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1、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 t
2、o 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 stockin
3、g 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
4、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 o
5、perations 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, increasin
6、g 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 meth
7、odologies, 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 a
8、lmost 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
9、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
10、 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
11、 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
12、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 formula
13、s 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 remainde
14、r 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 bo
15、und 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 inv
16、entory 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
17、 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 comf
18、ortable 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 s
19、ervice 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 no
20、t 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
21、=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 backorde
22、rs 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
23、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
24、 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
25、（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 Zip
26、kin, 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
27、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, formul
28、ation (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 p
29、arts. 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 lea
30、ding 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 go
31、al 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 sear
32、ch 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 perf
33、orms 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 funct
34、ional 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 standar
35、d 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 reor
36、der 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 pro
37、blem 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 an
38、nual 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 c
39、ategories, 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
40、/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 th
41、e 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 quant
42、ity 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 l
43、evel 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 u
44、se of the traditional EOQ formula (4.18)for some K0, 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
45、 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
46、 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 .
