Comparing strategies for addressing delivery shortages in
stochastic demand settings
Michael A. Haughton
a,*, Alan J. Stenger
baDepartment of Management Studies, University of the West Indies, Mona Campus, Kingston 7, Jamaica, W.I. b501J Business Administration Building, The Smeal College of Business Administration, The Pennsylvania State University,
University Park, PA 16802-3005, USA
Received 21 November 1996; received in revised form 11 September 1998; accepted 15 October 1998
Abstract
In logistics networks involving one supply point (depot) and several geographically dispersed demand points (e.g., retail stores), delivery shortages will result if the design of delivery routes ignores random period-to-period ¯uctuations in customer demands. Delivery shortages may be costly enough for the depot to seek strategies to prevent them. A requirement for rational comparison of strategies is quantifying their eects on total supply chain costs. Accurate distance prediction models are developed to help satisfy this prerequisite for the transportation cost element. These models are integrated into a comparison of strate-gies on the basis of how these stratestrate-gies aect inventory and transportation. The focus of ®ndings from the comparison involves identifying the information cost thresholds for accepting/rejecting a demand-respon-sive strategy. The study's implications for choosing a strategy are presented.#1999 Elsevier Science Ltd. All rights reserved.
1. Introduction
A core activity in many logistics networks is the delivery of goods from one echelon to another by road transport vehicles. When the delivering echelon is a single facility (e.g., a wholesale depot) and the receiving echelon is a set of geographically dispersed facilities (e.g., retailers), the logistics problem is sometimes formulated and solved as the classical vehicle routing problem (VRP). That is, the objective is to ®nd a set of delivery routes that simultaneously satis®es demand at each retail outlet and minimizes total transportation costs. Given the plethora of heuristics and related software for solving VRPs, the depot's logistics problem would seem straightforward. It might not be. If, for example, demand ¯uctuates from day to day, then the cost-minimizing routes for a
1366-5545/99/$ - see front matter#1999 Elsevier Science Ltd. All rights reserved. P I I : S 1 3 6 6 - 5 5 4 5 ( 9 8 ) 0 0 0 2 1 - 0
RESEARCH PART E
Transportation Research Part E 35 (1999) 25±41
particular day need not be appropriate for the following day or for any other day. The following day's ¯uctuations may be such that the previous day's routes fail to satisfy demand at all retail outlets; i.e., at least one retail outlet experiences a delivery shortage. Based on the economics of a particular situation, the depot could choose to ignore the daily ¯uctuations by maintaining the routes that would be appropriate if there were no ¯uctuations. This would involve providing on each route the delivery capacity that is just sucient to meet the exact mean demand on that route. However, if delivery shortages are costly, then the managerial challenge becomes: ``How can delivery shortages be overcome, and what are the resource implications of the strategies for overcoming them?'' This is the central question that the present research aims to answer, and the corresponding issues are important in many practical inventory/distribution settings. The dis-tribution of beverages and other retail grocery items are just two of the many applications, and examples of speci®c case studies of the named products appear in, respectively, Benton and Rossetti (1992) and Waters (1989).
Clearly, one strategy involves solving the VRP daily to ®nd the cost-minimizing routes for each day; i.e., route reoptimization. Compared to the ``do-nothing'' option of ignoring demand ¯uc-tuations, this demand responsive strategy not only eliminates delivery shortages (assuming ade-quate inventory at the depot), it also requires either the same or less transportation resources. These improvements require an ecient information system to support the operation of reopti-mization. For example, each day, the depot must enter/load the demand data to its VRP algo-rithm, run the algoalgo-rithm, then disseminate the relevant output information to the appropriate personnel to ensure that the correct routes are followed. Further, the daily demand data must be received early enough for these tasks to be completed before delivery vehicles are dispatched. The present research quanti®es the aforementioned improvements to provide a base for answering the question: ``at what information cost would it be cost-eective to use route reoptimization for overcoming delivery shortages?'' For the purposes of this research, information (system) cost is assumed to include not only the costs just mentioned as necessary for eecting route reoptimiza-tion but also the indirect costs of ineciencies resulting from drivers having to frequently change their delivery routes.
These direct and indirect costs of route reoptimization may make it desirable to use a ®xed set of routes each day in eliminating delivery shortages. This strategy requires the inventory buer on each route (the excess of vehicle capacity or quantity delivered over average demand on the route) to be adequate for the highest likely level of demand. Average demand on each route must therefore be lower than what it would need to be if demand did not ¯uctuate. Consequently, the number of vehicles (routes) and the total distance they travel will be greater, resulting in higher transportation costs. So, relative to route reoptimization, this static routing strategy, basically substitutes transportation and inventory for information. Quantifying this tradeo under a wide range of values for the key factors (including the extent of demand variability) is one of the tasks this research.
