Production, Manufacturing and Logistics
Two-storage inventory model with lot-size dependent fuzzy lead-time under possibility constraints via genetic algorithm

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Abstract

Multi-item inventory models with stock dependent demand and two storage facilities are developed in a fuzzy environment where processing time of each unit is fuzzy and the processing time of a lot is correlated with its size. These are order-quantity reorder-point models with back-ordering if required. Here possibility and crisp constraints on investment and capacity of the small storehouse respectively are considered. The models are formulated as fuzzy chance constrained programming problem and is solved via generalized reduced gradient (GRG) technique when crisp equivalent of the constraints are available. A genetic algorithm (GA) is developed based on fuzzy simulation and entropy where region of search space gradually decreases to a small neighborhood of the optima and it is used to solve the models whenever the equivalent crisp form of the constraint is not available. The models are illustrated with some numerical examples and some sensitivity analyses have been done. For some particular cases results observed via GRG and GA are compared.

Introduction

In the existing literature, inventory models are generally developed under the assumption of constant or stochastic lead-time [1], [9], [20]. But in real life situations, it is normally vague and imprecise, i.e., fuzzy in nature. Again the processing time of large orders is, in many industries, longer than that of small orders. While waiting time and machine setup time are usually independent of lot size, the actual processing time portion of lead time could well depend on the lot size in many industries which manufacture discrete parts or products. This renders supply lead times in such settings to be increasing with the order size. Yet existing inventory control research papers do not explicitly incorporate lot-size dependent fuzzy lead time.

In an inventory management, different types of demand are considered – constant, stock-dependent, time-dependent, probabilistic demand etc. According to Levin et al. [11], “the presence of inventory has a motivational effect on the people around it”. In the present competitive market, the inventory/stock is decoratively displayed through electronic media to attract the customers and thus to push the sale. Schary and Baker [19] and Wolfe [22] also established the impact of product availability for simulating demand. Mandal and Phaujder [14], Datta and Pal [4] and others considered linear form of stock-dependent demand, i.e., D = c + dq, where D, q represent demand and stock level respectively, c, d are two constants, so chosen to best fit the demand function, where as Urban [21], Giri et al. [7], Mandal and Maiti [15] and others took the demand of the form D = dqβ, where β is a constant, so chosen to best fit the demand function.

Now-a-days due to globalization of market with the introduction of multinationals in the business, there is a trend among the business houses specially middle order retailers and small retailers of different multi-national products to compete with each other for sale and as a result, they use the decorative showroom at the selling point to boost their items in addition to a separate warehouse for storage. Moreover, in the important market places like super markets, municipality markets, etc., it is almost impossible to have a big showroom/shop due to the scarcity of space and very high rent. Normally, moderate and big business houses operates through two rented houses – one, smaller in size and situated at the heart of the market place and other with large capacity little away from the market place. During last two decades, two warehouse inventory models have been developed and solved by many researchers [3], [10], [18]. Till now, all two-storage problems have been formulated with one own house and another rented one. But normally, both are rented houses, rent at the market place being higher than that at far-away places.

It has been recognized that one’s ability to make precise statement concerning different parameters of an inventory model diminishes with increasing complexities of the environment. As a result it may not be possible to define the different inventory parameters as well as the constraints precisely. During controlling period of inventory, the resource constraints may be possibilistic in nature and it may happen that the constraints on resources will satisfy in almost all cases except possibly for a very few cases, where they may be allowed to violate. In stochastic environment, for the solution of this type of problems Mohon [17] proposed a ‘here and now’ approach, i.e., the chance constrained programming approach in which a minimum probability level for satisfying each of the constraint is specified. Similarly possibilistic constraints also may be defined [12].

Here, order-quantity reorder-point models having two-storage facilities with back-ordering, if required are considered for multi items. Demand rate of an item is assumed as stock dependent, while the lead time is fuzzy in nature. It is further assumed that immediately after the arrival of an order, the installation stock always exceeds the reorder level, so at most one order will be outstanding at any time for an item. The processing time of single-unit order is fuzzy in nature, L˜1; the processing time of a lot of Q units is partly constant L0 (which includes allowances of material handling), and partly assumed to be multiple of L˜1, QβL˜1, where the value of the positive parameter β indicates the extent of economies of scale and/or learning effects, if any, in production speed. Two rented storehouses are used for storage – one (say RW1) at the heart of the market place and the other (say RW2) little away from the market place. At the beginning, items are stored at both RW1 and RW2. The items are initially sold from RW1 and are filled up from RW2 in continuous release pattern. Here, the size of RW1 is finite but that of RW2 has unlimited capacity. Unit costs of the items and the capital for investment are also fuzzy in nature. Hence there are two constraints, one on the storage space of RW1 and other on the total investment amount. Since investment amount and purchase cost are fuzzy in nature, this constraint represents a fuzzy event and should be satisfied with some degree of possibility in optimistic sense. Here it is assumed to be satisfied with at least some possibility α. With these assumptions, the problem is formulated to maximize the optimistic return of the average profit under the above-mentioned space and investment constraints. A fuzzy simulation based genetic algorithm using entropy theory is developed where search space is gradually reduced as number of iteration increases. This algorithm (named as FSRRGA) is used to solve the models. For some particular cases, the models are solved via GRG technique also and results are compared with the results observed via FSRRGA. Some numerical examples are presented to illustrate the models and some sensitivity analyses have been performed.

