This paper considers a two-echelon distribution inventory system with a central ... inventory systems it is usually assumed that all stochastic customer demand ...
Handling Direct Upstream Demand in Multi-Echelon Inventory Systems
Sven Axsäter Fredrik Olsson Patrik Tydesjö Lund University, Sweden
Abstract This paper considers a two-echelon distribution inventory system with a central warehouse and a number of retailers. There is direct customer demand also at the warehouse. If this upstream demand requires a high service, it may be advantageous to give higher priority to customer orders compared to retailer replenishments. We suggest and evaluate two techniques for handling this situation. One technique means that we have an extra separate stock point for the direct customer demand at the warehouse. The other technique means that we apply a so-called critical level policy at the warehouse, i.e., if the stock on hand at the warehouse is less than or equal to the critical level, retailer replenishments are backordered.
Key words: Inventory management, two-level, stochastic, upstream customer demand
1
1. Introduction Consider the two-echelon inventory system in Figure 1. There are a number of local sites (retailers) that replenish their stocks from a single central warehouse. The warehouse, in turn, replenishes its stock from an outside supplier. When studying such inventory systems it is usually assumed that all stochastic customer demand takes place at the retailers. For overviews of research based on this assumption, see Axsäter (1993, 2003), Federgruen (1993), Diks et al. (1996), and Van Houtum et al. (1996). See also some recent textbooks: Sherbrooke (1992), Silver et al. (1998), Axsäter (2000b), and Zipkin (2000).
warehouse retailers • • •
Figure 1
Two-echelon distribution inventory system
Assume that we, still given the assumption of customer demand at the retailers only, consider some class of control policies, and that we within that class optimize the control for some reasonable cost structure. As has been noticed by many researchers (see the references above), the resulting control will nearly always result in very low stocks at the central warehouse. A typical solution can mean that the fill rate at the retailers is about 95%, while it should be only around 60% at the warehouse. In other words, limited stochastic delays at the warehouse are not that serious for the retailer replenishments.
2 In this paper, however, we shall assume that the demand at the warehouse is not only the replenishments from the retailers but also direct customer demand. The direct customers can very well require a very high service level. If this is the case we face a dilemma. At the warehouse there are now two types of demand that have very different service requirements. The purpose of this paper is to discuss and evaluate techniques for overcoming this problem. The outline of the paper is as follows. In Section 2 we give a detailed problem formulation. Section 3 describes the considered approaches. A numerical study is presented in Section 4. Finally, we give some concluding remarks in Section 5.
2. Problem formulation We consider the distribution inventory system in Figure 1, and assume that all sites use continuous review installation stock (R, Q)-policies with given batch quantities when replenishing their stocks. When the inventory position (stock on hand + outstanding orders – backorders) declines to or below R a number of batches are ordered so that the resulting inventory position is in the interval (R, R + Q]. Customer orders at the retailers and at the warehouse, and retailer orders at the warehouse that cannot be met directly are backordered and delivered later. If an order can only be met partly, a partial delivery will take place. The remaining part of the order is backordered. The direct customer demands at all sites are independent compound Poisson processes, i.e., the customers arrive according to a Poisson process, and each customer demands a stochastic discrete number of units. There are standard holding and shortage costs at all sites. Note that the shortage cost at the warehouse is only for direct customer demand. There are no shortage costs for delayed retailer orders. However, such delays will, of course, affect the delays at the retailers. We wish to minimize the expected total holding and backorder costs. We can affect these costs not only by choosing different reorder points, but also by giving different priorities to retailer orders and direct customer demand at the warehouse.
3 Let us introduce the following basic notation: N = number of retailers, L0 = constant lead-time for an order to arrive at the warehouse from the outside supplier, Li = constant transportation time for an order to arrive at retailer i from the warehouse (i = 1, 2, ..., N), λi = customer arrival intensity at site i (i = 0, 1, ..., N),
fi,j = probability for customer demand quantity j at site i (i = 0, 1, ..., N), fi,j = 0 for j < 1. (We assume that not all demand quantities are multiples of some unit larger than one.), Qi = given batch size at site i (i = 0, 1, ..., N), Ri = reorder point at site i (i = 0, 1, ..., N), hi = holding cost per unit and time unit at site i (i = 0, 1, ..., N), pi = shortage cost per unit and time unit for direct customer demand at site i (i = 0, 1, ..., N).
