minus 181198762 + 4920Subject to exp (minus1800 (41198761 + 31198762))

ge 0561198761 ge 01198762 ge 0

(40)

Using any optimization software its local solution can bedetermined Here we have solved this by using LINGO andits solution is 1198761 = 08053035119864 minus 04 and 1198762 = 0 and thecorresponding objective value is 4920 Each value of 120572 givesa corresponding solution

6 Conclusion

This paper presents a solution procedure for solving SPIFPMin which the the storage space in the constraint is a FRVconsidered under two cases that is normal distributionand exponential distribution Also the demand is a FRVwhose mean and variance are fuzzy numbers in the objectivefunction of the given model Initially some theorems areproposed to transform the fuzzy model into an equivalentcrisp model Then the resultant model is solved by using theLINGO software When the storage space in the constraintis normally distributed then the optimal solution is 1198761 =5016959 and 1198762 = 0 and the corresponding objectivevalue is 5120678 and for exponentially distributed randomvariable its solution is 1198761 = 08053035119864 minus 04 and 1198762 = 0and the corresponding objective value is 4920 for one valueof 120572 However more approximate solution can be obtainedby taking more numbers of values of 120572 For the futurescope of the present work triangular fuzzy number can bereplaced by other fuzzy numbers and the demand for eachitems can be considered as other types of nonindependentFRVs instead of normally distributed independent FRVsA multiconstrained optimization model can be developedof our present work and our model can be extended toother mathematical models To analyze the effect of changesin the optimal solution with respect to change in variousparameters sensitivity analysis can be carried out

Notations

119876119895 Order quantity for item 119895119895 Demand as a fuzzy random variable for item 119895119888119895 Purchase cost per unit of item 119895119901119895 Selling price per unit of item 119895119904119895 Salvage value per unit of item 119895119886119895 Area required for each unit of item 119895119895 Fuzzy profit function for product 119895 Fuzzy total profit function for all products Available storage space a fuzzy random variable

Advances in Fuzzy Systems 9

Competing Interests

The authors declare that they have no competing interests

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] J J Buckley Fuzzy Probabilities New Approach and Applica-tions Physica Heidelberg Germany 2003

[3] H Kwakernaak ldquoFuzzy random variablesmdashI definitions andtheoremsrdquo Information Sciences vol 15 no 1 pp 1ndash29 1978

[4] H-J Zimmermann ldquoFuzzymathematical programmingrdquoCom-puters amp Operations Research vol 10 no 4 pp 291ndash298 1983

[5] S Nanda and K Kar ldquoConvex fuzzy mappingsrdquo Fuzzy Sets andSystems vol 48 no 1 pp 129ndash132 1992

[6] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986

[7] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy models forthe newsboy problemrdquo International Journal of Production Eco-nomics vol 45 no 1ndash3 pp 435ndash441 1996

[8] K Das T K Roy and M Maiti ldquoMulti-item stochastic andfuzzy-stochastic inventory models under two restrictionsrdquoComputers and Operations Research vol 31 no 11 pp 1793ndash1806 2004

[9] S Nanda G Panda and J K Dash ldquoA new methodology forcrisp equivalent of fuzzy chance constrained programmingproblemrdquo Fuzzy Optimization and Decision Making vol 7 no1 pp 59ndash74 2008

[10] N K Mahapatra and M Maiti ldquoDecision process for multiob-jective multi-item production-inventory system via interactivefuzzy satisficing techniquerdquo Computers amp Mathematics withApplications vol 49 no 5-6 pp 805ndash821 2005

[11] H-C Chang J-S Yao and L-Y Ouyang ldquoFuzzy mixtureinventory model involving fuzzy random variable lead timedemand and fuzzy total demandrdquo European Journal of Opera-tional Research vol 169 no 1 pp 65ndash80 2006

[12] Z Shao and X Ji ldquoFuzzy multi-product constraint newsboyproblemrdquoAppliedMathematics and Computation vol 180 no 1pp 7ndash15 2006

[13] C-M Lee ldquoA Bayesian approach to determine the value ofinformation in the newsboy problemrdquo International Journal ofProduction Economics vol 112 no 1 pp 391ndash402 2008

[14] J C Hayya T P Harrison and D C Chatfield ldquoA solutionfor the intractable inventory model when both demand andlead time are stochasticrdquo International Journal of ProductionEconomics vol 122 no 2 pp 595ndash605 2009

