A base stock inventory model with possibility of rushing part of order∗ Harry Groenevelt • Nils Rudi Simon School, University of Rochester, Rochester, NY 14627
[email protected] •
[email protected] December 2003
Abstract This article studies a situation where a firm outsources production to a distant manufacturer. The firm has two possible freight modes; a slow mode and a fast but more expensive mode. We formulate a dynamic model and use it to analyze how the firm can combine these two freight modes to enjoy the lower cost of the slow mode while using the fast mode as a hedge against cases in which the demand during the production lead time is high. The structure of the optimal solution of this dual freight mode strategy is in the form of a nested order-up-to policy. Several properties of the solution are characterized and these are compared to the case of deterministic demand and the two pure mode strategies. The analytical findings are extensively illustrated by numerical investigations throughout the paper.
1
Introduction
To succeed in today’s markets, firms are facing increasing pressure on price as well as on their responsiveness to volatile market conditions. Price pressure has led to increased sourcing from low-cost countries, primarily from the Far East. A notable consequence has been increased lead times, which, in turn, lead to reduced responsiveness to the market. To counter this, several Supply Chain initiatives have evolved, such as the use of faster freight, Quick Response, variety postponement, and assembly-to-order based on component commonality. ∗
We would like to thank Sigrid Lise Non˚ as, Asmund Olstad, Michal Tzur and Yu-Sheng Zheng for helpful discussions.
Helpful feedback was provided by seminar participants at INSEAD, New York University, University of Oslo and University of Rochester, as well as from participants at the 2001 Multi-echelon conference at Berkeley on an older version of the paper.
1
Consider a firm that outsources production to a distant low-cost country. The first part of the lead time will consist of production (from placing an order to the order being ready for shipping), and the second part of the lead time will consist of shipping (from the order being ready for shipping until it is available to the firm). The firm can choose from two alternative modes of shipping: a low-cost slow mode (e.g., sea freight) or a fast but more expensive mode (e.g., air freight). An issue of interest is then how to characterize the advantages and disadvantages of the two freight modes, as well as how to prescribe the choice between them. A third alternative is to combine the two freight modes. One can then postpone the decision of the mix of the freight modes until the demand during the first (production) part of the lead time is known. So then, if the demand during the production lead time is rather low, the firm is likely to possess sufficient inventory to cover the demand during the lead time of the slow freight mode and use of the expensive freight mode can be avoided. On the other hand, if the demand during the production lead time is rather high, some (or all) of the produced quantity can be rushed using the fast freight mode to cover shortages that would otherwise be likely to occur. The additional demand information can be used to improve the performance over each of the two static pure freight mode strategies. Hence, using the dual freight strategy provides a hedge for cases in which the demand during the production lead time is rather high. Combining the two freight modes makes the firm able to enjoy low freight cost on a large portion of the total demand while still being responsive through the use of the fast freight mode when needed. Several questions are of interest to firms facing such decisions: How does the opportunity of rushing part of an order affect the decision of optimal inventory policy? To what degree should the firm rely on the fast freight mode? What is the impact in terms of cost reduction of using the dual freight mode strategy? This paper addresses, among others, these issues. The scenario studied in this paper is closely related to existing work on the use of express orders and expediting. The classical serial multistage model of Clark and Scarf (1960) can already be interpreted as giving the decision-maker some form of control over the total lead time: by leaving units at a stage for one or more periods, the lead time (i.e. the time it takes the unit to reach the most downstream stage and be available to satisfy final customer demand) can be extended, while sending the unit on to the next stage minimizes the total lead time. Of course in Clark and Scarf’s model this means that the longer a unit stays in the upper stages (the longer the lead time) the more expenses are incurred. This shortcoming is addressed by Lawson and Porteus (2000), who extend the Clark and Scarf model by allowing expediting at each stage (i.e., a unit can either be sent to the next stage in the regular lead time of one period, or it can be expedited at some additional cost and reach the next stage in
2
zero periods). They show that a “top-down base stock policy” is optimal for their model. However, as Lawson and Porteus state in their conclusion, their model cannot accurately represent the situation we examine, where the two supply modes differ in their lead time by an amount different from a single review period. We refer to Lawson and Porteus for a discussion of additional multi-echelon papers with expediting. There are also a number of single echelon models with emergency replenishment or multiple supply sources. A key difference between these papers and ours is that while we consider a single (production) quantity, some of which can be rushed, models in the literature assume two or more independent supply alternatives with different lead times and costs. The initial work on the use of express orders, such as Barankin (1961) and Daniel (1962), are periodic review models where the lead time of the regular order is one period while the express order has zero lead time. Fukuda (1964) generalizes this to a model with two supply sources where the lead times differ by one period, and a model with three supply sources with lead times of three consecutive integers and orders are placed every other period. Whittemore and Saunders (1977) consider the more general case where the two lead times can take any value that is a multiple of the review period length. Their resulting optimal policy is extremely complex