3. Different approaches We shall consider three different approaches for handling direct customer demand at the warehouse. These approaches are described in Sections 3.1-3.3, respectively.
3.1 First come – first served at the warehouse
This approach means simply that we give the same priority to all orders at the warehouse. All orders are served according to a first come – first served policy, i.e., replenishment orders from the retailers are treated in the same way as the direct customer demand. We note that the resulting inventory system can be modeled as a standard system without customer demand at the warehouse. We simply add one retailer, which we denote retailer d. See Figure 2. The demand at retailer d is the direct customer demand, i.e., λd = λ0, and fd,j = f0,j. The lead-time is zero, i.e., Ld = 0. Retailer
4 d applies the policy Rd = - 1 and Qd = 1. This means that each customer demand at retailer d triggers a corresponding warehouse order of the same size from retailer d. The inventory situation at the warehouse is consequently not affected. This means that the holding costs at the warehouse and the delays for retailer replenishments are unchanged. By setting pd = p0, the shortage costs at retailer d are identical to the shortage costs for the direct customer demand in the original system. Recall that Ld = 0. Because of the chosen policy there will never be any stock on hand at retailer d, so no holding costs are incurred.
warehouse retailers • • •
retailer d
Figure 2
Two-echelon distribution inventory system with an artificial retailer for direct customer demand
Because we are able to use a model without direct customer demand at the warehouse, we can use the exact technique in Axsäter (2000a) for determination of the optimal reorder points. We simply add the constraint that Rd = - 1.
3.2 Separate stock for direct customer demand at the warehouse
The second considered approach, which is related to the model in Section 3.1, means that we give higher priority to the direct demand by again using an artificial retailer d with zero lead-time. This artificial retailer satisfies the direct warehouse demand and replenishes its stock from the warehouse. The batch quantity is still Qd = 1, but the
5 artificial retailer can now carry stock, i.e., Rd is optimized. The direct demand will now get a better service than the orders from the retailers provided Rd ≥ 0. A retailer order is satisfied directly only if the warehouse has stock on hand. A direct demand is also always satisfied directly if the warehouse has stock on hand because Ld = 0. However, a direct demand can also be satisfied directly if there is no warehouse stock, provided that the artificial retailer has stock on hand. We assume that the possible stock at the artificial retailer is situated in the warehouse, so we have hd = h0. The shortage costs associated with the direct demand are accounted for at the artificial retailer, so we have pd = p0. As in Section 3.1 the holding costs at the warehouse and the costs at the other retailers are the same, because the reorder point at retailer d does not affect the demand process at the warehouse. Furthermore, as in Section 3.1 the resulting model has no direct customer demand at the warehouse, and we can use the exact technique in Axsäter (2000a) for determination of the optimal reorder points.
3.3 Critical level policy at the warehouse
Our third approach is to use a so-called critical level policy. This means that we define a nonnegative critical level c for stock on hand at the warehouse. If the stock on hand at the warehouse is positive but less than or equal to c only direct customer demand is satisfied directly while retailer orders are backordered. The direct customer demand is satisfied on a first come – first served basis, i.e., backorders are satisfied first. When the stock on hand exceeds c all possible backorders (retailer orders and direct demand) are served first. This is done on a first come – first served basis. It is clear that the considered policy like the policy in Section 3.2 will give higher priority to direct demand as long as the critical level c is strictly positive. When using this policy we wish to optimize the critical level and the reorder points jointly. A critical level policy has been applied in a related but different context in Axsäter et al. (2004). See also Dekker et al. (1997), and Ha (1997).
6 A disadvantage with the considered critical level policy is that there seems to be no easily available technique for exact evaluation and optimization of the policy in the considered setting. In the numerical study in Section 4 we have therefore used simulation.
4. Numerical results The considered approaches for handling direct customer demand at the warehouse have been compared in a numerical study comprising 16 sample problems. All problems have N = 2 identical retailers. The transportation times are L0 = L1 = L2 = 1, and the given batch quantities Q0 = 4, and Q1 = Q2 = 2. The holding costs are h0 = h1 = h2 = 1, and the retailer backorder costs p1 = p2 = 10. Furthermore we considered all 16 combinations of: p0 = 5, 10, 20, or 40, and four types of customer demand. The demand processes were generated in the following way. The demand size has a geometric distribution (the same at all locations), f i , j = x (1 − x ) j−1 . Two values of x were used, x = 0.8, and x = 0.6. (The average demand size is 1/x. A lower x means that the stochastic variations are larger.) For each x, the customer arrival intensities at the retailers were chosen as λ 1 = λ 2 = x . This means that the average demand per time unit at each retailer is µ 1 = µ 2 = 1 . For the direct customer demand at the warehouse we considered for each x, λ 0 = 2 x , and λ 0 = 4x corresponding to the average demands per time unit µ 0 = 2 , and µ 0 = 4 respectively.
