Component part stocking policies

Component Part Stocking Policies Ricardo Ernst

Georgetown University

David E Pyke The Amos Tuck School of Business Administration, Dartmouth College,

Hanover, New Hampshire 03755

This article presents research designed to aid firms who assemble many components into a final product. We assume that purchase quantities are fixed, and that all parts and components are assembled at one stage in a short time. Demand for the final product is represented by a stationary independent and identically distributed random variable; and unmet demand is backordered. Ordering is done on a periodic review basis. We develop infinite horizon, approximate expected cost, and expected service level functions, and we present an algorithm for finding approximately minimum cost reorder points for each part subject to a service level constraint. Extensive results on the accuracy of the approximations are presented. Due to the size of the problem, we present only limited results on the performance of the optimization algorithm.

1. INTRODUCTION

In recent times just-in-time purchasing has become widely used. Large firms negotiate long-term contracts with one or two suppliers of a given part or component, and the suppliers deliver small quantities frequently. Some firms even negotiate price reductions over time, and develop very close, cooperative re- lationships with their supplier network. Many manufacturers, however, do not have the luxury of long-term contracts and frequent deliveries. They represent a relatively small portion of their suppliers’ businesses and must settle for quantity discount policies and delivery schedules that their suppliers dictate.

One of the authors has worked with such a firm. This particular firm assembles 60 components into a single product. Some of the components are very simple, inexpensive items, such as clips and bolts. However, some components are very expensive, custom units, such as specially made Plexiglas sheets and custom pumps. This firm is a new manufacturing startup which is attempting to take to market the innovative ideas of the founder. Since the firm is so new, and the market so undeveloped, the purchased parts make up an extremely small portion of their suppliers’ businesses. Large quantity discounts and long lead times are the rule; in fact, in many cases, suppliers dictate fixed order quantities. This

Naval Research Logistics, Vol. 39, pp. 509-529 (1992) Copyright 0 1992 by John Wiley & Sons, Inc. CCC 0894-069X/92/040509-21$04.00

510 Naval Research Logistics, Vol. 39 (1992)

particular firm was interested in knowing when to order parts and components, given that the purchase quantities were fixed.

We have spoken with other manufacturing managers who, although working for much larger firms, indicated that they face similar issues in purchasing. Lead times are long; lead times among different components vary widely; and demand for the final product is random. Large discounts for quantity purchases imply fixed quantity ordering in the sense that the optimal order quantity is often the minimum quantity to which the discount applies. In other words, it is costly to either increase or decrease the order quantity.

This article presents research designed to aid these firms. We assume that purchase quantities are fixed. [In general, these fixed quantities could be the economic order quantity (EOQ), the quantity discount breakpoint, or some supplier-dictated fixed quantity.] We assume that, as in the start-up firm just mentioned, there is only one level of assembly. In other words, all parts and components are assembled at one stage in a short time. (In the sequel, we use the terms “part” and “component” interchangeably.) One product is assembled. Demand for the product is represented by a stationary independent and identically distributed (iid) random variable; and unmet demand is backordered. Ordering is done on a periodic review basis. We develop infinite horizon, approximate expected cost, and expected service level functions, and present an algorithm for finding approximately minimum cost reorder points for each part, subject to a fill rate constraint for the final product.

The rest of this article is organized as follows. In Section 2, we present a review of the relevant literature. In Section 3, we introduce the model, with assumptions and notation. In Section 4, we present our approximations and the optimization algorithm. Computational results and testing are presented in Sec- tion 5 ; conclusions and extensions are discussed in Section 6.

2. LITERATURE REVIEW

Lambrecht, Muckstadt, and Luyten [ 191 introduce a dynamic programming model to obtain the optimal operating policy for multistage systems with uncertain demand, finite horizon, and periodic review. They modify the Clark and Scarf (8, 91 approach for serial systems to get generalized (s, S) policies. They show that these policies appear to be optimal for assembly-type systems. These policies perform well for low lead times and low demand variability; but all tests were performed on two-stage serial systems, rather than on assembly systems. Our model introduces the additional factor of a service level constraint for meeting demand; and we consider fixed purchase quantities, rather than variable ones suggested by (s, S ) policies.

Mamer and Smith [20] extend the so-called “tool kit” problem to incorporate general stock levels and inventory depletion. The tool kit problem assumes that multiple parts make up a field-serviceable unit and attempts to determine the optimal stocking policy for a repair kit carried by a service person. Visits to the warehouse generally are allowed after each service visit. Mamer and Smith allow dependent demand among parts and usage rates that are greater than one part per field unit. They develop an objective function which serves as a lower bound for the service level, and they find minimum cost stock levels to bring the lower

Ernst and Pyke: Stocking Policies 51 1

bound up to the desired level. Thus they guarantee that the service level constraint will be satisfied, but at a somewhat higher cost than the true optimum. In this model the repair truck can return to the store after t periods. In our problem, it is not possible to replenish all parts at the same time (due to fixed quantities, stochastic demand, and different lead times).

Cohen et al. [lo] consider a variant of the repair kit problem, in which a firm must stock multiple parts that compose a field-serviceable product. They use a demand weighted service objective at the product level, and assume that part demand is independent across parts. Demands arrive in two forms: normal replenishment orders from lower echelons in a multilevel distribution network, and emergency orders. Near-optimal (s, S ) policies are derived for all parts. In this article we are faced with demand at the part level that is identical for all parts; and we employ a service constraint at the product level.