in nature. Related models are studied in Gross and Soriano (1972), where in each review period the firm chooses only one of the supply modes, and Chiang and Gutierrez (1996), who extend this to allow the express replenishment to have a non-zero lead time that is shorter than the review period, but the cost expressions are approximations. Moinzadeh and Nahmias (1988) formulate a continuous review model where regular orders are placed according to a standard reorder point model and express orders can be placed during the lead time of the regular order. They develop a heuristic approach where neither the reorder point nor the order quantity for express orders depends on the time remaining of the lead time of the regular order. Tagaras and Vlachos (2001) have a periodic review model that differs from Moinzadeh and Nahmias in that the orders are placed at specific review times while the quantities are variable. Moinzadeh and Schmidt (1991) analyze a continuous review (S-1, S) model with the possibility of express orders assuming Poisson demand. A common characteristic of all these models with two or more independent supply alternatives is that the problems are inherently complex to solve, which leads to complex optimal policies or to the necessity of simplification and approximations in order to achieve solutions, or both. In our differing model setup, we are able to get exact solutions as well as several interesting structural properties. Haggins and Olsen (2003) show that (s,S) policies are optimal in a discrete time, discrete demand model where “expediting” can be used to satisfy unmet demand in a period.
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A class of models that are different in setting but similar in spirit are the ones that combine cheap specific resources, that only can satisfy the corresponding demand classes, with more expensive flexible resources, that can satisfy any demand class. Examples include Fine and Freund (1990), Van Mieghem (1997) and Rudi and Zheng (1997). Van Mieghem and Rudi (2001) formulate and analyze a rather general framework for this type of problems in multi-period settings. Seifert, Thonemann and Hausman (2001) consider the combination between forward buying and trading in a more flexible spot market at a higher price in expectation. When modeling periodic review inventory systems, starting with Arrow, Harris and Marschak (1951) and Bellman, Glicksberg and Gross (1955), the most frequently used approach to account for inventory dependent costs is as a function of end-of-period inventory. Specifically, each positive unit of inventory at the end of a period incurs a unit holding cost and each negative unit of inventory (i.e., demands not met directly from inventory) incurs a unit penalty cost. Hadley and Whitin (1963) consider an approximation of the average inventory level during the period by assuming no demand uncertainty and Moses and Seshadri (1999) use the average of starting and ending period inventory levels as an approximation to the average inventory during the period. In this paper, we account for inventory dependent costs in continuous time. Hadley and Whitin (1963) formulate several similar periodic review models with Poisson and Normal demand. However, they do not obtain analytical solutions. In a recent paper, Rao (2002), independently of our paper, studies a model similar to our pure freight mode scenario. His focus is, however, on studying worst-case performance of heuristic ways of setting the length of the review period with extensions to joint replenishment and multi echelon scenarios. The remainder of the paper is organized as follows. Section 2 describes the model scenario, its cost accounting and appropriate demand processes. For demand processes of Normal increments, Section 3 analyzes the use of a pure (single) freight mode and Section 4 analyzes the dual freight mode problem. In Section 5 we treat the case of Compound Poisson demands and Section 6 gives concluding remarks.
2
Model
We consider a periodic review model where the firm places a production order every T time units. Unmet demands are assumed to be backlogged. At the completion of production, which has lead time L1 , the firm decides how much to ship using the slow freight mode with lead time L2 and how much to ship using the fast freight mode with lead time l2 . The total lead time of the production and slow freight is denoted by L = L1 + L2 . The only assumption made on the relationship between the decision
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time epochs is T ≥ L2 − l2 , which ensures that all units of a production order arrive before the arrival of any units of the following production order. This is a rather mild assumption that will hold for many applications. The timeline of an order is illustrated in Figure 1.
Figure 1: Timeline for an order. Figure 2 illustrates a sample path of net inventory (i.e., physical inventory minus backorders) when a dual freight mode strategy is used. It also shows the total inventory position, express order inventory position and the net inventory path that would have occured without the use of express orders. Note that the use of the fast freight mode in the first inventory cycle eliminates the backorders and in the fourth and fifth cycles reduces backorders. We assume that the demand process D is stochastically increasing and has independent stationary increments. Demand processes satisfying this assumption include Normal increments with positive expectation (i.e., Brownian motion with positive drift) and Compound Poisson processes with positive expected demand increments. The random demand occuring in the time interval t1 to t2 (t1 included and t2 not) will be denoted by D[t1 ,t2 ) . Let µ denote the demand rate and σ 2 denote demand variance, both per time unit. We then have ED[0,t) = tµ, and V arD[0,t) = tσ 2 . To improve clarity of exposition, we will state the results in Sections 3 and 4 assuming that the orderup-to level is always achievable, i.e., the probability of the demand in an order interval being negative is negligible. 5
The firm incurs unit holding cost h and unit shortage cost p, both per time unit. It is assumed that the firm pays on delivery, which results in holding cost being charged on physical inventory. The use of the fast freight mode incurs an additional unit freight cost cf . The firm seeks to minimize the long-run expected controllable costs, which is equivalent to minimizing the expected controllable costs per order cycle.