7 Table 1 provides the results. For the first come – first served policy and the separate stock policy the exact results have been obtained by the technique in Axsäter (2000a). The evaluation and optimization of the critical level policy has been carried out by simulation.
Table 1
Numerical results for 16 test problems. N = 2, L0 = L1 = L2 = 1, Q0 = 4, Q1 = Q2 = 2, h0 = h1 = h2 = 1, p1 = p2 = 10, µ 1 = µ 2 = 1 . Standard deviation of simulation results in parenthesis.
DATA
RESULTS First come-first served
p0
µ0
x
Policy
Costs
R0,R1=R2
Separate stock Policy
Costs
R0,R1=R2,Rd
Critical level Policy
Costs
R0,R1=R2, c
5
2
0.8
3, 2
9.72
2, 2, 0
9.61
3, 2, 1
9.55 (0.02)
5
2
0.6
4, 2
12.34
4, 2, -1
12.34
4, 2, 0
12.32 (0.04)
5
4
0.8
6, 2
10.75
4, 2, 0
10.61
5, 2, 1
10.51 (0.03)
5
4
0.6
7, 2
13.59
5, 2, 0
13.58
6, 2, 1
13.43 (0.04)
10
2
0.8
4, 2
10.70
2, 2, 1
10.40
4, 2, 1
10.34 (0.03)
10
2
0.6
5, 2
13.46
4, 2, 0
13.40
5, 2, 1
13.25 (0.03)
10
4
0.8
7, 1
11.87
4, 2, 2
11.58
7, 2, 2
11.44 (0.04)
10
4
0.6
8, 2
15.03
6, 2, 1
14.88
8, 2, 2
14.71 (0.04)
20
2
0.8
6, 1
11.65
3, 2, 1
11.21
5, 2, 2
11.07 (0.02)
20
2
0.6
6, 2
14.78
4, 2, 1
14.50
6, 2, 2
14.37 (0.04)
20
4
0.8
9, 1
13.00
4, 2, 3
12.57
8, 2, 3
12.35 (0.03)
20
4
0.6
10, 2
16.60
6, 2, 2
16.30
9, 2, 3
16.04 (0.05)
40
2
0.8
7, 1
12.63
3, 2, 2
12.00
6, 2, 3
11.87 (0.02)
40
2
0.6
8, 2
16.15
4, 2, 2
15.63
7, 2, 3
15.46 (0.04)
40
4
0.8
10, 1
14.15
4, 2, 4
13.54
9, 2, 3
13.38 (0.04)
40
4
0.6
12, 2
18.23
6, 2, 4
17.62
11, 2, 4
17.38 (0.04)
It is clear from Table 1 that the policies giving higher priority to direct customer demand at the warehouse perform significantly better. The cost reduction is as expected
8 increasing with the backorder cost associated with the direct demand at the warehouse. The different types of demand give similar cost reductions. The critical level policy performs significantly better than the policy with a separate stock point for direct warehouse demand. Still the cost difference is not especially large. A problem with the separate stock policy seems to be that there is no coordination of the extra stock for direct demand and the size of the warehouse stock. The extra stock may be unnecessary or too large at times when the warehouse has plenty of stock on hand. On the other hand the separate stock policy has the advantage that it can be evaluated and optimized in the same way as a standard policy without special priority for direct demand.
5. Conclusions We have considered a two-echelon inventory system with direct customer demand also at the upper level. If this upstream direct demand requires a high service, it may be advantageous to give higher priority to customer orders than to replenishment orders from the retailers. We have evaluated two techniques for handling this situation. One technique means that we keep a separate extra stock for direct demand at the upstream level. The other technique is to apply a so-called critical level policy. Both techniques work well. The critical level policy gives slightly better results, but is, on the other hand, more difficult to evaluate and optimize.
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