Schmidt and Nahmias [22 ] consider an assembly system in which two components are combined to form one final product. They assume that there are no fixed costs of assembly or ordering; and they obtain the form of the optimal policy, which is essentially a base stock policy at assembly with an attempt to equalize inventory levels at components.

Yano and Carlson [25] consider safety stocks in assembly systems when there is a fixed production interval. They assume that the system uses a base stock policy, and that there are no fixed costs of ordering. They discover that only in certain. special cases is it cost effective to hold safety stock of components.

DeBodt and Graves [ l l] investigate a continuous review version of the assembly ordering problem. They consider joint replenishment of components and minimize expected holding and backorder costs. They use n,Q policies, in which each part’s order quantity is a multiple of Q. Clearly in our problem, it would be desirable to order using n,Q policies so that all components would be exposed to stockout at the same time. However, with many components, fixed quantities, different lead times, and stochastic demand it is impossible to have cycles for all components in sync.

Kumar [ 181 investigates component inventories when there is a fixed production interval and uncertain supplier lead times. He assumes one-for-one ordering for components, and measures inventory cost in terms of time spent in stock prior to assembly. Delay costs are linear in the time assembly is delayed. He finds that adding a component increases the cost for all other components. He solves for the optimal time to place an order from component suppliers by trading off the cost of holding inventory with the cost of delay.

Hopp and Spearman [16] consider a system in which multiple components are assembled into a product in a very short time. They assume that a final assembly schedule dictates production timing and solve for safety lead times when lead times are stochastic. Lot-for-lot purchasing is assumed. They provide two for- mulations of the problem: The first minimizes total inventory carrying cost and lateness costs, and the second minimizes carrying costs subject to a service constraint.

Other research on safety stock in assembly systems includes Baker. Powell, and Pyke [2], Bhatnagar and Chandra [6], and Hopp and Simon [15].

There is a stream of research on MRP purchase decisions. Specifically, these articles discuss purchase lot sizing issues when quantity discounts are in effect.

5 12 Naval Research Logistics, Vol. 39 (1992)

The lessons from this research are that time phasing of requirements is useful when demand is not stationary, that the EOQ is a not particularly good lot sizing choice, since it does not utilize time phasing information, and that the EOQ improves in performance as demand uncertainty increases. Benton [4], Benton and Whybark [ 5 ] , Benton [3], and Christoph and LaForge [7] are representative articles in this area.

Wijngaard and Wortmann [24] review MRP and inventories specifically to understand the capabilities of current MRP-I1 software with regard to safety stock, safety time, and hedging. Base stock control is assumed at all stages of production. There is no explicit treatment of mating delay, or the delay which results from one component being ready while its mate is not.

In conclusion, there has been no research to our knowledge that investigates component ordering in fixed batches (rather than lot for lot) when the components will not necessarily arrive in the same period, and when the absence of one component will prohibit assembly operations. In addition, we employ a product level service constraint and we develop approximations that can effi- ciently deal with many components.

3. THE MODEL

In this section we present our assumptions, notation, and the optimization problem.

Assumptions

We assume the following.

1. Components are assembled to order in a short time. For purposes of this study we assume that the production lead time is zero. In reality, it is very short. However, for some other firms, the lead time will be longer. Subsequent work should consider longer production lead times and the issue of storing assembled products as well.

2. One product is assembled from rn parts or components. 3. Product demand is a random variable with a known, stationary distribution, and is

4. Lead times for components are constant, but differ across components. (Lead times

5 . Ordering decisions are made at the end of each period. 6. Ordering quantities are fixed. These quantities may be dictated by the supplier, by

the EOQ, or by some quantity discount computation. 7. Production cannot begin until all components are available. 8. Unmet demand is backordered. 9. The order of events in a period is as follows: order receipt, demand, order placement.

10. The usage rate is scaled to be one part per product. 11. The order quantities are sufficiently large relative to mean demand so that backorders

independent across periods.

were from 2 to 8 weeks in the startup firm.)

do not approach infinity.

Notation We define the following notation. For random variables with the time subscript

t , the absence of t implies that the random variable is in steady state.

rn = number of parts. I,, = inventory on hand of part i at the start of period t .

Ernst and Pyke: Stocking Policies 513

I,, = approximate inventory on hand of part i at the start of period I. to be defined

I = min{l,. I , . . . . . lm}. I = min(1,. I ? , . . . , im}.

Q, = fixed order quantity for part i. T,, = I,, + S,(Q,) Q, = inventory on hand of part i in period 1. after orders from

suppliers arrive: S, (Q, ) = 1 if a batch of Q, arrives in period t , and S,(Q,) = 0 otherwise.

in Section 4.

~1

B, = backorders of the product at the start of period t .

Y, = demand for the product in period t . Y*' = demand for the product for x periods; i.e. the x-fold convolution of the single

period demand. L, = procurement lead time for part i. p = mean demand for the product per period.

F ~ . ~ = mean demand over the lead time of part i. u = standard deviation of demand for the product per period. a,, = standard deviation of demand over the lead time of part i If,, = inventory position of part i in period 1. If,, = inventory on hand + on or-

P, = production quantity in period 1. P, = Y, + B, if Y, + B, 5 min, T,,: and P, =

h, = holding cost per unit per year for part i. A, = fixed cost per order for part i. Z, = random variable representing the undershoot of the reorder point for part i.

der - backorders of the product.

min, T,, if Y, + B, > min, t,,.

This is the amount below the reorder point when an order is placed. E(Z, ) = expected undershoot.

n = number of periods per year. R, = reorder point for part i. p = required service level for the product.

