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A Model for Resource Constrained Production and Inventory Management

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A Model for Resource Constrained Production and Inventory Management
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  A Model for Resource Constrained Production and Inventory Management Kurt zyxwvuts retthauer, Bala Shetty, Siddhartha zyxw yam, nd Susan White zyx eparment of Business Analysis and Research, Texas A&M University, College Station, zyxwvu X 77843 ABSTRACT We present a general model for multi-item production and inventory management problems that include a resource restriction. The decision variables in the model can take on a variety of interpretations, but zyxwvu ill typically represent cycle times, production batch sizes, number of production runs, or order quantities for each item. Weconsider environments where item demand rates are approximately constant and performing an activity such as producing a batch of a product or placing an order results in the consumption of a scarce resource that is shared among the items. Some examples of shared resources include limited machine capacity, a restriction on the amount of money that can be tied up in stock, or limited storage capacity. We focus on the case where the decision variables must be integer valued or selected from a discrete set of choices, such as when an integer number of production runs is desired for each item, or in order quantity problems where the items come in pack sizes containing more than one unit and, therefore, the order quantities must be an integer multiple of the pack sizes. Wedevelop a heuristic and a branch and bound algorithm for solving the problem. The branch and bound algorithm includes reoptimization procedures and the heuristic to improve its performance. Computational testing indicates that the algorithms are effective for solving the general model. Subject Areas: Mathematical Programming and ProductiodOperations Management. INTRODUCTION We present a general model for multi-item production and inventory management problems that include a resource restriction and decision variables that must be integer valued or selected from a discrete set of choices. The decision variables will typically represent cycle times, production batch sizes, number of production runs, or order quantities for each item. We consider environments where performing any of these activities results in the consumption of a scarce resource that is shared among the items. For example, there may be limited machine capacity when items are manufactured z 6] [16] [18], a restriction on the amount of money that can be tied up in stock [8] [9] [12] [13] [17], or limited storage capacity for the items [9] [Ill [12] [16]. These situations all require including a single constraint in the model to enforce the resource restriction. The objective function of the model includes reciprocal, linear, and constant terms which often arise from setup (or ordering) costs, inventory holding costs, and purchase costs. The model also includes lower and upper bounds on the decision variables. This allows for additional non-shared restrictions that affect each item individually, such as requiring a minimum and/or maximum number of production runs or a supplier specifying a minimum allowable order quantity. zyx 561 Decision Sciences Volume 25 Number 4 Printed in the U.S.A.  562 Resource Constrained Production z Specific examples of our integer model have received zyx ome ttention in the literature. Consider, for instance, a setting such as the one described in Sundararaghavan and Ahmed zyxwvut   zyxw 81 where zyxwv   different products zyxw re produced in a common facility. One of the problems addressed involves determining the minimum cost integer number of batches to produce for each item with a restriction on the number of days the facility is available for production. Sundararaghavan and Ahmed assume processing times are independent of the batch sizes and the products are usable only after a complete batch has been produced. Their model is appropriate for producing many chemical products like resins, paints, inks, pharmaceuticals, and dyes [18]. As another example, consider “basic period” approaches to the economic lot scheduling problem (ELSP) where the integer variables represent integer multiples of a basic period cycle time. Elmaghraby z 7] presents a formulation of the ELSP that includes a single feasibility constraint and can be stated in the form of our general model. In a problem unrelated to production and inventory management, optimal allocation in stratified sampling [4] [51 [lo] [ll] is a special case of our general model where the integer variables represent sample sizes. In specific hierarchical production planning models with continuous variables [ 13 [2], the cost function has the same structure as our general model and the additional constraint becomes necessary to ensure consistency between the aggregate production plan and a more detailed production plan. Other integer variable production and inventory management problems with a single resource constraint representing a restriction on inventory investment include Goyal [8], Lawrence and Schaefer [13], and Smith and Schaefer [17]. Goyal considers situations where a product must be packaged into a number of container sizes after manufacture. Lawrence and Schaefer and Smith and Schaefer develop models to determine the size of the spare parts inventory for the maintenance center of a repairable system. The resource constrained production and inventory management model proposed in this paper is formulated as a nonlinear integer program. To the best of our knowledge, there is no method in the literature for solving this problem to optimality. Its continuous relaxation, however, zyxwv an be solved easily via manipulations of the Kuhn-Tucker conditions when the bounds on the decision variables are ignored. Hax and Candea [9], Ibaraki and Katoh [ll], Johnson and Montgomery [12], Maloney and Klein [14], Ventura and Klein [19], and Ziegler [20] all present continuous variable models and solution techniques that are based on satisfying the Kuhn-Tucker conditions. None of these studies address the integer problem and, therefore, we develop a heuristic that provides good integer solutions to the general model and a branch and bound algorithm that provides optimal integer solutions. Our solution methodology for the continuous variable subproblem at each node of the branch and bound tree generalizes the above mentioned approaches to handle lower and upper bounds on the variables. A branch and bound algorithm for determining the optimal integer solution is desirable because the heuristic solution can, in some cases, yield much higher costs than the optimal solution (see, for example, Table zyxwvu ). The branch and bound algorithm seeks integer variable values that minimize average total cost per year while satisfying the resource constraint and the lower and upper bounds on the variables. We assume item demand rates are approximately constant. The form of the total cost function and other assumptions will be discussed further.  