Optimal base stock policies and truck capacity in a two-echelon system

Optimal Base Stock Policies and Truck Capacity in a Two-Echelon System

Ricardo Ernst Georgetown University

David F. Pyke Dartmouth College

With the recent trend toward just-in-time deliveries and reduction of inventories, many firms are reexamining their inventory and logistics policies. Some firms have dramatically altered their inventory, production, and shipping policies with the goal of reducing costs and improving service. Part of this restructuring may involve a specific contract with a trucking company, or it may entail establishing in-house shipping capabilities. This restructuring, however, raises new questions regarding the choice of optimal trucking capacity, shipping frequency, and inventory levels. In this study, we examine a two-level distribution system composed of a warehouse and a retailer. We assume that demand at the retailer is random. Since the warehouse has no advance notice of the size of the retailer order, inventory must be held there as well as a t the retailer. We examine inventory policies at both the warehouse and the retailer, and we explicitly consider the trucking capacity, and the frequency of deliveries from the warehouse to the retailer. Both linear and concave fixed transportation costs are examined. We find the optimal base stock policies at both locations, the optimal in-house or contracted regular truck capacity, and the optimal review period (or, equivalently, delivery frequency). For the case of normally distributed demand we provide analytical results and numerical examples that yield insight into systems of this type. Some of our results are counterintuitive. For instance, we find some cases in which the optimal truck capacity decreases as the variability of demand increases. In other cases the truck capacity increases with variability of demand. 0 1993 John Wiley & Sons, Inc.

1. INTRODUCTION

With the recent trend toward just-in-time deliveries and reduction of inventories, many firms are reexamining their inventory and logistics policies. Some firms are dramatically altering their inventory, production, and shipping policies with the goal of reducing costs and improving service. Part of this restructuring may involve a specific contract with a trucking company, or it may entail establishing in-house shipping capabilities. Both of these approaches tend to reduce the uncertainty surrounding time of delivery, thereby allowing for decreased inventory levels. However, they raise new questions regarding the choice of optimal (in-house) trucking capacity or contract volume with a trucking company, shipping frequency, and inventory levels.

Naval Research Logistics, Vol. 40, pp. 879-903 (1993) Copyright 0 1993 by John Wiley & Sons, Inc. CCC 0894-069X1931070879-25

880 Naval Reseurch Logistics, Vol. 40 (1993)

In this study, we examine a two-level distribution system composed of a warehouse and a retailer. Alternatively, we could see the two locations as being a component factory and an assembly plant. Because customer demand, or production schedules, may be largely unknown in advance, we assume that the demand at the lower echelon (hereafter, the retailer) is random. Thus, the retailer must hold inventory to buffer against this variability. Likewise, the warehouse, which has no advance notice of the size of the retailer order, must hold inventory. (The warehouse may be under separate ownership; or it may be part of the same firm as the retailer, but with a lack of real-time demand data. At this writing, only a few supply chains have information linkages that provide up-to-date data on realized demand at other locations. Bourland, Powell, and Pyke (51 investigate changes in inventory policies in light of up-to-date information that is transmitted via Electronic Data Interchange.)

We examine inventory policies at both the warehouse and the retailer, and we explicitly consider the contract volume, or the size of the fleet operated by the warehouse, and the frequency of deliveries from the warehouse to the retailer. Hereafter we denote in-house trucking to describe the fleet operated by the warehouse. By implication, we assume that it could refer to a specific contract with an outside trucking firm. We also denote trucking capacity to describe the volume in units of an in-house fleet or of a contract with a trucking firm. We assume that requests by the retailer must always be met. Therefore, if the trucking capacity is not sufficient to satisfy a request, the shipper must use a common carrier, at higher cost, to ship the difference.

We assume that the retailer faces stationary independent and identically distributed (iid) demand. Because the firm is charged a fixed transportation cost per shipment, the optimal form of the policy will have at least two parameters, perhaps of the (s, S) type. We choose to employ a base stock policy, however, because it is commonly used in practice, and because it affords analysis so that we can develop insights into these systems. Hence, we assume that the retailer orders using a periodic review, base stock policy: Every P days, the retailer orders enough to bring on-hand plus on-order inventory up to a level SR. (We are exploring more complex four-parameter policies in related research. See Henig and Gerchak [14] and Ernst and Pyke [ll].)

The warehouse faces demand only from the retailer. Thus, demand arrives at the warehouse in the form of retailer orders every P days, and the warehouse orders using a base stock policy with order-up-to-level Sw. For analytical simplicity we assume that the period used by the warehouse is also P. It is quite possible to relax this assumption. In fact, in reality the warehousing function often develops in response to the need to pool risk or to take advantage of economies of scale in ordering. For the current research, however, we restrict the analysis to one retailer, one product, and a common period in order to develop insights that will aid further research into more complex systems. We will see below that the literature on stochastic systems of this type is sparse. Research that considers both multiple products and multiple retailers is almost exclusively developed for the case of deterministic demand. The current research is designed to advance the sparse literature on the stochastic demand case, and as such begins with less complex models.

There is a fixed lead time LR to ship goods from the warehouse to the retailer, and a fixed lead time Lw to ship from the supplier to the warehouse. For

Ernst and Pyke: Optimal Base Stock Policies 88 1

simplicity we assume that 0 5 Lw 5 P. The retailer backorders excess demand, and the warehouse’s supplier has infinite supply. The retailer incurs a per-unit per-day cost for holding inventory and a per-unit per-day cost for backordering.

When the retailer places an order from the warehouse, the size of the shipment may be constrained by two factors. First, the warehouse may not have enough stock to completely fill the order. When this happens, we assume that goods are purchased at a premium. We can think of this premium as the cost of overtime production or of an expedited shipment from the warehouse’s supplier. Thus, the warehouse faces a trade-off between holding and shortage costs. Second, the trucking capacity may not be large enough to carry all the requested goods. In this case, we assume that the warehouse ships using a common carrier. We approximate the common carrier transport time by assuming that these shipments arrive at the retailer at the same time as regular shipments.

Shipping costs have several components. (See Daganzo [lo].) First, we assume a fixed cost per shipment, i.e., a cost that is incurred every time the truck is dispatched. There are several components of this cost. It includes a cost that is fixed for a given trip, regardless of contents or distance, which is composed in part of the cost of stopping a truck and keeping it idle while it is being loaded and unloaded. A cost per truck-mile is also incorporated. This includes the cost of the driver and other costs that are associated with distance, regardless of contents. We model this cost both as a linear, and as a concave, function of truck capacity. Second, we allow for a cost per unit shipped, which can depend on the weight, cube, or both. This cost may be quite small in relation to the fixed cost. In addition, we assume that the common carrier charges both a fixed cost per trip and a cost per unit of goods shipped. Common carrier costs often are composed solely of a cost per unit shipped, which is based on weight andl or cube. It is possible in our model to set any of the component costs to zero; however, for generality, we explicitly include each component. We assume that trucking capacity is a continuous decision variable. However, the model can be applied to a set of discrete choices of truck sizes and number of trucks.