Eq. (1) is presented as a schematic portrayal of the analytical framework used. In this schematic model, the average daily values of unsold inventory, delivery shortage, required number of motor vehicles, and the travel distance, are denotedI,S,M, andD, respectively, while the total daily cost of information/communication is denoted W. The subscript for these terms identify the routing approach:k=0 if demand ¯uctuations are ignored in the design of the routes,kis designated asF
used. For ease of exposition, and without loss of generality, only the route reoptimization strat-egy is assumed to have an information cost; i.e., W0=WF=0. The average per day total cost of
inventory, transportation, and information is Ck, and the relevant unit costs or cost coecients
are denoted as H, B, V, and T in Eq. (1). Table 1 explains these coecients, together with the other variables and related assumptions of the analytical framework.
CkHIkBSkVMkTDkWk: 1
Using this framework, the basic research task involves calibrating the schematic model (for each of the three approaches) as functions of the key factors and then using the calibrated models to derive some generalizable results about how the approaches compare with each other. It should be noted that the setting of interest in the present research is the supply chain. As such, the cost coecients in the analytical model are conceptualized as supply chain costs, regardless of which echelon in the supply chain incurs them. Therefore, it does not matter whether a cost is incurred by the depot or by the retailers, or even whether the depot is a distinct legal entity from the retailers. For ultimately, what matters is that the total supply chain cost of getting a product to the ®nal user supports, or at least does not harm, that product's competitiveness.
The work to answer the primary question of the research is reported on in the next ®ve sections of the paper. The ®rst of these sections (Section 2) reviews the segment of the literature that
Table 1
Key variable de®nitions and assumptions
Q The capacity of each delivery vehicle (number of units of product)
Q The arti®cial capacity of each delivery vehicle
L The length (in days) of the planning horizon under consideration
mi The mean demand of customer (retailer)ion days when customeri; places an order; i.e., the mean nonzero demand of customeri;(=E[mi]) is the mean nonzero demand per customer. Each customer's nonzero demands are normally distributed with (mean, standard deviation)=(mi,omi), and there is no demand correlation, either across customers or over time
p The probability with which each customer places an order each day
! The coecient of variation in the nonzero demand of each customer
The coecient of variation in mean nonzero demand across customers N The number of customers in the region served by the depot
A The area of the (rectangular) region served by the depot r The length:width ratio of the region served by the depot
The average Euclidean distance of customers from the depot H The per day cost of holding one unit of the product in inventory B The per day cost of a delivery shortage of one unit of the product T The per mile cost of transportation
V The per day cost of dispatching a delivery vehicle (regardless of distance traveled)
Ik Average inventory at the end of each day if the routing approach isk(kis designated as either 0,F, orR, depending on whether the approach is, respectively, ``do-nothing'', ®xed routes with buer inventories, or, route reoptimization
Sk Average delivery shortage per day if the routing approach isk(Only the ``do-nothing'' option incurs shortages, soSF=SR=0)
Mk Average per day number of vehicles required if the routing approach isk Dk Average per day distance traveled if the routing approach isk
focuses on vehicle routing with stochastic customer demands. The review sets the background for identifying the four main contributions of the present research. Section 3 explains the research design. Section 4 presents the calibrated models, along with the related results from the model calibration process. Section 5 discusses the principal ®ndings from the use of the models to com-pare the approaches for dealing with delivery shortages. Section 6 concludes the paper and iden-ti®es some possible research to build on the present results.
2. Literature review
The problem of managing vehicle routing and dispatch operations under conditions of sto-chastic demands, commonly called the stosto-chastic vehicle routing problem (SVRP), has received two broad types of treatment in the literature. One type involves developing heuristics for a priori optimization; i.e., designing, before knowing the exact demands by customers, a ®xed set of routes that minimizes the expected transportation cost. This cost includes the cost of making emergency deliveries to redress shortages. This line of research utilizes the principles of chance-constrained programming presented by Charnes and Cooper (1959, 1963) and can be traced back to Tillman (1969). It includes the seminal work by Stewart and Golden (1983), key contributions from Bertsimas et al. (1990) and Jaillet (1988), as well as more recent studies by, for example, Bertsimas et al. (1995); Gendreau et al. (1995); Savelsbergh and Goetschalkx (1995).