[15] J K Dash G Panda and S Nanda ldquoChance constrained prog-ramming problem under different fuzzy distributionsrdquo Inter-national Journal of Optimization Theory Methods and Applica-tions vol 1 no 1 pp 58ndash71 2009

[16] J-S Hu H Zheng R-Q Xu Y-P Ji and C-Y Guo ldquoSupplychain coordination for fuzzy random newsboy problem withimperfect qualityrdquo International Journal of Approximate Reason-ing vol 51 no 7 pp 771ndash784 2010

[17] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[18] R-H Su D-Y Yang and W L Pearn ldquoDecision-making ina single-period inventory environment with fuzzy demandrdquo

Expert Systems with Applications vol 38 no 3 pp 1909ndash19162011

[19] K A M Kotb and H A Fergany ldquoMulti-item EOQmodel withboth demand-dependent unit cost and varying leading time viageometric programmingrdquoAppliedMathematics vol 2 no 5 pp551ndash555 2011

[20] O Dey and D Chakraborty ldquoA fuzzy random continuousreview inventory systemrdquo International Journal of ProductionEconomics vol 132 no 1 pp 101ndash106 2011

[21] R Banerjee and S Banerjee ldquoSolution of a probabilistic inven-tory model with chance constraints a general fuzzy program-ming and intuitionistic fuzzy optimization approachrdquo Interna-tional Journal of Pure and Applied Sciences and Technology vol9 no 1 pp 20ndash38 2012

[22] M Nagare and P Dutta ldquoOn solving single-period inventorymodel under hybrid uncertaintyrdquo International Journal of Eco-nomics and Management Sciences vol 6 no 4 pp 290ndash2952012

[23] M A Jahantigh S Khezerloo A A Hosseinzadeh and MKhezerloo ldquoSolutions of fuzzy linear systems using rankingfunctionrdquo International Journal of AppliedOperational Researchvol 2 pp 67ndash75 2012

[24] A Karpagam and P Sumathi ldquoNew approach to solve fuzzy lin-ear programming problems by the ranking functionrdquo BonfringInternational Journal of Data Mining vol 4 no 4 pp 22ndash252014

[25] D Dutta and P Kumar ldquoFuzzy inventory model without short-age using trapezoidal fuzzy number with sensitivity analysisrdquoIOSR Journal of Mathematics vol 4 no 3 pp 32ndash37 2012

[26] S Ding ldquoUncertain multi-product newsboy problem withchance constraintrdquo Applied Mathematics and Computation vol223 pp 139ndash146 2013

[27] H Zolfagharinia and K P S Isotupa ldquoErratum on lsquoOptimalinventory control of empty containers in inland transportationsystemrsquo Int J Prod Econ 133(1) (2011) 451ndash457rdquo InternationalJournal of Production Economics vol 141 no 1 pp 43ndash436 2013

[28] P Majumder U K Bera and M Maiti ldquoAn EPQ model ofdeteriorating items under partial trade credit financing anddemand declining market in crisp and fuzzy environmentrdquoProcedia Computer Science vol 45 pp 780ndash789 2015

[29] S M Mousavi J Sadeghi S T A Niaki N Alikar A Bahre-ininejad and H S C Metselaar ldquoTwo parameter-tuned meta-heuristics for a discounted inventory control problem in a fuzzyenvironmentrdquo Information Sciences vol 276 pp 42ndash62 2014

[30] H N Soni and M Joshi ldquoA periodic review inventory modelwith controllable lead time and backorder rate in fuzzy-stocha-stic environmentrdquo Fuzzy Information and Engineering vol 7 no1 pp 101ndash114 2015

[31] R S Kumar and A Goswami ldquoA fuzzy random EPQ modelfor imperfect quality items with possibility and necessity con-straintsrdquo Applied Soft Computing vol 34 pp 838ndash850 2015

[32] N Kazemi E Shekarian L E C Barron and E U OluguldquoIncorporating human learning into a fuzzy EOQ inventorymodel with backordersrdquo Computers amp Industrial Engineeringvol 87 pp 540ndash542 2015

[33] J K Dash and A Sahoo ldquoOptimal solution for a single periodinventorymodel with fuzzy cost and demand as a fuzzy randomvariablerdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 28 no 3 pp 1195ndash1203 2015

[34] M Nagare P Dutta and N Cheikhrouhou ldquoOptimal orderingpolicy for newsvendor models with bidirectional changes in

