Open Journal of Applied Sciences, 2012, 2, 188192 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 productionrelated 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, [810]). 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 189 revenue. 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 it r cp : 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; cu it : unit inventory holding cost for item i in time pe riod t. cq The Decision Variables it p q : production of item i during time period t; it u : inventory of item i at end of time period t; it The 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. 11 max TI itititititit itit it ti rducpp cqq cuu t d (1) subject to 1 , I ik itkt i ap bkt (2) ,1 , i titititit qpqudi (3) ,,0 , it it it pqu 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 it b Ld Lb 11 max TI itititititit itit it ti rducpp cqq cuu (5) subject to 1 , ik itkt i ap bkt (6) ,1 , i titititit qpqudi t (7) , it it dLd it (8) , kt kt bLbkt (9) ,,0 , it it it pqu it (10) The nonfuzzy production planning problem can be re written in the following equivalent form: 11 max TI itititititit itit it ti rducpp cqq cuu (11) subject to 1 , I ik itkt i ap bkt (12) ,1 , i titititit qpqudi t (13) , it itit hd Hit (14) , kt ktkt lbL kt (15) ,,0 , it it it pqu it (16) where are lower and upper bounds on it respectively and are lower and upper bounds on respectively. and it it hH d and it it lL it b 3. Qualitative Analysis of Basic Notions for the Problems (11)(16) Let ,, and ,, itit ktkt hHlL 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) IT it itit XhHlL pqu 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 ty TI K UhHlLRX hHlL 3.2. The Solvability Set The solvability set of problems (11)(16), which is de noted by V, is defined by ,,, roblem (11)(16) has optimal solution. VhHlL U 3.3. The Stability Set of the First Kind Suppose that with a corresponding αoptimal solution **** ,,,hHlL V *** ,, it it it pqu for problems (11)(16) together with the αlevel optimal parameters ** , kt it bd . 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 it Spqu is defined by ** of problems (1 ,,, , it kt db ik *** ***,,,,, is optimal solution1)(16) ,, with corresponding level optimal parameters it it it it it it hHlLVpqu Sp qu t 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 ++ ++ I ktik itkt i ititit it itit kt ktktkt ktkt itititit itit ititititit it LFZa pb qpqud bH hb dL ld pqu where 11 max TI itititititit itit it ti rducpp cqq cuu . The KuhnTucker necessary optimality conditions for problems (11)(16) are as follows: 0, 0, 0 , it itit LFLFLF it up q t t t it 1 0 , I ik itkt i ap bkt ,1 0 , i titititit qpqud i 0, 0 , it ititit hdd Hit 0, 0 , kt ktktkt lb bLkt ,,0 , it it it pqu it 10 , I ktik itkt i ap bkt ,1 0 ,, iti titititit qpqudik 0 , kt ktkt bH k 0 , kt ktkt hb kt 0 , it itit dL it 0 , it itit ld it 0, 0, 0 , ititit itit it pqu ,,,,,,,,0 ,, ktit ktktit it it ititikt where all the relations of the above system are evaluated at the αoptimal solution with the corre sponding αlevel optimal parameters *** ,, it it it pqu ** ,. it kt db ,,,, , 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 rcuk Table 2 contains the values of aik, 1,2,3,i t1, 2, 3. itit itit We 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 kt db 12 dd 32 d 1 dd d and 121 bb 2 b 3 bb b let α = 0.5, problems (11)(16) can be written as follows: 1121 21 31 12 22 32 max 29 2 4 91 Zd d d d 11 11 31 31 22 22 8 4 378 116 u q duq duq 21 12 32 8 7 u u u 12 32 6 89 q q q 21 2,5,6d 11 1, 3, 6d 3,4,6 Table 1. Values of triangle fuzzy parameters. 