The second line of SVRP research addresses a posteriori solutions such as route reoptimiza-tion; i.e., selecting suitable routing adjustments after exact customer demands become known. Some recent examples of representative studies from this line of research are Bertsimas (1992), and the previously cited studies by Benton and Rossetti (1992), Bertsimas et al. (1995), and Savelsbergh and Goetschalkx (1995). The two types of SVRP research, though far less volumi-nous than research on the deterministic version of the problem, is still of signi®cant interest. The number of recent papers on the topic re¯ects that interest. This is also emphasized by the litera-ture reviews in Gendreau et al. (1996), and in Bertsimas and Simchi-Levi (1996), both of which are more extensive than what can be presented here.
The present review identi®es four concerns regarding issues that have been either ignored or inadequately treated in the existing literature. The ®rst concern is that the treatment of inventory, even where it is a key issue (see, for example, Dror and Trudeau, 1988 and Trudeau and Dror, 1992), does not permit ready quanti®cation of how inventory impacts the cost comparison of the dierent routing strategies for addressing delivery shortages. Thus, because inventory was outside the scope of assumptions in previous comparisons, the conclusions cannot be taken at face value. A case in point is that Bertsimas (1992) and Bertsimas et al. (1995) conjecture, using travel dis-tance comparisons, that stable routes based on a priori optimization may be competitive with route reoptimization, especially given the latter's potentially high combined costs of information/ communication and route instability. The obvious diculty with this conjecture is that, for a given service level, greater route stability requires more inventory, making it costlier and therefore a less competitive alternative to route reoptimization.
emergency deliveries might be undesirable enough for the depot to, instead, contemplate a strat-egy of completely preventing delivery shortages; i.e., ®xed routes with in-vehicle buer inven-tories. The third concern, closely related to the second, is that the ``do-nothing'' option has not been explored. The motivation for studying that option in the present study is the maxim that it is prudent to not solve a problem and incur the consequences if the costs of solving that problem far exceed those consequences. So, analyzing that option permits quanti®able answers to the man-agerially relevant question of whether, in a given situation, the bene®t of eliminating delivery shortages is worth the eort.
The fourth concern is that to estimate the transportation cost, one must ®rst run a VRP heuristic to get the required travel distance and number of vehicles for a given strategy. Having a formula that accurately estimates these inputs to the cost calculation can make the process of analysis more computationally ecient. True, the deterministic side of VRP research has yielded many useful distance prediction formulae that are adaptable to the SVRP, and these include Daganzo (1984) and Robuste et al. (1990). Nevertheless, the problem de®nitions and assumptions in previous research are dierent enough from those of this research to warrant a separate model development eort here. Among the dierences is the type of routing heuristics used.
3. Research design
The main elements of the research design involved selecting the factors that aect the relative desirability of each of the three approaches for dealing with demand ¯uctuations, and determin-ing, for combinations of the values or levels of these factors, the transportation and inventory impacts of each approach. The resulting data were then used to model the eect of those factors on transportation and inventory costs. Probabilistic simulation was necessary in generating the data for modeling the transportation impacts, while analytical derivation based on established principles of probability theory automatically yielded the models for the inventory impacts. The presentation that follows therefore focuses on the more challenging task of designing the research to study the transportation impacts.
Of the maximum possible 14,400 combinations of the factors, 4115 could not be used in the study. Those with (!, p)=(0, 1) depict no variability so there was no point in analyzing them. Other combinations were dropped either because the mean demand of at least one customer exceeded the vehicle capacity, making the corresponding VRP infeasible, or because it was impossible to design the routes to prevent delivery shortages. For each of the 10,285 usable combinations the experiment was replicated ®ve times to yield a total of 51,425 observations. Each replicate represented a random repositioning of the customer locations in the area served by the depot, under the assumption that the x and y location coordinates are both uniformly dis-tributed. This method of replication ensured that the resulting models account for variations in how customer locations are juxtaposed in the area served by a depot.
To obtain the empirical data on transportation cost, mathematical programming formulations and accompanying vehicle routing heuristics had to be applied to solve three VRPs for each of the 51,425 data points: one for the scenario where the depot ignores demand ¯uctuations in designing the routes; a second for the scenario where ®xed routes are designed with in-vehicle inventory buer to prevent delivery shortage; and a third for the scenario where reoptimization is used. For this latter scenario, a sample of 1000 random demand outcomes were generated for each of the 51,425 data points, and the corresponding VRP solved to estimate the mean travel distance and vehicle requirements for route reoptimization. Because the largest observed sampling error was 3.83% of the estimate (the average was 2.02%), the sample size of 1000 was considered adequate. For the ®rst and third scenarios, the VRPs were solved by a combination of two heuristics: the modi®ed Clarke±Wright method (Clarke and Wright, 1964) due to Paessens (1988) and the gen-eralized insertion procedure with stringing and unstringing (GENIUS) proposed by Gendreau et al. (1992). For the second scenario, the procedure was to formulate thedeterministic equivalentof the SVRP and then solve it with the previously stated combination of VRP heuristics. The validity of this procedure, formally introduced in Golden and Yee (1979) is well established in the literature. This deterministic formulation, the formulation of the SVRP, and other details of the procedure can be found in Stewart and Golden (1983).