To illustrate transportation costs consider the following example. Assume that the warehouse has 45 units on hand, the truck capacity is 40, and a demand from the retailer arrives for 60 units. Then the warehouse will ship 40 units via regular transportation, and place an emergency order for 15 units. When these units arrive, they and the remaining 5 units of warehouse inventory are shipped via common carrier to the retailer. Regular and common-carrier shipments arrive at the retailer in the same day. As this example illustrates, it is useless to have truck capacity greater than Sw, since at most Sw units can be shipped in regular mode. We now review relevant literature for this problem.

Jackson [ 161 describes the just-in-time (JIT) production system and highlights the effects for logistics managers. He discusses one significant implication of JIT systems for logistics-namely, the need to fully integrate procurement, production, and distribution. Conflicting demands are evident, since JIT aims to reduce inventories by, in part, utilizing frequent small shipments; logistics managers, on the other hand, aim to take advantage of economies of scale in transportation.

Herron [15] comments on situations in which expediting is preferred to stock- outs, and describes the trade-off among transportation costs, inventory costs, and stockout costs in determining the best shipping frequency.

882 Naval Research Logistics, Vol. 40 (1993)

Blumenfeld, Burns, Diltz, and Daganzo [3] address the trade-offs between transportation, inventory, and production costs in a deterministic setting. They examine a number of network configurations, including one origin shipping to one destination, one origin shipping to many destinations, many origins shipping to many destinations, and shipping via a consolidation terminal. In the more complex cases, they show how the network can be decomposed into subnetworks that can be analyzed more efficiently.

Blumenfeld, Hall, and Jordan [4] extend the previous work to the stochastic demand case. One item is produced by a supplier, shipped to a manufacturer, and consumed according to a random process. An order-up-to policy is used at the manufacturer, and the supplier is perfectly responsive to consumption. Therefore, total inventory in the system is constant. Fixed costs per shipment are charged for both regular and common carrier shipments. Illustrations and sensitivity analyses are developed for the case of normally distributed demand.

Benjamin [2] considers a joint production, transportation, inventory problem with deterministic demand. The issues are similar to those addressed in Blu- menfeld et al. but Benjamin allows for supply capacity constraints.

Yano and Gerchak (261 examine a system similar to the one we model in which a location, call it a retailer, is supplied by a source with infinite supply. The retailer follows a base stock policy, and fixed costs per truck movement are incurred for both regular and common carrier shipments. Shortage costs are charged on a per-unit basis. The transportation lead time is zero, and emergency shipments always supply the difference between requests and truck capacity. The decision variables are the order-up-to level at the retailer, the integer number of trucks (there will be at least one truck), and the period length for the retailer inventory policy. They find that the time between shipments is less than the time corresponding to the analogous economic order quantity (EOQ) value. They also find conditions under which it is desirable from a cost perspective to have more than one truck. These include high demand variability, high emergency shipment costs, low inventory holding costs relative to regular transportation costs, and reasonably low shortage costs relative to inventory holding costs.

Our research extends Yano and Gerchak to incorporate the inventory policy at the warehouse and to include per-item shipping costs. The decision variables are order-up-to levels at both locations, truck capacity, and period length. In contrast to Yano and Gerchak, we assume that truck capacity is a continuous rather than integer variable.

Henig and Gerchak [ 141 extend Yano and Gerchak to explore the optimization of a similar system in a general discounted costs framework. They assume that truck capacity is continuous and consider holding and shortage costs at the retailer, and a per-unit cost of emergency shipment. They derive the form of the optimal policy-a four-parameter policy-for this scenario and then address suboptimal base stock policies. Ernst and Pyke [ll] develop Markov models for the optimal policy (as defined by Henig and Gerchak) and for two suboptimal policies. These models allow comparison of the easy-to-use suboptimal policies with the optimal four-parameter policy. The Markov models are used to dem- onstrate the shape of the three cost functions and to suggest optimization algorithms.

Ernst and Pyke: Optimal Base Stock Policies 883

The vehicle routing problem is somewhat related to the issue we consider in this article. Ball, Golden, Assad, and Bodin [ l ] examine the problem of finding an optimal fleet size in conjunction with determining the vehicle routes. A common carrier ships all products that are not designated for the firm’s fleet. They provide two formulations for the optimization problem and illustrate the results with a case example from a client firm. Demand is taken to be deterministic.

Burns, Hall, Blumenfeld, and Daganzo [7] address the problem of minimizing transportation and inventory costs for a supplier distributing products to many customers. They compare direct shipping (distributing to each customer separately) with peddling (distributing to many customers via the same truck). De- mand is again deterministic and the results indicate that trucks should be dispatched full when peddling. Peddling is less expensive than direct shipping when products are costly, customers are distant or densely populated, or the carrying charge is high.

In the stochastic version of the vehicle routing problem the basic issue is to determine the best vehicle routes for multiple retailers who face stochastic demand. The goal generally is to minimize expected travel time or distance, subject to a constraint on the probability of completing the routes. Articles by Cook and Russell [9], Golden and Yee [13], and Tillman [19] are representative. This research area does not incorporate inventory costs or higher cost emergency shipments, which we consider in this article.

The integrated inventory-vehicle routine problem with random demands has been studied by Federgruen and Zipkin [12]. They examine a one-period problem with multiple retailers and random demands. The objective is to choose the optimal vehicle routes and shipment quantities, so as to minimize inventory, transportation, and shortage costs. Their model differs from ours in that we find the optimal review period; and, in contrast to our model, they allow backorders to endure until the next shipping period. (We require that all requested units be delivered, even if an emergency shipment is necessary.) In addition, vehicle routing decisions are not necessary for our one-warehouse, one-retailer problem.

Finally, Bregman, Ritzman, and Krajewski [6] consider the problem of inventory control and transportation costs, and Chandra and Fisher [S] investigate coordinating production and transportation decisions, both for the deterministic case. A good introduction to the entire problem area is provided by Daganzo

The contribution of our research is threefold. First, we develop simple models and optimal policies that extend the Yano and Gerchak article to a more complex situation. Second, we see this research as another step toward even more complex environments possibly involving multiple retailers and vehicle routing capability under stochastic demand. Finally, we develop insight into the nature of inventory policies and truck capacities.

For instance, we develop conditions under which it is optimal to have no trucking capacity in house, but rather to do all shipping on an emergency (or common carrier) basis. These conditions tend to confirm one’s intuition. How- ever, we also present numerical results that are somewhat counterintuitive in this regard. As an example, we examine a case in which the cost of shipping via common carrier is twice as high as the cost of regular shipment, in both per-

[lo].


shipment and per-unit costs, but the optimal truck capacity is zero. In other words, even though the per-shipment and per-unit cost of common-carrier shipments is twice as high as having some fixed trucking capacity in house (or a fixed contract), the optimal choice is to do all shipping using a common carrier. We discuss these, and other results, in Section 3.

