Production, Manufacturing and LogisticsA two warehouse inventory model for a deteriorating item with partially/fully backlogged shortage and fuzzy lead time
Introduction
The classical inventory models are mainly developed with the single storage facility. But, in the field of inventory management, when a purchase (or production) of large amount of units of an item that can not be stored in its existing storage (viz., own warehouse – OW) at the market place due to limited capacity is made, then excess units are stocked in a rented warehouse (RW) which is located at some distance away from OW. The RW is of infinite capacity, i.e. it can be made large as it may be required as per the situation. Here, infinite capacity means that RW is sufficiently large. In practice, management goes for large purchase at a time when either it gets an attractive price discount for bulk purchase or the acquisition cost is higher than the holding cost in RW. The actual service to the customer is done at OW only. As the emptied OW (fully or partially) creates negative impression amongst the customers, OW should remain full with goods as far as possible. For this reason, at certain interval of time t1, which is assumed as a variable, the stock at OW is replenished from the stock of RW. Further, units at RW are first exhausted fully.
A basic two-warehouse model was presented by Hartely (1976), Hartely presented a basic two-warehouse model, in which the cost of transporting a unit from rented warehouse (RW) to own warehouse (OW) was not considered. Sarma (1983) extended Hartely’s model by introducing the transportation cost. Goswami and Chaudhuri (1992) showed the generalization of Sarma’s model by considering a linear demand, the equal shipment cycle and the transportation cost depending on the quantity to be transported. Bhunia and Maiti (1994) formulated and solved a deterministic inventory model with inventory level-dependent consumption rate for two warehouses. Later, Bhunia and Maiti (1998) also extended their model by taking into consideration inventory deterioration under the assumption of continuous release. On the other hand, Lee and Ma (2000) proposed an optimal policy for deteriorating items with two-warehouses and time-dependent demands.
Again, real world is full of uncertainties in non-stochastic sense, though few inventory models have been developed under such uncertainty. Yao and Lee, 1996, Yao et al., 1998a, Yao and Lee, 1998b formulated several inventory models in fuzzy environment with fuzzy demand, fuzzy production quantity, fuzzy ordering quantity, fuzzy shortage cost, etc. Lee and Yao, 1999, Chiang et al., 2005 also solved two fuzzy inventory problems without and with backorders using classical centroid formula and signed distance method, respectively, for defuzzyfication.
Normally, in inventory analysis, lead time is considered to be stochastic with a probability distribution. To form a probability distribution, past record/data are required. But, with the advent of new companies and innovative product in the market due to globalization, it is not always possible to have old/earlier record/data. Hence, in such cases, lead time is assumed to be imprecise. In this area, Dey et al. (2005) proposed an interactive method for a single warehouse inventory control system with fuzzy lead time and dynamic demand. Till now, none have considered fuzzy lead time in two warehouse inventory system, which is, now-a-day, a reality in business sector.
In this paper, for the first time, fuzzy lead time has been introduced in a two-warehouse inventory control system and ordering cost has been assumed to be partly lead time dependent. Here, the holding cost at RW decreases with the increase of its distance from the market place, which is also another reality. This was not considered by earlier researchers in two warehouse models. Here, a method via interval mathematics has been proposed to deal decision making problems with fuzzy parameters. Moreover, two two-warehouse inventory models with fully and partially backlogged shortages have been formulated and some interesting conclusions about the results of two models are derived. The focus of this research paper is the introduction of fuzzy lead time in two-warehouse inventory models for the optimum decisions of new companies and/or innovative products.
