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The general optimal market area model with uncertain and nonstationary demand
Pergamon 096&8349(95)00002-X
Laorion Science, Vol. 3, No. 1, pp. 25-38, 15% Ekvier Science Ltd Printed in Great Britain. 09664349195 $9.50 + 0.00
THE GENERAL OPTIMAL MARKET AREA MODEL WITH UNCERTAIN AND NONSTATIONARY DEMAND SCOTT WEBSTER University of Wisconsin-Madison,
and AMIT GUPTA
School of Business, 975 University Avenue, Madison, WI 53706, U.S.A. (Receioed 12 February 1993)
Abstract-Facility location problems involve determining the number of facilities to open, the location of facilities, and the market area that each facility will serve. While the environment is seldom stable, algorithms for solving models that incorporate change and uncertainty are rare and limited in scope. Consequently, the common approach in practice is to use algorithms for static models with perhaps some scenario analysis to accommodate change and uncertainty. We introduce an analytically tractable model that includes dynamic and uncertain demand. By studying how the model can be solved by solving a related model with stationary demand, we are able to derive guidelines on how to apply static analysis tools to problems with dynamic demand. While the analysis is contingent upon the underlying model, evidence on the model’s robustness suggests that the results represent plausible guidelines for problems that extend well beyond the model. Keywords: Facility location, market area model, distribution planning.
1. INTRODUCTION
Facility location problems involve determining the number of facilities to open, the location of facilities, and the market area that each facility will serve. These problems exhibit a basic cost tradeoff that is due to economies of scale in facility operation and the fact that inbound transportation rates are lower than outbound transportation rates; as the number of facilities increases, transportation costs tend to decrease and facility operation costs tend to increase. Furthermore, they involve a decision that is not easily modified (it is generally not practical to be continually changing the configuration). That is, the common real world facility location problem is one of incorporating anticipated changes in the environment along with the degree of uncertainty in future expectations and identifying a configuration that, while perhaps phased-in over a period of time, is expected to be effective over a given planning horizon. Research on facility location problems has been fairly successful in the sense that algorithms for solving various models of these problems are widely available. However, many of the algorithms with the scope and size capabilities most suitable for industry assume a known and unchanging environment . . . an assumption that rarely holds in practice; see, e.g. Ballou (1985). Consequently, the common approach in practice is to analyze facility location problems under the assumption that the environment is stable, with dynamics and uncertainty captured to a limited degree through the analysis of alternative scenarios. One problem with this approach is that it is very much an art with little in the way of guidelines on how to proceed effectively. One possible line of research that can help to improve this situation is to work on developing algorithms for solving models that capture dynamics and uncertainty. This 25
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SCOTT WEBSTER and AMIT GUPTA
paper describes an entirely different approach; our goal is to provide a first step in the development of guidelines on how to best use the well-developed tools of static analysis when the environment is changing and uncertain. Towards this end, we (1) extend the general optimal market area model to include nonstationary and stochastic demand, (2) identify theoretical properties that link the nonstationary and stochastic -model with an equivalent static model, and (3) use these properties to formulate guidelines on how to apply tools of static analysis to facility location problems with nonstationary and stochastic demand. While the guidelines stem from the underlying model, there is evidence in the literature that the model is quite robust. Thus, the guidelines represent plausible hypotheses on the use of static facility location tools when the environment is changing and uncertain. Work that helps to improve our ability to develop good solutions to facility location problems is important. Distribution cost represents a significant fraction of sales in developed economies and the importance of designing effective distribution systems is increasing. Recent surveys show distribution costs as a percentage of sales to be 7.5% in the United States (Davis, 1988) 11.6% in the United Kingdom (Survey of Distribution Costs, 1985), and 21.1% in Australia (Gilmour, 1985). Importance is increasing due to a rapidly changing environment. In particular, we note the globalization of the marketplace and a trend towards using speed as a competitive weapon (e.g. shortened times from order to delivery). Both of these factors are putting pressure on the need to redesign or expand existing distribution systems. The next section contains a review of related literature. The methodology is reviewed and the analysis is presented in Section 3. Managerial implications of the analysis are summarized in Section 4 and Section 5 concludes the paper.
2. RELATED
LITERATURE
Facility location problems have been considered by a number of researchers who have employed a variety of solution techniques. The approaches include mathematical programming models, dynamic .programming models, and heuristic algorithms. There are several ways to classify the approaches that have been taken by the various researchers. For the purposes of this paper, we will distinguish among the different approaches by considering static models and dynamic models. One common approach to solving facility location problems has been to model the problem as a static problem, with dynamics not being captured in the model at all. The effect of dynamics in such models is considered by performing sensitivity analysis on the static model. Within the static model framework, the approaches may be further distinguished by considering discrete demand versus continuous demand models. Discrete demand static problems have been considered by Cooper (1963, 1972), Chen (1983), Van Roy (1986), Daganzo (1987, 1988), Ghosh and Craig (1991), Hakimi and Kuo (1991), Densham and Rushton (1988, 1992), and Koroglu (1992). Continuous demand static problems have been considered by Drezner and Wesolowsky (1980), Marucheck and Aly (1981), Cavalier and Sherali (1986), Ghoneim and Wirasinghe (1987), and Erlenkotter (1989), among others. The literature on dynamic demand models may be distinguished according to discrete demand versus continuous demand as well. Discrete demand dynamic models have been studied by Ballou (1968), Wesolowsky and Truscott (1975), Schilling (1980), and Drezner
The general optimal market area model with uncertain and nonstationary demand
21
and Wesolowsky (1991) among others. The continuous demand dynamic modelling approach has been utilized by Campbell (1990). Ballou (1968) shows how a dynamic programming model would be used for determining the optimal warehouse location in a dynamic decision problem. He applies dynamic programming to find a dynamic optimum warehouse location-relocation plan. However, the solution’methodology is limited to very small problems. Wesolowksy and Truscott (1975) solve a problem of locating facilities among demand points, where some of the demand points could themselves serve as facility sites. They find that ‘myopic’ relocation plans, which are based on current demands only, are not optimal, and their multi-period model provides a better solution to the problem. They present two methods of solution: (1) a mixed integer programming (MIP) approach, and (2) a dynamic programming approach. It is shown that the MIP approach is feasible for small problems only, whereas the dynamic programming approach can solve relatively larger problems but its state space has to be of ‘manageable’ size. Schilling (1980) considers a dynamic location modelling approach for public sector facilities. It is pointed out that private sector locational decisions are typically based on monetary criteria: minimization of costs, or maximization of profits. These criteria may not be appropriate for public-sector facilities where the decisions may be based on some measure of social welfare. Schilling proposes a general methodology using multi-objective analysis to plan public-sector facility systems in a dynamic environment. The methodology leads to a set of ‘efficient alternatives’ from which the decision maker selects a solution based on her planning strategies. Drezner and Wesolowsky (1991) investigate the problem of locating a facility among a given set of demand points when the demand at each point changes with time in a linear fashion. They determine the location of a single facility in the planning horizon but allow the location to be relocated over time. They also determine the time points at which the facility should be relocated. This model assumes linearity in demand growth as well as the location and relocation of a single facility only. Campbell (1990) develops an approximation model to locate terminals for serving a fixed region with uniform and increasing demand. He claims that since determining an optimal strategy requires knowledge of future demand, myopic strategies may be nearly optimal. He considers three different strategies and shows that myopic strategies are nearly optimal unless relocation costs are large. Thus, in this model, the dynamic demand problem is really treated as a small multi-period problem. The algorithms for dynamic models discussed above are either limited to small problems, or limited in the choice of demand growth rates, or limited to the location of a single facility. In addition, the models assume that demand is deterministic. We note, however, that dynamic and uncertain demand has been considered in the capacity planning literature. Bean, Higle, and Smith (1992) consider the problem of meeting a stochastically growing demand for capacity over an infinite horizon. They allow a demand process that is either a transformed Brownian motion, or a regenerative birth-death process. They also allow general cost structures. The resulting stochastic dynamic problem is transformed to an equivalent static deterministic problem to determine the optimal capacity expansion periods. All costs are discounted at an interest rate smaller than the original, in proportion to the uncertainty in demand. The problem formulation assumes that the location of facilities (and, as a result, transportation cost) is unimportant, and capacity expansion to meet the demand is the only criterion used to minimize the costs.
