مشکل موجودی تصادفی افقی محدود با منابع دوگانه: محدوده نزدیک و ابتکارات

We consider a class of multi-periodic non-stationary stochastic single-product inventory planning problems where two procurement modes can be used at each period: a first order with immediate delivery and a second order with a single-period delivery delay. Clearly, the slow delivery mode is less expensive than the fast. We develop a discounted backlog model, with non-stationary procurement, inventory holding and backlog penalty costs proportional to the ordered quantities, inventory levels and number of backlogged units respectively. The demands are defined as non-stationary and independent random variables. We partially characterize the optimal ordering policy structure and we develop theoretical bounds and heuristic approximations for this optimal policy. Efficiency of these approximations is illustrated via extensive numerical experiments.

Nowadays, in supply chains, the gap between procurement costs from different suppliers may be important, especially in the case where suppliers production sites are located in different countries. This is particularly due to the difference in raw materials and workforce costs between the production sites. Often, the difference in procurement costs goes hand in hand with the difference in delivery lead times: the longer the delivery lead time, the lower the unit ordering cost. In this paper, we model this issue in a stochastic multi-periodic inventory planning framework with backlog where a retailer can order twice at each period of the horizon: the first order is made using a fast procurement mode with immediate delivery, while the second order is made using a slow, but cheaper procurement mode that has a single-period delivery lead time. Unsatisfied orders at a given period are backlogged to be fulfilled in the next period. We associate this dual procurement process with a discounted cost inventory model with procurement, inventory holding and backlog penalty costs, which are non-stationary and proportional to the ordered quantities, inventory levels and number of backlogged units respectively. The random demands are assumed to be independent with possibly non-stationary probability distributions. From a global perspective, two main research trends are related to this class of problems. A first trend of papers analyzes the theoretical properties holding for such multi-ordering inventory systems (as the convexity of the cost function or the theoretical structure of the optimal policies) and studies, mainly under time-stationarity assumption, how classical inventory policies for single ordering systems (periodical review or continuous review policy) can be adapted to such multi-ordering systems. A second stream of research considers heuristics for non-stationary multi-periodic single ordering problems. This paper can be viewed in some sense at the crossing of these research streams. Indeed, first we extend and adapt some known theoretical results holding for non-stationary single ordering mode inventory models to the class of slow/fast ordering modes setting. Then using the methodology provided in (Morton and Pentico, 1995 and Anupindi et al., 1996) for a single procurement mode inventory model with lost sales, we generalize their approach to our two delivery modes backlog inventory setting. We show how the optimal ordering policy can be approximated by a classical non-stationary order-up-to inventory policy and we give numerical computation procedure in order to estimate heuristic approximations for the order-up-to levels, based on upper and lower bounds. A numerical analysis is provided in order to compare the approximate policies obtained from these bounds to the optimal solution numerically computed by a stochastic dynamic programming approach. The rest of this paper is structured as follows. In Section 2, we provide a literature review for the considered inventory setting. In Section 3, we introduce the model and the parameters. In Section 4, we first present an equivalent formulation via a classical cost transformation and we provide theoretical optimality conditions. The theoretical upper and lower bounds and associated heuristic approximations are presented in Sections 5 and 6. In Section 7, we provide a numerical study showing efficiency of the proposed approximations. Finally, we conclude and give some new research perspectives.

In this paper we have developed a procurement and inventory model in which two procurement modes are possible. We have provided closed-form expressions for the upper and lower bounds on the optimal values of the decision variables, and then using a heuristic, we have used these bounds in order to develop an approximate value for each of the optimal decision variables. The numerical results were satisfactory. Future areas for research include developing additional theoretical bounds or heuristic bounds under less restrictive assumptions for the two procurement mode system. It is also obvious that another improvement that should be made to the model is the procedure with which the upper and lower bounds are used to determine the approximate solutions. As a final improvement, we could include an information-updating process that may enhance the knowledge of the demand distribution.