Section snippets

Possibility in fuzzy environment

Let X be a classical set of objects, called the universe, whose generic elements are denoted by x. Membership in a classical subset A of X is often viewed as a characteristic function μA from X to {0, 1} such thatμA(x)=1forxA,0otherwise.Here {0, 1} is called a valuation set. If the valuation set is allowed to be the real interval [0, 1], A is called a fuzzy set and to distinguish from classical set, it is denoted by A˜. In this case characteristic function μA is called membership function of A˜

Fuzzy chance constrained programming

A general single-objective mathematical programming should have the following form:maxf(x,ξ)subject togi(x,ξ)0,i=1,2,,n,where x is a decision vector, ξ is a vector of crisp parameters, f(x, ξ) is the return function, gi(x, ξ) are constraint functions, i = 1, 2,  , n. In the above problem when ξ is a fuzzy vector ξ˜ (i.e., a vector of fuzzy numbers), then return function f(x,ξ˜) and constraint functions gi(x,ξ˜) are imprecise in nature and can be represented by two fuzzy numbers whose membership

Fuzzy simulation based region reducing genetic algorithm

Genetic algorithms are exhaustive search algorithms based on the mechanics of natural selection and genesis (crossover, mutation etc.) and have been developed by Holland, his colleagues and students at the University of Michigan (cf. [8]). Because of its generality and other advantages over conventional optimization methods it has been successfully applied to different decision making problems.

Generally a GA starts with a single population [8], [16], randomly generated in the search space. One

Assumptions and notations for the proposed models

The following notations and assumptions are used in developing the models.

  • (i)

    Inventory system involves N items and two rented warehouses RW1 and RW2. Location of RW1 is at the heart of the market place and RW2 little away from the market place. The holding cost at RW1 is higher than that at RW2.

  • (ii)

    Storage area of RW1 is W units but RW2 has unlimited capacity.

  • (iii)

    R is the capital for investment.

  • (iv)

    Z is the profit per unit time.

  • (v)

    α1, α2 denote the confidence levels for profit function and investment constraint

Model development and analysis

In the development of the model, we assume that at the beginning of every cycle company purchases an amount Qi units of ith item. Among these units at first shortages of previous cycle (if any) are fulfilled, then RW1 is fulfilled and the remaining units are stored at RW2. Demands of the item are met using the stocks of RW1 and are continuously filled up from RW2. When stock level drops to reorder level ri, order for next cycle placed (cf. Fig. 7) and order reaches after time Li, when inventory

Numerical illustration

For the present models, the algorithm FSRRGA is developed following Bessaou and Siarry [2] and according to them a good convergence of the algorithm requires the tuning of some of its characteristic parameters (cf. [2, p. 70]). To solve the present model following parametric values for the algorithm are considered. This set of parametric values was also considered by Bessaou and Siarry [2].pc=0.6,pm(0)=0.9,M=5,N=10,Maxgen1=100,Maxgen2=100,Maxgen3=10.For this parametric set, the algorithm FSRRGA

Discussion

It is observed in Table 3, Table 5 that profits via both the techniques GRG and FSRRGA are all most same. But GRG is applicable to only those problems whose crisp equivalents are available. Also it is observed from above tables that profit for γ = 1 is more compared to profit for γ = 0.5. It happens because for γ = 1 demand is much more compared to γ = 0.5. From Table 8 it is observed that as βi increases profit decreases for γ = 1 but remains unchanged for γ = 0.5. The explanation for this phenomenon is

Conclusion

For the first time a fuzzy simulation based genetic algorithm using entropy theory has been developed where the search space gradually decreases to a small neighborhood of optimal solution and is used to solve non-linear optimization problems with possibility constraints. As illustration, for the first time the said technique has been applied to realistic inventory models developed in fuzzy environment with possibilistic resource constraints. This paper modelled and explored the implications of

Acknowledgements

The authors would like to thank AICTE, New Delhi for financial support of the project under which present research paper has been prepared.

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