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SCOTT WEBSTER and AMIT GLJPTA
Uncertain demand has also been considered by ReVelle and Hogan (1989) who introduce a probabilistic version of the maximal covering location problem which they call the maximum availability location problem (MALP). Marianov and ReVelle (1992) incorporate the probability of being able to meet demand into a maximal covering formulation for the location of fire stations and the joint allocation of trucks and engines to those stations. Although the formulation is specifically developed for fire protection services, it may be applied to many types of emergency services that operate in a dynamic setting. In a related line of research, Friesz, Miller, and Tobin (1988) present heuristic algorithms for locating a firm’s production facilities while simultaneously determining production levels with the assumption that existing firms as well as new entrants act in accordance with an appropriate model of spatial equilibrium. Miller, Tobin, and Friesz (1992) address the facility location problem of a firm entering an industry and competing on a discrete network with several other oligopolistic competitors. Their location model combines game theory and a hierarchical mathematical programming approach with sensitivity analysis of variational inequalities. It should be noted that the methodology used in the above two papers is markedly different from the other papers referenced here, and as such, the results are not easily transferable to the other models. 3. ANALYSIS
3.1 Overview of methodology Erlenkotter (1989) describes a model of a distribution system with uniform and unchanging demand. This model, which is known as the general optimal market area (GOMA) model, has a number of desirable properties. First, the operating cost of the distribution system can be expressed as a continuous function that is amenable to analysis. Second, a closed form expression for the optimal market area per facility exists for a wide range of facility operation cost structures and product transportation cost structures. Third, while it is true that demand is often lumpy in practice, researchers have found that basic properties derived from the model extend to models with much more complex demand spaces. For example, Geoffrion (1976) describes how a special case of the GOMA model is used in a facility location study of a consumer products manufacturer which distributes product nationwide out of a single plant in California. Approximately 65 potential warehouse sites and 130 customer zones were identified for the purposes of the study. The sensitivity of the optimal number of warehouses to small changes in problem parameters as determined by an integer programming model is found to be nearly identical to the sensitivity results predicted by the GOMA model. Erlenkotter (1989) reports a comparison of results generated by an integer programming model and the GOMA model for a system comprised of 100 customers, each with a demand of one unit per period. Each demand point is a potential warehouse site and transportation cost is proportional to Euclidean distance. The optimal number of warehouses for the system is calculated for different fixed warehousing rates. Very similar results are found when the system is approximated by the appropriate GOMA model (i.e. eight cases are tested; the results are the same in six cases and differ by one warehouse in two cases). These two studies suggest that the model is suited to the discovery of fundamental insights into problem characteristics that, due to a large quantity of problem specific data, are difficult to achieve through analysis of detailed numerical models (e.g. mathematical programming models).
The general optimal market area model with uncertain and nonstationary demand
29
A generalization of the GOMA model is used in our analysis. We first extend the GOMA model to accommodate dynamic and uncertain demand and then use the model to study how a problem with changing demand can be solved by solving a related problem with stationary demand. 3.2 The model We begin by reviewing the GOMA model with static demand. As noted by Erlenkotter (1989), this model has a long and varied history in the literature as it has been discovered and rediscovered in a number of different forms by researchers from a variety of disciplines. Assumptions
(1) Demand is uniform over an infinite plane with density D > 0 per square mile. (2) Given market area x served by a facility, facility operating cost as a function of volume, Dx, is c,(Dx)” + UDXwhere c1 > 0 and u > 0. Economies of scale are reflected in cl(Dx)LI, where 0 $ c1< 1, with smaller values of tl implying greater relative economies of scale. (3) Unit transportation cost as a function of miles travelled, 6, is cd8 where c > 0 and /I > 0. Economies of distance exist when /? < 1. (4) Various regular two-dimensional market shapes may be specified (e.g. hexagon, circle, diamond). (5) Various distance norms may apply (e.g. Euclidean, rectilinear). (6) The objective is to minimize cost. For many combinations of fi, market shapwnd distance norm, transportation cost is proportional to the market area raised to the /?/2 power; that is, average transportation cost per unit is crx @I*where z is the conJigurationfactor. The value of z depends on fi, the market shape, and the distance norm. For example, with a circular market shape and the Euclidean distance norm, r = 27+‘2/(2 + fl).
(1)
(See Erlenkotter (1989) for other values of 7.) When /? = 1, T is simply the average distance from the market centre to customers for a one square mile market area. For ease of notation, let c2 = cz. As in Erlenkotter (1989), we focus on the GOMA model defined over the parameter set such that per unit transportation cost is proportional to x8’*. Thus 7 the average facility operation and transportation cost per square mile is f(x) = cl D”x’- ’ + vD + c2 Dx@‘*
(2)
and the minimum cost solution is x* = {[2(1 - a)c&?c2] [Da/D]}2’[8+2(1-a)1.
(3)
The GOMA model can be extended to include stochastic demand. First, for the stationary case, let D be a random variable for demand. Then, expression (2) is a random variable for distribution cost per square mile and the market area that will minimize expected cost is x* = {[2(1 - CX)C~/~C~](E[D=]/E[D])}*‘[~+*(~ -‘)I.
(4)
Now, for the case of nonstationary demand, let D(t) for t E [0, 7’l be a second order (i.e. E[D(t)*] < co) diffusion process with drift and dispersion coefficients that are continuous with respect to demand rate and time. The sample space of D(t) is assumed to be a subset
SCOTT
30
WEBSTER
and AMIT GUPTA
of the positive real numbers for all t and the inflation rates for facility operation costs and transportation costs are assumed to be the same. Thus, f(X, t) = clD(t)Y1
+ uD(t) + c,D(t)xfl’2
(5)
is a random variable for the net of inflation distribution cost rate per square mile at time t given market area x. Given a planning horizon I = [0, T] and a real discount rate r (i.e. nominal rate less inflation), the average discounted distribution cost per square mile is f(x, t)e-“dt.
E
(6)
sI The minimum expected cost solution is x* = {[2(1 - a)c1/~c2][D1/D2]}2’[a’2(1-a)1
(7)
where D, = E
D(t)‘e-*‘dt
(8)
o(t)emrLdt.
(9)
sI and D, = E sI 3.3 Results
There exists a point in time during the planning horizon, say t’E I, such that the solution that minimizes the expected cost rate at time t’is optimal for the problem with nonstationary demand o(t) for tel. This notion is expressed below in Theorem 1. Theorem 1. The nonstationary demand problem is equivalent to minimizing the expected distribution cost rate at some point in time during the planning horizon.
Proof:
Defining h(t) = re-“/(l
- eerT)
(10)
and expanding on (6), we get E
sI
f(x, t)e-*‘dt = [(l - e-rT)/r] = [(l - ewrT)/r]
sI sZ
Ecf(x,
WW~
(c,E[D(t)“]x”-
’ + vE[D(t)] + c2 E[D(t)]xs/2}h(t)dt.
