Directory UMM :Data Elmu:jurnal:T:Transportation Research_Logistics & Transportation Review:Vol35.Issue1.MAr1999:
TRANSPORTATION
RESEARCH
PART E
Transportation Research Part E 35 (1999) 25±41
Comparing strategies for addressing delivery shortages in
stochastic demand settings
Michael A. Haughton a,*, Alan J. Stenger b
a
Department of Management Studies, University of the West Indies, Mona Campus, Kingston 7, Jamaica, W.I.
501J Business Administration Building, The Smeal College of Business Administration, The Pennsylvania State University,
University Park, PA 16802-3005, USA
b
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 strategies on the basis of how these strategies aect inventory and transportation. The focus of ®ndings from the
comparison involves identifying the information cost thresholds for accepting/rejecting a demand-responsive 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
* Corresponding author. Tel.: (876) 977-3775; Fax: (876) 977-3829; E-mail: mhoughton@unimona.edu.jm
1366-5545/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved.
PII : S 1366-5545(98 ) 0 0 0 2 1 - 0
26
M.A. Haughton, A.J. Stenger / 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 distribution 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 ¯uctuations, this demand responsive strategy not only eliminates delivery shortages (assuming adequate 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 reoptimization. For example, each day, the depot must enter/load the demand data to its VRP algorithm, run the algorithm, 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 reoptimization 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 denoted I, S, M, and D, 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, k is designated as F
if the strategy is ®xed routes with in-vehicle buer inventories, and as R if route reoptimization is
M.A. Haughton, A.J. Stenger / Transportation Research Part E 35 (1999) 25±41
27
used. For ease of exposition, and without loss of generality, only the route reoptimization strategy 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.
Ck HIk BSk VMk TDk Wk :
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
Q
L
mi
p
!
N
A
r
H
B
T
V
Ik
Sk
Mk
Dk
Ck
The capacity of each delivery vehicle (number of units of product)
The arti®cial capacity of each delivery vehicle
The length (in days) of the planning horizon under consideration
The mean demand of customer (retailer) i on days when customer i; places an order; i.e., the mean nonzero
demand of customer i; (=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
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
The number of customers in the region served by the depot
The area of the (rectangular) region served by the depot
The length:width ratio of the region served by the depot
The average Euclidean distance of customers from the depot
The per day cost of holding one unit of the product in inventory
The per day cost of a delivery shortage of one unit of the product
The per mile cost of transportation
The per day cost of dispatching a delivery vehicle (regardless of distance traveled)
Average inventory at the end of each day if the routing approach is k (k is designated as either 0, F, or R,
depending on whether the approach is, respectively, ``do-nothing'', ®xed routes with buer inventories, or,
route reoptimization
Average delivery shortage per day if the routing approach is k (Only the ``do-nothing'' option incurs
shortages, so SF=SR=0)
Average per day number of vehicles required if the routing approach is k
Average per day distance traveled if the routing approach is k
The average per day total cost of inventory and transportation if the routing approach is k
28
M.A. Haughton, A.J. Stenger / Transportation Research Part E 35 (1999) 25±41
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 compare the approaches for dealing with delivery shortages. Section 6 concludes the paper and identi®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 stochastic demands, commonly called the stochastic 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 chanceconstrained 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 reoptimization; 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 voluminous 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 literature 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 distance 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.
The second concern is that the optimum solution using a priori optimization methods allows a
nonzero probability of delivery shortage on each route. This necessitates emergency deliveries to
redress delivery shortages. However, it is conceivable that the route instability involved in making
M.A. Haughton, A.J. Stenger / Transportation Research Part E 35 (1999) 25±41
29
emergency deliveries might be undesirable enough for the depot to, instead, contemplate a strategy of completely preventing delivery shortages; i.e., ®xed routes with in-vehicle buer inventories. 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 managerially 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 determining, 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.
Table 2 lists the nine factors that were examined, and their corresponding values. As the table
shows, three of these were not varied in the study. They did not need to be since their in¯uence
can be modeled without the need for data from simulation experiments. As an example, the way a
service region's area (A) aects travel distance is well established in the literature; see, for example, Larson and Odoni (1981). The selection basis for the chosen factors is quite logical. One
would expect, for example, that since the coecient of variation in each customer's nonzero
demand (!) and the probability with which each customer places orders ( p) are indicators of
demand ¯uctuations, their selection is warranted. The coecient of variation in mean demand
across customers () is similarly justi®ed. The ®ve modi®ed Poisson distributions used to depict
this were chosen to portray the fact that in test cases used for VRP research, demand across
customers is positively skewed. The number of customers (or customer density of the service
region) and vehicle capacity were considered because they aect both travel distance and inventory. The service region's length to width ratio (or its degree of elongation) is also a determinant
of travel distance; see, for example, Kwon et al. (1995).