Optimization Problem

We are interested in minimizing the total inventory holding and ordering costs for all parts, subject to a fill rate constraint for the product. We choose a fill rate constraint rather than a backorder cost since it is often difficult to specify a cost of stockout, particularly for a startup firm. The decision variables are R = {Rl , R2, . . . , Rm}. We first present some useful transfer equations.

Transfer Equations

The transfer equation for backorders is

B , + l = max{(B, + Y, - P,) , 0)

so that backorders increase by unmet demand until stocks are sufficient to satisfy backorders. Then B, = 0.

For inventory on hand, the transfer equation is

For inventory position:

ZPj(,+l) = Pi, + Q, - Y,, if ZP,, > R, and ZP,, - Y, 5 R i ,

otherwise.


The optimization problem is then

min total annual cost = TC(R,, R 2 , . . . , R,) R

s.t. E{1 - ( Y - I + ) + / Y } I p, (2)

where x+ = max(0, x}. The first term of (1) is the end-of-period holding cost, and the second term is the ordering cost. In (2), ( Y - I + ) + is the unmet demand in a randomly selected period. Thus, ( Y - I + ) + / Y is the proportion of demand not met, and 1 - ( Y - I + ) + / Y is the proportion of demand met, or the fill rate.

There are several aspects of this problem that significantly complicate the analysis. First, note that Z,, the inventory on hand for part i, is a function of not only its own reorder point and order quantity, but also of all other parts (via P,), as well as of product demand. When demand occurs, if there is no inventory of part i available, production cannot occur, and thus, inventory on hand for parts j # i will not decrease. As an illustration see Figure 1. Inventory on hand for part 2 does not follow the traditional curve, since the stockout of part 1 stops production. Thus, inventory on hand for part 2 is constant until part 1 becomes available. Since we assume that production time is zero, the inventory drops sharply when part 1 becomes available. (If the order for part 1 arrives before it stocks out, the effect of the dependence disappears.)

Second, the fill rate function depends on the inventory of all parts as well. Figure 2 illustrates the complications in fill rate. In this example, part 2 would have stocked out if part 1 had been available. However, the stockout of part 1 halts production, thereby allowing the part 2 order to arrive prior to production beginning again. The effect is an increased observed service level for part 2 over what would have been obtained in a single-part case.

Third, finding I + depends on the synchronization among parts. Consider for instance the following example. Two identical parts, each with the following distribution of li, compose a product.

Inventory level Probability -1 0.5

0 0.4 1 0.1

If the parts were independent, E{I+} would be 0.01. However, the parts are dependent, since the demand for each is exactly the product demand. The impact of the dependence between parts on E{I+} varies with the way ordering is carried out for the two parts. Complete synchronization of ordering (both parts ordered at the same time and with identical inventory positions), which would be desirable, implies that E{I+} = 0.10. Multiple parts that have different parameters complicate the analysis further.


Part 1

Inventory Level

Part 2

Inventory Level

I Time' I I I

I I I I I I ! I

Time * Figure 1.

To derive exact analytical expressions for the inventory on hand for a given part or for the product fill rate is extremely difficult. One could model the system as a Markov process, but the state space would explode rapidly with increasing numbers of parts or with long lead times. Hence we derive approximations and compare them to simulation in order to test their accuracy.

4. APPROXIMATIONS

In this section we present our approximations for the components of the cost function, and for the fill rate.


Part 1

Part 2

Inventory + Level

Time

I I

R ,

I++ L2

Figure 2.

Ordering Cost

The expected annual ordering cost is

The fixed holding cost A includes the cost of postage, purchase order forms, phone calls, perhaps a portion of the shipping cost, and other costs that vary with the number of orders, but not with the size of the order. In the short run, the cost of the purchasing agent’s time is probably fixed. An incremental order


will not increase out-of-pocket costs to the firm. However, many firms are beginning to use activity-based costing (ABC), and therefore view all costs as variable. In the ABC context, the purchasing agent's time may be included in A;. Also, if there is an opportunity cost of spending time on ordering, that cost definitely should be included in A;. We note that with the advent of electronic linkages between firms, fixed ordering costs are declining.

Holding Cost

The key random variable is 1,. One important aspect of the approximation of 1; may be the triangular region illustrated for part 2 in Figure 1. However, our tests indicate that the effects of this region are minimal when the required fill rate is high. Hence, we ignore the times during which a part is available but production is halted due to a stockout of another part. As /3 decreases, these effects increase, and our approximations degrade. Results are discussed in Sec- tion 5 . We note in passing that we tried applying results from renewal theory as well as from queueing theory to approximate the triangular region, but the approximations did not improve significantly.

We report results for three different holding-cost approximations based on three approximations of l i , which we denote fi, !;, and !!, respectively. First, we examine a simple EOQ-type approach that is computationally trivial. Second, we account for the end of the cycle effects. Third, we use a uniform distribution to approximate the inventory position. We develop these in order, and we suppress the part subscript i for ease of notation.

Approximation 1: EOQ

This approximation assumes deterministic demand. When R > pL, we assume no stockout in the cycle. (A cycle is defined as the time between two successive replenishments of batches.) Thus, the minimum inventory for the cycle is approximately R - E ( Z ) - pL, or the reorder point less the undershoot decreased by the mean lead-time demand. The maximum inventory for the cycle is Q + R - E(2) - pL. When R 5 p L , which is unlikely for high fill rate, we adjust the standard backorder EOQ model (Hax and Candea [14]) for the undershoot. The approximation is

If R > pL average inventory = f' = Q/2 + R - E ( Z ) - pL. (4)

If R I pL average inventory = f' = (Q + R - E ( Z ) - P ~ ) ~ / ~ Q . ( 5 )

The distribution of Z is based on the excess random variable of a renewal process, and its expected value E ( Z ) is approximately ((a' + $)/p - 1)/2 (Silver and Peterson [23]).