31 d 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 . ,,itk 11 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 191 subject to 11213111 32pp pb , , 112131 21 465pppb 122131 31 456pppb, 122232 12 32pp pb , , 122232 22 465pppb 122232 32 456pppb, .5, .5 1,0 11111111 0,qpqud 2,0212121210,qpqud 3,031313131 0,qpqud 1,112121212 0,qpqud 2,122222222 0,qpqud 3,132323232 0,qpqud 11 3.5 5.5,b 21 3.5 5.5,b 31 2.5 5,b 12 3.5 6,b 22 4.5 7,b 32 34b 11 24d.5,21 3.5 5.5,d 31 3.5 5,d 12 2.5 3.5,d 22 35d.5, 32 57d So, we get the following results by using any software package for solving linear pr ogram ming problem: 1132 12 22 11 21 311121 1.375,0.5, 1.5,0.5, 0.625, 3.5, 2,3.5, ppqqu uud d 31 12223211 21 31 1222 3.5,5.5,7.5, 4.125, 5.5,2.5, 3.5,4.5. dd dd b bbbb 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 ktkt UhHlL ikt hHh 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. Candea, “Production and Inventory rid Pro , J. O. McClain and L. J matical Programming Approache s to Capacity Constrained MRP Systems: Review, Formulation and Problem Reduc tion,” Management Science, Vol. 29, No. 15, 1983, pp. 11261141. [2] A. C. Hax and Management,” PrenticeHall, Englewood, 1984. [3] B. Kim and S. Kim, “Extended Model for a Hyb duction Planning Approach,” International Journal of Pro duction Economics, Vol. 73, No. 2, 2001, pp. 165173. doi:10.1016/S09255273(00)001729 [4] J. Galbraith, “Designing Complex Organizations,” Addi rciaSabater and F. C. Lario, n, “A Hybrid Fuzzy ka, K. Soric and V. V. Rosenzweig, “Production “Evaluating the Impact of Operating Environ ethi, H. Yan and Q. Zang, “Optimal and Hierar Yano and H. L. Lee, “Lot Sizing with Random ade, “Fuzzy Sets and Systems, The Analysis of Basic Notions in Parametric Programming, II (Parameters in the Objective sonWesley, Boston, 1973. [5] J. Mula, R. Poler, J. P. Ga “Models for Production Planning under Uncertainty: A Review,” International Journal of Production Economics, Vol. 103, No. 1, 2006, pp. 271285. [6] A. Jamalnia and M. Ali Aoukhakia Goal Programming Approach with Different Goal Priori ties to Aggregate Production Planning,” Computers & Industrial Engineering, Vol. 56, No. 4, 2009, pp. 1474 1486. [7] L. Zrin Planning Problem with Sequence Dependent Setups as a Bilevel Programming Problem,” European Journal of Operational Research, Vol. 187, No. 3, 2008, pp. 1504 1512. [8] C. Ho, ments on MRP System Nervousness,” International Jour nal of Production Research, Vol. 27, No. 7, 1989, pp. 1115 1135. [9] S. P. S chical Controls in Dynamic Stochastic Manufacturing Systems: A Survey,” Manufacturing and Service Opera tions Management Journal, Vol. 4, No. 2, 2002, pp. 133 170. [10] C. A. Yields: A Review,” Operations Research, Vol. 43, No. 2, 1995, pp. 311334. [11] D. Dubois and H. Pr ory and Applications,” Academic Press, New York, 1980. [12] D. Dubois and H. Prade, “Operations on Fuzzy Num bers,” International Journal of System Science, Vol. 9, No. 6, 1978, pp. 613626. [13] M. Osman, “Qualitative Copyright © 2012 SciRes. OJAppS
S. A. ABASS Copyright © 2012 SciRes. OJAppS 192 Function),” Applied Mathematics, Vol. 22, No. 5, 1977, pp. 333348. [14] M. osman and A. H. ElBanna, “Stability of Multiobjec tive Nonlinear Programming Problems with Fuzzy Pa rameters,” Mathematics and Computers in Simulation, Vol. 35, No. 4, 1993, pp. 321326.