Figure 2: Example of inventory path. We use the following base case example throughout the paper. The time between placing production orders is T = 1. Lead times are: L1 = 0.3 for production, L2 = 0.6 for slow freight, and l2 = 0.3 for fast freight. Per time unit inventory related costs are holding cost per unit of inventory h = 1 and penalty cost per unit of backlog p = 10, and the additional unit freight cost of using fast freight is cf = 0.5. Finally, the expected demand per time unit is µ = 16 with standard deviation per time unit σ = 4. While Figure 2 uses the Poisson demand process to illustrate a possible inventory path of this example, for the numerical illustrations in the remainder of the paper we will employ a demand process where the increments are Normally distributed. The detailed treatment of the model is done for demand processes of Normal increments in Sections 3 and 4; analytical results for the Compound Poisson process are analogous and are summarized in Section 5.
3
Pure freight modes
We will here analyze the case of using only one of the two alternative freight modes.
Let sub-
script/superscript s denote the slower freight mode and subscript/superscript f denote the fast freight mode. Define x+ = min(0, x). Consider the slower freight mode. At time t0 (the beginning of an order cycle), 6
the firm orders up to quantity S. This order will then affect the inventory level (or, more precisely, the net inventory level) during the time interval [t0 + L, t0 + L + T ). The inventory level at any time t ∈ [t0 + L, t0 + L + T ) is then given by S − D[t0 ,t) , where a negative inventory represents backlogs. At
time t, the firm will then incur holding/stockout cost at rate G S, D[t0 ,t) , where G (y, d) = h (y − d)+ + p (d − y)+ . The cost affected by a specific order as a function of the order up to level S for the slow freight mode can then be written as Cs (S) = E =
Z
Z
L+T
G S, D[0,t) dt
L L+T
EG S, D[0,t) dt.
L
(1)
Correspondingly, the cost affected by a specific order as a function of the order-up-to level S for the fast freight mode can be written as Cf (S) = cf µT +
Z
L1 +l+T L1 +l
EG S, D[0,t) dt.
Without loss of generality, we consider the slower freight mode. The following proposition provides the optimality condition for the ordering policy: PROPOSITION 1 The expected cycle cost given in (1) is optimized by S s , which is the value of S such that 1 T
Z
L+T L
Pr D[0,t) < S dt =
p . p+h
(2)
PROOF: All proofs are given in the appendix. 2 Hadley and Whitin (1963) in their Appendix 4 give a (rather lengthy) expression for the left-hand side of (2) that involves only evaluations of the standard normal cdf and pdf at various points, (as well as other elementary operations) that can readily be implemented in a computer program. The optimality condition (2) allows some interesting insights. Under an optimal solution, the inventory is, in expectation, positive p/(p + h) of the time and the inventory is, in expectation, negative the remainder h/(p + h) of the time. This provides an easy to apply rule of thumb for setting the inventory policy. Note that the optimality condition resembles the optimality condition of the classic newsvendor problem. The difference is that, while the newsvendor solution prescribes a probability of satisfying all demands, this solution specifies the expected proportion of time that arriving demands can be satisfied directly from inventory, also refered to as the fill rate. 1
1
We refer to Groenevelt and Rudi (2002) for discussion of settings where the probability of demand between two
consecutive ordering instants being negative is not negligible.
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The next proposition provides additional properties and insights into the analysis of Cs (S). PROPOSITION 2 Let Cˆs (S) represent the deterministic case of Cs (S) (i.e., no demand uncertainty) with optimizer Sˆs . We have the following results:
p (a) The optimizer of Cˆs (S) is given by Sˆs = µL+ p+h µT with corresponding cost Cˆs Sˆs =
h
i
h
1 hp 2 2 h+p µT .
i
(b) limS→−∞ Cs (S) − Cˆs (S) = 0, and limS→∞ Cs (S) − Cˆs (S) = 0. (c) limS→−∞ Cs0 (S) = limS→−∞ Cˆs0 (S) = −pT , and limS→∞ Cs (S) = limS→∞ Cˆs (S) = hT . (d) Define the right-hand side linear unit Normal loss function as R (u) =
R∞ u
(x − u) φ (x) dx. Then,
for demand processes with Normal increments, Cs (S) = Cˆs (S) + (h + p)
Z
L+T
√ tσR
L
|S − µt| √ dt. tσ
(3)
Proposition 2d implies the following Corollary. COROLLARY 1 Cs (S) is increasing in the standard deviation σ for any S. It follows as special cases
that Cs (S s ) is increasing in σ and that Cs (S s ) ≥ Cˆs Sˆs . Proposition 2 and Corollary 1 offer several insights into the behavior of Cs (S).