Approximation 2: End-of-Cycle Effects

Since the EOQ ignores the variability of demand, we introduce another approximation that specifically accounts for the possibility of stockout at the end


of the replenishment cycle. Define two phases of the replenishment cycle. Phase 1 is the time from the start of the cycle until inventory crosses the reorder point; and phase 2 is the time just after crossing the reorder point until the end of the cycle. (See Figure 3.)

First consider phase 1. Inventory on hand for a particular part at the start of the cycle is R - E ( Z ) - pL + Q as above. Then at the end of the first period of the cycle, inventory decreases by one more period’s demand. Thus, inventory at the start of phase 1 (assuming end of day costing) is approximately R - E(2) - pL+l + Q. Phase 1 ends when inventory crosses the reorder point, or when inventory on hand is approximately R - E ( 2 ) . Therefore the average inventory during phase 1 is

R - E ( 2 ) + ( Q - P L . + I ) / ~ . (6) Phase 1 lasts approximately p - L time periods, where L is the lead time,

and p = Q / F , or the cycle length. Inventory at the end of the first day of phase 2 is R - E ( 2 ) - Y . Then, it

decreases by one period’s demand each day. (This reasoning assumes other parts have stock available, and production occurs. The approximation will certainly degrade for low fill rates.) Thus, inventory on hand at the end of period j is approximately

Therefore, the approximate average inventory per period for approximation 2 is

f 2 = [R - E ( Z ) + ( Q - P L + I ) / ~ I ( ( P - L) /P)

1 Time ’

Figure 3.


If R - E ( 2 ) I 0, we could alter the approximation slightly. Inventory on hand at the end of the first day of the cycle is approximately R - E(2) + Q - p L - Y . Then it decreases by demand each period, with the approximate average inventory on hand per period equal to

However, our tests indicate (to our surprise) that (8) is quite cases, even if p < L , i .e.. more than one order outstanding.

+ } . (9)

accurate in all

Approximation 3: Uniform Distribution

Hadley and Whitin [13] have shown that in continuous review (Q, R ) systems, the inventory position at any point in time follows a uniform distribution on the interval ( R + 1. R + Q). For periodic review (nQ, R ) systems, the inventory position follows a uniform distribution in any period after the review. In our system, with fixed order quantities, it is not necessarily true that the inventory position in any period will follow a uniform distribution. If the undershoot of the reorder point is greater than Q , for instance, the inventory position will not return to R + 1 after the review. (nQ systems would order a multiple of Q to guarantee that the inventory position would return to at least R + 1.) We observed the distribution of the inventory position using data from a number of simulations and discovered that the inventory position in any period appears to be uniform on the interval ( R + 1, R + Q). We also ran tests of the fit of the uniform distribution (Rohatgi [21]) for three cases using data from our simulation, and found that we could not reject the hypothesis that the observations are uniformly distributed on the interval ( R + 1, R + Q). Detailed results are presented in Table 1.

Therefore the inventory on hand at the start of a period is approximately y - Y * L . where Y * L is the L-fold convolution of demand and y is a uniform

Table 1. Fit of the inventory position to the uniform distribution. Kolmoeoroff-Smirnov Chi-square

Case test statistic (Rohatgi [21, p. 5391) test statistic (df)

Q = 50 R = O 0.00 133 Q = 158 R = 18 0.001 11 Q = 548 R = 69 0.000876

51.224 (49)

152.405 (157)

598.716 (547)

Critical Values Alpha

20% 0.003384 10% 0.003858 63.1671 (50)

118.498 (100) 5% 0.004301 67.5048 (SO)

124.342 (100)


random variable on the interval ( R + 1, R + Q ) . Thus, expected inventory on hand at the end of a period is

13 = E { ( y - Y*L - Y ) + } = E { ( y - Y*L+1)+}. (10)

The computational burden for approximations 2 and 3 is clearly heavier than for approximation 1. We report results of the accuracy of the approximations in Section 5 . The approximate total cost using f1j2 and f 3 will be denoted TC'(R), TC2(R), and TC3(R), respectively.

Service Level

We discuss two approximations for the fill rate, one that is easy to compute, and one that is more time consuming. The first assumes that parts are independent of each other, and takes a simple product of individual fill rates. The second employs the uniform approximation as described above.

Approximation 1: Independence

For an individual part ( Q , R ) system, the fill rate is 1 - ,!?{shortage per cycle}/ Q . For normally distributed demand the fill rate is

where G,(k,) = $;, ( x - k i ) f ( x ) dx is the unit normal linear loss function, uLi is the standard deviation of demand over the lead time of part i, and k, is the safety factor (so kiuLi is the average safety stock just before a replenishment order arrives). (See Silver and Peterson [23].) However, when there are un- dershoots, we adjust the formula as follows:

T, = 1 - ( G u ( k ; ) ~ x ~ i ) / Q i , (12)

where the safety factor ki = (Ri - E(2) - pLi)/uxti . and uX'; is the approximate standard deviation of lead-time demand plus the undershoot, so uxvi = (& +

and

02 = (1/12)[4E(Y3)/p - 3 ( E ( Y 2 ) / p ) 2 - 11 (13)

(Silver and Peterson [23]). The first approximation simply takes the product of all part fill rates:

This approximation will degrade as the interference among parts increases. Spe- cifically, if parts have low fill rates, it is more likely that a part which would have stocked out at a given time will not actually stock out, since production has been interrupted by another stocked-out part. This observation, in con- junction with Figure 2, indicates that (14) provides a lower bound on the true fill rate. In other words, the only effect of the interference is that actual part fill rates improve.


