电子表格实现配送中心的库存控制毕业论文外文翻译

外文文献翻译Spreadsheet Implementable Inventory Controlfor a Distribution CenterAbstractThis paper develops and tests a simple procedure for establishing stocking rules for a multi-component distribution center that supplies spare parts for an equipment maintenance operation. Our basic formulation seeks to minimize inventory investment subject to constraints on average service level and replenishment frequency. We simplify this formulation by classifying parts according to a new ABC methodology and applying heuristics to the classical (Q, r) model that lead to closed-form expressions for the stocking parameters. Our numerical results show that:(1) the proposed ABC scheme does not introduce large errors provided that it is done in a manner that reflects the key parameters in the model, and (2) any of a number of simple reorder point heuristics can provide the basis for an effective spreadsheet implementable system for controlling inventory in a complex multi-component environment as long as the service level is checked against the exact formulaKey Words: multi-component inventory control, spare parts, ABC methodology, heuristics1. Motivation and backgroundFirms that support after-market operations are facing an increasingly challenging environment. As customer expectations rise and product complexity increases, they are being forced to stock an enormous variety of service parts. At the same time, product life cycles are becoming shorter due to rapid technology advancement, increasing the likelihood of obsolescence. Under these conditions advanced inventory management methodologies can significantly reduce the inventory cost of achieving a given level of customer service. However, many firms in this type of situation still have not invested in advanced inventory management methodologies, largely due to the difficulty of implementing existing methodologies. According to a benchmarking study of service parts logistics by Cohen, Zheng and Agrawal (1997), most firms are still using simple, easy-to-understand inventory management techniques. Indeed, the survey indicates that almost all (11 out of 12) of the participating companies use some form of simple ABC method for parts classification. Although ABC methods are simple, they are not always effective for two main reasons. First , traditional ABC methods classify parts based only on the dollar value of the sales, which tends to allocate a large portion of the capital investment to expensive parts. Second, the ABC disaggregation does not by itself eliminate the need for optimization of stocking parameters within each group. Since most companies lack the necessary optimization capacity, they often make use of ad hoc and unreliable heuristics. In this paper, we formulate the inventory control problem as one of minimizing inventory investment subject to constraints on average service level and replenishment frequency . We generate a simple solution procedure by first developing a parts classification scheme that is more effective than traditional ABC methods because it is sensitive to key attributes of the parts in a manner consistent with the optimization model. Within each category, we constrain service and order frequency uniformly, and then use various approximations to compute stocking parameters. We show by numerical examples that the proposed classification scheme does not introduce large errors. We also show that a variety of simple heuristics for the stocking parameters can be used effectively within this framework, as long as the exact formula for service level is used to adjust the formulas to achieve target performance. All of the calculations involved in implementing this system can easily be done within a simple spreadsheet. Hence, the methods proposed here are eminently implementable in practical settings and represent viable alternatives to methods currently in use. The remainder of the paper is organized as follows. In Section 2, we present the model. In Sections 3-5, we derive a simple rule for categorizing parts and closed-form expressions for computing order quantities and reorder points. In Section 6, we evaluate the performanceof the heuristics relative to a lower bound on the optimal inventory investment for given service/order frequency targets. The paper concludes in Section 7.2. Model formulationThe basic problem is to determine how to stock and replenish inventories of spare parts in a distribution center that supports a repair operation. We model this inventory problem qualitatively as a constrained optimization problem as follows:Minimize inventory investment (2.1)Subject to: average order frequency F (2.2) average service level S (2.3)where F and S are the target order frequency and service level as specified by the user. Notice that we are making use of a constrained model, instead of a cost-based model with costs for placing fixed orders and stockouts. In some sense, the two approaches are analogous, since the constraints can be brought into the objective function via Lagrange multipliers. However, we found our industrial clients more comfortable with specifying values for the constraints than the costs. This model gives them a clear framework for examining the inventory cost of achieving various levels of service and order frequency, and is therefore natural to use in decision support mode. Notice that we are making use of average service level, which means that having one part with a very high service level can compensate for another part with a low service level. This is not always appropriate (e.g., when customers order several different parts to complete a single repair operation and therefore their perceived service is not the average but rather the minimum among the parts in their order). However, as we will note below, additional constraints can be used to help the model address this situation. To make formulation (2.1)-(2.3) precise, we make use of the following notation:N=number of itemsCi=unit cost for item iDi=expected demand for item i per yearDtot=Dii=replenishment leadtime for item i (assumed constant)i=Dii, i=standard deviation of demand during lead time for item iQi=order quantity for item iri=recoder point for item iAi（ri ，Qi）=probability of stockout for item