When the de-
mand process is deterministic, the solution resembles the EOQ model with backlog adjusted for predetermined order interval and non-zero lead time. As S becomes sufficiently small, Cs (S) approaches the deterministic cost function Cˆs (S) with slope −pT , and, similarly, as S becomes sufficiently large, Cs (S) approaches the deterministic cost function Cˆs (S) with slope hT . We next turn to the relationship between the inventory holding cost h and the penalty cost p. Clearly, from (2), S s is increasing in p/(p + h). Letting p + h be fixed at its base value 11, Figure 3 illustrates the effect of varying p from 1 to 10 in steps of 1. Recall that in the standard newsvendor model with a symmetric demand distribution (e.g., the normal distribution), for a fixed sum of unit overage and unit underage costs the maximum expected opportunity cost (i.e., total cost of demand uncertainty consisting of overage and underage costs) is achieved when setting these two equal (with the corresponding optimal order quantity being equal to the expected demand). Our model closely resembles the newsvendor model with one fundamental difference: the costs are integrated over a time interval where the standard deviation of cumulative demand since the last order was placed is increasing in the time since its placement. This means that the demand variability (in terms of cumulative demand since the last order was placed) is low early in the cycle when the inventory is positive (just after receiving an order), relative to later in the cycle when the inventory level is more likely to be negative (just 8
before receiving an order). Consequently, the firm is more vulnerable to a high p (i.e., p > h) than to an equally high h, while in the standard newsvendor model, with symmetric demand distribution around the mean, their effects are symmetric. This effect is observed in Figure 3; the expected cost at optimality is lower for p = 1 (Cs (13.8) = 11.0), than for p = 10 (Cs (31.6) = 12.3).
Figure 3: Effect on slow freight mode expected cost function of varying p from 1 to 10 keeping p+h = 11 fixed. The curved arrow connects the various minima.
Figure 4: Effect on slow freight mode expected cost function of varying σ. Finally, the expected cost is increasing in the demand variability, which conforms to our intuition. We would here like to emphasize the compact expression (3) which isolates the expected cost due to demand uncertainty from the cost under deterministic demand. The cost of uncertainty in the base case example is 5.0, or an 68% increase over the cost when having no demand uncertainty. Figure 4 illustrates the effect of standard deviation on Cs (S) and on S s by varying σ from 0 to 12 in steps of 2.
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Following up the discussion of the relationship between p and h, when these take their base example values, S s is adjusted up from Sˆs . Figure 4 indicates that this adjustment is amplified as σ increases. The next lemma compare the two freight modes: LEMMA 1 The two pure freight modes are related as follows:
(a) Cˆs Sˆs = Cˆf Sˆf − cf µT . (b) limS→−∞ [Cf (S) − Cs (S)] = cf µT − pµ (L2 − l2 ) T , and limS→∞ [Cf (S) − Cs (S)] = cf µT + hµ (L2 − l2 ) T . (c) S f < S s .
(d) Assume that the demand process has Normal increments. Cs (S s ) ≤ Cf S f (h + p)
Z
L+T
√ tσR
L
|S s − µt| √ dt ≤ cf µT + (h + p) tσ
Z
L1 +l2 +T L1 +l2
iff
S f − µt √ dt. tσR √
tσ
As one might expect, the shorter lead time of the fast freight mode does not lead to any benefits over the slow freight mode, only of extra freight costs. Further, we are able to characterize insightful relationships in terms of the behavior of the cost functions, as well as the optimal policies, between the two freight modes. We refer to Groenevelt and Rudi (2002) for a more in-depth treatment of the pure freight modes.