Note that E[D(t)] and E[D(t)“] are continuous on I [by the It6 transformation formula; see Karlin and Taylor (19Sl)] and h(t) 2 0 on I. Consequently, ELf(x, t)] is continuous with respect to t and, from the Mean Value Theorem, it follows that there exists t’~(0, T) such that Ecf(x, t)]h(t)dt = ECf(x,‘t’)]
sI
h(t)dt sI
= Ecf(x, t’)].
The general optimal market area model with uncertain and nonstationary
demand
31
Hence, the problem of minimizing E jI f(x, t)r-“dt is equivalent to minimizing Ecf(x, t’)]. 0 Theorem 1 indicates that for each problem with nonstationary demand, there exists a closely related problem with stationary demand. This relationship forms the basis for analysis on how algorithms for static models can be used to solve problems where change is expected during the planning horizon. Two convenient statistics for characterizing the distribution of demand for the equivalent static demand problem are t’ and E[D(t’)]. From expressions (7) through (9) and Theorem 1, there exists L’EI such that E[D(t’)]/E[D(t’)“] = { E j1 D(t)e- ”dt },/{E j1 DWe-“dtj,
(11)
from which it follows that t’ and E[D(t’)] are not affected by changes in cl, c2, u, 8, distance norm and market area shape, but depend only upon T, I, a, and D(t). If a = 0, then rewriting expression (1 l), E[D(t’)] = E
[D(t)re-“/(l
- e-“‘)]dt
sI =
sI
ECW)lWW
(12)
where h(t) (defined in (10)) is the probability density function for a conditional exponential random variable with mean l/r and the condition that the value of the random variable is less than ZYIn this case, E[D(t’)] is a weighted average of the expected demand rate during the horizon. The weight curve as a function oft matches the conditional exponential probability density function, which is downward sloping. Therefore, expected demand early in the horizon is weighted heavier than expected demand late in the horizon and, interpreting t’ as the minimum of those values satisfying (ll), t’ varies inversely with r. Furthermore, the degree of uncertainty in demand has no effect on E[D(t’)] and, when r = 0, E[D(t’)] is simply the average demand per period during the horizon (i.e. h(t) = l/T when r = 0). Uncertainty in demand makes a difference when economies of scale in facility operation are less than extreme (i.e. a # 0). For consideration of nonzero a, we first define h(r)
=
s
E[D(t)]h(t)dt.
I
(13)
Thus, p,,(r) = E[D(t’)] when a = 0. For example, ~~(0) is the average demand per period during the horizon. E[D(t’)]/E[D(t’)“] = E
D(t)e-“dt/E s
D(t)“e-“dt s
= E d D@)h(,,/j-,
E&).lh(r)dl
2 E II D(Qh(Q/jI
(E[D(t)])“h(t)dt
by Jensen’s inequality (equality iff Var[D(t)] = 0 for all t)
32
SCOTT WEBSTER and AMIT GUPTA
diicouat
rate
Fig. 1. Effect of discount rate on E[D(t’)] for exponential demand rate (a = 0.05, D, = 10, T = 15).
E[D(t)]h(t)dt =
O1 by Jensen’s inequality (equality iff 1 E[D(t)] is constant for all t)
pJr)l -a.
(14)
If Var[D(t)] = 0 for all t~l, then E[D(t’)]/E[D(t’P] = (E[D(t’)])’ --01
(15)
~CWI 2 PDb9.
(16)
and it follows that
Therefore, for dynamic deterministic demand, E[D(t’)] for CC> 0 is greater than E[D(t’)] for a = 0. If demand is stochastic, then the effect of increasing c1is problem dependent. 3.4 Example demand functions The preceding results apply to general demand processes. In this subsection, E[D(t’)] is derived for several specific demand processes, beginning with geometric Brownian motion [see Bean, Higle and Smith (1992) for an alternative example of demand modelled as a function of Brownian motion]. Let dD = aDdt + oDdz where dz is an increment of a stochastic process z(t) that obeys Brownian motion. Then by W’s lemma, D(t) = D, ,@- S’lW+az@)_
(17)
D(t) is a log-normal random variable with mean, E[D(t)] = DOeat
and variance
(18)
The general optimal market area model with uncertain and nonstationary demand
33
Fig. 2. Effect of GL and variance rate on E[D(t’)] for exponential demand rate (a = 0.05, D, = 10, T = 15).
Var[&t)]
= (Doe”)2(eoz’- 1).
(1%
If a = 0, then (11) can be written as E[D(t’)] = D,[r/(r
- a)][(1 - eWCr-‘)r)/(l - eWrT)].
(20)
The effect of changes in r on E[D(t’)] is illustrated for a sample problem in Fig. 1. Recall that Var[D(t)] does not affect E[D(t’)] when a = 0. If r = 0, then setting E[D(t’)] = E[D(t’)qj(D,/D,), noting E[D(t)“] = (DOe’*~e-“““(‘-“)‘2, solving for t’, and substituting into (18), we find E[D(t’)] = Do{[fA//a] [(e”lT- l)/(e*r - l)]}“‘[(’-“l)(a+aa”2)1
(21)
where 0 = au - aa2(1 - a)/2.
(22)
The effect of changes in a and a2 on E[D(t’)] is illustrated for a sample problem in Fig. 2. Figure 2 shows that E[D(t’)] (and consequently t’) decreases with a2. If the value of a in the sample problem is reversed (i.e. declining demand), then E[D(t’)] increases and t’ decreases with a2. Now suppose E[D(t)] = D, + at” where n belongs to the set of natural numbers and first consider the case of a = 0. Then E[D(t’)] = Do + a(n!/r” - [T” + nT”-‘/r
If T = co, then the expression simplifies to
+ ... + n!T/r"-']/(erT- l)}.
(23)
SCOTT WEBSTER and AMIT GUPTA
34
16.0 -
discount rate
Fig. 3. Effect of discount rate on E[D(t’)] for linear demand rate (a = 0.5, D, = 10, T = 15).
E[D(t’)] = D, + a(n!/r”).
(24)
Expression (23) simplifies to E[D(t’)] = D, + aT[l/rT
- l/(erT - l)]
(25)
when n = 1. The effect of changes in r on E[D(t’)] in expression (25) is illustrated for a sample problem in Fig. 3. Now suppose that c1# 0 so that the variance rate will affect E[D(t’)] and consider the case where expected demand is linear with time (i.e. n = 1). Let Y(0) = D, and dY = adt + adz where dz is an increment of a stochastic process z(t) that obeys Brownian motion. Then by ItG’s lemma, Y(t) = D, + at + oz(t).
(26)
Y(t) is a normal random variable with mean E[Y(t)] = D, + at
(27)
Var[Y(t)] = a2t.
(28)
D(t) = Max{ Y(t), O+>
(29)
and variance
We define
where O+ is a value slightly larger than zero. For cases where the probability that Y(t) < 0 is small (e.g. D,, + at 2 3at’12) for all t, EC&t)] x E[Y(t)] and Var[D(t)] w Var[Y(t)]. By the second order approximation of the Taylor series expansion about the mean, E[D(t)“] x (D, + atp - a(1 - a)&/[2(D,
+ at)2 -“I.
(30)
The general optimal market area model with uncertain and nonstationary demand
35
Fig. 4. Effect of c( and variance rate on E[D(t’)] for linear demand rate (a = 0.5, D, = 10, T = 15).
