Fuzzy multi-item economic production quantity model under space constraint: A geometric programming approach
Introduction
A basic assumption in the classical EPQ model is that the production set-up cost is fixed [23]. In addition the model also implicitly assumes that items produced are of perfect quality [10], [13]. While the set-up time, hence set-up cost, will be fixed in short term, it will tend to decrease in the long term because of the possibility of investment in new machineries that are highly flexible, e.g. flexible manufacturing systems [16]. In a paper, Van Beek and Van Puttin [24] addressed extensively the issue of flexibility improvement production and inventory management under various scenarios, while the issues of process reliability, quality improvement and set-up time reduction discussed by Porteus [18], [19], Rosenblatt and Lee [20] and Zangwill [27]. Cheng [5] proposed a general equation to model the relationship between production set-up cost and process reliability and flexibility. Cheng [6] also introduced the demand dependent unit production cost. The economic production quantity model for items with imperfect quality discussed by Ben-Daya [3], Goyal et al. [8] and Salameh and Jaber [22]. Cao et al. [4] describe an examination of inventory and production cost in a revised EMQ/JIt production-run model.
It is often difficult to precise the actual costs and others parameters of the inventory problem. They fluctuate depending upon different aspects. So the inventory cost parameters such as holding cost, set-up cost, shortage cost are assumed to be flexible (i.e. fuzzy in nature). In 1965 Zadeh [26] first gave the concept of fuzzy set theory. This theory now has made an entry into the inventory control systems. Park [17] examined the EOQ formula in the fuzzy set theoretic perspective associating the fuzziness with the cost data. Roy and Maiti [21] solved a single objective fuzzy EOQ model using GP technique.
GP method is an effective method to solve a typical non-linear programming problem. It has certain advantages over the other optimization methods. Duffin et al. [7] discussed the basic theories on GP with engineering application in their books. Another famous book on GP and its application appeared in 1976 [2]. Worral and Hall [25] analysed the inventory models with some constraints and solved by GP technique. Hariri and Abou-el-ata [9] and Abou-el-ata and Kotb [1] used GP to solve multi-item inventory problems. Jung and Klain [14] developed single item inventory problems and solved by GP method.
In this paper we propose an economic production quantity model with flexibility and reliability consideration of production process and demand dependent unit production cost with fuzzy parameters. The model is associated with available limited storage space constraint. The inventory related costs, storage spaces and other parameters are taken here as triangular fuzzy numbers. The problem is solved by modified geometric programming method. Finally, the model is illustrated by numerical.
Section snippets
Notations
- TC(Di, Si, qi, ri)
total average cost of production and inventory carrying cost per unit time
- n
number of items
- Parameters for the ith (i = 1, 2, … , n) item are
- Si
set-up cost per batch (a decision variable)
- Di
demand rate (a decision variable)
- qi
production quantity per batch (a decision variable)
- ri
production process reliability (a decision variable)
- Hi
inventory carrying cost per item per unit time
- fi(Si, ri)
total cost of interest and depreciation for a production process per production cycle
- pi
unit demand
Prerequisite mathematics
Fuzzy sets first introduced by Zadeh [26] in 1965 as a mathematical way of representing impreciseness or vagueness in everyday life.
Fuzzy set: A fuzzy set in a universe of discourse X is defined as the following set of pairs . Here is a mapping called the membership function of the fuzzy set and (x) is called the membership value or degree of membership of x ∈ X in the fuzzy set . The larger (x) is the stronger the grade of membership form in .
Fuzzy model
In general the shape parameters for cost of interest and depreciation and unit production cost, cost parameter for the objective function and storage spaces are not precisely known. So, here we have assumed that all the parameters (ai, bi, ci, αi, βi, Hi) and storage spaces (wi, w) are fuzzy in nature. Hence, these are expressed as triangular fuzzy number. Then the above crisp inventory model (2.3.6) reduces to
Geometric programming (GP) technique
The method of geometric programming for solving the problem is described in the following form.
Primal program:The coefficient of each term of the objective function is positive. Hence this is the posynomial primal geometric programming problem.
Dual program:
Applying modified geometric programming (GP) technique according to Hariri and Abou-El-Ata [9], the dual problem
Numerical example
A manufacturing company produces two types of machines. The machines are produced in lots. The demand rate of each machine is uniform over time and can be assumed to be deterministic. The pertinent data for the machines is given in Table 1.
With imprecise shape parameters are given in Table 2.
Determine the demand rates (D1, D2), set-up cost (S1, S2), production quantity (q1, q2) and production process reliability (r1, r2) of each machines and also find optimal total average
Conclusion
We have solved fuzzy multi-item EPQ model with fuzzy parameters by modified GP techniques. This technique can be applied to solve the different decision making problems in inventory control and other engineering and management sciences.
Acknowledgement
This research was supported by CSIR junior research fellowship in the Department of Mathematics, Bengal Engineering and Science University, Shibpur. This support is greatfully acknowledged.
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