Approximation 2: Uniform Distribution

The second approximation uses the uniform approximation of the inventory position. Recall that the inventory on hand of part i at the start of a period is approximately ( y, - Y * L r ) where y, is a uniform random variable on the interval (R , + 1, R, + Q,). Denote d, = (7, - Y * L f ) + . Let d = min{dl, d 2 , . . . , d,} in a randomly selected period. Then the fill rate for the product will be approximately

@'(R) = 1 - ( Y - d ) + / Y . (15)

The computational burden of the second approximation far exceeds that of the first.

Optimization Algorithm

We now present an algorithm designed to find near-optimal reorder points for all parts, given a fixed order quantity for each part. If any set of parts is identical, we treat the set as one part in the algorithm. They will therefore have identical reorder points and order quantities, and interference will be eliminated. The algorithm increments the reorder point of the part that provides the maximum increase in service for the minimum increase in cost.

We use the algorithm for a given total cost approximation TC"(R), u = 1, 2, or 3, and a given service approximation @"(R), u = 1 or 2 .

Optimization Algorithm

1 . Treat all parts that have identical parameters as one 2. Set Q, for all i . using the EOQ, the quantity discount order quantity computation,

or the fixed order quantity set by the supplier. 3. Compute k, using (12), and then R,, for all i so that R, is the minimum value such

that the independent T, P p . Round down noninteger values of R,. The rounding and our observations on the independent service level imply that the obtained product service level at this step is less than or equal to 0.

4. Compute W(R) = the approximate service level. I f @"(R) 2 p go to 8; else go to 5.

5. Compute R, = [TC"(R, + 1, R,J - TC"(R)]I[@"(Rk + 1, R,,,) - @,"(R)] for all k . where R,Fk represents the vector of reorder points for all parts other than k . Thus, increment the reorder point for k , holding all others fixed, compute total cost and service; and then repeat for all k

6. Let a = argmin, R,. 7. R, = R, + 1. Go to 4. 8. stop.

5. EXPERIMENTAL RESULTS

In this section we present our tests of the approximations and the optimization algorithm, and the results of sensitivity analyses.

To test the approximations, we ran a total of nearly 1,000 cases. For each case, we first computed the quantity discount value of Q for each part. This value is assumed to be the fixed order size. Next, we selected parameters from


Table 2. Parameters for testing the approximations. No. of parts

1 2 5 20 40

Cases per

L 1 , 4 , o r 8 1 , 4 , o r 8 1 , 2 , . . . or10 1 , 2 , . . . or10 1 , 2 , . . . or10

For all cases:

approximation 120 120 60 40 20

h = l A = 1 or 10 /3 = 0.85, 0.90, 0.95, or 0.99

( L 4 = (10, 11, (10,3), (20, 11, (20, 3), Or (20, 6)

the data in Table 2, ran the optimization algorithm to the near-optimal R value for each part, and then ran a simulation at those values. The simulation ran for 10,000 days following a 250-day start-up phase. We investigated products composed of 1 ,2 ,5 ,20 , and 40 parts. Product demand followed a normal distribution with parameters as given in Table 2. We tested required fill rates of 0.85, 0.90, 0.95, and 0.99. For products composed of 5 , 20, and 40 parts, we only tested the first two holding cost approximations, since the uniform approximation is fairly time consuming to compute. Tables 3-5 present the overall average, minimum, and maximum percentage errors, respectively, in holding cost and total cost for the three approximations. They also present the absolute errors in the two fill rate approximations.

It is clear from these results that the approximations are quite accurate. Av- erage error in holding cost is under 2.6% for approximation 2 (end-of-cycle effects), and is under 1.5% for approximation 3 (uniform). Even the EOQ approximation shows average errors under 9% for holding cost, although maximum errors in holding cost for the EOQ approximation are quite high. What is of interest to managers, however, is the total cost error. Once again, all approximations are quite accurate, although the maximum total cost errors for the EOQ approximation are around 10%. If one is willing to compute the end- of-cycle approximation or the uniform approximation, the error in total cost should be less than 1.2% on average, with no total cost error greater than 2.3%. For the EOQ approximation, average errors in total cost increase slightly as p increases, whereas the opposite effect is evident for the other two approximations.

Errors in the fill rate approximations are quite small as well. Average absolute errors are less than 0.01, and maximum errors are less than 0.07. Errors diminish as p increases. As we would expect, the uniform approximation is better than the independence approximation. However, on average, even the independence approximation performs quite well.

Table 3. Average errors (all cases). Independent Uniform EOQ End of cycle Uniform Cases

Required pcr Holding Total Holding Total Holding Total fill fill fill rate cell cost (%) cost (%) cost (%) cost (%) cost (%) cost (%) rate (abs.) rate (ahs.)

