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 Open Journal of Applied Sciences, 2012, 2, 188-192 doi:10.4236/ojapps.2012.23028 Published Online September 2012 (http :/ /www.SciRP.org/journal/ojapps) Stability of Production Planning Problem with Fuzzy Parameters Samir Abdou Abass Department of Mathematics and Theoretical Physics, Nuclear Research Center, Atomic Energy Authority, Cairo, Egypt Email: samir.abdou@gmail.com Received June 3, 2012; revised July 5, 2012; accepted July 16, 2012 ABSTRACT The traditional production planning model based upon the famous linear programming formulation has been well known in the literature. The consideration of uncertainty in manufacturing systems supposes a great advance. Models for production planning which do not recognize the uncertainty can be expected to generate inferior planning decisions as compared to models that explicitly account the uncertainty. This paper deals with production planning problem with fuzzy parameters in both of the objective function and constraints. We have a planning problem to maximize revenues net of the production inventory and lost sales cost. The existing results concerning the qualitative and quantitative analysis of basic notions in parametric production planning problem with fuzzy parameters. These notions are the set of feasible parameters, the solvability set and the stability set of the first kind. Keywords: Production Planning; Stability; Linear Programmin g; Fuzzy Parameters 1. Introduction The classical linear programming (LP) models for pro-duction planning have been around for many years. A typical formulation of the LP planning models has the objective minimizing the total production-related costs, such as variable production costs, inventory costs, and shortage costs, over the fixed planning horizon [1,2]. The usual constraints employed are: 1) inventory balance equations for making the inventory and/or shortages bal-anced with those from the previous period, production quantity, and the demand quantity; 2) capacity con-straints which ensure the total workload for each re-source not exceed the capacity in each period [3]. The LP model considers the limited availability of the resources (labor, machine, etc.) through the capacity constraints. In a real production system, such capacity constraints may not correct. Galbraith [4] defined uncer-tainty as the difference between the amount of informa-tion required to perform a task and the amount of infor-mation already possessed. Mula et al. [5] presented an exhaustive literature survey about models for production planning under uncertainty. Abouzar Jamalnia and M. Ali Soukhakian [6] introduced a hybrid fuzzy multi ob-jective nonlinear prog ramming model with differen t goal priorities. Zrinka et al. [7] introduced the production planning problem as a bilevel programming problem. In the real world, there are many forms of uncertainty that affect production process. Ho [8] categorizes them into two groups: 1) environmental uncertainty and 2) system uncertainty. Environmental uncertainty includes uncer-tainties beyond the production process, such as demand uncertainty and supply uncertainty. System uncertainty is related to uncertainties within the production process, such as operation yield uncertainty, production lead time uncertainty, quality uncertainty, failure of production system and changes to product structure, to mention some. In this paper, we will use the first category of un-certainty. The literature in production planning under uncertainty is vast. Different approaches have been pro-posed to cope with different forms of uncertainty (see, for example, [8-10]). This paper is organized as follows. In next section, a model of production planning problem to maximize revenues net with fuzzy parameters is formulated. Sec-tion 3 presents a qualitative analysis of some basic no-tions for the problem of concern. An illustrative numeri-cal example is provided in Section 4. Finally, Section 5 contains the concluding remarks. 