Thus a two warehouse inventory model with shortages for a deteriorating item and imprecise lead time has been formulated and an inventory policy is proposed for maximum profit. The customer’s demand is assumed to be selling price dependent. The retailer possesses two warehouses – OW (of finite capacity) and RW (infinite capacity i.e. sufficiently large). OW is situated in the main marketing place and RW at a distance from OW. The item is purchased in a lot and first OW is filled up with the item and then excess units are kept in RW. Demands are met from market warehouse (OW) and during sale, OW is filled up by shifting the units from the retailer’s distance warehouse (RW) in bulk release pattern at some time interval (which is assumed as a variable) till RW is emptied. As retailer’s RW is situated outside of the market place, the holding cost at RW and transportation cost from RW to OW depend on the distance between them. The shortages at OW are allowed and backlogged partially (in Model-1)/fully (in Model-2). There is a time gap (uncertain in non-stochastic sense) between the placement of order and receipt of it. The imprecise lead time is represented by a fuzzy number. It is first transformed to a corresponding interval number and then following interval mathematics, the single objective function for average profit over the time cycle is changed to respective multi-objective functions. Analytically it is shown that models with partially/fully backlogged shortages (i.e. Model-1/Model-2) possesses Pareto-optimal solutions. Also, these objectives for Model-1 and Model-2 are optimized numerically and solved for a set of Pareto-optimal solution with the help of Global Criteria Method using Generalized Reduced Gradient technique. Some sensitivity analyses on the inventory model due to the change in some parameters like rate of deterioration and rate of lost sale are presented.
Section snippets
Preliminaries
Definition 1 Fuzzy number A fuzzy subset of real number R with membership function is called a fuzzy number if is normal, i.e. there exist an element x0 such that ; is convex, i.e. for all x1, x2 ∈ R and λ ∈ [0, 1]; is upper semi-continuous; and is bounded, here .
Example 1 Trapezoidal fuzzy number
Trapezoidal fuzzy number (TrFN) is an example of fuzzy number with the membership function , a continuous mapping and is defined by
Assumptions and notations
The following assumptions and notations have been used in developing two-warehouse inventory model.
Model description
A retailer brings his/her item from outside supplier and store them first in OW and then excess units in RW. The demand of the customers are met from OW and that vacant place at OW are filled up by transferring the units from RW to OW in a certain interval of time t1. Initially S units are purchased out of which W units are kept in OW and (S − W) units in RW. At first, the stocks at OW declines due to customer’s demand and deterioration of the item. After time t1, Q units from RW are transported
Solution methodology
The models (Model-1 and 2) represented by (29), (31) are multi-objective models which are solved by Global Criteria (GC) Method with the help of Generalized Reduced Gradient technique.
Global Criteria Method to solve Model-1 (and Model-2):
The Multi-Objective Non-linear Integer Programming (MONLIP) problems are solved by Global Criteria Method converting it to a single objective optimization problem. The solution procedure is as follows.
- Step-1:
For integer value of n, solve the multi-objective
Numerical example and solution
Illustration 1. To illustrate the proposed inventory Model-1, following input data are considered. λ = 676, μ = 2, ψ = 1, ω = 0.325, H = $1.15, δ = 0.5, θ1 = 0.05, θ2 = 0.04, α = 3000, β = 0.4, γ = 0.6, a = 4000, b = 1000, W = 150, p = $56, m = 1.92, cs = $9, cg = $11, cd = $8.10. Lead time is a Trapezoidal Fuzzy Number [0.59, 0.61, 0.69, 0.71], which is transformed to an interval number [0.6, 0.7], i.e. LL = 0.6, LR = 0.7.
Optimum results:
The problem (29) is a non-linear integer programming problem. For optimum value of (29), at first
Conclusion
The present paper proposes a solution procedure for a two warehouse inventory system with selling price-dependent demand rate and deterioration. Here, shortages are allowed and backlogged partially/fully. In real life, ordering cost decreases with the increase of lead time. Lead time is taken as imprecise via a fuzzy number. The fuzzy number is described here by linear type membership function. Fuzzy number describing lead time is then approximated to an interval number. Following this, the
Acknowledgement
The authors express their thanks to the referees for comments which improve the quality of this paper.
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