This article is structured as follows. In Section 2 we describe the model and discuss the nature of optimal policies. We address linear and concave transportation costs in that section. We illustrate the use of the model in Section 3 by presenting a number of examples and the insights that follow. Finally, we sum- marize and discuss extensions in Section 4.

2. THE MODEL AND OPTIMAL POLICIES

In this section we describe the model of the two-level system. Recall that the truck capacity will always be less than Sw units since the truck carries regular transportation units and can carry at most Sw units. (It is possible to ship more than S R units since unmet demand is backordered.) We now introduce required notation.

2.1. Notation

costs

CB = backorder cost per unit ($/mitiday) CHw = holding cost at the warehouse ($/unit/day) CHR = holding cost at the retailer ($/mitiday)

C R = regular unit shipping cost ($/unit) Cc = common carrier unit shipping cost ($/unit)

g( T ) = fixed cost per shipment as a function of the regular truck size ($/shipment) K E = fixed cost per common carrier shipment ($/shipment) CE = marginal cost of obtaining units when the warehouse is stocked out ($/unit)

Demand

D, = demand In day 1

D"' = demand over y days with density f,(.), cdf F,(.) , mean p, and variance ut LR = transportation time from the warehouse to the retailer Lw = lead time to the warehouse

Decision Variables

S R = order-up-to level at the retailer Sw = order-up-to level at the warehouse

T = truck capacity in units P = period length in days

2.2. The Model

We now develop a cost function for inventory (holding and shortage) and shipping costs. Because we are assuming that Lw 5 P , the warehouse must buffer for P days of demand. Thus the supply lead time does not affect Sw, and


for simplicity we can disregard L , and denote L R as L. (We note that the close synchronization of the warehouse and the retailer will not be possible in a more complex setting of multiple retailers, unless the warehouse “pushes” products to the retailers. We leave this coordination problem for further research.) In this section we examine both linear and concave transportation costs. Finally, we will refer to a cycle as the time between successive replenishments-P days, in this case.

Because common-carrier shipments arrive in the same day as regular shipments, the retailer costs are the usual base-stock expressions. Thus, the per-day expected holding cost at the retailer is

Using end-of-day costs, the inventory position (on hand plus on order less backorders) begins the cycle at SR. By the end of the jth day, it decreases to S R less j days’ demand. The inventory on hand is L days’ demand less, because the pipeline clears after L days. Summing over P yields the total inventory on hand over the P-day cycle. We divide by P to obtain the average number of units of inventory on hand over the cycle. Then multiplying by C H R yields expected holding cost per day. By similar argument the expected daily backorder cost is

The warehouse holding cost follows from the fact that the rational warehouse manager will wait to order up to Sw until Lw days before the retailer’s order arrives. Thus, inventory at the warehouse resides at Sw less P days’ demand (the size of the retailer’s last order), spikes up when the order arrives, and immediately decreases to Sw less a new realization of P days’ demand. (See Figure 1.) Thus, the expected daily warehouse holding cost is

C H w TSw (S, - x ) f p ( x ) dx. J O

( 3 )

If the warehouse manager chooses not to delay ordering until Lw days before the retailer request is due, ( 3 ) can be modified accordingly.

If the warehouse stocks out, units must be purchased at a premium. The incremental cost of those units is CE. Thus, the expected daily stockout cost at the warehouse is

In (4) we divide by P to obtain the cost per day.

that is charged on a per-shipment basis. The per-day cost is The regular truck cost is composed first of a fixed cost per unit of truck capacity

‘fl P{D*p > 0). P

886 Naval Research Logistics. Vol. 40 (1993)

Inventoq Level

s,

I I I * Time

Figure 1. Warehouse Inventory

If we assume, as we do throughout the article, that there is essentially no probability of negative demand, we can let P{D"p > 0} = 1. Hence the truck will travel once each P days (if T > 0), and ( 5 ) becomes g( T)IP.

Additional regular trucking costs are charged per unit of goods shipped:

The first term represents the expected number of units shipped via regular transportation if demand is less than or equal to the truck size, charged at C R per unit. We assume only one regular shipment in a period of P days. Because demand may exceed the size of the truck, we include an additional term in the expected per-day regular unit transportation cost.

Common-carrier transportation costs consist of a fixed cost per shipment and a per-unit cost. The expected per-day fixed cost per shipment is

KE j; f p ( X ) dx. P

The expected per-day per unit common-carrier shipping cost is

+ i,; (x - T ) f p ( X ) dx.

(7)

As noted above, it is not uncommon for K E to be zero, while Cc is significantly greater than CR.


Thus, the optimization problem (I) is

Min (over Sw, SR, T , P)

We note in passing that the expected utilization of the trucking capacity, T , if T > 0 is

2.3. Properties of the Cost Function

For a given P , (I) separates into three optimization problems, in that solving them separately achieves a global optimum. This separation follows from our assumptions regarding transportation (common-carrier shipments arrive in the same day as regular shipments) and the retailer ordering policy (a base stock policy passes up demand in an unfiltered way to the warehouse). If the retailer were to use another policy, such as the four-parameter policy developed in Henig and Gerchak [14], (I) would not separate as conveniently. The three optimization problems are denoted Ia, Ib, and Ic:

(Ib) Min TC,, (over Sw) = CHw (SW - x) fp(x) dx il:"


(Ic) Min TC,, (over 7') = gm P{D*' > 0} + % / o T x f p ( x ) dx P

We can, therefore, solve easily for the optimal retail and warehouse base stock

It is easy to show that TCI, is convex in S R and that the optimal S R , S i , for levels, and focus our attention to a greater degree on trucking capacity.

problem Ia solves

Likewise, TCIb is convex in Sw and the optimal order-up-to level for the warehouse, S$, solves

2.3.1. Transportation Cost

It is generally agreed that the fixed cost per shipment is a complex function of the truck capacity. For instance, Daganzo [lo, pp. 42-46] discusses the relationship between shipment size and the cost per shipment of transportation and handling. This function is subadditive and increasing, but not concave. Several levels of complexity are open to those modeling these systems. One level is to explicitly model fixed cost per shipment as a subadditive increasing step function. Daganzo, however, suggests ignoring handling costs if shipment sizes are likely to be larger than a pallet, thereby eliminating steps in the function due to pallet handling costs. In addition, he notes that there is a large step increase in cost as shipment size increases from the maximum capacity of one vehicle to the next unit shipped ( T to T + 1, in our notation), because an entire second vehicle must then be employed. As a result, most firms will select the smallest number of vehicles possible. These considerations give rise to a second, less complex, approach: approximating the step function with a continuous concave function, which is appropriate especially when demand is low relative to truck size. A third approach is to use a yet less complex approximation-for example, a linear function. Herein we employ two approximation schemes; g( T ) is linear in T and g( T ) is concave in T. For each case we examine the shape of the total transportation cost function and develop an optimization algorithm.

