Analysis of an inventory system under supply uncertainty

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Abstract

In this paper, we analyze a periodic review, single-item inventory model under supply uncertainty. The objective is to minimize expected holding and backorder costs over a finite planning horizon under the supply constraints. The uncertainty in supply is modeled using a three-point probability mass function. The supply is either completely available, partially available, or the supply is unavailable. Machine breakdowns, shortages in the capacity of the supplier, strikes, etc., are possible causes of uncertainty in supply. We demonstrate various properties of the expected cost function, and show the optimality of order-up-to type policies using a stochastic dynamic programming formulation. Under the assumption of a Bernoulli-type supply process, in which the supply is either completely available or unavailable, and when the demand is deterministic and dynamic, we provide a newsboy-like formula which explicitly characterizes the optimal order-up-to levels. An algorithm is given that computes the optimal inventory levels over the planning horizon. Extensions and computational analysis are presented for the case where the partial supply availability has positive probability of occurrence.

Introduction

In a real-life shipment problem, considering realization of sure delivery times and/or receipt of exact quantity ordered may not be proper assumptions. There are many reasons, such as capacity restrictions or non-deterministic transit times, for not satisfying due dates with desired quantities as planned. The aim of this article is to consider revised ordering quantities, if the desired quantity at the desired time is not available. The policy may not be applicable unless real-time delivery monitoring is achieved, but still it gives a benchmark to any other solution proposed.

In most of the production/inventory models that involve uncertainties in the environment, the attention has been focused on the probabilistic modeling of the customer demand side. Therefore, up until the recent years the uncertainties in the supply side have not received the amount of treatment they deserved. Especially, in the last decade, widespread application of just-in-time (JIT) practices in supply situations increased the importance of modeling the supplier uncertainties. Several factors, such as the shortages in material availability, unexpected machine breakdowns, process adjustments, strikes, etc., make the treatment of supply uncertainty an important issue in the analysis of inventory problems. For a recent review of supply/yield uncertainty models, please refer to Yano and Lee [1].

Supply uncertainty may take several different forms. Silver [2] considers an EOQ model where the quantity received is a random proportion of the quantity requisitioned. He obtains modified EOQ formulas under such a supply uncertainty model. In the spirit of the proportional supply uncertainty, Shih [3] and Ehrhardt and Taube [4] consider single-period models where the demand in the period is a random variable. Henig and Gerchak [5] analyze a periodic review model where the quantity received is a random multiple of the order size, and show the optimality of an order-up-to type policy.

Ciarallo et al. [6] analyze a stochastic demand production/inventory model where the available capacity in a given period is a random variable. They show the optimality of order-up-to type policies, but the generality of their model leads to intractable computations of the order-up-to levels. Under a similar capacity uncertainty model and average cost per period criterion Güllü [7] proposes a solution for the optimal order-up-to level by constructing an analogy with queueing systems.

Parlar and Berkin [8] propose an EOQ-type formulation where the supply is available or disrupted for random durations in the planning horizon. Karaesmen et al. [9] extend their model to incorporate correlations between supply availability and disruption durations. Parlar et al. [10] consider a periodic review model with set-up costs using a Markovian supply availability structure in which the supply is either available or completely unavailable. They show the optimality of (S, s) policies where s depends on the supply state in the previous period.

In this article we consider a single-item periodic review inventory system. In our model we assume that the demands in successive periods are deterministic and dynamic numbers. We assume linear holding and backorder costs and the objective is to minimize total expected costs over the planning horizon. We suppose that in any period the supply is either fully available, partially available or completely unavailable. In other words, the quantity ordered is either fully or partially satisfied, or completely fails to be fulfilled. Partial availability in our context means that, if an amount of u is ordered and u>Q for some predetermined Q, then there is a positive probability that only the amount Q will be shipped by the supplier. The amount Q is called the partial availability level. It can be thought as the capacity allocated to us by the supplier. Availability random variables are stationary and independent from one period to another. The assumption on the supply structure may not directly follow some real practices. However, the authors' of this article observed a real-life case, where the buyer is in a position of revising the quantity ordered at the end of each period. Hence, if a delivery is not made at a certain period, the total quantity to be delivered next period is computed taking the supply information into account, rather than leaving that late shipment in transit.

In this article we make three contributions. Under the above assumptions we present a supply uncertainty model integrated with a dynamic demand structure. We demonstrate the optimality of a non-stationary order-up-to policy and provide useful properties of it. As the main contribution that makes our work different from the previous research, we provide a simple newsboy-like formula for computing the optimal order-up-to levels over the planning horizon for the case of two-point stationary supply availability (supply is either fully available or completely fails). Finally, for the case of partial availability, we present computational results and a conjecture on the form of the optimal policy parameters.

The rest of the article is organized as follows. In Section 2we present our cost and supply model and analyze the form of the optimal ordering policy. In Section 3, under the assumption of two-point supply availability, the optimal parameters of the policy, the order-up-to levels are explicitly characterized and a one-pass algorithm is presented for computing the levels. It turns out that the optimal order-up-to level for any period n is equal to the cumulative demand of the following Kn periods. These levels can be determined by simply checking a newsboy-like ratio. In Section 4, we present our computational findings for the case of partial supply availability. We conclude the study, and discuss some extensions in Section 5.

Section snippets

The model and structural properties

Notation
Nnumber of periods in the planning horizon.
Dndemand in period n for n=1, 2,…, N.
piprobability that the supply is unavailable (i=1), fully available (i=2), partially available (i=3) in a given period, 0⩽pi⩽1,p1+p2+p3=1.
Qpartial availability level on supply.
S(u)realized supply if an amount u is requisitioned;
S(u)=0withprobabilityp1,uwithprobabilityp2,min(Q,u)withprobabilityp3.
hholding cost per unit per period.
bbackorder cost per unit per period.

Let Ln(y) denote the expected single-period

Solution in the case of two-point supply availability

In this section, we suppose that the supply structure can be characterized by a two-point distribution: supply is either fully available or unavailable (that is, p3=0). By using , we can write Eq. (2)asCn(I)=p1Gn(I)+p2miny⩾IGn(y).In the rest of the section, passing through various steps we will show that the optimal order-up-to level for period n is equal to the cumulative demand of Kn periods (including the demand for period n),yn*=i=nn+Kn−1Diforsome1⩽Kn⩽N−n+1.Moreover, progressing backwards,

Partial supply availability case: Computational results

Unfortunately, the compact solution obtained in the previous section cannot be extended easily to resolve the case where the partial availability probability is non-zero. Wang and Gerchak [13], in a similar setting (with some more general aspects, but under proportional supply availability) observe that, in a multi-period situation the realized “capacity” values will affect the order-up-to levels. Hence, possibility of obtaining a compact solution is very unlikely. In order to observe the

Extensions and conclusion

In this article we analyzed a deterministic demand inventory problem under supply uncertainty. For the case of two-point supply availability, we obtained a simple formula which determines the policy parameters of an optimal order-up-to level policy. Our formula would provide guidance as to the appropriate amount of inventory to stock in the face of uncertainties in the supply process. An efficient dynamic programming formulation was applied in the presence of relatively more general stochastic

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