The essence of the procedure is that, instead of directly solving the more dicult SVRP, one can formulate its deterministic equivalent and solve it with VRP heuristics that are designed for
Table 2
Experimental values of main eect variables
Factor/variable Levels/values
Q 3 5 0.30Nm 0.40Nm 0.60Nm Nm
L Not experimented with (its in¯uence can be modeled without the need for experimental data)
{=E[mi]} Held constant at 1 unit throughout the study
p 0.50 0.55 0.65 0.80 0.95 1.00
! 0.00 0.05 0.10 0.20 0.30
* 0.50 0.33 0.20 0.11 0.06
N 20 40 80 100
A Held constant at 10,000 square miles throughout the study
r 1 2 4 6
* The values forare based on the use of ®ve dierent modi®ed Poisson distributions: [{Poisson (=1) +x}/(1 +x)]; withx
problems with stable customer demands. This involves using the parameters of customers' demand distributions and depot management's policy (zero delivery shortage in this study) to compute anarti®cialcapacity (Q), and let the heuristic work withQ instead of the actual capacity (Q) in solving the VRP. The basic function of an arti®cial capacity in the SVRP is to ensure that the mean demand on each route isless thanthe actual capacity (Q) of the vehicle.Q is the highest mean demand a route can have. The excess of Q over Q is the inventory buer for preventing delivery shortages. For moderate demand variability; e.g., (!, p, )=(0.1, 0.1, 0.85), a typical ratio forQ toQis approximately 0.76 across the full range of possible values forQ. Golden and Yee (1979) showed that if the variance to mean ratio of demand on any given route is some constant, , then the arti®cial capacity required to restrict each route's probability of a delivery shortage tois given by Eq. (2). In this equation,Z1ÿis the 100(1ÿ) percentile of the standard
normal distribution of route demand. Since this research examines the case where the goal is to limit each route's probability of delivery shortage to zero, a Z value (or safety factor) of 4 was used; i.e., to cover route demand up to four standard deviations above the mean. In truth, aZof 4 allows a nonzero probability of delivery shortage equal to 0.000032, but this probability is considered small enough to be essentially zero. Lower safety factors, e.g., 3, were not used because they would yield delivery shortage probabilities that would be too high for the strategy to be regarded as one that virtually eliminates delivery shortages.
For the present research, it can be shown that the expected value ofis given by the expression in Eq. (3), where the terms are as de®ned in Table 1
!21ÿp 21: 3
4. Research results: The models
The formulas that were found to provide the most accurate predictions of daily travel distances are Eqs. (4)±(6). The models capture the characteristics that elongation (the expression involving
DF2
In each equation, the ®rst parenthetical term, rounded up to the next integer, is the daily number of vehicles, Mk. Some other noteworthy features of the models follow. First is that the
do-nothing and route reoptimization approaches require the same number of vehicles, and that if p=1, their travel distances are identical. At p< 1, the customer to customer component of travel distance is lower for route reoptimization. This is because, on average, route reopti-mization requires a visit to Np customers, while the do-nothing option requires a visit to all N
customers each day. Another feature is that because Q <Q, the strategy of ®xed routes with in-vehicle buer inventories requires more vehicles and involves longer travel distances than route reoptimization. These features are signi®cant in the overall comparison of the three routing approaches, a matter to be dealt with after the following presentation of the inventory models.
Starting with the inventory models for the ``do-nothing'' option of supplying each route with exactly the expected demand on that route, the expected delivery shortage per period (day), S0,
and the expected inventory at the end of each day,I0, are:
S0
Comparative predictive accuracy of distance prediction models
Model Prediction errors measured as
I0
XM0
M1
XL
t1
q
0
qÿqt;Mf qt;Mdqt;M
8
<
:
9
=
; 0
@
1
A
1
L
: 8
In these expressions, M identi®es each route (M1;2;. . .;M0), qt,Mis the cumulative quantity
demanded on route M up to day t, f(qt,M) is the probability density function (p.d.f.) of that
demand,q*is its expected value, and all other terms are as de®ned in Table 1. Of note is that both
S0andI0increase with increases in the length of the planning horizon (L). The explanation is that
on each route the ¯uctuations of demand around its mean increase with time while the quantity delivered remains at the mean. The clear implication is that the depot's average per day shortage and inventory costs will rise as long as it continues to neglect demand ¯uctuations in its route design. Fig. 1 depicts this fact for the case of shortages using (Q,, p, !, , N)=(7, 1, 1, 0.05, 0.05, 100). The vertical axis scales S0 so that it is expressed as the average per day fraction of
expected total demand (Npm) that is unmet.