30
M.A. Haughton, A.J. Stenger / Transportation Research Part E 35 (1999) 25±41
Table 2
Experimental values of main eect variables
Factor/variable
Q
L
{=E [mi]}
p
!
*
N
A
r
Levels/values
3
5
0.30Nm
0.40Nm
Not experimented with (its in¯uence can be modeled without
the need for experimental data)
Held constant at 1 unit throughout the study
0.50
0.55
0.65
0.80
0.00
0.05
0.10
0.20
0.50
0.33
0.20
0.11
20
40
80
100
Held constant at 10,000 square miles throughout the study
1
2
4
6
0.60Nm
Nm
0.95
0.30
0.06
1.00
*
The values for are based on the use of ®ve dierent modi®ed Poisson distributions: [{Poisson (=1) + x}/(1 + x)]; with x
taking the values 1, 2, 4, 8, and 16.
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 distributed. 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 generalized insertion procedure with stringing and unstringing (GENIUS) proposed by Gendreau et
al. (1992). For the second scenario, the procedure was to formulate the deterministic equivalent of
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
M.A. Haughton, A.J. Stenger / Transportation Research Part E 35 (1999) 25±41
31
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
and let the heuristic work with Q instead of the actual capacity
compute an arti®cial capacity (Q),
(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 is less than the 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 for Q to Q is approximately 0.76 across the full range of possible values for Q. 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 to is given by Eq. (2). In this equation, Z1ÿ is the 100(1 ÿ ) 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, a Z of
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.
q
1
2
2Q Z1ÿ ÿ 2 Z41ÿ 4QZ21ÿ :
2
Q
2
For the present research, it can be shown that the expected value of is given by the expression in
Eq. (3), where the terms are as de®ned in Table 1
!2 1 ÿ p 2 1:
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
r) increases the customer to customer component of travel distance (the second compound term
of each expression) at a decreasing rate, and that this component becomes smaller with smaller
vehicle capacities (Q). Several competing models that involved dierent ways of accounting for
these characteristics were tried. Table 3 presents the best three of these alternatives and comparative statistics on their predictive accuracy vis-aÁ-vis the chosen models. The comparisons were
consistent across Eqs. (4)±(6) so the results are presented in terms of Eq. (4) only. These statistics
constituted the crux of the model selection rationale, and were the obtained by using regression
procedures for building the models on a randomly chosen half of the data and extrapolating them
to the other half. The logic of this standard model validation technique can be found in Neter et
al. (1990), p. 465.
p r 1
Np
p 0:38
p
1:28 A
N ;
1ÿ
4
D0 2
Q
Q
2 r
32
M.A. Haughton, A.J. Stenger / Transportation Research Part E 35 (1999) 25±41
Table 3
Comparative predictive accuracy of distance prediction models
Model
Prediction errors measured as
p r 1
Np
p 0:38
p
D0 2
1:28 A
N
1ÿ
Q
Q
2 r
s
p r 1
Np
p
D0 2
N0:41
1:09 A p
1ÿ
Q
Q
2 r
2 !
p r 1
Np
p
D0 2
0:99 A p
N0:43
1ÿ
Q
Q
2 r
p s
p
Np
2 r
p
D0 2
N0:41
1:11 A p
1ÿ
Q
Q
r 1
MAPE (%)
Miles per customer
2.81
1.75
3.29
2.21
3.47
2.68
4.32
4.47
Note: MAPE=Mean absolute percentage error.
p r 1
Np
p 0:38
p
1:28 A
N ;
1ÿ
DF 2
Q
Q
2 r
5
p r 1
Np
p
DR 2
1:28 A
Np0:38 :
1ÿ
Q
Q
2 r
6
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 reoptimization 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:
0 8
91
1
M
<
= 1
L
0
X BX
C
S0
;
7
qt;M ÿ q f qt;M dqt;M A
@
:
;
L
M1 t1
q
M.A. Haughton, A.J. Stenger / Transportation Research Part E 35 (1999) 25±41
91
0 8 q
M0
= 1
L
RESEARCH
PART E
Transportation Research Part E 35 (1999) 25±41
Comparing strategies for addressing delivery shortages in
stochastic demand settings
Michael A. Haughton a,*, Alan J. Stenger b
a
Department of Management Studies, University of the West Indies, Mona Campus, Kingston 7, Jamaica, W.I.