4
Dual freight mode
We now move on to consider the case of dual freight modes, i.e., the combination of the slow but cheaper and the fast but more expensive modes. We assume that the retailer places production orders at times −T , 0, T , etc. – each time to bring the inventory position up to the level S. Hence the production order placed at time 0 will be for D[−T,0) units of product. At time L1 the firm decides how to allocate the produced quantity between the two freight modes, knowing both the production order quantity D[−T,0) and the inventory position excluding the most recent production order S − D[−T,L1 ) . This is the relevant inventory position for the fast freight order, since the production order placed at time 0 will not be available to the retailer before time L unless the retailer decides to use the fast freight mode. Let q be the quantity shipped using the fast freight mode. The (random) cost incurred during this order cycle is then cf q +
Z
L L1 +l2
G q + S, D[−T,t) dt + 10
Z
L1 +l2 +T L
G S, D[0,t) dt.
(4)
Next, we consider the optimal use of the fast freight mode. PROPOSITION 3 For an arbitrary S, the following use of the fast freight mode is optimal at time L1 : If cf ≥ (L2 − l2 )p then q ∗ = 0, otherwise
∗
∗
q = min D[−T,0) , z − S + D[−T,L1 )
+
,
(5)
where z ∗ is the solution to 1 L2 − l 2
Z
L L1 +l2
Pr D[L1 ,t) < z dt =
p−
cf L2 −l2
p+h
,
(6)
with respect to z. Notice that the optimal fast freight mode order-up-to quantity z ∗ does not depend on the production order-up-to quantity S. This implies that the problem of finding an optimal combination of fast shipping policy and production policy can be reduced to the sequential process of first finding z ∗ and then deciding on S ∗ (given z ∗ ). Furthermore, the unconstrained optimality condition (6) has a nice intuitive interpretation. The RHS of (6) is the familiar ratio p/(p + h) adjusted down due to the additional unit cost of the fast freight mode cf . In the case of the pure freight mode, the total shipping cost is independent of the inventory policy followed (assuming all demand is satisfied), so the per unit shipping cost does not play a role in determining the optimal policy. The objective function to consider when deciding S can then be expressed as ∗
C(S) = cf Eq +
Z
L L1 +l2
∗
EG S + q , D[−T,t) dt +
Z
L1 +l2 +T L
EG S, D[0,t) dt.
(7)
We are now ready to characterize the optimal S. PROPOSITION 4 The expected cycle cost given in (7) is optimized by S ∗ , which is the unique value of S such that
R L1 +l2 +T
Pr D[0,t) < S dt
1 p L . = R L T p+h + L1 +l2 Pr D[−T,t) < S, D[L1 ,t) ≥ z ∗ + Pr D[0,t) < S, D[L1 ,t) < z ∗ dt
(8)
An intuitive explanation for (8) is the following. Consider the marginal unit of product included in S. This unit will decrease the penalty cost rate incurred at time t during the cycle by p, unless the inventory at time t is already positive, in which case the marginal unit will increase the holding cost rate incurred at time t by h. Hence, if we denote the physical inventory at time t by I(t), we can write the impact of the marginal unit at time t as −p + (p + h) Pr (I(t) > 0). It is not hard to see that I(t) > 0 ⇔ D[0,t) < S when L ≤ t < L1 + l2 + T . This follows since at time L the entire production 11
order placed at time 0 will have arrived at the retailer, regardless of the fast shipping decision made at time L1 . The situation is more complex when L1 + l2 ≤ t < L. Some reflection shows that in this case we have I (t) > 0 ⇔ D[L1 +l2 ,t) < S − D[−T,L1 +l2 ) + q ∗ , since the RHS of this last inequality is the amount of inventory available at time L1 + l2 (after the delivery of q ∗ fast shipped units) and no further delivery takes place until time L > t. Using (5) we obtain I (t) > 0 ⇔ 0 < S − D[−T,t) + q ∗
⇔ 0 < min S − D[0,t) , max S − D[−T,t) , z ∗ − D[L1 ,t) ⇔
z ∗ > D[L1 ,t) and S > D[0,t)
or
z ∗ ≤ D[L1 ,t) and S > D[−T,t) .