The Taylor series expansion applies because the sample space of D(t) is contained by the set of positive real numbers. If Y= 0, then simplifying E[D(t’)] = (D,/D,)E[D(t’)7 using (27), we find (E[D(t’)])“-’
- A(E[D(t’)])“-’
+ AD,(E[D(t’)])“-3
z [2a((D, + aT)“+ 1 - D”,+ ‘) - 0]/[2a’(~ + l)T(D, + aT/2)]
(31)
A = a(1 - a)a2/2a,
(32)
where
6 = (a + l)a’{(l - a)[(D, + UT)” - D”,] - aD,[Dt-’
- (Do + C-AT)“-‘]}.
(33)
A closed form solution for E[D(t’)] is not available. However, (31) can be solved for o2 and plotted to illustrate the relationship among a, c2, and E[D(t’)] (see Fig. 4). Figure 4 shows that E[D(t’)] (and consequently t’) decreases with 0’. If the value of a in the sample problem is reversed (i.e. declining demand), then E[D(t’)] increases and t’ decreases with 0’. 4. DISCUSSION
Expression (11) provides a way to determine a time period for analysis using a static model algorithm when the real problem is characterized by changing demand. ‘Roughly speaking, one should identify the time period such that the ratio of average demand to the average of demand raised to the a power during the period is equal to the ratio of discounted average demand to the discounted average demand raised to the a power during
36
SCOTT WEBSTER and AMIT GUPTA
the planning horizon. A practitioner may choose to use simulation to estimate the time period for a specific problem when it is not possible or convenient to derive t’ analytically (e.g. simulate demand over time to estimate the ratio for the planning horizon and for different time periods). Expression (11) also leads to a number of more general insights that relate to use of static analysis tools on problems with dynamic demand. 1. If economies of scale in facility operation are extreme (i.e. c1= 0), then the level of uncertainty in demand can be ignored in the analysis. This is because distribution cost per period is directly proportional to the demand per period. 2. If economies of scale in facility operation are extreme and the real discount rate is zero, then the dynamic demand problem may be analyzed using a static model algorithm with demand equal to the dverage demand during the planning horizon. The period to period dynamics of demand are not relevant. In fact, this approach of forecasting future conditions over a reasonable planning horizon and analyzing as a static problem using the average of these data has been suggested as one possible pragmatic alternative (Ballou, 1985). 3. Under the assumption of extreme economies of scale in facility operation, the period to period dynamics of demand become relevant when the real discount rate is greater than zero. Instead of using a simple average of demand in a static model algorithm, one should use a weighted average of demand with demand early in the horizon weighted heavier than demand later in the horizon. The demand weight as a function of time is h(t) = re-“Ml - eKrT). 4. If demand is deterministic, yet changing, then the value of E[D(t’)] with CI> 0 is larger than E[D(t’)] with a = 0. In brief, this result derives from the fact that average facility operation cost during the horizon is less than (respectively, equal to) the cost of operating a facility with a constant demand set to the average demand during the horizon when c1> 0 (respectively, c1= 0). For the case of a # 0, the value oft’ was found to decrease with demand rate variance 5. in the example demand functions. While it is not clear whether this is true of some general class of demand functions, the result is consistent with intuition (e.g. one tends to take a more myopic view as uncertainty increases). 6. A number of problem parameters do not affect the time period for static analysis. These are the facility operation cost coefficient (c,), the transportation cost coefficient (c), the variable warehousing cost rate (v), the degree of economies of distance in transportation cost (B), the distance norm and the market area shape. One issue that has not been addressed is the selection of the planning horizon length. Relevant factors for this decision include the ease with which the configuration may be updated and the degree of expected stability in the environment over time. Clearly, quick and inexpensive implementation of a new configuration favours a short planning horizon. A shorter planning horizon is also appropriate when the stability of the environment over time is suspect. In the interest of efficiency and customer satisfaction, a fundamental change in the environment (e.g. the arrival of a new competitor, the invention of a new product) can prompt a change in the configuration. Therefore, it is not prudent in this situation to design a system that is expected to be highly efficient 10 years hence, for example. It is possible that this aspect of the problem, which is basically concerned with the question of what is the foreseeable future, is best left to managerial judgment.
The general optimal market area model with uncertain and nonstationary demand
31
5. CONCLUSION
Erlenkotter (1989) suggests that much remains to be discovered about the potential of the GOMA model as a tool for analyzing facility location problems. In this paper, we use a dynamic demand variation of the GOMA model to explore the problem of designing a distribution system when the demand rate is expected to change over time. Under rather general conditions for the distribution of demand over time, we show that the dynamic demand problem is equivalent to minimizing the expected distribution cost rate at some point during the planning horizon. This result provides the foundation for subsequent analysis. A number of results on the relationship between the dynamic demand problem and its equivalent stationary demand problem are derived, both for general demand functions and several specific demand functions. All of the results are based on an analysis of a model with a simple demand space. The applicability of the results to models with other demand spaces is an open question. The critical issue concerns the degree to which the GOMA model captures the essential elements of more complex problem types. There is evidence that the GOMA model is successful in this regard. Thus, the results represent plausible hypotheses for more complex facility location problems. A natural extension of this research is to evaluate conditions (i.e. alternative demand spaces) under which these hypotheses break down. Acknowledgement-We Theorem 1.
wish to acknowledge the suggestions of one of the referees that led to a simpler proof of
REFERENCES Ballou, R. H. (1968) Dynamic warehouse location analysis. Journal of MarketingResearch, 5, 271-276. Ballou, R. H. (1985) Business logistics management: planning and control. Englewood Cliffs, NJ: Prentice Hall. Bean, J. C., Higle, J. L. & Smith, R. L. (1992) Capacity expansion under stochastic demands. Operations Research, 40, S210-S216. Campbell, J. F. (1990) Locating transportation terminals to serve an expanding demand. Transportation Research, 24B, 173-192.
Cavalier, T. M. & Sherali, H. D. (1986) Euclidean distance location-allocation problems with uniform demands over convex polygons. Transportation Science, 20, 107-l 16. Chen, R. (1983) Solution of minisum and minimax location-allocation problems with Euclidean distances. Naval Research Logistics Quarterly, 30, 449459.
Cooper, L. (1963) Location-allocation problems. Operations Research, 11, 331-343. Cooper, L. (1972) The transportation-location problem. Operations Research, 20, 94-108. Daganzo, C. F. (1987) The break-bulk role of terminals in many-to-many logistic networks. Operations Research, 35, 543-555.
Daganzo, C. F. (1988) A comparison of in-vehicle and out-of-vehicle freight consolidation strategies. Transportation Research, 22B, 173-180.
Davis, H. W. (1988) Physical distribution costs: performance in selected industries, 1988. In Annual Conference Proceedings, Volume 1 (pp. 73-81). Boston, MA: Council of Logistics Management. Densham, P. J. & Rushton, G. (1988) Strategies for solving large location-allocation problems by heuristic methods. Environment and Planning A, 24,289-304. Densham, P. J. & Rushton, G. (1992) A more efficient heuristic for solving large p-median problems. Papers in Regional Science:
The Journal of the RSAI, 11, 307-330.
Drezner, Z. & Wesolowsky, G. 0. (1980) Optimal location of a facility relative to area demands. Naval Research Logistics Quarterly, 21, 199-206.
Drezner, Z. & Wesolowsky, G. 0. (1991) Facility location when demand is time dependent. Naval Research Logistics Quarterly, 38, 763-771.
Erlenkotter, D. (1981) A comparative study of approaches to dynamic location problems. European Journal of Operational Research, 6, 138-143.
Erlenkotter, D. (1989) The general optimal market area model. Annals of OperationsResearch, 18, 45-70.