It is worth remarking that the formulas forS0andI0can be evaluated exactly, even if the p.d.f
is not well de®ned; i.e., if there is ¯uctuation, both in whether a customer places an order (p< 1) and in the size of nonzero orders (!> 0). In such cases, if both the number of periods in the planning horizon (L) and the number of customers on each route are small, the required volume of calculations is quite manageable. On the other hand, if eitherLor the number of customers on each route is large, then clearly, one can appeal to the Central Limit Theorem and use the normal approximation to obtain very accurate estimates.
For the strategy of ®xed routes with in-vehicle buer inventory of Q±Q per route, calculation of the average inventory at the end of each day is more straightforward; as Eq. (9) shows, it is merely the product of the buer on each route and the number of routes. Note that as per earlier description of this strategy, its expected delivery shortage is zero (SF=0). Route reoptimization's
expected delivery shortage is also zero but it carries the buer at the depot to facilitate delivery of the exact and known demands each day. This average end-of-day inventory is given by Eq. (10), and like the corresponding quantity for the strategy of ®xed routes, it is based on a safety factor of 4. Given the transportation and inventory models in Eqs. (4)±(10) the key results can now be discussed.
IF QÿQMF; 9
IR 4
Np !21ÿp 212
p
: 10
5. Discussion of results
The ensuing discussion is based primarily on a series of pairwise comparisons of the three approaches using Eqs. (4)±(10). The primary question addressed was what information cost (WR)
would make it cost-eective to prevent delivery shortages using route reoptimization. In answer-ing that question by ®rst comparanswer-ing route reoptimization to the use of in-vehicle buer inventory, the generally much larger transportation and inventory requirements of the latter was con-spicuous. Table 4 shows the greater resource requirements, using a test case that might be typical: 100 customers and vehicles with a capacity of seven times the mean nonzero demand per custo-mer. It shows, for example, that even under quite moderate demand ¯uctuations, using the in-vehicle buer inventory strategy instead of route reoptimization can require a 77% increase in the required number of vehicles (this is based onr=1 and it increases slightly asris increased). Thus, for route reoptimization to be preferred, the ratio of its daily information cost to daily per vehicle dispatch cost (V) cannot exceed 77% of its required number of vehicles (MR). Putting this in the
perspective of the scenario from which the value was obtained,WRwould have to be no greater
than 10V. More generally, this threshold can be stated asMFÿMR. (See Appendix A.)
Table 5 shows the threshold values (MFÿMR) for some additional scenarios. The fact that the
threshold varies inversely with vehicle capacity is an understandable result that can be viewed as a demonstration of the inventory consolidation principle. Speci®cally, larger vehicles mean larger groups of customers can be served together. For a given service level, this requires a smaller overall inventory buer, and hence fewer additional vehicles than when there are a large number
Table 4
Ratios of resource requirements for ®xed routes with in-vehicle buer inventory to resource requirements for route reoptimization
Resource Level of demand variability
Low Moderate High
(!,,p)=(0.05,0.05, 1) (!,,p)=(0.1,0.1,0.85) (!,,p)=(0.3,0.3,0.5)
Travel distance 1.05 1.49 2.23
Vehicles 1.03 1.77 2.81
Inventory 3.93 4.70 4.75
of small customer groupings. However, with pragmatic restrictions in the number of delivery stops a driver can make, perhaps because of work rules, it might be dicult for the in-vehicle buer inventory strategy to realistically reduce its margin of inferiority through customer-dense routes.
A further point regarding the diculty of justifying this strategy is that, by ignoring inventory carrying and travel distance costs to facilitate exposition, the calculation of the thresholds are biased in its favor. Thus, once these are considered, then information cost can exceed the thresholds in Table 5, and still make route reoptimization the preferred strategy. The rationale for presenting the information costs thresholds as relative to Vinstead of to one of the ignored costs is based on a conjecture that in practice, Vis likely to be large enough to make aggregate vehicle dispatch cost the major element of total cost. In any case, the other ratios can be easily obtained to address instances in which the aggregate cost of either inventory or travel distance is likely to dominate. The important point though is that the results suggest that it is only condi-tions of low demand ¯uctuacondi-tions or very high information costs that the in-vehicle buer inven-tory strategy is likely to be a desirable way of addressing delivery shortages. Given this inference, the remainder of the discussion will focus on the comparison of route reoptimization with the approach of ignoring demand ¯uctuations in route design.