If we assume that demand follows a normal distribution, computation of TC is simplified by transforming (1)-(8) into functions of the standard normal distribution. (See Silver and Peterson [18] and Johnson and Montgomery [17].) In the sequel, we will rely on the assumption of normal demands so that we can develop insight into these systems. We also assume that the parameters of the distribution are such that there is essentially no probability of zero or negative demand.


2.3.2. Linear Regular Transportation Cost

We first approximate the fixed cost of a regular shipment g( T ) by a linear function, KRL( T ) , where KRL is the fixed cost per unit of truck capacity charged on a per-shipment basis. We find

dTC1c - KRL + CR [l - FP(T)] + cc [Fp(T) - 11 - - KE f P ( T ) (11) dT P P P P

and

Because f P ( T ) IP L 0, the second derivative is positive if

Therefore, for a given P, TCI, is concave, then convex, as T increases: TC,, is convex when

and concave otherwise. (See Figure 2.)

2.3.3. The Optimal Truck Capacity, T*, for Linear Fixed Transportation Cost

When all parameters are positive, it is evident from (5)-(8) that TCI, + tfi,

as T-+ w; i.e., TCI, is convex increased as T-, a. [See Figures 2(a), 2(b), 2(d), 2(e), and 2(f).] Therefore, we employ the following algorithm to solve Problem Ic .

IT

Figure 2a. Transportation Cost for Parameters A4/B1


Figure 2b. Transportation Cost for Parameters A4/B2

Algorithm T L 1. Solve for T (using the first derivative):

If there are two solutions, let

i.e., the value in the convex portion of the function. If there is one solution, let T = F. Otherwise, let TC,,(Tl = 30. -

2. If TC,, ( T = 0) < TC,,( T ) , let T* = 0; otherwise, let T* = T.

Notice using (11) that because Fp(0) = fp (0) = 0, we have TCIc increasing at T = 0 if Cc < CR + K R L . Also, using (13), TCI, is convex at T = 0 if

We may comment, therefore, that for a large fixed cost of regular shipments and a small fixed cost of common-carrier shipments, TCIc is convex increasing at T = 0, and thus T* = 0 [see Figure 2(d)]. Also, large demand variability (&) and expensive regular transportation (C, + K R L ) imply that T* = 0. (Recall that we are assuming normal demand with negligible probability of zero or negative demand, so crp will not be larger than roughly pp13.) Finally, if Cc > CR + KRL, i.e., if the unit common carrier transportation cost is greater than

Figure 2c. Transportation Cost for KRL = 0


'T I

Figure 2d. Transportation Cost Convex Increasing at T = 0

the total (fixed and per unit) regular transportation cost, TCIc is decreasing at T = 0, and regular truck capacity will be purchased. [See Figure 2(f).]

We now examine the optimal T for Cc = CR = 0. Other special cases are explored in the Appendix.

When there is no incremental per unit cost of shipping, we can draw several conclusions. (This case follows Yano and Gerchak [20], because they assume a fixed cost per truck movement, but no incremental cost associated with additional units on the truck.) First, assume that K E > K R L . Then from (ll),

which is greater than zero for large T and for small T. If K , is not significantly larger than K R L , TCIc is increasing for all T > 0, and T* = 0 [Figures 2(d) and 2(e)]. If K E is sufficiently greater than K R L , on the other hand, the first derivative is negative at T = 0, and the optimal Twill be greater than zero. [See Figure 2(f), for instance.] In this case, we see from (12) that

which implies that the function is concave if T < p p , and convex if T > pP. Hence in Algorithm T L , we set the first derivative to zero and solve. Because the cost at T = 0 is higher than the cost at some positive T, we use the value of T that is greater than p p . That is, we find T > p P such that f p( T) = K R L I

K E .

TC IC

Figure 2e.

I 'T

Transportation Cost for Parameters A3


TCk

I 'T

Figure 2f. Transportation Cost for Parameters A l , A2, A5

2.3.4. Concave Regular Transportation Cost

We now examine the case in which g ( T ) is a concave function of T. Let g( T ) = KRC(TU), 0 < a < 1 , where K R C is a scaling factor. We then find

and

- = - a2 TcIc KRC a(a - l )Ta-* - $ f P ( T ) a l ? P

Because a < 1 , the first term of the second derivative is less than or equal to 0. So for T < p P , TCIc is concave unless Cc is quite large. For T > p P , TCIc is concave if

Thus for small values of K E and large values of K R C , TCIc is always concave. Now from (15) it is clear that as T + w, TCIc + m. Therefore, for small values of K E and large values of K R C , the function is concave increasing, and T* = 0.

On the other hand, TCIc is convex for T > ( u p if


2.3.5. The Optimal Truck Capacity, T*, for Concave Fixed Transportation Cost

In this case, we employ a similar algorithm to Algorithm TL.

Algorithm Tc 1 . Solve for T (using the first derivative):

F p ( T ) ( C C - CR) - f p ( T ) K E = ( C , - C,) - K R C ~ T m - ' . (19)

If there are two solution_s, let T. Otherwise, let TC,,(E) = to.

be the larger value. If there is one solution, let r =

- 2. If TC,,(T = 0) < TC,,(T), let T* = 0; otherwise, let T* = T.

We now examine the optimal T for Cc = CR = 0. From (15 ) ,

and from

Therefore

-- d2 TcIc - - K R C a(a - 1)Ta-2 + K, [ + f P ( T ) ] . (21) dz-2 P p f f P

the function is concave if T < p p . If for all T ,

- K R C a(1 - a)Ta-2 > $ [y f P ( I ) ] , P

the function is also concave when T > p p , and so T* = 0. Again, this condition will hold for large values of KRC relative to KE, and large 02.

From (20) it is evident that for T significantly less than p p , f P ( T ) will generally be close to zero. Thus, except for very unusual values of KE, up, and/or p p , (20) will be greater than zero, and the function will be increasing. The implication is that for a truck size that is small relative to mean demand it is less expensive to use a common carrier rather than to have in-house trucking capacity. This result is intuitive since, with such a small truck capacity in house, the common carrier will be employed in nearly every cycle anyway. At T = p p it is likely for the function to be decreasing. For example, let p p = 10, up = 3, a = 0.5. Then, at T = p p , to have a decreasing function is equivalent to KRLaTa-l - KEfp(T) < 0; which is equivalent to having KRL(0.5)(10-0.5) < K E / ( 6 up), or KEIKRL > 1.2. This latter condition is not unlikely, implying that a truck size larger than or equal to the mean demand over the cycle may be desirable.

2.4. The Optimal Period, P* We have calculated many examples of TCI, where for each P , we compute

the optimal values of Sw, SR, and T. In each case we observe that the function is convex in P with a unique minimum. However, we have not been able to


prove convexity. It is evident from (1)-(8) that as P+ m, total costs will increase. Our heuristic algorithm, therefore, increments P until costs are clearly increasing and very high. Specifically, we start with P = 1, compute Sg, S$, and T" (using Algorithm TL or Tc) , and record the total cost TCI ( P = 1). Then we increment P and repeat until the total cost gets very high. The algorithm, therefore, is performing a one-dimensional search on P.