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SCOTT WEBSTER and AMIT GUPTA
Friesz, T. L., Miller, T. & Tobin, R. L. (1988) Algorithms for spatially competitive network facility-location. Environment and Planning B: Planning and Design, 15, 191-203. Geoffrion, A. M. (1976) The purpose of mathematical programming is insight, not numbers. Interfaces, 7,81-92. Ghoneim, N. S. A. & Wirasinghe, S. C. (1987) Optimum zone configuration for planned urban commuter rail lines. Transportation Science, 21, 106114. Ghosh, A. 8~ Craig, C. S. (1991) FRANSYS: a franchise distribution system location model. Journal ofRetailing, 67,466489.
Gilmour, P. (1985) Managing distribution in Australia in the mid-1980s. Graduate School of Management, Macquarie University, North Ryde, NSW. Hakimi, S. K. & Kuo, C. C. (1991) On a general network location-allocation problem. European Journal of Operational Research, 55, 3145.
Karlin, S. & Taylor, H. M. (1981) A second course in stochastic processes. New York: Academic Press. Koroglu, D. (1992) Possible locations for new universities in Turkey. Socio-Economic Planning Science, 26,27-42. Marianov, V. & ReVeIle, C. (1992) A probabilistic fire-protection siting model with joint vehicle reliability requirements. Papers in Regional Science: The Journal of the RSAI, 71, 217-242. Marucheck, A. S. & Aly, A. A. (1981) An efficient algorithm for the location-allocation problems with rectangular regions. Naval Reserch Logistics Quarterb, 28, 309-323. Miller, T., Tobin, R. L. & Friesz, T. L. (1992) Network facility-location models in Staclkelberg-NashhCournot spatial competition. Papers in Regional Science: The Journal of the RSAI, 71, 277-291. ReVelle, C. & Hogan, K. (1989) The maximum availability location problem. Transportation Science, 23, 192-200. Schilling, D. A. (1980) Dynamic location modeling for public sector facilities: a multicriteria approach. Decision Sciences, 5, l-7.
Survey of distribution costs (1985). London: The Centre for Physical Distribution Management. Van Roy, T. J. (1986) A cross decomposition algorithm for capacitated facility location. Operations Research, 34, 145-163.
Wesolowsky, G. 0. & Truscott, W. G. (1975) The multiperiod location-allocation facilities. Management Science, 22, 57-65.
problem with relocation of
Laorion Science, Vol. 3, No. 1, pp. 25-38, 15% Ekvier Science Ltd Printed in Great Britain. 09664349195 $9.50 + 0.00
THE GENERAL OPTIMAL MARKET AREA MODEL WITH UNCERTAIN AND NONSTATIONARY DEMAND SCOTT WEBSTER University of Wisconsin-Madison,
and AMIT GUPTA
School of Business, 975 University Avenue, Madison, WI 53706, U.S.A. (Receioed 12 February 1993)
Abstract-Facility location problems involve determining the number of facilities to open, the location of facilities, and the market area that each facility will serve. While the environment is seldom stable, algorithms for solving models that incorporate change and uncertainty are rare and limited in scope. Consequently, the common approach in practice is to use algorithms for static models with perhaps some scenario analysis to accommodate change and uncertainty. We introduce an analytically tractable model that includes dynamic and uncertain demand. By studying how the model can be solved by solving a related model with stationary demand, we are able to derive guidelines on how to apply static analysis tools to problems with dynamic demand. While the analysis is contingent upon the underlying model, evidence on the model’s robustness suggests that the results represent plausible guidelines for problems that extend well beyond the model. Keywords: Facility location, market area model, distribution planning.
1. INTRODUCTION
Facility location problems involve determining the number of facilities to open, the location of facilities, and the market area that each facility will serve. These problems exhibit a basic cost tradeoff that is due to economies of scale in facility operation and the fact that inbound transportation rates are lower than outbound transportation rates; as the number of facilities increases, transportation costs tend to decrease and facility operation costs tend to increase. Furthermore, they involve a decision that is not easily modified (it is generally not practical to be continually changing the configuration). That is, the common real world facility location problem is one of incorporating anticipated changes in the environment along with the degree of uncertainty in future expectations and identifying a configuration that, while perhaps phased-in over a period of time, is expected to be effective over a given planning horizon. Research on facility location problems has been fairly successful in the sense that algorithms for solving various models of these problems are widely available. However, many of the algorithms with the scope and size capabilities most suitable for industry assume a known and unchanging environment . . . an assumption that rarely holds in practice; see, e.g. Ballou (1985). Consequently, the common approach in practice is to analyze facility location problems under the assumption that the environment is stable, with dynamics and uncertainty captured to a limited degree through the analysis of alternative scenarios. One problem with this approach is that it is very much an art with little in the way of guidelines on how to proceed effectively. One possible line of research that can help to improve this situation is to work on developing algorithms for solving models that capture dynamics and uncertainty. This 25
26
SCOTT WEBSTER and AMIT GUPTA
paper describes an entirely different approach; our goal is to provide a first step in the development of guidelines on how to best use the well-developed tools of static analysis when the environment is changing and uncertain. Towards this end, we (1) extend the general optimal market area model to include nonstationary and stochastic demand, (2) identify theoretical properties that link the nonstationary and stochastic -model with an equivalent static model, and (3) use these properties to formulate guidelines on how to apply tools of static analysis to facility location problems with nonstationary and stochastic demand. While the guidelines stem from the underlying model, there is evidence in the literature that the model is quite robust. Thus, the guidelines represent plausible hypotheses on the use of static facility location tools when the environment is changing and uncertain. Work that helps to improve our ability to develop good solutions to facility location problems is important. Distribution cost represents a significant fraction of sales in developed economies and the importance of designing effective distribution systems is increasing. Recent surveys show distribution costs as a percentage of sales to be 7.5% in the United States (Davis, 1988) 11.6% in the United Kingdom (Survey of Distribution Costs, 1985), and 21.1% in Australia (Gilmour, 1985). Importance is increasing due to a rapidly changing environment. In particular, we note the globalization of the marketplace and a trend towards using speed as a competitive weapon (e.g. shortened times from order to delivery). Both of these factors are putting pressure on the need to redesign or expand existing distribution systems. The next section contains a review of related literature. The methodology is reviewed and the analysis is presented in Section 3. Managerial implications of the analysis are summarized in Section 4 and Section 5 concludes the paper.