Hence, the integrand of the last integral in (8) equals Pr (I (t) > 0) for L1 + l2 ≤ t < L, and (8) says nothing other than 1 T
Z
L1 +l2 +T
Pr (I(t) ≥ 0) dt = L1 +l2
p . p+h
Note that if cf ≥ (L2 − l2 ) p, then z ∗ = −∞ and the optimality condition (8) reduces to the optimality condition of the pure slow mode (2) and S ∗ = S s . We then move on to characterize properties of the dual freight mode problem. PROPOSITION 5 Let Cˆ (S, z) represent the deterministic case of C (S) (i.e., no demand uncertainty)
with optimizers Sˆ∗ , zˆ∗ and let Cˆ (S) = Cˆ (S, zˆ∗ ). We have the following results: (a) If cf < min ((L2 − l2 ) p, (T − (L2 − l2 )) h), then the optimizers of Cˆ (S, z) are given by c
Sˆ∗ = µL +
f p − L2 −l p 2 µT − µ (L2 − l2 ) p+h p+h
and ∗
zˆ = µl2 +
p−
cf L2 −l2
p+h
µ (L2 − l2 ) ,
otherwise it is optimal to only use the slow mode. The corresponding cost is given by 1 hp 2 ((L2 − l2 ) p − cf )+ ((T − (L2 − l2 )) h − cf )+ Cˆ Sˆ∗ , zˆ∗ = T µ− µ. 2h+p h+p
(b) limS→−∞ C (S) = limS→−∞ Cˆ (S) = limS→−∞ Cs (S)−(p (L2 − l2 ) − cf )+ µT , and limS→∞ C (S) = limS→∞ Cˆ (S) = limS→∞ Cs (S). (c) limS→−∞ C 0 (S) = limS→−∞ Cˆ 0 (S) = −pT , and limS→∞ C 0 (S) = limS→∞ Cˆ 0 (S) = hT .
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(d) min (Cs (S) , Cf (S)) ≥ C (S) ≥ Cs (S) − (p (L2 − l2 ) − cf )+ µT and Cf0 (S) ≥ C 0 (S) ≥ Cs0 (S). (e) S s ≥ S ∗ ≥ S f ≥ z ∗ . Proposition 5 offers several interesting insights into the behavior of C (S). The case of no demand uncertainty has an interesting relationship to its equivalent when using only the slow freight mode in terms of the adjustment of Sˆs . Also, we have that the deterministic costs at optimality for the dual freight mode and the slow freight mode are related by ((L − l ) p − c )+ ((T − (L − l )) h − c )+ 2 2 2 2 f f Cˆ Sˆ∗ , zˆ∗ = Cˆs Sˆs − µ, h+p
where the second term on the right-hand side represents the cost savings per cycle achieved by going from only using the slow freight mode to using a dual freight mode. Proposition 5b shows that the expected cost approaches the cost of the deterministic problem as S gets sufficiently small/large. Further, note that for sufficiently small S, the quantity rushed using the fast mode is equal to the order quantity that cycle (in expectation µT ). This results in a reduction in penalty cost of p (L2 − l2 ) µT compared to only using the slow freight mode, at an extra fast freight expense of cf µT per cycle. For sufficiently large S, the fast freight mode will never be used and the cost of the dual strategy approaches the cost of only using the slow freight mode. Much like Cs (S), the slope of C (S) approaches −pT as S becomes sufficiently small and hT as S becomes sufficiently large.
Figure 5: Expected cost dual freight mode compared to its deterministic case and the two pure freight modes. We then turn to illustrating the analytical findings using the base example much like we did for the pure freight modes. The relationships between C (S), Cˆ (S), Cs (S) and Cf (S) are illustrated in Figure 13
5. Comparing the dual freight mode to the slow freight mode, we see that its optimum C (29.1) = 10.7 represents an expected cost saving of 1.6 or 13%. Breaking up this cost difference, we see that it consists of the difference between the deterministic costs of Cˆs (28.9) − Cˆ (25.3) = 7.3 − 6.5 = 0.8 h
i
h
i
and the difference between the costs of uncertainty of Cs (31.6) − Cˆs (28.9) − C (29.1) − Cˆ (25.3) = [12.3 − 7.3]−[10.7 − 6.5] = 0.8. Further, the optimal base stock level of the dual freight mode S ∗ = 29.1 is 2.5 less than its slow freight mode counterpart. This is because the option of using the fast freight for (part of) the production order reduces the need for a high base stock level.
Figure 6: Effect on dual freight mode optimal decisions and corresponding expected cost of varying p from 1 to 10 keeping p + h = 11 fixed. The effect of the relationship between p and h on the expected cost at optimality is illustrated in Figure 6, keeping p + h = 11 fixed and varying p from 1 to 10. Also, the optimal decision variables S ∗ and z ∗ , as well as the expected quantity shipped using the fast freight mode Eq ∗ , are plotted to illustrate how they are affected by p/(p + h). Much like the slow freight mode illustrated in Figure 3, C (S ∗ ) attains its maximum value when p and h are close to equal. As p increases, the consequence of an underage becomes larger relative to the consequence of an overage, which causes S ∗ and z ∗ to increase with p. For p ≤ 5/3, it is, by the condition of Proposition 3, optimal not to use the fast freight mode at all (i.e., q ∗ = 0 and z ∗ = −∞). When 5/3 < p ≤ 2.9, z ∗ is increasing more rapidly than S ∗ with a corresponding increase in Eq ∗ . For the majority of p-values (i.e., 2.9 < p), however, Eq ∗ is decreasing even though z ∗ is increasing. This is due to the fact that z ∗ increases at a slower rate than S ∗ for these values of p, i.e., an increase in S ∗ will lead to a corresponding increase in the inventory level at the time of making the decision of how much to ship by the fast freight mode.