Bretthauer, Shetty, zyxwvu yam, and White zyxwv 63 z THE GENERAL RESOURCE CONSTRAINED MODEL zy he general resource constrained production and inventory management model will be formulated as subject to where zyxwvut   Xi n (P) Min C(ai + bpi + zyx i xi , i= zyxwvut   li s xi I zyxwv i, i= 1, ..., n, (3) xi a discrete variable, i = 1, , n, 4) the decision variable, i=l, ..., n (for now, its interpretation will be left general), number of items, cost coefficients for item i ai, b, c120), total amount of the limited resource available (f>O), consumption of the limited resource per unit of item i (di>o), proportion of xi consuming the limited resource (OceiSl), lower bound on decision variable xi (li>o, li=O will be discussed), and upper bound on decision variable zyx i (ui>li, ui== will be discussed). The interpretation of the decision variables, the cost terms in the objective func- tion, and the resource constraint will depend on the particular environment being modeled. We assume item demand rates are approximately constant and the cost pa- rameters ai, b, and ci are time-invariant and independent of xi for all i. Any additional assumptions will depend on the application being considered. Of course, the product of the two parameters di nd ei n the resource constraint could be replaced by a single parameter. Also, the solution techniques presented in later sections can be easily modified to solve problem (P) hen the resource constraint is an equality instead of an inequality. Next we show how the general model can be specialized for four different applications. THE NUMBER OF BATCHES PROBLEM WITH LIMITED MACHINE CAPACITY As discussed in a previous section, Sundararaghavan and Ahmed [ 181 study a problem that requires determining the minimum cost number of batches to produce for each item when there is limited machine capacity. The decision variable xi represents the integer number of batches of item i to produce and the general model can be written as:  564 zyxwvusrq esource Constrained Production z subject to liIxiIui, zyx = zyxwvuts , ..., n, xi integer, i = zyx , , n, where xi = the number of batches of item i to be produced, n = number of items being produced, Si = setup cost per batch of item i, Hi = annual inventory carrying charge per unit of item i, Di units demanded per year for item i Ti = machine time consumed per batch of zyx tem i, and Tt = total machine time available. In the objective function of problem (NB), zyxw y lSpi represents yearly setup costs and Zy .lHiDi/(2xi) epresents average yearly inventory holding costs. Note that cost parameter ui from problem (P) is zero. Also, because every batch of product i is assumed to consume Ti units of whine time, we set el=l for i=l, , n. The production batch size for item i, qi, is easily computed given xi via qi=Di xi, i= 1, . n. Order Quantities with an Inventory Investment Limit Various authors have addressed production and inventory management problems with an inventory investment constraint and integer variables [8] [13] [17]. We consider an environment where items are ordered and there is a restriction on inventory investment. As discussed by Silver and Peterson [16], some products are sold in pack sizes that contain more than one unit and, therefore, order quantities must be selected from an integer multiple of the pack sizes. The general model presented earlier can be written as n (OQ Min c(PiDi + HPpi /2 + AiDi xi), i=l subject to n i= 1  Bretthuuer, Shetty, Syam, and White li 5 xi zyxwvu   i, zyxwvu = zyxwvut , zyx .., n, 565 xi a discrete variable, i = 1, .. , n, where xi = order quantity of item i, n = number of items, Pi = unit variable purchase cost of item i Di = units demanded per year for item i, H = annual inventory carrying charge zyx s a percentage of unit purchase cost P, Aj = fixed cost of ordering item i, ei = a parameter satisfying O<eill, and F = upper limit on inventory investment. In the objective hnction of problem OQ), CY=,PiDi represents yearly item purchase costs, Zn ,HPpi / 2 represents average yearly inventory carrying costs, and Cn=lAiDi xi represents yearly fixed ordering costs. We assume instantaneous replenishment and no backorders. Different values of the parameter ei yield different interpretations of the resource constraint. Setting el=l for all i imposes an upper limit on maximum inventory investment. This upper limit may be overly restrictive because it implicitly assumes that orders for all n items are received simultaneously. If we use e,=S instead of z l=l for all i, then the resource constraint can be interpreted as imposing an upper limit on average inventory investment instead of maximum inventory investment. Basic Period Approaches to the Economic Lot Scheduling Problem The economic lot scheduling problem (ELSP) nvolves scheduling the production of multiple items in a common facility on a repetitive basis. Basic period approaches to the ELSP limit the choice of cycle times for the items to integer multiples of a basic period which is long enough to accommodate the production of all items. Elmaghraby [7] presents the following formulation of the basic period ELSP. n (ELSP) Min X(Si ( Wxi) + HiDi( 1 - Di Ei)>Wxi 2), i= 1 subject to n C(Gi + DiWxi /Ei) 5 W, zyx = 1 1 <xi 5 zyxwv i i = 1, ..., n, xi integer, i = 1, ,. , n,
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