iBi(ri ，Qi)=expected number of backorders for item i at any time For ease of modeling, we assume demand occurs one at a time for all part numbers. If parts are actually ordered in batches, (e.g., sucker cups are always used in sets of 12), then we assume that all parameters and decision variables are also defined in units of batches. We let pi()and Pi()represent the probability mass function and cumulative distributionfunction of demand during replenishment lead time for item i. Because parts are for repairs that must eventually be completed, we assume demands that cannot be filled from stock must be backordered. We can now express the objective function mathematically by writing the expected onhand inventory (i.e., inventory position minus on-order inventory plus backorders) for item i at any point in time as （2.4）where Bi (ri，Qi ) is the expected number of backorders for item i at any point in time which can be computed as （2.5）Where （2.6）Note that Bi(v) represents the time-weighted backorders arising from lead time demand in excess of v. The conditions required for the above expressions to hold are fairly general, provided Qi and ri are integers and inventory position is uniformly distributed on ri+1,，ri+Qil (see Zipkin, 1986). The order frequency constraint is also straightforward to express, since an order of quantity Qi implies that the average number of orders per year for item i is Di/QiSo, we can write the order frequency constraint as （2.7）Finally, we can express the service constraint mathematically by noting that average service level is 1-Ai (ri，Qi), where Ai (ri，Qi ) is the probability of stockout at any point in time (i.e., the exact definition of fill rate), and can be computed as （2.8）Where （2.9）Note that ai (v) can be viewed as the expected lead time demand in excess of v.With these, formulation (2.1)-(2.3) can be written as:Minimize (2.10)Subject to: (2.11) (2.12) (2.13) (2.14) : integers (2.15)Note that we have added constraints (2.13)-(2.15) as restrictions on the allowable values of Qi and ri. Constraints (2.14) and (2.15) are obvious, merely representing the reality of discrete parts. Constraint (2.13) requires reorder points to be at least as large as some preset numbers. For instance, we might set ri=Bi, which would ensure that service level for allpart numbers is at least 50% (approximately). This is one way of preventing the use of the average service criterion from leading to unreasonably low service levels for some parts. Formulation (2.10)-(2.15) is a large scale (depending on the number of parts in the system) nonlinear, integer optimization problem. Even solving the Lagrangian relaxation requires solution of a large optimization problem. Hence, to meet our goal of a spreadsheet implementable system, we must simplify it somehow. We do this in two stages. First, we derive a classification scheme that divides parts into categories within which service level will be made constant. This vastly reduces the size of the optimization problem by limiting the search to only a few service levels instead of one for every part in the system. Second, we specify approximations for the expressions in (2.10)-(2.15) in order to generate simple closed-form expressions for the order quantities and reorder points. We put these together in a simple spreadsheet that performs a simple search on the few remaining variables to find a low cost solution that satisfies the constraints in (2.10)-(2.15).3. ABC classification of partsWe begin by determining a classification scheme. To do this, we observe that many approaches to the (Q, r) problem lead to the following functional form for reorder points (3.16)where ki is set in various ways. For instance, using a service-constrained approach with Type Si, given by the probability that there is no stockout during lead 1993), in a single product model leads to this form with ki=zsi where zsi time (Nahmias, is the standard normal ordinate such that (zsi)=Si .In a more sophisticated analysis of the multi-product case with the assumptions that average inventory can be approximated by。Ri-i+Qi/2 and the Type I formula is used to compute average service level, Hopp, Spearman and Zhang (1997) derive an expression for reorder points that is of this form with (3.17)where is a Lagrange multiplier corresponding to the average service constraint. Note that by itself this expression does not yield the simple formula we need for spreadsheet implementation because it still requires solution of a large-scale optimization problem to compute the optimal Lagrange multiplier, . But it does suggest a categorization scheme, since items with higher values of Di/li/ci2 are given higher ki，values and hence higher service levels for given values of ci, li , and Di .Therefore, instead of ranking parts according to unit cost or annual cost, as is typically done in ABC analysis, we propose ranking them according to the ratio Di/li/ci2 and then dividing into ABC groupings. This gives us the basis for a classification scheme. To make it into a methodology, we must decide on the number of categories to use. Obviously, the more categories, the better the accuracy (the maximum number of categories is the number of items). Since traditional ABC methods typically use three categories, we choose to follow suit and also term our categories A, B, and C. That is, we sort the part numbers in ascending order according to their Di/li/ci2 values, and assign, for instance, the first 20% of the part numbers to group A, the next 30% of the part numbers to group B, and the remaining 50% to group C. Parts in group A will be given the lowest service, parts in group B will be assigned moderate service, and parts in group C will have the highest service.4. HeuristicsThe above classification scheme groups parts according to a ratio that determines the choice of reorder point. Of course, we must also set order quantities for the parts in the system. However, there is literature supporting the accuracy of methods that set order quantity and reorder point separately (see e.g., Zheng, 1992, Hopp, Spearman and Zhang, 1997). If we do this, then it is possible to solve for all of the order quantities subject to an order frequency constraint and then use the classification scheme