14
Figure 7: Effect on dual freight mode optimal decisions and corresponding expected cost of varying σ. Similarly, the effect of demand variability is illustrated in Figure 7 by plotting how the optimal decisions and corresponding expected cost depend on σ. In terms of the cost of uncertainty, comparing Figure 7 with Figure 4 shows that C (S ∗ ) increases more slowly in σ than Cs (S s ). Further, S ∗ is increasing at a faster rate than z ∗ in σ. The intuition being this is that S ∗ is to cover the uncertainty of a longer time span (i.e., L + T ) than is z ∗ (i.e., L2 ). There are two main effects of σ on Eq ∗ : (i) keeping S and z fixed, a higher σ would increase the expected difference between z and the inventory level at the time of making the decision of how many units to ship using the fast freight mode, driving Eq ∗ up, and (ii) S ∗ increases at a faster rate than z ∗ in σ, which means that for a given quantity to be shipped by the fast freight mode, the demand during the production lead time needs to be larger. From Figure 7 we see that the first of these effects dominates the second for smaller σ’s and vice versa for larger σ’s. We then turn to illustrating how the additional cost of using the fast freight affects the attractiveness of the dual freight mode in Figure 8. Note here that even if cf = 0, the dual freight mode will dominate the fast freight mode as it is a way to keep the inventory level “closer” to the mean by two effects: (i) by two inventory replenishments during a cycle, and (ii) by the postponement of the decision of what proportion of the production order to receive using the fast freight mode. As cf becomes large, the dual mode policy and its cost approach the case of only using the slow transportation mode, and for cf ≥ 3 they coincide by the condition of Proposition 3. As expected, to retain the optimal service level (i.e., positive inventory p/ (p + h) of the time), S ∗ is increasing and z ∗ is decreasing in cf , with a resulting decrease of Eq ∗ in cf . In Figure 9 we illustrate the effect of the shipping lead time l2 of the fast freight mode. Intuitively, we would expect the expected cost of the dual freight mode C (S ∗ ) to be increasing in the fast freight 15
Figure 8: Effect on dual freight mode optimal decisions and corresponding expected cost of varying the additional shipping cost of the fast freight mode cf . lead time l2 due to the loss of responsiveness. For small values of the fast freight lead time (l2 < 0.06), however, C (S ∗ ) is actually decreasing. This can be explained as follows. If the fast freight lead time is very small, fast shipped units tend to arrive before they are really needed to reduce shortage costs and the result is higher holding costs. Then C (S ∗ ) is increasing in the lead time of the fast freight for 0.06 ≤ l2 < 11/20. When l2 ≥ 11/20, it is, by the condition of Proposition 3, optimal not to use the fast freight mode at all (i.e., q ∗ = 0 and z ∗ = −∞), and as a result C (S ∗ ) = Cs (S s ).
Figure 9: Effect on dual freight mode optimal decisions and corresponding expected cost of varying the shipping lead time of the fast freight mode l2 .
16
Figure 10: Effect on dual freight mode optimal decisions and corresponding expected cost of varying the production lead time L1 , keeping the total lead times L1 + L2 and L1 + l2 fixed. Finally, in Figure 10 we illustrate the effect of the value of additional information in terms of how long the decision of what to ship using fast freight can be postponed. Specifically, we keep the total lead times, i.e., L1 + L2 and L1 + l2 , fixed while varying L1 from 0 to 0.6. First note that S ∗ is quite insensitive to changes in L1 and that z ∗ is decreasing close to the rate that l2 µ is decreasing in L1 . Consequently, Eq ∗ is quite insensitive to changes in L1 . The realized q ∗ will, however, be quite different as L1 changes as the firm is making the decision of the quantity shipped by fast freight based on better information. Consequently, the expected cost of the dual mode at optimality C (S ∗ ) decreases from 11.1 when L1 = 0 to 10.2 when L1 = 0.6, or an 8.1% decrease.