501J Business Administration Building, The Smeal College of Business Administration, The Pennsylvania State University,
University Park, PA 16802-3005, USA
b
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 strategies on the basis of how these strategies aect inventory and transportation. The focus of ®ndings from the
comparison involves identifying the information cost thresholds for accepting/rejecting a demand-responsive 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
* Corresponding author. Tel.: (876) 977-3775; Fax: (876) 977-3829; E-mail: mhoughton@unimona.edu.jm
1366-5545/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved.
PII : S 1366-5545(98 ) 0 0 0 2 1 - 0
26
M.A. Haughton, A.J. Stenger / 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 distribution 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 ¯uctuations, this demand responsive strategy not only eliminates delivery shortages (assuming adequate 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 reoptimization. For example, each day, the depot must enter/load the demand data to its VRP algorithm, run the algorithm, 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 reoptimization 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 denoted I, S, M, and D, 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, k is designated as F
if the strategy is ®xed routes with in-vehicle buer inventories, and as R if route reoptimization is
M.A. Haughton, A.J. Stenger / Transportation Research Part E 35 (1999) 25±41
27
used. For ease of exposition, and without loss of generality, only the route reoptimization strategy 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.
Ck HIk BSk VMk TDk Wk :
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
Q
L
mi
p
!
N
A
r
H
B
T
V
Ik
Sk
Mk
Dk
Ck
The capacity of each delivery vehicle (number of units of product)
The arti®cial capacity of each delivery vehicle
The length (in days) of the planning horizon under consideration
The mean demand of customer (retailer) i on days when customer i; places an order; i.e., the mean nonzero
demand of customer i; (=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
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
The number of customers in the region served by the depot
The area of the (rectangular) region served by the depot
The length:width ratio of the region served by the depot
The average Euclidean distance of customers from the depot
The per day cost of holding one unit of the product in inventory
The per day cost of a delivery shortage of one unit of the product
The per mile cost of transportation
The per day cost of dispatching a delivery vehicle (regardless of distance traveled)
Average inventory at the end of each day if the routing approach is k (k is designated as either 0, F, or R,
depending on whether the approach is, respectively, ``do-nothing'', ®xed routes with buer inventories, or,
route reoptimization
Average delivery shortage per day if the routing approach is k (Only the ``do-nothing'' option incurs
shortages, so SF=SR=0)
Average per day number of vehicles required if the routing approach is k
Average per day distance traveled if the routing approach is k
The average per day total cost of inventory and transportation if the routing approach is k
28
M.A. Haughton, A.J. Stenger / Transportation Research Part E 35 (1999) 25±41
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 compare the approaches for dealing with delivery shortages. Section 6 concludes the paper and identi®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 stochastic demands, commonly called the stochastic 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 chanceconstrained 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 reoptimization; 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 voluminous 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 literature 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 distance 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.
The second concern is that the optimum solution using a priori optimization methods allows a
nonzero probability of delivery shortage on each route. This necessitates emergency deliveries to
redress delivery shortages. However, it is conceivable that the route instability involved in making
M.A. Haughton, A.J. Stenger / Transportation Research Part E 35 (1999) 25±41
29
emergency deliveries might be undesirable enough for the depot to, instead, contemplate a strategy of completely preventing delivery shortages; i.e., ®xed routes with in-vehicle buer inventories. 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 managerially 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 determining, 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.
Table 2 lists the nine factors that were examined, and their corresponding values. As the table
shows, three of these were not varied in the study. They did not need to be since their in¯uence
can be modeled without the need for data from simulation experiments. As an example, the way a
service region's area (A) aects travel distance is well established in the literature; see, for example, Larson and Odoni (1981). The selection basis for the chosen factors is quite logical. One
would expect, for example, that since the coecient of variation in each customer's nonzero
demand (!) and the probability with which each customer places orders ( p) are indicators of
demand ¯uctuations, their selection is warranted. The coecient of variation in mean demand
across customers () is similarly justi®ed. The ®ve modi®ed Poisson distributions used to depict
this were chosen to portray the fact that in test cases used for VRP research, demand across
customers is positively skewed. The number of customers (or customer density of the service
region) and vehicle capacity were considered because they aect both travel distance and inventory. The service region's length to width ratio (or its degree of elongation) is also a determinant
of travel distance; see, for example, Kwon et al. (1995).