2. Problem Formulation For some production planning problems we have the option of not meeting all demand in each time period. Indeed, there might not be sufficient resources to meet all demand. In this case, the optimization problem is to de- cide what demand to meet and how. We assume that de- mand that cannot be met in a period is lost, thus reducing Copyright © 2012 SciRes. OJAppS S. A. ABASS 189revenue. First we gi ve t he notion: T: number of time periods; I: number of items; K: number of resources; kt : fuzzy parameters represents the amount of re-source k available in time period t; bit : fuzzy parameters represent the demand for item i in time period t; ditrcp: unit revenue for item i in time period t; it : unit variable cost of production for item i in time period t; it : unit cost of not meeting demand for item i in time period t; cuit : unit inventory holding cost for item i in time pe-riod t. cqThe Decision Variables itpq: production of item i during time period t; itu: inventory of item i at end of time period t; itThe optimization model of production planning prob-lem to maximize revenues net with fuzzy parameters is as follows: : unmet demand of item i during time period t. 11max TIitititititit itit ittirducpp cqq cuu  td (1) subject to 1 ,Iik itktiap bkt (2) ,1 ,i tititititqpqudi (3) ,,0 ,it it itpqu it (4) The -level set of the fuzzy numbers are defined as the ordinary set and respec-tively for which the degree of their membership function exceeds the level   [0,1]. This definition is introdu ced by Dubois and Prade [11,12]. For a certain degree , problems (1)-(4) can be understood as the following non-fuzzy production planning problem: and kt itbLdLb11max TIitititititit itit ittirducpp cqq cuu  (5) subject to 1 ,Iik itktiap bkt (6) ,1 ,i tititititqpqudi t (7)  , it itdLd it (8)  , kt ktbLbkt (9) ,,0 ,it it itpqu it (10) The nonfuzzy production planning problem can be re-written in the following equivalent form: 11max TIitititititit itit ittirducpp cqq cuu  (11) subject to 1 ,Iik itktiap bkt (12) ,1 ,i tititititqpqudit (13) , it itithd Hit (14) , kt ktktlbL kt (15) ,,0 ,it it itpqu it (16) where are lower and upper bounds on it respectively and are lower and upper bounds on respectively. and it ithH dand it itlLitb3. Qualitative Analysis of Basic Notions for the Problems (11)-(16) Let ,, and ,,itit ktkthHlL ikt are assumed to be pa-rameters rather than constants. The decision space of problem (11)-(16) can be defined as follows: 3,,,,,,satisfies the constraints (12)-(16)ITit ititXhHlLpqu Rit In what follows we are give the definitions of some basic notions for the problem (11)-(16). Such notions are the set of feasible parameters, the solvability set and the stability set of the first kind (see [13,l4]). 3.1. The Set of Feasible Parameters The set of feasible parameters of the prob lems (11)-( 16), which is denot e d by U, is defined by: 2,,,,,, is not emp tyTI KUhHlLRX hHlL 3.2. The Solvability Set The solvability set of problems (11)-(16), which is de-noted by V, is defined by ,,,problem (11)-(16) has -optimal solution.VhHlLU 3.3. The Stability Set of the First Kind Suppose that with a corresponding α-optimal solution ****,,,hHlL V***,,it it itpqu for problems (11)-(16) together with the α-level optimal parameters **,kt itbd . Copyright © 2012 SciRes. OJAppS S. A. ABASS Copyright © 2012 SciRes. OJAppS 190 The stability set of the first kind of problems (11)-(16) that is denoted by ***,,it it itSpqu is defined by ** of problems (1,,, ,it ktdb ik******,,,,, is -optimal solution1)-(16),, with corresponding -level optimal parameters it it itit it ithHlLVpquSp qut 3.4. Determination of the Stability Set of the First Kind The Lagrange function of