2. RELATED
LITERATURE
Facility location problems have been considered by a number of researchers who have employed a variety of solution techniques. The approaches include mathematical programming models, dynamic .programming models, and heuristic algorithms. There are several ways to classify the approaches that have been taken by the various researchers. For the purposes of this paper, we will distinguish among the different approaches by considering static models and dynamic models. One common approach to solving facility location problems has been to model the problem as a static problem, with dynamics not being captured in the model at all. The effect of dynamics in such models is considered by performing sensitivity analysis on the static model. Within the static model framework, the approaches may be further distinguished by considering discrete demand versus continuous demand models. Discrete demand static problems have been considered by Cooper (1963, 1972), Chen (1983), Van Roy (1986), Daganzo (1987, 1988), Ghosh and Craig (1991), Hakimi and Kuo (1991), Densham and Rushton (1988, 1992), and Koroglu (1992). Continuous demand static problems have been considered by Drezner and Wesolowsky (1980), Marucheck and Aly (1981), Cavalier and Sherali (1986), Ghoneim and Wirasinghe (1987), and Erlenkotter (1989), among others. The literature on dynamic demand models may be distinguished according to discrete demand versus continuous demand as well. Discrete demand dynamic models have been studied by Ballou (1968), Wesolowsky and Truscott (1975), Schilling (1980), and Drezner
The general optimal market area model with uncertain and nonstationary demand
21
and Wesolowsky (1991) among others. The continuous demand dynamic modelling approach has been utilized by Campbell (1990). Ballou (1968) shows how a dynamic programming model would be used for determining the optimal warehouse location in a dynamic decision problem. He applies dynamic programming to find a dynamic optimum warehouse location-relocation plan. However, the solution’methodology is limited to very small problems. Wesolowksy and Truscott (1975) solve a problem of locating facilities among demand points, where some of the demand points could themselves serve as facility sites. They find that ‘myopic’ relocation plans, which are based on current demands only, are not optimal, and their multi-period model provides a better solution to the problem. They present two methods of solution: (1) a mixed integer programming (MIP) approach, and (2) a dynamic programming approach. It is shown that the MIP approach is feasible for small problems only, whereas the dynamic programming approach can solve relatively larger problems but its state space has to be of ‘manageable’ size. Schilling (1980) considers a dynamic location modelling approach for public sector facilities. It is pointed out that private sector locational decisions are typically based on monetary criteria: minimization of costs, or maximization of profits. These criteria may not be appropriate for public-sector facilities where the decisions may be based on some measure of social welfare. Schilling proposes a general methodology using multi-objective analysis to plan public-sector facility systems in a dynamic environment. The methodology leads to a set of ‘efficient alternatives’ from which the decision maker selects a solution based on her planning strategies. Drezner and Wesolowsky (1991) investigate the problem of locating a facility among a given set of demand points when the demand at each point changes with time in a linear fashion. They determine the location of a single facility in the planning horizon but allow the location to be relocated over time. They also determine the time points at which the facility should be relocated. This model assumes linearity in demand growth as well as the location and relocation of a single facility only. Campbell (1990) develops an approximation model to locate terminals for serving a fixed region with uniform and increasing demand. He claims that since determining an optimal strategy requires knowledge of future demand, myopic strategies may be nearly optimal. He considers three different strategies and shows that myopic strategies are nearly optimal unless relocation costs are large. Thus, in this model, the dynamic demand problem is really treated as a small multi-period problem. The algorithms for dynamic models discussed above are either limited to small problems, or limited in the choice of demand growth rates, or limited to the location of a single facility. In addition, the models assume that demand is deterministic. We note, however, that dynamic and uncertain demand has been considered in the capacity planning literature. Bean, Higle, and Smith (1992) consider the problem of meeting a stochastically growing demand for capacity over an infinite horizon. They allow a demand process that is either a transformed Brownian motion, or a regenerative birth-death process. They also allow general cost structures. The resulting stochastic dynamic problem is transformed to an equivalent static deterministic problem to determine the optimal capacity expansion periods. All costs are discounted at an interest rate smaller than the original, in proportion to the uncertainty in demand. The problem formulation assumes that the location of facilities (and, as a result, transportation cost) is unimportant, and capacity expansion to meet the demand is the only criterion used to minimize the costs.
28
SCOTT WEBSTER and AMIT GLJPTA
Uncertain demand has also been considered by ReVelle and Hogan (1989) who introduce a probabilistic version of the maximal covering location problem which they call the maximum availability location problem (MALP). Marianov and ReVelle (1992) incorporate the probability of being able to meet demand into a maximal covering formulation for the location of fire stations and the joint allocation of trucks and engines to those stations. Although the formulation is specifically developed for fire protection services, it may be applied to many types of emergency services that operate in a dynamic setting. In a related line of research, Friesz, Miller, and Tobin (1988) present heuristic algorithms for locating a firm’s production facilities while simultaneously determining production levels with the assumption that existing firms as well as new entrants act in accordance with an appropriate model of spatial equilibrium. Miller, Tobin, and Friesz (1992) address the facility location problem of a firm entering an industry and competing on a discrete network with several other oligopolistic competitors. Their location model combines game theory and a hierarchical mathematical programming approach with sensitivity analysis of variational inequalities. It should be noted that the methodology used in the above two papers is markedly different from the other papers referenced here, and as such, the results are not easily transferable to the other models. 3. ANALYSIS
3.1 Overview of methodology Erlenkotter (1989) describes a model of a distribution system with uniform and unchanging demand. This model, which is known as the general optimal market area (GOMA) model, has a number of desirable properties. First, the operating cost of the distribution system can be expressed as a continuous function that is amenable to analysis. Second, a closed form expression for the optimal market area per facility exists for a wide range of facility operation cost structures and product transportation cost structures. Third, while it is true that demand is often lumpy in practice, researchers have found that basic properties derived from the model extend to models with much more complex demand spaces. For example, Geoffrion (1976) describes how a special case of the GOMA model is used in a facility location study of a consumer products manufacturer which distributes product nationwide out of a single plant in California. Approximately 65 potential warehouse sites and 130 customer zones were identified for the purposes of the study. The sensitivity of the optimal number of warehouses to small changes in problem parameters as determined by an integer programming model is found to be nearly identical to the sensitivity results predicted by the GOMA model. Erlenkotter (1989) reports a comparison of results generated by an integer programming model and the GOMA model for a system comprised of 100 customers, each with a demand of one unit per period. Each demand point is a potential warehouse site and transportation cost is proportional to Euclidean distance. The optimal number of warehouses for the system is calculated for different fixed warehousing rates. Very similar results are found when the system is approximated by the appropriate GOMA model (i.e. eight cases are tested; the results are the same in six cases and differ by one warehouse in two cases). These two studies suggest that the model is suited to the discovery of fundamental insights into problem characteristics that, due to a large quantity of problem specific data, are difficult to achieve through analysis of detailed numerical models (e.g. mathematical programming models).
The general optimal market area model with uncertain and nonstationary demand
29
A generalization of the GOMA model is used in our analysis. We first extend the GOMA model to accommodate dynamic and uncertain demand and then use the model to study how a problem with changing demand can be solved by solving a related problem with stationary demand. 3.2 The model We begin by reviewing the GOMA model with static demand. As noted by Erlenkotter (1989), this model has a long and varied history in the literature as it has been discovered and rediscovered in a number of different forms by researchers from a variety of disciplines. Assumptions
(1) Demand is uniform over an infinite plane with density D > 0 per square mile. (2) Given market area x served by a facility, facility operating cost as a function of volume, Dx, is c,(Dx)” + UDXwhere c1 > 0 and u > 0. Economies of scale are reflected in cl(Dx)LI, where 0 $ c1< 1, with smaller values of tl implying greater relative economies of scale. (3) Unit transportation cost as a function of miles travelled, 6, is cd8 where c > 0 and /I > 0. Economies of distance exist when /? < 1. (4) Various regular two-dimensional market shapes may be specified (e.g. hexagon, circle, diamond). (5) Various distance norms may apply (e.g. Euclidean, rectilinear). (6) The objective is to minimize cost. For many combinations of fi, market shapwnd distance norm, transportation cost is proportional to the market area raised to the /?/2 power; that is, average transportation cost per unit is crx @I*where z is the conJigurationfactor. The value of z depends on fi, the market shape, and the distance norm. For example, with a circular market shape and the Euclidean distance norm, r = 27+‘2/(2 + fl).