30
M.A. Haughton, A.J. Stenger / Transportation Research Part E 35 (1999) 25±41
Table 2
Experimental values of main eect variables
Factor/variable
Q
L
{=E [mi]}
p
!
*
N
A
r
Levels/values
3
5
0.30Nm
0.40Nm
Not experimented with (its in¯uence can be modeled without
the need for experimental data)
Held constant at 1 unit throughout the study
0.50
0.55
0.65
0.80
0.00
0.05
0.10
0.20
0.50
0.33
0.20
0.11
20
40
80
100
Held constant at 10,000 square miles throughout the study
1
2
4
6
0.60Nm
Nm
0.95
0.30
0.06
1.00
*
The values for are based on the use of ®ve dierent modi®ed Poisson distributions: [{Poisson (=1) + x}/(1 + x)]; with x
taking the values 1, 2, 4, 8, and 16.
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 distributed. 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 generalized insertion procedure with stringing and unstringing (GENIUS) proposed by Gendreau et
al. (1992). For the second scenario, the procedure was to formulate the deterministic equivalent of
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
M.A. Haughton, A.J. Stenger / Transportation Research Part E 35 (1999) 25±41
31
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
and let the heuristic work with Q instead of the actual capacity
compute an arti®cial capacity (Q),
(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 is less than the 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 for Q to Q is approximately 0.76 across the full range of possible values for Q. 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 to is given by Eq. (2). In this equation, Z1ÿ is the 100(1 ÿ ) 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, a Z of
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.
q
1
2
2Q Z1ÿ ÿ 2 Z41ÿ 4QZ21ÿ :
2
Q
2
For the present research, it can be shown that the expected value of is given by the expression in
Eq. (3), where the terms are as de®ned in Table 1
!2 1 ÿ p 2 1:
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
r) increases the customer to customer component of travel distance (the second compound term
of each expression) at a decreasing rate, and that this component becomes smaller with smaller
vehicle capacities (Q). Several competing models that involved dierent ways of accounting for
these characteristics were tried. Table 3 presents the best three of these alternatives and comparative statistics on their predictive accuracy vis-aÁ-vis the chosen models. The comparisons were
consistent across Eqs. (4)±(6) so the results are presented in terms of Eq. (4) only. These statistics
constituted the crux of the model selection rationale, and were the obtained by using regression
procedures for building the models on a randomly chosen half of the data and extrapolating them
to the other half. The logic of this standard model validation technique can be found in Neter et
al. (1990), p. 465.
p r 1
Np
p 0:38
p
1:28 A
N ;
1ÿ
4
D0 2
Q
Q
2 r
32
M.A. Haughton, A.J. Stenger / Transportation Research Part E 35 (1999) 25±41
Table 3
Comparative predictive accuracy of distance prediction models
Model
Prediction errors measured as
p r 1
Np
p 0:38
p
D0 2
1:28 A
N
1ÿ
Q
Q
2 r
s
p r 1
Np
p
D0 2
N0:41
1:09 A p
1ÿ
Q
Q
2 r
2 !
p r 1
Np
p
D0 2
0:99 A p
N0:43
1ÿ
Q
Q
2 r
p s
p
Np
2 r
p
D0 2
N0:41
1:11 A p
1ÿ
Q
Q
r 1
MAPE (%)
Miles per customer
2.81
1.75
3.29
2.21
3.47
2.68
4.32
4.47
Note: MAPE=Mean absolute percentage error.
p r 1
Np
p 0:38
p
1:28 A
N ;
1ÿ
DF 2
Q
Q
2 r
5
p r 1
Np
p
DR 2
1:28 A
Np0:38 :
1ÿ
Q
Q
2 r
6
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 reoptimization 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:
0 8
91
1
M
<
= 1
L
0
X BX
C
S0
;
7
qt;M ÿ q f qt;M dqt;M A
@
:
;
L
M1 t1
q
M.A. Haughton, A.J. Stenger / Transportation Research Part E 35 (1999) 25±41
91
0 8 q
M0
= 1
L