Stability of Production Planning Problem with Fuzzy Parameters

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.


Introduction
The classical linear programming (LP) models for production 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 balanced with those from the previous period, production quantity, and the demand quantity; 2) capacity constraints which ensure the total workload for each resource 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 uncertainty as the difference between the amount of information required to perform a task and the amount of information 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 objective nonlinear programming model with different 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 uncertainties 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 uncertainty.The literature in production planning under uncertainty is vast.Different approaches have been proposed to cope with different forms of uncertainty (see, for example, [8][9][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.Section 3 presents a qualitative analysis of some basic notions for the problem of concern.An illustrative numerical example is provided in Section 4. Finally, Section 5 contains the concluding remarks.

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 decide what demand to meet and how.We assume that demand that cannot be met in a period is lost, thus reducing revenue.First we give the notion: T The optimization model of production planning problem to maximize revenues net with fuzzy parameters is as follows: , , , 0 , The -level set of the fuzzy numbers are defined as the ordinary set and respectively for which the degree of their membership function exceeds the level   [0,1].This definition is introduced by Dubois and Prade [11,12].For a certain degree , problems ( 1)-( 4) can be understood as the following nonfuzzy production planning problem: , , , 0 , The nonfuzzy production planning problem can be rewritten in the following equivalent form: , , , , , 0 ,  11)-( 16) can be defined as follows: , , , satisfies the constraints ( 12)-( 16) 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]).

The Set of Feasible Parameters
The set of feasible parameters of the problems ( 11)-( 16), which is denoted by U, is defined by:

The Solvability Set
The solvability set of problems ( 11)-( 16), which is denoted by V, is defined by -( 16) has -optimal solution .

The Stability Set of the First Kind
Suppose that with a corresponding α-optimal solution , , , , , it it it p q u for problems ( 11)-( 16) together with the α-level optimal parameters   The stability set of the first kind of problems ( 11

Determination of the Stability Set of the First Kind
The Lagrange function of problems ( 11)-( 16) can be written as follows: The Kuhn-Tucker necessary optimality conditions for problems (11)-( 16) are as follows: , , , , , , , , 0 , , where all the relations of the above system are evaluated at the α-optimal solution with the corre-sponding α-level optimal parameters

Illustrative Example
Let us consider the following production planning problem 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 , We assume that the membership functions to the triangle fuzzy numbers is take the following form: 11)-( 16) can be written as follows: 32 and all other variables are equal to zero.Objective function value is equal to 41.5.The sets of feasible parameters, solvability set and the stability set of the first kind are calculated as follows:

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 resources 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 uncertainty, 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 get the following results by using any software package for solving linear programming problem: