Modeling and Solving Production Planning Problem under Uncertainty: A Case Study

Data in many real life engineering and economical problems suffer from inexactness. In the real world there are many forms of uncertainty that affect production processes. Uncertainty always exists in practical engineering problems. in order to deal with the uncertain optimization problems, fuzzy and stochastic approaches are commonly used to describe the imprecise characteristics. Herein we assume some intervals in which the data can simultaneously and independently perturb. In this study production planning related data of AlAraby firm fo r electric sets in Egypt was collected. A production planning model based on linear programming (LP) was formulated. This formulat ion based on the outcomes of collected data. The data includes the amount of required and availab le resources, the demand, the cost of production, the cost of unmet demand, the cost of inventory holding and the revenue. in this work, the objective is to maximize the revenues net of the production, inventory and lost sales costs. The general LP model was solved by using software named W in QSB.


Introduction
The objective of the production planning problem is the maximization of the revenue. For solving it, the demand and resource levels are assumed to be fixed and given. But a production planning problem exists because there are limited production resources that cannot be stored from period to period. Also the planning problem starts with a specification of customer demand that is to be met by the production plan. In most contexts, future demand is partially known. So one relies on a forecast for the future demand but the forecast is inaccurate. This leads to that demand that cannot be met in a period is lost, thus reducing revenue. So a production planning problem to maximize revenues net of the production, inventory and lost sales costs is suggested by Stephen C. Graves [1]. For solving it , the coefficients of the objective function, the inequalities and the equalities constraints are assumed to be known numbers.
In real world, the data are imp recise that is the data can be represented as uncertainty. In the real world there are many fo rms of u ncert aint y that affect p ro duct ion p rocesses. Uncertainty always exits in practical engineering problems. In order to deal with the uncertain optimization problems, fu zzy and stochastic app roaches are co mmon ly used to d es crib e the imp recis e ch aracteris tics . In sto ch astic programming [20] the uncertain coefficients are regarded as random variables and their probability distributions are assumed to be known. In fuzzy programming [21] the constraints and objective function are viewed as fuzzy sets and their membership functions also need to be known. In these two kinds of approaches, the membership functions and probability distributions play important roles. It is sometimes difficult to specify an appropriate membership function or accurate probability distribution in an uncertain environment [22]. The interval analysis method was developed to model the uncertainty in uncertain optimization problems, in which the bounds of the uncertain coefficients are only required, not necessarily knowing the probability distributions or membership functions. So for the first time, we will deal the production planning problem with interval data as uncertainty in both of the objective function and constraints.

Review Literature
Production planning is a complicated task which requires cooperation among mult iple functional units in an organization. For solving the p roblem of production planning, powerfu l optimization models have been constructed by means of the fo rmulat ion of mathematical programming. We refer to [1,7] for an overview of optimization techniques in this field.
Galbraith [3] defines uncertainty as the difference between the amount of information required to perform a task and the amount of informat ion already possessed. we refer to [13] for applying a fuzzy linear programming method for solving model related to aggregate production planning problem with mult iple objectives. also we refer to [19] for applying a stochastic linear programming method for solving model related to supply chain planning with bi-level linear mu ltip le objective programming. As the recent contribution for the problem of concern can be viewed as fo llo ws: R. Svend [23] d iscusses how the linear programming model can be used as the basis for a company's operational planning, in practice. F. G. Juan Carlos et al. [24] developed a new model for production planning using fuzzy sets in order to use classical mathematical programming techniques to reach an optimal solution over a mu ltiple criteria context.
Production planning problem that involve the lost of future demand have been investigated by Stephen C. Graves [1]. This problem deals with precise coefficients in both of objective function, the inequalities and the equalities constraints. While the data in the real world are imp recise then the input data can be represented as uncertainty.
In our work, we develop a new planning problem to minimize the lost demands and thus maximize revenues. For the first time, we construct the production planning problem with interval nu mbers as uncertainty in both of the objective function and constraints. After that we will treat the uncertain of objective function and constraints. In section 5, parametric study for the treat ment problem is introduced. Finally, for illustration, a production planning examp le is given, where the integer programming optimal solution is determined using the WinQSB software package.

Solution Algorithm
In this sub section, we describe a solution algorith m for solving problem (1)- (14).
Algorith m steps: Step 1: Construct the production planning problem with interval nu mbers in both of the ob jective function and constraints as in (1)- (14).
Step 4: solve the problem by WinQSB software package.

Methodology Adopted
In this study, production related data of Al-Araby firm for electric sets in Egypt was collected. Data was collected, classified and analy zed statistically. Linear programming model was formulated based on the outcomes of the analyzed data. The data were collected through the use of questionnaires and oral interview among employees in the firm. The data that were collected include the following: The firm is planning for production of three items: refrigerators, ovens and washing machines. The manufacturer of each item requires three resources. These resources are number of workers, over time and varying inventory. There are two time periods for production. The length of each time period is six months. The amount of the three resources required of production of the three items is illustrated in table (1). The amounts of availab le resource and demand for each item are represented by interval numbers as illustrated in tables (2) and (3) respectively. The cost of production of the three items in a given two time periods is illustrated in table (4). The unit cost of unmet demand of the three items in a g iven two time periods is illustrated in table (5). The unit inventory holding costs of the three items in a given two time periods is illustrated in table (6). The unit revenue for each item in each time period is illustrated in table (7).     The decision variables are as follows : With these variables, it is possible to formulate the production planning problem that maximizes revenues net of the production inventory and lost sales cost with interval numbers as follo ws: Maximize Z = ( ) 11 11 11 3000 ,

The Optimization Approach
Based on the proposed approach of Jiang [8] for t reating interval nu mber, we will treat the uncertainty of model (1)- (14)  100u − (15) in interval mathemat ics, The uncertain objective function (1) can be transformed into two objective optimizat ion problems as follows: where m is called the midpoint value, w is called the rad ius of interval nu mber and the two functions

Second: Treatment of the uncertain constraints
The possibility degree of interval nu mber represents certain degree that one interval nu mber is larger or s maller than another. The set of inequality constraints (2)-(7) can be written as 11 21 31 11 11 As in interval linear programming [18], the possibility degree of the constraints (20) can be written as follows 11 11 21 The equality constraints (8)

The Deterministic form of Problem (1)-(14)
The linear co mbination method [2,5,6] is adopted with the mu lti object ive optimizat ion. In mult i objective optimization, applying the linear comb ination method to integrate the objective function is relatively easy provided that the preferences of the objective functions are available. Then two objective function (18)  Table (8) shows the comparison between the results of our approach which is based on uncertainty case and one's of Stephen C. Graves approach [1] which is based on the deterministic case.
It is clear that the results obtained from our approach are better than the results obtained by Stephen C. Graves especially for the objective function value.

Conclusions
In this paper, However, as a point for future research, a co mparison study is needed between the interval and fuzzy programming to tackle the production planning problem, where each of fuzzy programming and interval programming are two forms of uncertainty. This point fo r future research is to determine which of interval and fuzzy programming is more suitable for problem of concern.