For this comparison, as with the previously presented comparison, the main task was to determine highest level of information cost at which route reoptimization would still be preferred to its alternative. This is expressed as the ratio of information cost to shortage cost, and, as the derivation in the appendix shows, the threshold value for this ratio evaluates toS0. Table 6 shows
the threshold ratios for several scenarios. The data show, for example, that under moderate demand variability, if the vehicle capacity is 7, then a depot cannot justi®ably reject route reoptimization in favor of the do-nothing approach for a 20-day period unless WR exceeds a
threshold of 16.76Bm; (S0=16.76). In other words, if the scenario on which Table 6 is based
characterizes a situation where delivery shortages cost $0.01 per unit per day, and is, say, 10 units, then the depot should reject route reoptimization if its use will incur information costs exceeding $1.68 per day.
As is evident in Table 6, the thresholds vary inversely with vehicle capacity but move in the same direction as the length of the planning horizon. The explanation for the former is inferable from the discussion of Table 5. It is that a small number of customer-dense routes (highQ) yields a lower total shortage than a large number of thinly populated routes (low Q). Naturally, there-fore, higher values ofQ, by reducing the severity of the delivery shortage problem, means that the cost of information for overcoming the problem has to be lower. The explanation for the latter is
Table 5
Threshold ratios for information cost to vehicle dispatch cost
Level of demand variability
Vehicle capacity (Q) Low Moderate High
7 1 10 15
14 1 3 5
21 1 1 2
based on an earlier point that longer neglect of demand ¯uctuations in route design worsens the delivery shortage situation. The higher threshold is thus indicating that as time progresses, it becomes harder to justify that neglect, or alternatively, easier to justify route reoptimization as an alternative. For simplicity, cost of holding inventory (H) and the travel distance cost (T) were temporarily ignored in computing the thresholds in Table 6. Ignoring Tis unfavorable for route reoptimiza-tion since as menreoptimiza-tioned in explaining the travel distance models, its travel distance never exceeds that of the do-nothing option. Ignoring the former is not as clear-cut since the inventory for route reoptimization sometimes exceeds the inventory for the do-nothing approach. This generally occurs unless the latter approach is sustained for long periods and the vehicle capacity is low. When this occurs, then depending on the ratio ofHtoT, route reoptimization can still be the less preferred option even when WR falls belowBS0. Conversely, when route reoptimization has the
smaller inventory, one can say that it is the unequivocally better approach ifWRfalls belowBS0.
The analogous derivation of the threshold for the ratio ofHtoTis presented in the appendix and Table 7 shows the results for the same scenarios used in Table 6. Scenarios in which route reoptimization has the lower inventory are ¯agged. The others show the corresponding threshold values forH/T. For those scenarios, the interpretation is that ifWRfalls belowBS0and H/Tfalls
below the ratio of (D0-DR) to (IR-I0) then route reoptimization is more cost eective. Conversely,
ifboth conditions are unmet then the depot would be better o ignoring demand ¯uctuations in
Table 6
Threshold ratios for information cost to shortage cost
Level of demand variability
Low Moderate High
Vehicle capacity (Q) 1 10 20 1 10 20 1 10 20
7 0.79 1.78 2.44 5.44 12.21 16.76 6.52 14.66 20.12
14 0.58 1.29 1.78 4.01 9.01 12.36 4.52 10.16 13.95
21 0.45 1.00 1.38 3.31 7.43 10.19 3.96 8.89 12.20
Notes: (1) The bold numbers represents the lenght of planning horizon (L). (2) Data are based on (,N,r)=(1, 100, 1). (3) Uses the same numerical de®nitions of ``Low'', ``Moderate'', and ``High'' as Table 4. (4) To get results for other values of, ®gures in the table would have to bemultipliedby.
Table 7
Threshold ratios for holding costs to transportation costs
Level of demand variability
Low Moderate High
Vehicle capacity (Q) 1 10 20 1 10 20 1 10 20
7 0.00 0.00 *** 5.71 20.53 *** 12.30 24.78 77.51
14 0.00 0.00 0.00 4.52 8.39 19.80 10.17 14.75 21.15
21 0.00 0.00 0.00 4.10 6.39 10.21 9.61 13.05 17.17
designing its delivery routes. Taking the scenario where the vehicle capacity is 7, and demand variability is high, the interpretation is that the depot can unreservedly justify ignoring demand ¯uctuations in its route design for 20-day periods only if WR/B exceeds 20.12 and H/Texceeds
77.51.