1 . Set i = 1, 2. Set P = i. 3.

= 30. Set /3 = large number (say 40).

Solve la , Ib, Ic. Compute TC*(P = i, (Sg. S&, T*)IP = i). To compute T * , use Algorithm TI. or T, where appropriate. -

If T c * ( P = i, (Si , S&, T*)IP = i) < TC set - TC = TC*(P = i, (S;, S&, T*)jP = i) SK = SZlP = i , sw = S&IP = i, T = T*IP = i, P = i.

- - - -

Go to 4. 4. 5 .

If i < /3 let i = i + 1 and gz to-2;:therwise go to 5. The optimal values are SK, Sw, T , P.

In our experiments we found that the optimal P was 8 or less. However, for most cases we checked values of p up to 40, or even 100, to be sure we had found the optimal solution.

3. EXPERIMENTAL RESULTS AND INSIGHTS

3.1. Linear Transportation Cost

We first discuss results for the case of linear transportation cost. We examined 16 cases to develop some insight into the operation of this two-echelon system. The parameters are listed in Tables 1-3, and include a wide variety of cost and demand values. Specifically, we examined fixed common-carrier shipping cost at 2, 3, and 20 times the regular fixed cost; and we examined per-unit common- carrier cost of shipping at 2, 3, and 4 times the regular per unit shipping cost. We also include a case in which the fixed component of the common carrier cost is zero, but the per-unit common-carrier cost is four times that of in-house trucks. (We know from the Appendix conditions for which T* = 0 when K E = 0. To avoid the trivial case, we chose parameters A5 to narrowly violate these conditions.) We analyzed high and low holding and shortage costs. Finally, we used high and low variability of demand.

Table 1. Transportation cost parameters.

K R L O ~ KRC KE CR CC A1 5 15 2 8 A2 5 10 3 9 A3 5 10 2 4 A4 5 100 2 4 A5 5 0 2 8

Ernst and Pyke: Optimal Base Stock Policies

Table 2. Holding and shortage cost parameters.

CHR CHw CC CR B1 20 15 1000 250 B2 0.3 0.2 40 20

895

The results are summarized in Tables 4 and 5 . We highlight several interesting points. First, the optimal truck size is zero for parameters A3 ( K R L = 5, K E = 10, CR = 2, Cc = 4). It is interesting that, even though common-carrier costs are twice regular costs, the optimal decision is to have no in-house or regularly contracted truck capacity. The optimal truck size is also zero for cases 14 and 16 (involving parameters A4). The comparison of the A3 cases with the A4 cases is enlightening. The only difference in the two parameter sets is the fixed common carrier cost KE (10 or 100). Parameter set A3 has such a low fixed common carrier cost ( K E = 10) that the cost of always shipping in common- carrier mode is lower than the cost at the optimal positive truck size [Figure 2(e)]. However, in parameter set A4, the fixed common-carrier cost is high ( K E = 100) and we see optimal truck capacities that are positive only when holding and shortage costs are very high [Figure 2(a) or 2(b)].

One interpretation of these results is the following. When T* = 0 and the optimal period length is eight days, for instance, the firm will hire common- carrier trucking capacity every eight days. That policy, however, sounds very similar to regularly contracted trucking. The difference is that to use in-house or regular trucking capacity, as we have defined it, implies a limit ( T ) on the capacity. Demand that exceeds T must be shipped in a higher-cost common- carrier mode. Therefore, to have common-carrier trucking on a regular basis as implied by T* = 0 and a fixed P*, suggests that unlimited trucking capacity is supplied on a regular basis. That is, the cost structure does not change regardless of the volume shipped in a given period. The firm pays a premium (of K E -

K,, and CE - C,) for this flexibility. In today’s deregulated trucking environment, common-carrier costs are generally no more than 2.5 times regular costs, which suggests that a firm may be able to negotiate for this type of flexible capacity.

When T* > 0, we see some cases for which the optimal truck capacity (in house, or fixed contract) is greater than the mean demand (examples 5-8, 13, 15). In other cases, however, we see optimal truck capacity (when T* > 0) less than or equal to the mean demand (examples 1-4 and 17-20).

To examine these results in more detail, we first highlight examples 13 and 15. These two examples are identical except for variability of demand. When demand variability increases from (T = 1 to u = 3 (examples 15-13), the optimal

Table 3. Demand (per day) and lead time parameters.

c1 10 3 2 1 c2 10 1 2 1


Table 4. Results for linear transportation function. Truck

Case Parameters SW S R T TC, P utilization

Ex. 1 Ex. 2 Ex. 3 Ex. 4 Ex. 5 Ex. 6 Ex. 7 Ex. 8 Ex. 9 Ex. 10 Ex. 11 Ex. 12 Ex. 13 Ex. 14 Ex. 15 Ex. 16 Ex. 17 Ex. 18 Ex. 19 Ex. 20

A1 A1 A2 A2 A1 A1 A2 A2 A3 A3 A3 A3 A4 A4 A4 A4 A5 A5 A5 A5

B1 B2 B1 B2 B1 B2 B1 B2 B1 B2 B1 B2 B1 B2 B1 B2 B1 B2 B1 B2

C1 16.5 37.5 C1 41.3 61.4 C1 16.6 37.5 C1 29.9 51.3 C2 12.2 32.5 C2 23.3 43.8 C2 12.2 32.5 C2 23.3 43.8 C1 16.5 37.5 C1 29.9 51.3 C2 12.2 32.5 C2 23.3 43.7 C1 16.5 37.5 C1 95.0 112.0 C2 12.2 32.5 C2 85.0 103.9 C1 16.5 37.5 C1 17.8 41.3 C2 12.2 32.5 C2 12.6 33.8

10.0 27.9 9.0

17.8 10.9 20.9 10.6 20.4

0 0 0 0 14 0 12 0 7.1 7.1 9 9

395.1 86.2

402.3 94.9

181.4 78.5

190.3 87.7

360.2 52.9

153.5 48.6

410.4 71.5

186.1 65.8

384.6 80.0

174.9 73.3

1 3 1 2 1 2 1 2 1 2 1 2 1 8 1 8 1 1 1 1

88.0% 95.7% 91.4% 95.4% 90.8% 94.6% 92.7% 96.1% -

- - -

70.5%

83.3%

96.3% 96.3% 99.0% 99.0%

-

-

truck size increases. This result seems intuitive. Now examine similar pairs in the first eight examples, and examples 17-20 (example 1 vs 5 , 2 vs 6, etc.). As variability increases (example 5 vs example 1, for instance), the optimal truck size actually decreases. This pattern holds for each of the other pairs in the

Table 5. Results for linear transportation function.