(1)
(See Erlenkotter (1989) for other values of 7.) When /? = 1, T is simply the average distance from the market centre to customers for a one square mile market area. For ease of notation, let c2 = cz. As in Erlenkotter (1989), we focus on the GOMA model defined over the parameter set such that per unit transportation cost is proportional to x8’*. Thus 7 the average facility operation and transportation cost per square mile is f(x) = cl D”x’- ’ + vD + c2 Dx@‘*
(2)
and the minimum cost solution is x* = {[2(1 - a)c&?c2] [Da/D]}2’[8+2(1-a)1.
(3)
The GOMA model can be extended to include stochastic demand. First, for the stationary case, let D be a random variable for demand. Then, expression (2) is a random variable for distribution cost per square mile and the market area that will minimize expected cost is x* = {[2(1 - CX)C~/~C~](E[D=]/E[D])}*‘[~+*(~ -‘)I.
(4)
Now, for the case of nonstationary demand, let D(t) for t E [0, 7’l be a second order (i.e. E[D(t)*] < co) diffusion process with drift and dispersion coefficients that are continuous with respect to demand rate and time. The sample space of D(t) is assumed to be a subset
SCOTT
30
WEBSTER
and AMIT GUPTA
of the positive real numbers for all t and the inflation rates for facility operation costs and transportation costs are assumed to be the same. Thus, f(X, t) = clD(t)Y1
+ uD(t) + c,D(t)xfl’2
(5)
is a random variable for the net of inflation distribution cost rate per square mile at time t given market area x. Given a planning horizon I = [0, T] and a real discount rate r (i.e. nominal rate less inflation), the average discounted distribution cost per square mile is f(x, t)e-“dt.
E
(6)
sI The minimum expected cost solution is x* = {[2(1 - a)c1/~c2][D1/D2]}2’[a’2(1-a)1
(7)
where D, = E
D(t)‘e-*‘dt
(8)
o(t)emrLdt.
(9)
sI and D, = E sI 3.3 Results
There exists a point in time during the planning horizon, say t’E I, such that the solution that minimizes the expected cost rate at time t’is optimal for the problem with nonstationary demand o(t) for tel. This notion is expressed below in Theorem 1. Theorem 1. The nonstationary demand problem is equivalent to minimizing the expected distribution cost rate at some point in time during the planning horizon.
Proof:
Defining h(t) = re-“/(l
- eerT)
(10)
and expanding on (6), we get E
sI
f(x, t)e-*‘dt = [(l - e-rT)/r] = [(l - ewrT)/r]
sI sZ
Ecf(x,
WW~
(c,E[D(t)“]x”-
’ + vE[D(t)] + c2 E[D(t)]xs/2}h(t)dt.
Note that E[D(t)] and E[D(t)“] are continuous on I [by the It6 transformation formula; see Karlin and Taylor (19Sl)] and h(t) 2 0 on I. Consequently, ELf(x, t)] is continuous with respect to t and, from the Mean Value Theorem, it follows that there exists t’~(0, T) such that Ecf(x, t)]h(t)dt = ECf(x,‘t’)]
sI
h(t)dt sI
= Ecf(x, t’)].
The general optimal market area model with uncertain and nonstationary
demand
31
Hence, the problem of minimizing E jI f(x, t)r-“dt is equivalent to minimizing Ecf(x, t’)]. 0 Theorem 1 indicates that for each problem with nonstationary demand, there exists a closely related problem with stationary demand. This relationship forms the basis for analysis on how algorithms for static models can be used to solve problems where change is expected during the planning horizon. Two convenient statistics for characterizing the distribution of demand for the equivalent static demand problem are t’ and E[D(t’)]. From expressions (7) through (9) and Theorem 1, there exists L’EI such that E[D(t’)]/E[D(t’)“] = { E j1 D(t)e- ”dt },/{E j1 DWe-“dtj,
(11)
from which it follows that t’ and E[D(t’)] are not affected by changes in cl, c2, u, 8, distance norm and market area shape, but depend only upon T, I, a, and D(t). If a = 0, then rewriting expression (1 l), E[D(t’)] = E
[D(t)re-“/(l
- e-“‘)]dt
sI =
sI
ECW)lWW
(12)
where h(t) (defined in (10)) is the probability density function for a conditional exponential random variable with mean l/r and the condition that the value of the random variable is less than ZYIn this case, E[D(t’)] is a weighted average of the expected demand rate during the horizon. The weight curve as a function oft matches the conditional exponential probability density function, which is downward sloping. Therefore, expected demand early in the horizon is weighted heavier than expected demand late in the horizon and, interpreting t’ as the minimum of those values satisfying (ll), t’ varies inversely with r. Furthermore, the degree of uncertainty in demand has no effect on E[D(t’)] and, when r = 0, E[D(t’)] is simply the average demand per period during the horizon (i.e. h(t) = l/T when r = 0). Uncertainty in demand makes a difference when economies of scale in facility operation are less than extreme (i.e. a # 0). For consideration of nonzero a, we first define h(r)
=
s
E[D(t)]h(t)dt.
I
(13)
Thus, p,,(r) = E[D(t’)] when a = 0. For example, ~~(0) is the average demand per period during the horizon. E[D(t’)]/E[D(t’)“] = E
D(t)e-“dt/E s
D(t)“e-“dt s
= E d D@)h(,,/j-,
E&).lh(r)dl
2 E II D(Qh(Q/jI
(E[D(t)])“h(t)dt
by Jensen’s inequality (equality iff Var[D(t)] = 0 for all t)
32
SCOTT WEBSTER and AMIT GUPTA
diicouat
rate
Fig. 1. Effect of discount rate on E[D(t’)] for exponential demand rate (a = 0.05, D, = 10, T = 15).
E[D(t)]h(t)dt =
O1 by Jensen’s inequality (equality iff 1 E[D(t)] is constant for all t)
pJr)l -a.
(14)
If Var[D(t)] = 0 for all t~l, then E[D(t’)]/E[D(t’P] = (E[D(t’)])’ --01
(15)
~CWI 2 PDb9.
(16)
and it follows that
Therefore, for dynamic deterministic demand, E[D(t’)] for CC> 0 is greater than E[D(t’)] for a = 0. If demand is stochastic, then the effect of increasing c1is problem dependent. 3.4 Example demand functions The preceding results apply to general demand processes. In this subsection, E[D(t’)] is derived for several specific demand processes, beginning with geometric Brownian motion [see Bean, Higle and Smith (1992) for an alternative example of demand modelled as a function of Brownian motion]. Let dD = aDdt + oDdz where dz is an increment of a stochastic process z(t) that obeys Brownian motion. Then by W’s lemma, D(t) = D, ,@- S’lW+az@)_
(17)
D(t) is a log-normal random variable with mean, E[D(t)] = DOeat
and variance
(18)
The general optimal market area model with uncertain and nonstationary demand
33
Fig. 2. Effect of GL and variance rate on E[D(t’)] for exponential demand rate (a = 0.05, D, = 10, T = 15).
Var[&t)]
= (Doe”)2(eoz’- 1).
(1%
If a = 0, then (11) can be written as E[D(t’)] = D,[r/(r
- a)][(1 - eWCr-‘)r)/(l - eWrT)].
(20)
The effect of changes in r on E[D(t’)] is illustrated for a sample problem in Fig. 1. Recall that Var[D(t)] does not affect E[D(t’)] when a = 0. If r = 0, then setting E[D(t’)] = E[D(t’)qj(D,/D,), noting E[D(t)“] = (DOe’*~e-“““(‘-“)‘2, solving for t’, and substituting into (18), we find E[D(t’)] = Do{[fA//a] [(e”lT- l)/(e*r - l)]}“‘[(’-“l)(a+aa”2)1
(21)
where 0 = au - aa2(1 - a)/2.