When the cost coecients do not strictly meet the stated conditions, the choice of routing approach for this or any other scenario is not as immediately obvious. Determining the best choice would have to be done by multiplying Eqs. (4)±(10) by the appropriate cost coecients to determine the total costs. Table 8 illustrates this with input data based on a realistic scenario involving the distribution of grocery items by wholesalers to retailers. Information on actual grocery distribution cases presented by Waters (1989), Benton and Rossetti (1992), and Carter et al. (1996) is the source of all these data, except the unit costs of inventory holding and delivery shortage, and the length of the planning horizon. The former is based on an estimated average price of $2.00 for a grocery item and assumes the typical 25% for inventory holding cost as a percentage of item price (see, e.g., Lambert and Stock, 1993 pp. 364±367). The shortage cost is set at 10% of item price. This is because the pro®t margin on sales is fairly well accepted as an approximation of the cost of shortage, and pro®t margins in excess of 10% do not seem likely (see, Lambert and Stock, 1993 p. 360). A planning horizon of 5 days is assumed.
Table 8 presents the particulars of the scenario (in terms of the input values) and the results. As the total costs show, the do-nothing option would be the least expensive approach. Table 9 extends the analysis of this scenario by exploring the sensitivity of the outcome to dierent pairs of values of the unit costs of inventory holding and delivery shortage. The outcome for each pair of these values is presented as the excess of the total cost of the do-nothing option over the total cost for each of route reoptimization. As the multiple occurrence of negative values shows, the do-nothing option can be a competitive alternative under certain circumstances. The obvious caveat is that the outcomes in Table 9, like any other set of outcomes, depend on the set of input values that are in eect. The outcome might be dierent with costs and other parameters that lie outside grocery distribution (and possibly even outside the grocery distribution context used here). One example, alluded to earlier, is that because the do-nothing option incurs avoidable customer-to-customer travel distances, the relative performance of that option worsens as the number of
Table 8
Comparative results for a sample problem
Inputs
Physical Units: (Q, L,m, p,o,d, N, A, r,r)=(10, 5, 100, 0.85, 0.10, 0.20, 100, 1002, 1.60, 40.72) Cost Coecients: (H, B, T, V, WR)=($0.50, $0.20, $1.00, $17, $32.00)
Outputs
Average costs per day Ck HIk BSk VMk TDk Wk
Do-nothing (use routes suitable for stable demand) $2,087 $392 $157 $153 $1,385 Fixed routes; with in-vehicle buer inventory $4,888 $2,830 $ÿ $255 $1,803
Route reoptimization $2,270 $752 $ÿ $153 $1,333 $32
Average physical units per day Ik Sk Mk Dk
Do-nothing (use routes suitable for stable demand) 784.11 784.11 9 1385 Fixed routes; with in-vehicle buer inventory 5661.00 0.00 15 1803
customers is increased. One would therefore expect a distributor serving 1000 retailers to be more likely to prefer route reoptimization than one serving only 50 customers, all else being equal. Given the likely wide variations in this and other input values across dierent distribution situa-tions, the sort of analysis summarized in Table 8 and Table 9 is best left to the analyst studying a particular situation. The requisite formulae are presented in this paper to facilitate such analysis. From the perspective of the do-nothing approach, two general conclusions came out of the type of the threshold analysis presented in Tables 6 and 7. One is that in a majority of the cases, the conditions for justifying the do-nothing approach seemed dicult to satisfy. This result may help to explain a perception that in practice, if demands are stochastic, then cases where routes are adjusted in response to the demand ¯uctuations are probably more likely to be encountered than cases where the routes remain ®xed. The other conclusion is that in cases where the condi-tions seemed attainable, these generally occurred at lower levels of demand variability and lower values ofL(length of planning horizon). The implication of this latter conclusion is that ignoring demand ¯uctuations in route design is an approach that cannot be completely dismissed. In parti-cular, the previously discussed relationship between the performance of that approach andL sug-gests that the approach can be suitably modi®ed to simultaneously retain the bene®ts of route stability and prevent the continuous increases in shortages and inventories. The modi®cation would require the depot to review the inventory and shortage situation at appropriate intervals and respond accordingly. The chosen interval would be the point at which leaving the routes unchanged for another day would cause the total cost of the do-nothing option to exceed the total cost of route reoptimization. The interval from a given scenario can be readily determined from the formulas presented. Table 10 shows these intervals for the same set of scenarios analyzed in Table 9.