Transport

Ex. 1 A1 B1 C1 Figure 2(f) Ex. 2 A1 B2 C1 Figure 2(f) Ex. 3 A2 B1 C1 Figure 2(f) Ex. 4 A2 B2 C1 Figure 2(f) Ex. 5 A1 B1 C2 Figure 2(f) Ex. 6 A1 B2 C2 Figure 2(f) Ex. 7 A2 B1 C2 Figure 2(f) Ex. 8 A2 B2 C2 Figure 2(f) Ex. 9 A3 B1 C1 Figure 2(e) Ex. 10 A3 B2 C1 Figure 2(e) Ex. 11 A3 B1 C2 Figure 2(e) Ex. 12 A3 B2 C2 Figure 2(e) Ex. 13 A4 B1 C1 Figure 2(a) Ex. 14 A4 B2 C1 Figure 2(b) Ex. 15 A4 B1 C2 Figure 2(a) Ex. 16 A4 B2 C2 Figure 2(b) Ex. 17 A5 B1 C1 Figure 2(f) Ex. 18 A5 B2 C1 Figure 2(f) Ex. 19 A5 B1 C2 Figure 2(f) Ex. 20 A5 B2 C2 Figure 2(f)

Case Parameters graph


examples 1-8 and 17-20, correcting for the fact that examples 2 and 6 have different period lengths. Our explanation for this phenomenon follows.

If we examine optimal truck capacity in terms of mean demand plus a certain number k of standard deviations of demand, the results are more consistent. In example 15, the optimal truck size is 12, which is 2 standard deviations above the mean demand. In example 13 (higher variability), the optimal truck size is 13 standard deviations above the mean. When variability increases, the buffer, in terms of standard deviations of demand, decreases. A similar analysis on the pairs in examples 1-8 reveals that the low-variability cases (5-8) have truck capacity of less than 1 standard deviation above the mean, while the high variability cases (1-4) have truck capacity less than 1 standard deviation below the mean. For examples 17-20, the low-variability cases have truck capacity Q standard deviation below the mean, while the high variability cases have truck capacity 3 standard deviations below the mean. Again, when variability increases, k decreases.

From the perspective of truck utilization, the results above are consistent with intuition. For low variability of demand, it is not too costly to buffer for high- demand events and still observe relatively high utilization. When variability is high, however, it would be very costly to incur the fixed cost of regular truck capacity if the firm wanted to buffer for the same high-demand events, i.e., equal k values for both cases. Hence, the optimal trade-off for high-variability cases is to have lower utilization, and incur common-carrier costs when demand is high.

We see little effect on P* as IT varies. In only one case (example 2 compared with example 6) do we see the optimal review period change. In this case, as IT increases from 1 to 3 , the optimal review period increases from 2 to 3. In general, the optimal review period matches the time associated with the analogous EOQ.

Finally, we comment on an interesting observation from the search for P* in examples 14 and 16. When P = 1 and 2, the optimal truck capacity is slightly larger than Pp. (For the low-variability case, example 16, the optimal capacity is 12 for P = 1, and 23 for P = 2. For the high-variability case, the optimal capacities are 14 and 25, respectively.) However, when P 2 3, the optimal capacity is 0 in each case. Because the firm pays a cost KRL per unit of capacity, the fixed cost of regular trucking exceeds the fixed (i.e., the portion that is fixed per shipment, K E ) cost of common carrier trucking. (For examples 16 and 14 the optimal T values, if T > 0, are 33 and 35, respectively. At K R L = $5, the firm pays well over the $100 fixed cost charged for common carrier shipments.) Here we see P* of 8, and T* of 0: All shipping is done using common carrier, but it happens infrequently.

3.2. Concave Transportation Cost

For the case of concave transportation cost, we used nearly identical parameters as in the linear case. For each case, we examined three values for a-0.2, 0.5, and 0.8. The linear function is simply the concave case with a = 1.0. The results are summarized in Table 6. For convenience, we have included the linear cose, labeled as the fourth instance of each example.

Most of the results are qualitatively similar to those for the linear case. Several observations are worth noting, however. First, the optimal truck size for param-


Table 6. Results for concave transportation function. Truck

Case Parameters a sw S R T TC, P utilization Ex. l(a) Ex. l(b) Ex. l(c) Ex. l(d) Ex. 2(a) Ex. 2(b) Ex. 2(c) Ex. 2(d) Ex. 3(a) Ex. 3(b) Ex. 3(c) Ex. 3(d) Ex. 4(a) Ex. 4(b) Ex. 4(c) Ex. 4(d) Ex. 5(a) Ex. 5(b) Ex. 5(c) Ex. 5(d) Ex. 6(a) Ex. 6(b) Ex. 6(c) Ex. 6(d) Ex. 7(a) Ex. 7(b) Ex. 7(c) Ex. 7(d) Ex. 8(a) Ex. 8(b) Ex. 8(c) Ex. 8(d) Ex. 9(a) Ex. 9(b) Ex. 9(c) Ex. 9(d) Ex. lO(a) Ex. 10(b) Ex. 10(c) Ex. 10(d) Ex. l l(a) Ex. l l(b) Ex. ll(c) Ex. l l(d) Ex. 12(a) Ex. 12(b) Ex. 12(c) Ex. 12(d) Ex. 13(a) Ex. 13(b) Ex. 13(c) Ex. 13(d) Ex. 14(a)

A1 B1 C1 0.2 16.5 A1 B1 C1 0.5 16.5 A1 B1 C1 0.8 16.5 A1 B1 C1 1.0 16.5 A1 B2 C1 0.2 29.9 A1 B2 C1 0.5 41.3 A1 B2 C1 0.8 52.4 A1 B2 C1 1.0 41.3 A2 B1 C1 0.2 16.6 A2 B1 C1 0.5 16.6 A2 B1 C1 0.8 16.6 A2 B1 C1 1.0 16.6 A2 B2 C1 0.2 29.9 A2 B2 C1 0.5 41.3 A2 B2 C1 0.8 52.4 A2 B2 C1 1.0 29.9 A1 B1 C2 0.2 12.2 A1 B1 C2 0.5 12.2 A1 B1 C2 0.8 12.2 A1 B1 C2 1.0 12.2 A1 B2 C2 0.2 23.3 A1 B2 C2 0.5 33.8 A1 B2 C2 0.8 44.1 A1 B2 C2 1.0 23.3 A2 B1 C2 0.2 12.2 A2 B1 C2 0.5 12.2 A2 B1 C2 0.8 12.2 A2 B1 C2 1.0 12.2 A2 B2 C2 0.2 23.3 A2 B2 C2 0.5 33.8 A2 B2 C2 0.8 44.1 A2 B2 C2 1.0 23.3 A3 B1 C1 0.2 16.7 A3 B1 C1 0.5 16.7 A3 B1 C1 0.8 16.7 A3 B1 C1 1.0 16.7 A3 B2 C1 0.2 29.9 A3 B2 C1 0.5 41.3 A3 B2 C1 0.8 29.9 A3 B2 C1 1.0 29.9 A3 B1 C2 0.2 12.2 A3 B1 C2 0.5 12.2 A3 B1 C2 0.8 12.2 A3 B1 C2 1.0 12.2 A3 B2 C2 0.2 23.3 A3 B2 C2 0.5 33.8 A3 B2 C2 0.8 23.3 A3 B2 C2 1.0 23.3 A4 B1 C1 0.2 16.5 A4 B1 C1 0.5 16.5 A4 B1 C1 0.8 16.5 A4 B1 C1 1.0 16.5 A4 B2 C1 0.2 29.9