0.85 90 2.81 3.50 2.53 1.16 1.48 0.67 0.0096 0.0055 0.90 90 8.15 3.77 1.65 0.79 0.88 0.42 0.0070 0.0041 0.95 90 8.14 3.97 1.09 0.55 0.52 0.27 0.0041 0.0025 0.99 90 7.84 4.06 0.70 0.38 0.33 0.19 0.0011 0.0007


Table 4. Minimum errors (all cases). Cases EOQ End of cycle Uniform Independent Uniform

Required per Holding Total Holding Total Holding Total fill fill fill rate cell cost (%) cost (%) cost (%) cost (%) cost (76) cost ('70) rate (abs.) rate (abs.) 0.85 90 2.68 1.27 0.05 0.06 0.01 0.00 0.0002 O.O(K)l 0.90 90 3.15 1.53 0.06 0.12 0.01 0.00 0.oooo O.O(KX) 0.95 90 3.42 1.70 0.04 0.02 0.02 0.01 O.oo00 O.(WK)I 0.99 90 3.51 1.81 0.04 0.04 0.03 0.01 O . m 0.Wx)

Table 5. Maximum errors (all cases). Cases EOQ End of cycle Uniform Independent Uniform

Required per Holding Total Holding Total Holding Total fill f i l l fill rate cell cost (%) cost (9%) cost (%) cost (%) cost (%) cost ('3%) rate (abs.) rate (abs.) 0.85 90 29.16 10.54 4.75 2.29 3.00 1.36 0.0687 0.0527 0.90 90 27.22 10.66 3.89 2.10 1.71 0.97 0.0453 0.0369 0.95 90 24.95 10.64 3.14 1.97 1.58 1.01 0.0209 0.0188 0.99 90 22.72 10.69 2.81 1.67 1.27 0.94 0.0070 0.0038

Tables 6-10 report average errors by number of parts. As the number of parts increases, errors in cost actually decrease slightly for the EOQ approximation, and remain substantially the same for the other two approximations. Errors in service increase as the number of parts increases, however. The uniform approximation is consistently more accurate than the independence approximation. A full set of results is available from the authors.

Table 6. Average errors (one part). Cases EOQ End of cycle Uniform Independent Uniform

Required per Holding Total Holding Total Holding Total fill fill f i l l rate cell cost (%) cost (%) cost (%) cost (%) cost (%) cost (%) rate (abs.) rate (abs.)

0.85 30 12.87 5.29 1.75 0.71 0.51 0.21 O.(M35 0.0018 0.90 30 11.87 5.29 1.27 0.56 0.47 0.21 0.0036 0.0019 0.95 30 11.01 5.30 1.05 0.51 0.43 0.22 0.0028 0.0017 0.99 30 9.95 5.23 0.89 0.47 0.41 0.22 0.ooo9 0.0007

Table 7. Average errors (two parts). Cases EOQ End of cycle Uniform Independent Uniform

Required per Holding Total Holding Total Holding Total fill fill fill rate cell cost (%) cost (%) cost (%) cost (%) cost (%) cost (%) rate (abs.) rate (abs.)

0.85 30 5.99 2.70 3.26 1.51 2.45 1.14 O.(W2 0.0026 0.90 30 6.81 3.19 1.84 0.89 1.28 0.63 0.0036 0.0020

0.99 30 7.06 3.62 0.61 0.34 0.25 0.15 O . m 0.0005 0.95 30 7.12 3.48 1.05 0.54 0.61 0.33 0.0028 0.0015

Table 8. Average errors (five parts). Cases EOQ End Of Independent Uniform

Required per Holding Total Holding Total fill fi l l fill rate cell cost (%) cost (%) cost (%) cost (%) rate (abs.) rate (abs.)

0.85 15 5.19 2.45 2.73 1.30 0.0091 0.0057 0.90 15 5.92 2.83 1.88 0.91 0.0066 0.0041 0.95 15 6.48 6.16 1.19 0.60 0.0031 0.0020 0.99 15 6.69 3.37 0.60 0.33 0.0008 0.0007


Table 9. Average errors (20 parts).

Cases EOQ End Of Independent Uniform Required per Holding Total Holding Total fi l l fill

fill rate cell cost (%) cost (%) cost (%) cost (%) rate (abs.) rate (abs.) 0.85 10 5.11 2.56 2.51 1.24 0.0274 0.0155 0.90 10 5.69 2.87 1.86 0.92 0.0172 0.010s 0.95 10 6.21 3.17 1.18 0.59 0.0083 0.0055 0.99 10 6.44 3.36 0.62 0.31 0.0025 0.0012

Table 10. Average error (40 parts).

Cases EOQ End Of cycle Independent Uniform Required per Holding Total Holding Total fill fill

fill rate cell cost (%) cost (%) cost (%) cost (%) rate (abs.) rate (abs.) 0.85 5 5.03 2.51 2.22 1.11 0.0426 0.0252 0.90 5 5.50 2.82 1.68 0.84 0.0280 0.0175 0.95 5 5.94 3.08 1.10 0.55 0.0144 0.0087 0.99 5 6.14 3.24 0.61 0.31 0.0035 0.0015

To test the effect of low service levels on the approximations, we compared the EOQ and end-of-cycle approximations to the simulation for a variety of values of p, and for two parts. (See Table 11.) It is clear that when p is low, both approximations significantly underestimate the true cost. As noted above, our aproximations ignore the times when another part is stocked out. Thus, the triangular region in Figure 1, which is larger for low service levels, is absent from the approximate inventory levels. It is interesting that the end-of-cycle approximation is not as accurate for very low p as the EOQ approximation. Errors in the fill rate approximations are quite reasonable for very low service. The figures in Table 11 are for one run per row only.

