سیستم های موجودی مصالح محدود همراه با تقاضای چند محصولی مستقل: شیوه های کنترل و محدودیت های نظری

In many practical problems inventory managers are confronted with aggregate constraints that result typically from limitations in workspace, workforce, investment or from targeted service levels. In this paper we discuss some multi-product inventory problems with independent items under one or multiple aggregate constraints. We analyze some recent and relevant references grouped into five categories: deterministic lead-time demand, news vendor, base-stock policy, (r, Q) policy and (s, S) policy. We investigate the proposed model formulations, the algorithmic approaches and benefits of a system approach versus an item approach. A multi-product wholesaler case study is presented. Finally we highlight the limitations from a practical viewpoint of these models and point out some possible direction for future improvements.

The issues addressed in this paper are concerns and problems encountered in practice by managers who are confronted with system wide goals on service level or costs. As such the company can have for example a strategy to achieve an overall fill rate service level of 97% for this year. This service level may be part of a service contract which has a financial impact in the form of costly penalties if this pre-set target service level is not achieved. In practice managers need to find solutions for the limited available capacity of several resources. The warehouse has a limited available space that is not easily surmountable without extra costs. The money available to invest in inventory also has its boundaries and is sometimes used as a direct key performance indicator. The limited available workforce capacity can be a reason to limit the number of orders, as each order requires a set of activities: administer, perform quality control, receive and put away the goods. So inventory managers have system wide limitations (space, money or workforce) or goals (service levels or costs), while the majority of classic inventory closed formulas focus on single items and are unable or inefficient to realize these conditions. Applying a single item approach to attain these goals is not a best practice, neither is it effective to satisfy the system's constraints. Nevertheless we see it being applied too often within companies, without realizing the loss in efficiency or in money this has as a consequence. An IT system that lacks the support for a system wide approach may however be another significant obstacle. We believe that it is unawareness of the existence of these system approaches, by a large number of managers, or the assumed insurmountable complexity of these approaches that prevents their widespread use. As a first example of the value of these system approaches, we want to refer to Sherbrooke (2004) who reports using a system approach on 1.414 spare parts resulted in a 46% reduction of inventory investment without a decrease in performance. We believe that a better understanding and insight of multi-product inventory problems with aggregate constraints should become common knowledge for the inventory manager, knowing that the first papers on these topics date back to the sixties and seventies. This will certainly help them to achieve their system goals and will have a positive impact on the key performance indicators. An optimal policy surface, see Gardner and Dannenbring (1979), is a practical tool to deduct the optimal link between system cost and system service, while fulfilling the system constraints. An optimal policy surface can be generated for each system based on its specific characteristics. In this paper we want to provide an overview of the relevant references for the considered policies together with some insights in the algorithms used. The usefulness in practice requires the possibility of handling large data sets and easy implementation, e.g. closed form expressions or the use of familiar software packages. Zipkin (2000) gives a broad overview of multi-product inventory management and its several aspects. An important observation is that multi-product systems and multi-location systems are fundamentally identical. We observe the following three categories of multi-product inventory problems: independent items with aggregate constraints, network of items and shared supply chain processes. The first category of independent items describes problems with distinct supply and demand processes and no supply–demand links between the items. Of course when there are no links at all between the items, each item can be treated individually. This is where we introduce one or multiple aggregate constraints on the whole set of items. These constraints are not network or supply chain process related but focus on available resources (space, investment and workforce) or system result (service level and cost). A second multi-product inventory category is a network of items with a supply–demand relationship such as: a series system, an assembly system, a distribution system, a tree system or a general system. Axsäter (2003) offers a good overview of multi-echelon serial and distribution inventory systems in supply chains. Song and Zipkin (2003) give a detailed review on the assembly-to-order systems, this is a system with last minute assembly. Finally there is a multi-product problem category where the items share the supply chain processes themselves. Two well known problems in this area are the joint-replenishment problem and the economic lot scheduling problem (ELSP). Axsäter (2006) discusses extensively both problems. In case of joint replenishment, a group of items should be replenished jointly as much as possible due to many reasons: joint setup costs, quantity discounts or coordinated transports. The ELSP on the opposite tries to spread the cyclic schedules for a number of items with constant demand and no backordering, due to a finite production rate and a minimized holding and ordering cost. In this paper we will focus on the first category of multi-product inventory problems with independent items. We consider several instances of this problem and the remainder of the text is organized according to the following inventory policies: • Deterministic leadtime demand. • Newsvendor: a single period model with a stochastic demand and penalty costs for ordering too much or too little. • Base-stock: an (r, Q) policy with Q=1, this is relevant when ordering costs are negligible compared with other costs. • (r, Q) policy: an order of size Q is placed as soon as the inventory position falls to or below the reorder point r. • (s, S) policy: an order is placed to reach the stock maximum level S as soon as stock falls to or below reorder point s.

Within a broad range of inventory policies we see a practical need for a system approach, rather than an item approach. This enables managers to realize their goals with an optimal mix between cost and service while confronted with limited resources such as workspace, workforce or investment. For managers this can be expressed in optimal system policy curves. Test cases with a significant number of items report a cost decrease from 10% up to 46%. Approximations, in performance measures and lead-time demand, are not always without risk. Lagrangian multipliers are often used as an initial approach in case of a single constraint. For the policies where one variable describes the behavior of an item we see that a step is made towards linear or quadratic programming. In case of discrete demand complexity is higher so lower bounds and heuristics are used. We presented a case study where a system approach was able to further improve the system wide results and incorporate multiple aggregate constraints such as investment and warehouse space. Future work might focus on exact formulation of performance measures and demand distributions. Within discrete demand we see referrals to compound Poisson demand as a next step and also the use of multiple constraints.