10 Advances in Fuzzy Systems

demand using expert judgmentrdquoOPSEARCH vol 53 no 3 pp620ndash647 2016

[35] P Raula S K Indrajitsingha P N Samanta and U K Misra ldquoAFuzzy InventoryModel for constant deteriorating item by usingGMIRmethod in which inventory parameters treated as HFNrdquoOpen Journal of Applied andTheoretical Mathematics (OJATM)vol 2 no 1 pp 13ndash20 2016

[36] I Sangal A Agarwal and S Rani ldquoA fuzzy environment inven-tory model with partial backlogging under learning effectrdquoInternational Journal of Computer Applications vol 137 no 6pp 25ndash32 2016

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Fuzzy Systems 9

Competing Interests

The authors declare that they have no competing interests

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] J J Buckley Fuzzy Probabilities New Approach and Applica-tions Physica Heidelberg Germany 2003

[3] H Kwakernaak ldquoFuzzy random variablesmdashI definitions andtheoremsrdquo Information Sciences vol 15 no 1 pp 1ndash29 1978

[4] H-J Zimmermann ldquoFuzzymathematical programmingrdquoCom-puters amp Operations Research vol 10 no 4 pp 291ndash298 1983

[5] S Nanda and K Kar ldquoConvex fuzzy mappingsrdquo Fuzzy Sets andSystems vol 48 no 1 pp 129ndash132 1992

[6] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986

[7] D Petrovic R Petrovic and M Vujosevic ldquoFuzzy models forthe newsboy problemrdquo International Journal of Production Eco-nomics vol 45 no 1ndash3 pp 435ndash441 1996

[8] K Das T K Roy and M Maiti ldquoMulti-item stochastic andfuzzy-stochastic inventory models under two restrictionsrdquoComputers and Operations Research vol 31 no 11 pp 1793ndash1806 2004

[9] S Nanda G Panda and J K Dash ldquoA new methodology forcrisp equivalent of fuzzy chance constrained programmingproblemrdquo Fuzzy Optimization and Decision Making vol 7 no1 pp 59ndash74 2008

[10] N K Mahapatra and M Maiti ldquoDecision process for multiob-jective multi-item production-inventory system via interactivefuzzy satisficing techniquerdquo Computers amp Mathematics withApplications vol 49 no 5-6 pp 805ndash821 2005

[11] H-C Chang J-S Yao and L-Y Ouyang ldquoFuzzy mixtureinventory model involving fuzzy random variable lead timedemand and fuzzy total demandrdquo European Journal of Opera-tional Research vol 169 no 1 pp 65ndash80 2006

[12] Z Shao and X Ji ldquoFuzzy multi-product constraint newsboyproblemrdquoAppliedMathematics and Computation vol 180 no 1pp 7ndash15 2006

[13] C-M Lee ldquoA Bayesian approach to determine the value ofinformation in the newsboy problemrdquo International Journal ofProduction Economics vol 112 no 1 pp 391ndash402 2008

[14] J C Hayya T P Harrison and D C Chatfield ldquoA solutionfor the intractable inventory model when both demand andlead time are stochasticrdquo International Journal of ProductionEconomics vol 122 no 2 pp 595ndash605 2009

[15] J K Dash G Panda and S Nanda ldquoChance constrained prog-ramming problem under different fuzzy distributionsrdquo Inter-national Journal of Optimization Theory Methods and Applica-tions vol 1 no 1 pp 58ndash71 2009

[16] J-S Hu H Zheng R-Q Xu Y-P Ji and C-Y Guo ldquoSupplychain coordination for fuzzy random newsboy problem withimperfect qualityrdquo International Journal of Approximate Reason-ing vol 51 no 7 pp 771ndash784 2010

[17] G Zhang ldquoThe multi-product newsboy problem with supplierquantity discounts and a budget constraintrdquo European Journalof Operational Research vol 206 no 2 pp 350ndash360 2010

[18] R-H Su D-Y Yang and W L Pearn ldquoDecision-making ina single-period inventory environment with fuzzy demandrdquo

Expert Systems with Applications vol 38 no 3 pp 1909ndash19162011

[19] K A M Kotb and H A Fergany ldquoMulti-item EOQmodel withboth demand-dependent unit cost and varying leading time viageometric programmingrdquoAppliedMathematics vol 2 no 5 pp551ndash555 2011

[20] O Dey and D Chakraborty ldquoA fuzzy random continuousreview inventory systemrdquo International Journal of ProductionEconomics vol 132 no 1 pp 101ndash106 2011