6. Conclusions
The present research addressed the issue of evaluating dierent ways of dealing with delivery shortages when the demands of customers in a logistic network are stochastic. In addition to the strategies of route reoptimization and using in-vehicle buer inventory, the research studied the approach of ignoring demand ¯uctuations as a baseline strategy (the do±nothing approach). Accurate distance prediction models were developed, and combined with established models from probability theory to present a framework for comparing the approaches in terms of inventory,
Table 9
Total cost of the do-nothing option minus total cost of route reoptimization
Delivery shortage cost (B) Inventory holding cost (H)
$0.20 $0.40 $0.60 $0.80 $1.00
$0.05 $(84.68) $(228.73) $(372.77) $(516.82) $(660.87)
$0.10 $(45.48) $(189.52) $(333.57) $(477.61) $(621.66)
$0.20 $32.93 $(111.11) $(255.16) $(399.20) $(543.25)
$0.40 $189.76 $45.71 $(98.34) $(242.38) $(386.43)
$0.80 $503.40 $359.36 $215.31 $71.26 $(72.78)
Note: (Q, L,m, p,o,d, N, A, r,r)=(10, 5, 100, 0.85, 0.10, 0.20, 100, 100 2 , 1.60, 40.72) (T, V, W
transportation, and information costs. In addition to the distance prediction models and the comparison framework, a key contribution of the research is that it determines the information cost thresholds for accepting/rejecting route reoptimization. The thresholds show that the approach of ignoring demand ¯uctuations in route design can be cost-eective under certain cir-cumstances. The models capture the intuitive result that the cost-eectiveness of this approach deteriorates over time. Based on this result, the models were used to illustrate guidelines for per-iodically reviewing the do-nothing approach to ensure that inventory costs do not become excessive through prolonged neglect of day-to-day demand ¯uctuations in vehicle routing decisions.
Several issues appear to be prime topics for extending the research. One is to attempt to estimate the magnitude of the route instability problem associated with route reoptimization. While its existence as one component of the cost of route reoptimization is well established, its magnitude remains unknown. Estimates of the magnitude of the problem will help the exploration of strategies for controlling it without seriously diminishing the inherent eciency of route reoptimization. Finally, a closely related topic is that known or accurately forecasted demand for more than one period into the future might facilitate the achievement of route sta-bility by buying the depot more time to plan multi-period routes. As in this research, a funda-mental question in future studies of this topic would have to be whether or not the resulting bene®ts are sucient to cover the cost of obtaining (near) perfect multi-period demand forecasts/ information.
APPENDIX A. Derivation of cost thresholds
The route reoptimization approach will not be inferior to the approach of using in-vehicle buer inventory on ®xed routes if:
HIRVMRTDRWR4HIFVMFTDF: A:1
This condition can be expressed as the upper limit on the ratio of route reoptimizations' infor-mation cost (WR) to the cost of vehicle dispatch (V); i.e.:
Table 10
Recommended intervals (days) for reviewing the do-nothing option
Delivery shortage cost (B) Inventory holding cost (H)
$0.20 $0.40 $0.60 $0.80 $1.00
$0.05 11 15 17 18 19
$0.10 6 13 14 16 17
$0.20 3 8 11 13 14
$0.40 1 4 6 8 10
$0.80 * 1 2 4 5
Note: (Q, L,m, p,o,d, N, A, r,r)=(10, 5, 100, 0.85, 0.10, 0.20, 100, 100 2 , 1.60, 40.72) (T, V, W
R)=$1.00, $17, $32.00). * No
WR
V 4 MFÿMR H
V IFÿIR T
V DFÿDR
: A:2
Because all the right hand terms are nonnegative, Eq. (A.2) indicates that themaximumvalue that the above ratio can attain without making route reoptimization inferior to the strategy of in-vehicle buer inventory on ®xed routes is at least(MFÿMR). The focus on the ratio ofWRtoV
instead of the ratio ofWRtoHorWRtoTimplicitly assumes thatHandTare small in relation
toV, and can therefore be ignored. Where that assumption may not hold, the equivalent thresh-old for the ratio ofWRto the other cost coecients can be derived similarly.
The route reoptimization approach will not be inferior to the approach of ignoring demand ¯uctuations in route design is superior to route reoptimization if:
HIRVMRTDRWR4HI0BS0VM0TD0: A:3
SinceMR=M0, then expressing Eq. (A.3) as the upper limit on the ratio ofWRtoByields:
WR
B 4 S0 H
B I0ÿIR T
B D0ÿDR
: A:4
On the right-hand side only (I0ÿIR) is not guaranteed to be nonnegative. If the left hand side
ratio isS0then satisfaction of Eq. (A.4) requires that the remaining right hand side expression be
nonnegative. This requirement can be expressed as:
H T4
D0ÿDR
IRÿI0
: A:5
This requirement is only de®ned for strictly positive values of (IRÿI0). Taking the requirement in
Eq. (A.4) along with de®nable results for Eq. (A.5), the interpretation is that if the ratio ofWRto
B does not exceed S0 and the condition in Eq. (A.5) is satis®ed then route reoptimization is
unequivocally less costly than ignoring demand ¯uctuations in route design. The equivalent threshold for the ratio of WRto the other cost coecients can be similarly derived.
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