problems (11)-(16) can be written as follows: 1,1++++Iktik itktiititit it ititkt ktktkt ktktitititit ititititititit itLFZa pbqpqudbH hbdL ldpqu   where 11max TIitititititit itit ittiZrducpp cqq cuu . The Kuhn-Tucker necessary optimality conditions for problems (11)-(16) are as follows: 0, 0, 0 ,it ititLFLFLF itup q   t ttit 10 ,Iik itktiap bkt ,1 0 ,i tititititqpqud i 0, 0 ,it ititithdd Hit  0, 0 ,kt ktktktlb bLkt ,,0 ,it it itpqu it 10 ,Iktik itktiap bkt ,1 0 ,,iti tititititqpqudik  0 ,kt ktktbH k 0 ,kt ktkthb kt 0 ,it ititdL it 0 ,it ititld it 0, 0, 0 ,ititit itit itpqu ,,,,,,,,0 ,,ktit ktktit it it ititikt  where all the relations of the above system are evaluated at the α-optimal solution with the corre-sponding α-level optimal parameters ***,,it it itpqu**,.it ktdb ,,,, ,ktit ktktititit,,, and it it it  and it are the La- grange multipliers. 4. Illustrative Example Let us consider the following production planning prob- lem to maximize net revenues with fuzzy parameters. Consider I = 3, K = 3 and T = 2. Table 1 contains the values of triangle fuzzy parameters kt, db 1, 2,3,i 1, 2,1, 2, 3tk .,, and 1,2,cp cq rcukTable 2 contains the values of aik, 1,2,3,i t1, 2, 3. itit ititWe a ss ume th at th e me mb er sh ip functions to the trian-gle fuzzy numbers is take the following form: and it ktdb12dd32d1dd d  and 121bb2 b3bb b  let α = 0.5, problems (11)-(16) can be written as follows: 1121 2131 1222 32max 2924 91Zd ddd11 1131 3122 228 4378116u qduqduq21123287uuu1232689qqq 21 2,5,6d 11 1, 3, 6d 3,4,6 Table 1. Values of triangle fuzzy parameters. 31d 12 2,3,4d 22 1,5, 6d 32 4,6,9d 11 2,5,6b 21 3,4,7b22 3,6,8b 31 1, 4,6b 12 2,5,7b 32 2,4,5b Table 2. Values of aik, cpit, cqit, rit and cuit . ,,itk11 3a 21 1a 31 2a 12 4a 22 6a 32 5a 13 4a 23 5a 33 6a 11 2cp 21 5cp  31 4cp 12 3cp 22 4cp  32 6cp 11 2cq 21 3cq  31 4cq 12 3cq 22 2cq  32 3cq 11 6cu 21 8cu  31 4cu 12 5cu 22 7cu  32 9cu 11 2r 21 1r 31 3r 12 2r 22 4r 32 2r S. A. ABASS 191subject to 1121311132pp pb , , 112131 21465pppb122131 31456pppb, 122232 1232pp pb , , 122232 22465pppb122232 32456pppb,.5,.5 1,0 111111110,qpqud  2,0212121210,qpqud  3,031313131 0,qpqud 1,112121212 0,qpqud 2,1222222220,qpqud 3,1323232320,qpqud  113.5 5.5,b 213.5 5.5,b 312.5 5,b123.5 6,b 224.5 7,b 3234b1124d.5,213.5 5.5,d 313.5 5,d122.5 3.5,d 2235d.5,3257dSo, we get the following results by using any software package for solving linear pr ogram ming problem: 1132 12 22 1121 3111211.375,0.5, 1.5,0.5, 0.625,3.5, 2,3.5,ppqquuud d  31 1222321121 31 12223.5,5.5,7.5, 4.125,5.5,2.5, 3.5,4.5.dd dd bbbbb  32 and all other variables are equal to zero. Objec-tive function value is equal to 41.5. The sets of feasible parameters, solvability set and the stab ility set of the first kind are calculated as follows: 3b112 233112233,,,0,,1,2, 3,1,2, ,,,,itit ktktUhHlL ikthHh HhHlLlLl L  5. Conclusion In this paper, we have discussed the concept of stability for the planning problem to maximize revenues net of the production inventory and lost sales cost. We used fuzzy parameters to represent both of amount of available re- sources and demand for item in period. We have defined and characterized some basic notions for the problem of concern. These notions are the set of feasible parameters, the solvability set and the stability set of the first kind. Although an extensive literature on models for produc- further research is identified as the development of new models that contain additional sources and types of un- certainty, such as supply lead times, transport times, quality uncertainty, failure of production system and changes to product structure. Also as a point of further research, an investigation of incorporation all types of uncertainty is needed . tion planning under uncertainty was reviewed, a need for REFERENCES [1] P. J. Billington. Thomas, “Mathe- D. 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