[21] R Banerjee and S Banerjee ldquoSolution of a probabilistic inven-tory model with chance constraints a general fuzzy program-ming and intuitionistic fuzzy optimization approachrdquo Interna-tional Journal of Pure and Applied Sciences and Technology vol9 no 1 pp 20ndash38 2012

[22] M Nagare and P Dutta ldquoOn solving single-period inventorymodel under hybrid uncertaintyrdquo International Journal of Eco-nomics and Management Sciences vol 6 no 4 pp 290ndash2952012

[23] M A Jahantigh S Khezerloo A A Hosseinzadeh and MKhezerloo ldquoSolutions of fuzzy linear systems using rankingfunctionrdquo International Journal of AppliedOperational Researchvol 2 pp 67ndash75 2012

[24] A Karpagam and P Sumathi ldquoNew approach to solve fuzzy lin-ear programming problems by the ranking functionrdquo BonfringInternational Journal of Data Mining vol 4 no 4 pp 22ndash252014

[25] D Dutta and P Kumar ldquoFuzzy inventory model without short-age using trapezoidal fuzzy number with sensitivity analysisrdquoIOSR Journal of Mathematics vol 4 no 3 pp 32ndash37 2012

[26] S Ding ldquoUncertain multi-product newsboy problem withchance constraintrdquo Applied Mathematics and Computation vol223 pp 139ndash146 2013

[27] H Zolfagharinia and K P S Isotupa ldquoErratum on lsquoOptimalinventory control of empty containers in inland transportationsystemrsquo Int J Prod Econ 133(1) (2011) 451ndash457rdquo InternationalJournal of Production Economics vol 141 no 1 pp 43ndash436 2013

[28] P Majumder U K Bera and M Maiti ldquoAn EPQ model ofdeteriorating items under partial trade credit financing anddemand declining market in crisp and fuzzy environmentrdquoProcedia Computer Science vol 45 pp 780ndash789 2015

[29] S M Mousavi J Sadeghi S T A Niaki N Alikar A Bahre-ininejad and H S C Metselaar ldquoTwo parameter-tuned meta-heuristics for a discounted inventory control problem in a fuzzyenvironmentrdquo Information Sciences vol 276 pp 42ndash62 2014

[30] H N Soni and M Joshi ldquoA periodic review inventory modelwith controllable lead time and backorder rate in fuzzy-stocha-stic environmentrdquo Fuzzy Information and Engineering vol 7 no1 pp 101ndash114 2015

[31] R S Kumar and A Goswami ldquoA fuzzy random EPQ modelfor imperfect quality items with possibility and necessity con-straintsrdquo Applied Soft Computing vol 34 pp 838ndash850 2015

[32] N Kazemi E Shekarian L E C Barron and E U OluguldquoIncorporating human learning into a fuzzy EOQ inventorymodel with backordersrdquo Computers amp Industrial Engineeringvol 87 pp 540ndash542 2015

[33] J K Dash and A Sahoo ldquoOptimal solution for a single periodinventorymodel with fuzzy cost and demand as a fuzzy randomvariablerdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 28 no 3 pp 1195ndash1203 2015

[34] M Nagare P Dutta and N Cheikhrouhou ldquoOptimal orderingpolicy for newsvendor models with bidirectional changes in

10 Advances in Fuzzy Systems

demand using expert judgmentrdquoOPSEARCH vol 53 no 3 pp620ndash647 2016

[35] P Raula S K Indrajitsingha P N Samanta and U K Misra ldquoAFuzzy InventoryModel for constant deteriorating item by usingGMIRmethod in which inventory parameters treated as HFNrdquoOpen Journal of Applied andTheoretical Mathematics (OJATM)vol 2 no 1 pp 13ndash20 2016

[36] I Sangal A Agarwal and S Rani ldquoA fuzzy environment inven-tory model with partial backlogging under learning effectrdquoInternational Journal of Computer Applications vol 137 no 6pp 25ndash32 2016

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

10 Advances in Fuzzy Systems

demand using expert judgmentrdquoOPSEARCH vol 53 no 3 pp620ndash647 2016

[35] P Raula S K Indrajitsingha P N Samanta and U K Misra ldquoAFuzzy InventoryModel for constant deteriorating item by usingGMIRmethod in which inventory parameters treated as HFNrdquoOpen Journal of Applied andTheoretical Mathematics (OJATM)vol 2 no 1 pp 13ndash20 2016

[36] I Sangal A Agarwal and S Rani ldquoA fuzzy environment inven-tory model with partial backlogging under learning effectrdquoInternational Journal of Computer Applications vol 137 no 6pp 25ndash32 2016

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014