In general, it is difficult to test the performance of the optimization algorithm because it is necessary to search over a wide set of policies. Hence, we restricted our tests to four cases, each with two parts. For each case, we ran the optimization algorithm to the approximately optimal Ri value for each part, using the uniform approximation of holding cost and fill rate. Then, using simulation, we searched

Table 11. Errors (two parts).

Cases EOQ End Of Independent Uniform Required per Holding Total Holding Total fi l l fill

fill rate cell cost (%) cost (%) cost (%) cost (%) rate (abs.) rate (abs.) 0.50 0.55 0.60 0.65 0.70 0.15 0.80 0.85 0.90 0.95 0.99

1 1 1 1 1 1 1 1 1 1 1

-20.92 - 14.80 -8.81 -3.99 0.43 3.91 6.48 8.19 9.21 9.61 9.51

- 7.42 - 5.37 -3.29 - 1.56 0.10 1 S O 2.62 3.47 4.07 4.44 4.61

-44.56 -35.09 - 26.14 - 18.78 - 12.92 - 8.74 - 5.71 -3.52 - 1.98 - 1.01 -0.43

- 16.24 - 12.89 -9.76 -7.16 - 5.07 - 3.56 -2.43 - 1.58 - 0.94 -0.53 -0.27

- 0.0014 - 0.0008 -0.0006 -0.0007 -0.0012 -0.0011 -0.0016 -0.0018 -0.0015 - 0.0002 -0.0004

0.0019 0.0020 0.0016 0.0008 O.oo00

-0.0002 -0.0009 -0.0012 - O.OOO9 0.0003

-0.0003


Table 12. Errors in the optimization algorithm (two parts). Cases

Required Per H o I d i n g Total Fill fill rate cell cost (%) cost (5%) rate (abs.)

0.85 1 0.87 0.38 0.0061 0.90 1 0.82 0.38 0.0054 0.95 1 0.77 0.37 0.0047 0.99 1 0.00 0.00 0.0000

For each cell: p = 1 0 u = 3 h , = 1 h 2 = 1 L , = 1 L , = 4 A , = 1 Az = 10

a large grid of R values for each part to find the “true” optimal. If the optimal R happened to be on an edge of the grid, we extended the grid accordingly. Hence, while we cannot prove that the policies found to be optimal are truly optimal, we are quite confident of the results. The results indicate that the algorithm performs very well, at least for this subset of cases. (See Table 12.) Errors in total cost from using the uniform approximation and the optimization algorithm, as opposed to using simulation and search, are less than 0.4%. And deviation in fill rate is less than 0.0061.

To add to these results, we also ran a series of sensitivity analyses. The results were consistent across parameter sets and numbers of parts, so we will report only a two-part case. The procedure was to use the simulation model, run the optimization heuristic until a 95% fill rate [using @(R)] was satisfied, and then vary Q and R for each part while holding the other part’s decision variables fixed. Thus, we can observe the effect of changes in Q and R on total cost and product service level.

First, we were interested in knowing what penalty in cost a user may incur if Q should vary from the fixed quantity. Figure 4 illustrates the effect on total cost of varying Q from the fixed quantity by 250. As is clear from the figure, the deviation in total cost is less than roughly 7%. These results are supported by analytical results on using EOQ in a stochastic demand environment in Zheng [26]. Apparently, fixed order quantities will not hurt performance if they are not too far from the EOQ. Figure 5 illustrates the increase in product fill rate by increasing Q for one part, while holding Q for the other part and both R values fixed.

Figure 6 illustrates the product fill rate effects of varying R for one part while holding the other parameters fixed. The function appears to be nondecreasing concave in R , with an asymptote. The key insight from this analysis is that increasing inventory for one part will benefit product service level, but only up to a point. When inventory for this part is high enough, product backorders are caused entirely by the other part(s). Also, below the chosen reorder point, the slope of the part with the smaller order quantity is steeper, indicating that service benefits from increasing that part’s inventory in that region will be greater. On the other hand, above the chosen reorder point, the service benefits of increasing that part’s reorder point disappear more quickly.


Cost Effects of Deviations from EOQ

Total Cost ($/period)

-50 -40 -30 -20 -10 0 10 20 30 40 50 EOQ

Figure 4.

Service Effects of Deviations from EOQ

S e r V I C e

L e V e I

-50 -40 -30 -20 -10 0 10 20 30 40 50 EOQ

I - Part 1 (R 85) + Part 2 (R 7 ) 1 Figure 5.

1

S

r 0.95

V I C e 0.9

L e " 0.85 e I

0.8

Ernst and Pyke: Stocking Policies

Service Effects in R

521

-15 -10 -5 0 5 10 15 R

1 - Part 1 (EOQ 85) -+- Part 2 (EOQ * 224) 1 Figure 6.

6. SUMMARY AND CONCLUSIONS

We have introduced several approximations for the problem of stocking component parts that are assembled into a single product. We discovered that even the simple approximations are reasonably accurate. The more involved approximations are very accurate, averaging less than 1.2% error in total cost. We also introduced an optimization algorithm which performed very well in our limited tests.

Future research sould investigate multiple products with component part commonality. There is a small, but growing, literature in component commonality that demonstrates the difficulties involved. (See, for example, Baker, Magazine, and Nuttle [l], Gerchak, Magazine, and Gamble [12], and Jonsson and Silver [ 171.) In addition, future research should consider nonzero production times. Therefore, the firm can hold stock of component parts, finished goods, or both. Again, there are some insights in the literature that should aid the analysis (Yano and Carlson [25]). Finally, it is important to examine both nonzero production times and component part commonality.