Ex. 14(b) A4 B2 C1 0.5 41.3

37.5 17.8 37.5 15.3 37.5 12.7 37.5 10.0 51.3 31.3 61.4 39.6 71.5 45.1 61.4 27.9 37.5 17.4 37.5 14.9 37.5 12.2 37.5 9.0 51.3 30.9 61.4 39.1 71.5 44.5 51.3 17.8 32.5 12.8 32.5 12.1 32.5 11.5 32.5 10.9 43.8 24.1 53.8 33.7 63.8 42.6 43.8 20.9 32.5 12.7 32.5 12.0 32.5 11.3 32.5 10.6 43.8 23.9 53.8 33.5 63.8 42.2 43.8 20.4 37.5 17.0 37.5 14.2 37.5 0 37.5 0 51.3 30.3 61.4 37.6 51.3 0 51.3 0 32.5 12.6 32.5 11.9 32.5 0 32.5 0 43.8 23.8 53.8 33.3 43.7 0 43.7 0 37.5 19.5 37.5 17.5 37.5 15.7 37.5 14 51.3 33.5 61.4 42.8

339.2 350.6 373.0 395.1 33.0 40.7 59.9 86.2

349.1 360.3 381.9 402.3 43.0 50.6 69.6 94.9

131.8 141.1 159.8 181.4 28.4 35.1 52.6 78.5 41.8 51.0 69.4 90.3 38.4 45.0 62.5 87.7

339.1 350.1 360.5 360.5 32.9 40.5 52.9 52.9

131.8 141.0 153.5 153.5 28.4 35.0 48.6 48.6

331.3 351.8 378.3 410.4 33.0 41 .O

1 1 1 1 2 3 4 3 1 1 1 1 2 3 4 2 1 1 1 1 2 3 4 2 1 1 1 1 2 3 4 2 1 1 1 1 2 3 2 2 1 1 1 1 2 3 2 2 1 1 1 1 2 3

56.3% 64.9% 76.2% 88.0% 63.9% 75.5% 87.2% 95.7% 57.3% 66.6% 78.7% 91.4% 64.7% 76.4% 88.2% 95.4% 78.1% 82.4% 86.9% 90.8% 83.1% 88.9% 93.7% 94.6% 78.9% 83.5% 88.4% 92.7% 83.6% 89.5% 94.4% 96.1% 58.6% 69.7% - -

66.0% 79.4% - -

79.1% 84.0% -

- 83.9% 90.0% - -

51.4% 57.1% 63.6% 70.5% 59.7% 70.1%


Table 6. (continued) Truck

Case Parameters sW SR T TC, P utilization Ex. 14(c) A4 B2 C1 0.8 63.2 Ex. 14(d) A4 B2 C1 1.0 95.0 Ex. 15(a) A4 B1 C2 0.2 12.2 Ex. 15(b) A4 B1 C2 0.5 12.2 Ex. 15(c) A4 B1 C2 0.8 12.2 Ex. 15(d) A4 B1 C2 1.0 12.2 Ex. 16(a) A4 B2 C2 0.2 23.3 Ex. 16(b) A4 B2 C2 0.5 33.8 Ex. 16(c) A4 B2 C2 0.8 44.1 Ex. 16(d) A4 B2 C2 1.0 85.0 Ex. 17(a) A5 B1 C1 0.2 16.5 Ex. 17(b) A5 B1 C1 0.5 16.5 Ex. 17(c) A5 B1 C1 0.8 16.5 Ex. 17(d) A5 B1 C1 1.0 16.5 Ex. 18(a) A5 B2 C1 0.2 29.9 Ex. 18(b) A5 B2 C1 0.5 41.3 Ex. 18(c) A5 B2 C1 0.8 63.2 Ex. 18(d) A5 B2 C1 1.0 95.0 Ex. 19(a) A5 B1 C2 0.2 12.2 Ex. 19(b) A5 B1 C2 0.5 12.2 Ex. 19(c) A5 B1 C2 0.8 12.2 Ex. 19(d) A5 B1 C2 1.0 12.2 Ex. 20(a) A5 B2 C2 0.2 23.3 Ex. 20(b) A5 B2 C2 0.5 33.8 Ex. 20(c) A5 B2 C2 0.8 44.1 Ex. 20(d) A5 B2 C2 1.0 85.0

81.6 60.7

32.5 13.4 32.5 12.9 32.5 12.4 32.5 12 43.8 24.9 53.8 34.9 63.8 44.4

37.5 16.3 37.5 13.6 37.5 10.6 37.5 7.1 51.3 29.7 61.4 37.8 81.6 42.9

112.0 7.1 32.5 12.0 32.5 11.2 32.5 10.2 32.5 9.0 43.8 23.1 53.8 32.5

103.9 9.0

112.0 0

103.9 0

63.8 30.7

61.5 5 71.5 8

313.8 1 141.5 1 161.7 1 186.1 1 28.4 2 35.2 3 53.3 4 65.8 8

339.0 1 349.6 1 368.8 1 384.6 1 32.9 2 40.4 3 58.9 4 80.0 1

131.7 1 140.5 1 157.3 1 179.9 1 28.4 2 34.9 3 51.9 3 73.3 1

82.1%

74.6% 77.8% 80.8% 83.3% 80.4% 86.0% 90.2%

61.2% 72.1% 85.5% 96.3% 67.3% 79.1% 90.4% 96.3% 83.3% 89.1% 95.0% 99.0% 86.4% 92.1% 96.4% 99.0%

-

-

eters A3 is zero only for a = 0.8. When regular transportation costs are lower, due to the concave function, it is economical to maintain in-house trucking capability. However, as the relationship between T and regular transportation cost becomes linear, it is optimal to use common carrier exclusively. A similar rationale explains why we now observe positive values of T* for all cases in parameters A4.

Next we discuss the relationship of P* to a. As a increases, P* either remains constant (example 1, for instance), or increases and then, perhaps, decreases (examples 2 and 14, for instance). When P* remains constant, its value is always 1. This result occurs for the B1 parameters, for which holding and shortage costs are so high that the optimal decision is to ship every period. For parameters B2, it is optimal to incur higher inventory (holding and shortage) costs and lower transportation cost by shipping less frequently. When the transportation cost parameter, a, is low (0.2), it is possible to incur the cost of a larger truck more frequently (P* = 2) and keep inventories low. As a increases ( a = 0.5), the optimal time between shipments increases, due to the higher fixed cost of regular shipments. Inventory costs increase as P increases, but the cost of shipping still forces less frequent shipments. When a is larger ( a = 0.8 or 1.0), incremental inventory costs (as P increases) begin to dominate the gains in transportation cost? forcing more frequent shipping and smaller trucks.