(22)
The effect of changes in a and a2 on E[D(t’)] is illustrated for a sample problem in Fig. 2. Figure 2 shows that E[D(t’)] (and consequently t’) decreases with a2. If the value of a in the sample problem is reversed (i.e. declining demand), then E[D(t’)] increases and t’ decreases with a2. Now suppose E[D(t)] = D, + at” where n belongs to the set of natural numbers and first consider the case of a = 0. Then E[D(t’)] = Do + a(n!/r” - [T” + nT”-‘/r
If T = co, then the expression simplifies to
+ ... + n!T/r"-']/(erT- l)}.
(23)
SCOTT WEBSTER and AMIT GUPTA
34
16.0 -
discount rate
Fig. 3. Effect of discount rate on E[D(t’)] for linear demand rate (a = 0.5, D, = 10, T = 15).
E[D(t’)] = D, + a(n!/r”).
(24)
Expression (23) simplifies to E[D(t’)] = D, + aT[l/rT
- l/(erT - l)]
(25)
when n = 1. The effect of changes in r on E[D(t’)] in expression (25) is illustrated for a sample problem in Fig. 3. Now suppose that c1# 0 so that the variance rate will affect E[D(t’)] and consider the case where expected demand is linear with time (i.e. n = 1). Let Y(0) = D, and dY = adt + adz where dz is an increment of a stochastic process z(t) that obeys Brownian motion. Then by ItG’s lemma, Y(t) = D, + at + oz(t).
(26)
Y(t) is a normal random variable with mean E[Y(t)] = D, + at
(27)
Var[Y(t)] = a2t.
(28)
D(t) = Max{ Y(t), O+>
(29)
and variance
We define
where O+ is a value slightly larger than zero. For cases where the probability that Y(t) < 0 is small (e.g. D,, + at 2 3at’12) for all t, EC&t)] x E[Y(t)] and Var[D(t)] w Var[Y(t)]. By the second order approximation of the Taylor series expansion about the mean, E[D(t)“] x (D, + atp - a(1 - a)&/[2(D,
+ at)2 -“I.
(30)
The general optimal market area model with uncertain and nonstationary demand
35
Fig. 4. Effect of c( and variance rate on E[D(t’)] for linear demand rate (a = 0.5, D, = 10, T = 15).
The Taylor series expansion applies because the sample space of D(t) is contained by the set of positive real numbers. If Y= 0, then simplifying E[D(t’)] = (D,/D,)E[D(t’)7 using (27), we find (E[D(t’)])“-’
- A(E[D(t’)])“-’
+ AD,(E[D(t’)])“-3
z [2a((D, + aT)“+ 1 - D”,+ ‘) - 0]/[2a’(~ + l)T(D, + aT/2)]
(31)
A = a(1 - a)a2/2a,
(32)
where
6 = (a + l)a’{(l - a)[(D, + UT)” - D”,] - aD,[Dt-’
- (Do + C-AT)“-‘]}.
(33)
A closed form solution for E[D(t’)] is not available. However, (31) can be solved for o2 and plotted to illustrate the relationship among a, c2, and E[D(t’)] (see Fig. 4). Figure 4 shows that E[D(t’)] (and consequently t’) decreases with 0’. If the value of a in the sample problem is reversed (i.e. declining demand), then E[D(t’)] increases and t’ decreases with 0’. 4. DISCUSSION
Expression (11) provides a way to determine a time period for analysis using a static model algorithm when the real problem is characterized by changing demand. ‘Roughly speaking, one should identify the time period such that the ratio of average demand to the average of demand raised to the a power during the period is equal to the ratio of discounted average demand to the discounted average demand raised to the a power during
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SCOTT WEBSTER and AMIT GUPTA
the planning horizon. A practitioner may choose to use simulation to estimate the time period for a specific problem when it is not possible or convenient to derive t’ analytically (e.g. simulate demand over time to estimate the ratio for the planning horizon and for different time periods). Expression (11) also leads to a number of more general insights that relate to use of static analysis tools on problems with dynamic demand. 1. If economies of scale in facility operation are extreme (i.e. c1= 0), then the level of uncertainty in demand can be ignored in the analysis. This is because distribution cost per period is directly proportional to the demand per period. 2. If economies of scale in facility operation are extreme and the real discount rate is zero, then the dynamic demand problem may be analyzed using a static model algorithm with demand equal to the dverage demand during the planning horizon. The period to period dynamics of demand are not relevant. In fact, this approach of forecasting future conditions over a reasonable planning horizon and analyzing as a static problem using the average of these data has been suggested as one possible pragmatic alternative (Ballou, 1985). 3. Under the assumption of extreme economies of scale in facility operation, the period to period dynamics of demand become relevant when the real discount rate is greater than zero. Instead of using a simple average of demand in a static model algorithm, one should use a weighted average of demand with demand early in the horizon weighted heavier than demand later in the horizon. The demand weight as a function of time is h(t) = re-“Ml - eKrT). 4. If demand is deterministic, yet changing, then the value of E[D(t’)] with CI> 0 is larger than E[D(t’)] with a = 0. In brief, this result derives from the fact that average facility operation cost during the horizon is less than (respectively, equal to) the cost of operating a facility with a constant demand set to the average demand during the horizon when c1> 0 (respectively, c1= 0). For the case of a # 0, the value oft’ was found to decrease with demand rate variance 5. in the example demand functions. While it is not clear whether this is true of some general class of demand functions, the result is consistent with intuition (e.g. one tends to take a more myopic view as uncertainty increases). 6. A number of problem parameters do not affect the time period for static analysis. These are the facility operation cost coefficient (c,), the transportation cost coefficient (c), the variable warehousing cost rate (v), the degree of economies of distance in transportation cost (B), the distance norm and the market area shape. One issue that has not been addressed is the selection of the planning horizon length. Relevant factors for this decision include the ease with which the configuration may be updated and the degree of expected stability in the environment over time. Clearly, quick and inexpensive implementation of a new configuration favours a short planning horizon. A shorter planning horizon is also appropriate when the stability of the environment over time is suspect. In the interest of efficiency and customer satisfaction, a fundamental change in the environment (e.g. the arrival of a new competitor, the invention of a new product) can prompt a change in the configuration. Therefore, it is not prudent in this situation to design a system that is expected to be highly efficient 10 years hence, for example. It is possible that this aspect of the problem, which is basically concerned with the question of what is the foreseeable future, is best left to managerial judgment.
The general optimal market area model with uncertain and nonstationary demand
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5. CONCLUSION
Erlenkotter (1989) suggests that much remains to be discovered about the potential of the GOMA model as a tool for analyzing facility location problems. In this paper, we use a dynamic demand variation of the GOMA model to explore the problem of designing a distribution system when the demand rate is expected to change over time. Under rather general conditions for the distribution of demand over time, we show that the dynamic demand problem is equivalent to minimizing the expected distribution cost rate at some point during the planning horizon. This result provides the foundation for subsequent analysis. A number of results on the relationship between the dynamic demand problem and its equivalent stationary demand problem are derived, both for general demand functions and several specific demand functions. All of the results are based on an analysis of a model with a simple demand space. The applicability of the results to models with other demand spaces is an open question. The critical issue concerns the degree to which the GOMA model captures the essential elements of more complex problem types. There is evidence that the GOMA model is successful in this regard. Thus, the results represent plausible hypotheses for more complex facility location problems. A natural extension of this research is to evaluate conditions (i.e. alternative demand spaces) under which these hypotheses break down. Acknowledgement-We Theorem 1.
wish to acknowledge the suggestions of one of the referees that led to a simpler proof of
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problem with relocation of