5
Compound Poisson demand process
In this section, we show how to calculate the optimal policies for the pure and dual freight mode problems when the demand process is a compound Poisson process. Because the compound Poisson process is non-decreasing, the issue of negative demand does not arise in this case. In addition, as we will see, numerical integration can be avoided completely (in the Normal increments case one needs to numerically evaluate several double integrals each time C(S) or C 0 (S) is calculated), and the formulas we derive below can be implemented quite efficiently. Let D[t1 ,t2 ) =
PN (t2 )
i=N (t1 )
Xi , where {N (t)} is a Poisson process with rate ν, and X1 , X2 , ... are independent
and identically distributed discrete positive random variables. We will write rk = Pr{Xi = k}, k = 0, 1, ..., so r0 = 0 by assumption, and of course we have νEXi = µ as the total expected demand per 17
time unit, consistent with the notation in the rest of the paper. Further, define ρ = |{k : rk > 0}|, i.e., ρ is the number of positive probabilities in the distribution of Xi . For t1
≤
H[t1 ,t2 ) (s) =
t2 R t2 t1
and integer values of s,
n
we define P[t1 ,t2 ) (s)
o
Pr D[0,t) < s dt, and J[t1 ,t2 ) (s) =
R t2 t1
h
E s − D[0,t)
i+
=
R t2 t1
n
o
Pr D[0,t) = s dt,
dt. These quantities are needed
in calculating the optimal policies and the cost associated with arbitrary policies for both the pure and dual freight mode problems. Note that P[t1 ,t2 ) (s) = H[t1 ,t2 ) (s) = J[t1 ,t2 ) (s) = 0 for all s < 0, and it is easy to verify that for s ≥ 0 H[t1 ,t2 ) (s) = H[t1 ,t2 ) (s − 1) + P[t1 ,t2 ) (s − 1), J[t1 ,t2 ) (s) = J[t1 ,t2 ) (s − 1) + H[t1 ,t2 ) (s). Hence if P[t1 ,t2 ) (k) can be calculated efficiently for all 0 ≤ k ≤ s, so can H[t1 ,t2 ) (k) and J[t1 ,t2 ) (k). n
o
Define for t ≥ 0 and integer values of s the probabilities wt (s) = Pr D[0,t) = s , and Wt (s) = n
o
Pr D[0,t) ≤ s . These probabilities can be found using the efficient and stable recursion in Adelson (1966), see also Tijms (1994): wt (0) = e−νt , X νt s−1 (s − k) rs−k wt (k), s k=0
wt (s) =
s = 1, 2, ... .
PROPOSITION 6 For t1 ≤ t2 and integer values of s we have s−1 X 1 rs−j P[t1 ,t2 ) (j). (wt1 (s) − wt2 (s)) + ν j=0
P[t1 ,t2 ) (s) =
(9)
Note that to find P[t1 ,t2 ) (k) for k = 0, ..., s requires at most O(s · min(ρ, s)) operations. Next, we will find the optimal base stock level when only the slow freight mode is used. The total cost per cycle is given by expression (1): Cs (S) =
Z
T +L L
EG S, D[0,t) dt
= (h + p)
Z
T +L L
E S − D[0,t)
+
dt + p
Z
1 = (h + p) J[L,T +L) (S) + pT µ L + T 2
T +L
L
ED[0,t) − S dt
−S .
Define S s to be the smallest value of S that minimizes Cs (S). To find the minimum cost, it is useful to look at ∆Cs (S) = Cs (S) − Cs (S − 1)
= (h + p) J[L,T +L) (S) − J[L,T +L) (S − 1) − pT = (h + p) H[L,T +L) (S) − pT. 18
For S ≤ 0, this reduces to ∆Cs (S) = −pT , and it follows immediately that S s ≥ 0. Clearly, ∆Cs (S) is non-decreasing in S , so Cs (S) is convex. Hence, S S is the unique value of S that satisfies ∆Cs (S) < 0 ≤ ∆Cs (S + 1) ⇔ H[L,T +L) (S)
0, and hence that Cs (S) is convex and the first order condition guarantees the global optimum. Equating to zero and rearranging gives the desired result. 2 PROOF OF PROPOSITION 2: The proof is provided in Groenevelt and Rudi (2002). PROOF OF COROLLARY 1: To prove the desired monotonicity result of Cs (S), it is sufficient to show that
√ |S − µt| √ tσR tσ
is increasing in σ. This follows readily since R (·) is decreasing in its argument. Further, since Cs0 (S s ) = 0, we have d ∂S s ∂Cs (S s ) ∂Cs (S s ) Cs (S s ) = Cs0 (S s ) + = > 0, dσ ∂σ ∂σ ∂σ proving the monotonicity result of Cs (S s ). Finally, the result Cs (S s ) ≥ Cˆs Sˆs follows from Cs (S s ) ≥
Cˆs (S s ) ≥ Cˆs Sˆs .2 PROOF OF Lemma 1: Parts (a), (b) and (d) of this Lemma follow directly from Proposition 2. Part (c) follows from Z
L+T L
Pr D[0,t) < S dt =
Z
L1 +l2 +T L1 +l2
Pr D[0,t+L2 −l2 ) < S dt