The relationship between T* and a is more straightforward, when we examine T* as a number of standard deviations k from mean demand. Using example 1 as an illustration, we see that when a = 0.2, T* is 2.6 standard deviations above mean demand. As a increases, k decreases to 1.8, 0.9, and 0, respectively. Likewise, in example 2, the values of k as a increases are 2.7, 1.9, 0.9, and - 0.4, respectively. In words, when incremental regular trucking capacity is less expensive, the firm can afford to maintain a truck capacity buffer for a larger portion of demand variability. Thus, for parameters A4 and a = 0.2, T* is more than 3.4 standard deviations above mean demand. For example 4 we can compare a = 0.2 with a = 1.0. In each case, P* = 2; but for a = 0.2, T* is 30.9, while for a = 1.0, T* is 17.8. Costly incremental regular transportation forces the optimal truck size to be considerably smaller, thereby increasing the use of common carrier shipping.

Finally, we note that as a increases, the expected utilization of the truck increases. This result is consistent with the T* result-because costly regular transportation implies smaller trucks, those trucks will be utilized more fully.

4. SUMMARY AND CONCLUSIONS

We have investigated the problem of setting optimal base stock policies for a warehouse and a retailer, in conjunction with finding the optimal truck capacity and shipping frequency. We found many cases in which the optimal truck capacity is zero, implying that the firm should do all shipping without a fixed contract or in-house capability. This result holds in some cases even when the cost of common carrier shipments is twice as high as the cost of regular shipments. We can interpret these results by noting that regular trucking capacity, as we have defined it, implies a limit ( T ) on the truck capacity. Demand that exceeds the capacity must be shipped in a more expensive mode. To have common-carrier trucking on a regular basis as implied by T* = 0 and a fixed P*, suggests that unlimited trucking capacity is supplied on a regular basis. That is, unit costs do not increase even if the volume shipped in a given period increases. The firm pays a premium (of K E - KRL and C E - C,) for this flexibility.

In some cases the optimal truck capacity (in house, or fixed contract) is greater than the mean demand. In other cases, however, the optimal truck capacity (even if T* > 0) is less than the mean demand. If we express truck capacity as mean demand plus k standard deviations of demand, we see that as variability of demand increases, k decreases.

When regular transportation costs are expressed as a concave function of truck capacity, we observe similar results. Optimal truck capacity is positive in most examples, however , because the incremental cost of additional capacity is less than in the linear case. We also observe that as the incremental cost of truck capacity increases, the value of k decreases. This result suggests that when incremental regular trucking capacity is more expensive, the firm can afford to maintain a truck capacity buffer for a smaller portion of demand variability.

Further research should relax the assumption that common carrier shipments arrive at the same time as regular shipments. Indeed, it would be valuable to consider multiple modes of transport with varying lead times and costs. Certainly many firms use such mechanisms to keep the lower echelon supplied, but the


assumption allows the problem to separate neatly. The authors are currently researching a similar problem in which there are no common carrier shipments, but the warehouse can ship more than the requested amount todhe retailer if there is available capacity of the truck.

Finally, it would be interesting and useful to consider multiple retailers. in this case demand at the retailers could be combined (via convolution) to generate an aggregate demand distribution for the warehouse. A direct extension of our model would require the warehouse to use emergency trucking, if necessary, to meet all retailer requests. Retailer policies, then, would not change. The warehouse, on the other hand, would face a new issue-how to optimally allocate inventory and truck capacity to different retailers if supply were not sufficient to satisfy all requests. The warehouse would then face a trade-off involving the cost of shipping in both regular and emergency modes to each retailer. The optimal choice could change each time such a shortage is encountered. In some cases, the optimal choice may be for the warehouse simply to use available inventory to completely satisfy the requests of the retailers with the highest incremental cost of emergency shipments. Then emergency shipments would be sent only to the locations with the lowest incremental emergency shipping costs. Other cases will involve more complicated trade-offs, however, depending on the cost structure of regular and emergency shipping to each retailer. Additional extensions could include truck routes that can supply multiple retailers, implying the use of vehicle routing algorithms.

APPENDIX

In this Appendix, we discuss several special cases of the optimal truck capacity, when g( T ) is linear in T. We first examine the case in which KRL = 0; then we investigate K , = 0. Finally, we examine K E = KRL = 0.

T* when KRL = 0

When KRL = 0, expression ( 5 ) = 0, and the first term of (6) approaches C R p p / P as T + m. Also the second terms of (6)-(8) each approach 0 as T + m. Thus, TCIc+ CRPpfP as T+ a, which implies, with (13), that TCIc is convex decreasing as T + m [see Figure 2(c)]. Therefore, if the cost at T = 0 is less than cRPp/P, T* will be 0. At T = 0, TCI, = K E / P + ( C c / P ) p p , which implies that if

the optimal truck capacity is 0. Otherwise, T* = m. Intuitively, it is clear that if the average cost of shipping units via regular

transportation ( C R p p ) is more than the fixed plus variable cost of shipping in common carrier mode ( K , + C c p p ) , it is better not to contract any regular transportation. In this case ( T * = 0), the firm buys no in-house trucking capacity, and does not contract for a specified amount of transportation on a regular basis.


Rather, all shipments are done on an exception basis, incurring the associated fixed and variable costs.

T” when KE = 0

If K E = 0, we see from (11) conditions under which T* will be 0. First note using (12) that

for Cc > CR. Then from ( l l ) , note that

- - K,, + - CR [I - Fp(T)] + - C C [Fp(T) - 11 = ( K R L + CR - Cc) /P dT P P P

when T = 0. So for K R L + CR > Cc, TCI, is increasing at T = 0. Thus, for K R L + C R > Cc > CR, or K R L > Cc - CR > 0 , TCI, is increasing convex at T = 0, and T* = 0 [Figure 2(d)]. Again, this result is somewhat intuitive: If the fixed cost of regular shipments is greater than the incremental unit cost of common carrier shipments, the firm should not buy fixed trucking capacity. Of course, as in other cases, it is likely to have T* = 0 even when this condition is violated.

T” when KE = KRL = 0

If K E = K R L = 0 , we again see conditions under which T* will be 0. Note that

for Cc > CR. Then from ( l l ) , note that

aTCk C R - [ l -

dT P

when T = 0. So for CR > Cc, TC,, is increasing convex at T = 0, and T* = 0. Again, this result is intuitive.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the helpful comments of two reviewers and the Associate Editor.

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Manuscript received April 2, 1992 Revised manuscript received March 30, 1993 Accepted May 19, 1993

Documents

Optimal base stock policies and truck capacity in a two-echelon system