to set reorder points subject to a specified service level for each category. Below we discuss the procedure for setting order quantities and then present several heuristics for setting reorder points.4.1. Computing order quantitiesWe can derive a closed-form approximation for each order quantity Qi for any given order frequency requirement by making use of the traditional EOQ formula (4.18)for some K>0, where we will determine the appropriate value of the "fixed cost" K so as to meet the desired order frequency. To satisfy the order frequency constraint, we need (4.19)Therefore (4.20)If we set (4.21)Since we restrict Qi 1, we refine this definition to: (4.22)Furthermore, we round Qi to the nearest integer. In most of the examples we have examined this rounding results in feasible solutions. Occasionally, however, an infeasible solution is found (i.e., the order frequency constraint is violated). When this happens, the decisionmaker can either live with the violation (i.e., because the order frequency target F was probably a rough goal anyway) or round the Q; values up to the next highest integers so that the order frequency constraint is guaranteed to be satisfied.4.2. Computing reorder pointsNow, given order quantity Qi and target service level S; we need a method for determining ri. To do this, we approximate demand as normally distributed with mean B; and variance .The normal distribution is often convenient for representing demand processes in practice. In systems where the Poisson distribution is appropriate because data on demand variability is unavailable, the normal distribution with mean and variance B; is a reasonable approximation, especially where lead time demands are large. Let Fi() and fi()be the cdf and pdf of the normal distribution with mean i and variance. To get simple expressions for the reorder points r;, we need to approximate the service level expression by a more tractable formula. We consider two approximations from the literature and then develop a combined "hybrid" measure from these approximations.4.2.1. Type 1 approximation. The simplest approximation of fill rate is the (Nahmias, 1993). Under this approximation, the service of part i can be expressed aswhich, along with our normal approximation of lead time demand enables us to solve for the reorder point that achieves this service level aswhere zsi is the standard normal ordinate such that (zsi)=Si. Type I service corresponds exactly to the fill rate (i.e., proportion of demands met from stock) when Qi=1 in the discrete case and tends to be close to the fill rate when S; is large. However, reorder points obtained from (4.24) usually achieve service levels higher than Si, since Type I service underestimates the true service level in general. Note that (4.24) can be represented in closed-form in a spreadsheet by making use of the built-in inverse normal function (i.e., for zsi) or by using the following polynomial approximation for the inverse normal distribution ramowitz and Stegun, 1964):If Si is greater than 0.5, we solve for 1-Si and reverse the sign.4.2.2. Type II approximation. A second approximation of fill rate is known as Type II service, which is defined as (Nahmias,1993). Research results have shown that the Type II service measure is a good approximation at moderate and high service levels. The reason is that Type II service differs from the exact expression for fill rate only through the omission of the term ai (ri+Qi ), which only becomes important when there is a significant probability that the lead time demand will be greater than ri+Qi, which does not happen unless service level is low. Because it drops the ai (ri+Qi) term, Type II service tends to underestimate the true fill rate. Therefore, any feasible solution resulting from the Type II service constraint will be feasible to the original problem. Using our normal approximation for lead time demand, along with the Type II service definition, we can writeUnfortunately, cannot solve in closed form for ri Therefore, to find ri such that for given Si and Qi , we further approximate and (i.e., use the Type I approximation within the Type II formula). This allows us to derive a closed-form expression for ri.(Since Type I service is accurate for high Si or when Qi is close to 1, we would expect this approximation to be accurate under these conditions as well.) This results in:AndSolving for ri we get4.2.3. Hybrid approximation. We can generate yet a third formula for the reorder points by examining the type II formula for ri closely. Notice that (1) the formula limits the choice of ri to be equal to either ri or , where ki is always positive, and (2) the formula is not monotonically increasing in Si (as intuitively it should be). Furthermore, in our numerical experiments we have observed that the constant k can be too large (i.e., the service measure is too conservative). Therefore, to compensate, we can combine the type I and Type II results into the following hybrid expression for the reorder points: (4.27)As we show in our numerical examples, this correction can make a significant difference, enabling the hybrid approximation to outperform the Type II approximation.5. Spreadsheet implementationWe can make use of the above classification scheme and heuristics in a spreadsheet to compute Qi and ri (Table 1). The basic procedure for doing this is summarized below and then illustrated with a numerical example in Section 6.1. Create a separate section with cells for target order frequency F, service level S, and group service targets, SA, SB, Sc. Create columns indexed by part number i for cost ci lead time li (in years), demand rate Di .Compute i=Di.li in a separate column.2. In another column compute Qi using where ROUND(x) means to round x to the nearest integer.Check the order freuency constraint by computing ; in another column. If ,then the Qi are feasible. Otherwise the user can either (a) live with the constraint violation, or (b) round up some of the Qi hanging the formula toIf all Qi values are rounded up, the order frequency constraint is guaranteed to be satisfied.4Divide the part numbers into three groups by computing in another column and sorting the part numbers in ascending order according to this index.5Assig