ACKNOWLEDGMENTS

We wish to thank Thomas Dyckhoff of Georgetown University and John Peairs, formerly of The Tuck School, for writing computer code and for helpful discussions concerning this research. We have also benefited from conversations with Steve Graves and Larry Robinson. The two referees provided many useful comments.


REFERENCES

[l] Baker, K.R., Magazine, M.J., and Nuttle, H.L.W., “The Effect of Commonality on Safety Stocks in a Simple Inventory Model,” Management Science, 32, 982-988 (1986).

[2] Baker, K.R., Powell, S.G., and Pyke, D.F., “Buffered and Unbuffered Assembly Systems with Variable Processing Times,” Journal of Manufacturing and Operations Management, to be published.

[3] Benton, W.C., “Multiple Price Breaks and Alterntive Purchase Lot-Sizing Proce- dures in Material Requirements Planning Systems,” International Journal of Pro- duction Research, 23(5), 1025-1047 (1985).

[4] Benton, W.C., “Purchase Lot Sizing Research for MRP Systems,” International Journal of Operations and Production Management, 6(1), 5-14 (1986).

[5] Benton, W.C., and Whybark, D.C., “Materials Requirements Planning (MRP) and Purchase Discounts,” Journal of Operations Management, 2(2), 137-143 (1982).

[6] Bhatnagar, R., and Chandra, P., “Variability in Assembly and Competing Systems: Effect on Performance and Recovery,” Working Paper No. 91-02-18, McGill Uni- versity, 1991.

[7] Christoph, O.B., and LaForge, R.L., “The Performance of MRP Purchase Lot- Size Procedures under Actual Multiple Purchase Discount Conditions,” Decision Sciences, 20, 348-358 (1989).

[8] Clark, A., and Scarf, H., “Optimal Policies for a Multi-Echelon Inventory Problem,” Management Science, 6 , 474 (1960).

[9] Clark, A., and Scarf, H. , “Approximate Solutions to a Simiple Multi-Echelon In- ventory Problem,” in K. Arrow, s. Karlin, and H. Scarf (Eds.), Studies in Applied Probability and Management Science, Stanford University Press, Stanford, 1962, Chap. 5.

[lo] Cohen, M.A., Kleindorfer, P.R., Lee, H.L., and Pyke, D.F., “Multi-Item Service Constrained (s, S) Policies for Spare Parts Logistics Systems,” Working Paper No. 229, The Amos Tuck School of Business Administration, Dartmouth College, 1989 Naval Research Logistics, to be published.

(111 DeBodt, M., and Graves, S.C., “Continuous Review Policies for a Multi-Echelon Inventory Problem with Stochastic Demand,” Management Science, 31( lo), 1286- 1299 (1985).

[12] Gerchak, Y., Magazine, M.J., and Gamble, A.B., “Component Commonality with Service Level Requirement,” Management Science, 34(6), 753-760 (1988).

(131 Hadley, G., and Whitin, T., Analysis of Inventory Systems, Prentice-Hall, Engle- wood Cliffs, NJ, 1963.

[14] Hax, A , , and Candea, D., Production and Inventory Management, Prentice-Hall, Englewood Cliffs, NJ, 1984.

(151 Hopp, W.J., and Simon, J.T., “Bounds and Heuristics for Assembly-Like Queues,” Queueing Systems, 4, 137-156 (1989).

[16] Hopp, W.J., and Spearman, M.L., “Setting Safety Leadtimes for Purchased Com- ponents in Assembly Systems,” Working Paper No. 89-01, Department of Industrial Engineering, Northwestern University, 1989.

[17] Jonsson, J., and Silver, E.A., “Common Component Inventory Problems with a Budget Constraint: Heuristics and Upper Bounds,” Engineering Costs and Produc- tion Economics, 18, 71-81 (1989).

[18] Kumar, A., “Component Inventory Costs in an Assembly Problem with Uncertain Supplier Lead-Times,” IIE Transactions, 21(2), 112-121 (1987).

[19] Lambrecht, M.R., Muckstadt, J.A., and Luyten, R., “Protective Stocks in Multi- Stage Production Systems,” International Journal of Production Research, 22(6),

[20] Mamer, J.W., and Smith, S.A., “A Lower Bound Heuristic for Selecting Multi- Item Inventories for a General Job Completion Objective,” Working Paper, UCLA Graduate School of Management, 1987.

[21] Rohatgi, V.K., An Introduction to Probability Theory and Mathematical Statistics, Wiley, New York, 1976.

1001-1025 (1984).


[22] Schmidt, C., and Nahmias, S., “Optimal Policy for a Two-Stage Assembly System under Random Demand,” Operations Research, 33(5). 1130-1145 (1985).

[23] Silver, E.A., and Peterson, R., Decision Systems for Inventory Management and Production Planning, 2nd ed. Wiley, New York, 1985.

[24] Wijngaard, J., and Wortmann, J.C., “MRP and Inventories,” European Journal of Operations Research, 20, 281-293 (1985).

[25] Yano, C.A., and Carlson, R.C., “Safety Stocks for Assembly Systems with Fixed Production Intervals,” Journal of Manufacturing and Operations Management, 1,

[26] Zheng, Y., “On Properties of Stochastic Inventory Systems,” Working Paper, De- cision Sciences Department, The Wharton School, University of Pennsylvania, 1990.

182-201 (1988).

Manuscript received 2 / 5 / 9 1 Manuscript revised 8/ 19/91 Manuscript accepted 11/19/91

Documents

Component part stocking policies