Ann Oper Res DOI 10.1007/s10479-008-0338-x
Single facility location and relocation problem with time dependent weights and discrete planning horizon Reza Zanjirani Farahani · Zvi Drezner · Nasrin Asgari
© Springer Science+Business Media, LLC 2008
Abstract In this paper a single facility location problem with multiple relocation opportunities is investigated. The weight associated with each demand point is a known function of time. We consider either rectilinear, or squared Euclidean, or Euclidean distances. Relocations can take place at pre-determined times. The objective function is to minimize the total location and relocation costs. An algorithm which finds the optimal locations, relocation times and the total cost, for all three types of distance measurements and various weight functions, is developed. Locations are found using constant weights, and relocations times are the solution to a Dynamic Programming or Binary Integer Programming (BIP) model. The time horizon can be finite or infinite. Keywords Location · Time dependent · Discrete horizon · Relocation 1 Introduction Facility location is a strategic management decision. Such a decision is usually made applying the current conditions such as population, infrastructure, service requirements and others (Drezner 1995b; Francis et al. 1992; Mirchandani and Francis 1990). Common location models deal with single and multiple facility location, covering, p-median, p-center problems, their applications and extensions. Many of these problems can be very difficult to solve. Thus, it is not surprising that so much work has focused on stochastic and deterministic problem formulations. While such formulations are reasonable research topics, they do not capture many of the characteristics of real-world location problems. R.Z. Farahani () Dept. of Industrial Engineering, Amirkabir University of Technology, Tehran, Iran e-mail:
[email protected] Z. Drezner Department of ISDS, College of Business and Economics, California State University, Fullerton, CA, USA N. Asgari Entrepreneurship Faculty, University of Tehran, Tehran, Iran
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The strategic nature of facility location problems requires that models consider some aspect of future uncertainty. Since the investment required by locating or relocating facilities is usually large, facilities are expected to remain operable for an extended time period. Thus, the problem of facility location truly involves an extended planning horizon. Decision makers must not only select locations which will effectively serve changing demand over time, but must also consider the timing of facility expansions and relocations over the long term (Daskin et al. 1992). The first paper which recognized the limited application of static and deterministic models was published by Ballou (1968). Ballou used a series of static deterministic optimal solutions to solve the dynamic problem of locating a single facility warehouse so as to maximize profits over a finite planning horizon. For each period in the specified horizon, he solved for the optimal warehouse location, establishing a set of potential “good” location sites. Dynamic programming is then used to determine the best schedule for opening a subset of these sites as an “optimal” location and relocation strategy for the planning period. This approach was later found to be sub-optimal by Sweeney and Tatham (1976) who improved on Ballou’s solution by extending the set of potential location sites. Their method finds the Ri best (rank order) solutions in each period t through an iterative procedure of solving integer programs with Benders’ decomposition. The number of solutions (Ri ) varies by period and is found through bounding the overall optimal solution value. The expanded set of potential location sites for each period is then used in a dynamic program to determine an optimal location and relocation strategy. Note that both of these papers allow for frequent facility relocation, but neither considers construction time or cost of relocation in the objective function. Wesolowsky (1973) extended the single facility location problem to a model where the facility location could change in a given time interval. In his paper, Wesolowsky presented an algorithm for calculating the optimal location when the costs change at specific times and locations were pre defined. Chand (1988) provided several decision/forecast horizon results for a single facility dynamic location/relocation problem; these results are helpful in finding optimal initial decisions for the infinite horizon problem by using information only for a finite horizon. Bastian and Volkmer (1992) and Andretta and Mason (1994) proposed a forward algorithm for the solution of a single facility dynamic location/relocation problem. In their paper, the problem is reformulated in terms of a shortest path problem in an acyclic network and provide a condition (which is both necessary and sufficient) for the existence of a finite forecast horizon for obtaining an optimal initial decision. A new situation arises when weights are time dependent, and vary as time progresses. The decision maker must take the time dependent weights into consideration when making the location decision. It is likely that the cost of maintaining the current location exceeds the cost of relocation of the facility to a better location. Drezner and Wesolowsky (1991) investigated the problem using continuous weight functions. They presented an optimal procedure for single facility location problem with a single relocation, when the weights are linear, wi (t) = ui + vi t , and the distances are rectilinear. In their approach the relocation time can be anywhere in the range of [0, T ]. Hormozi and Khumawala (1996) proposed an algorithm for optimizing the problems with weights having different predefined values in time and predefined locations. Using a mixed integer programming model and a dynamic programming approach the problem is subdivided into smaller simpler problems. Dynamic multifacility location models are also of interest. Scott (1971) examined dynamic extensions of the location-allocation problem in which multiple facilities are located one at a time at discrete, equally time epochs. Once located, facilities must remain in operation at the specified site. A sub-optimal myopic approach, as well as a standard dynamic programming approach are described. Wesolowsky and Truscott (1976) considered
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the problem when the weights vary but their effect is not continuous. Using dynamic programming, their procedure decides whether or not to locate the facility in a predetermined location at the end of each period. Tapiero (1971) further extended the dynamic locationallocation problem to include possible facility capacities and shipping costs. The optimal solution to this transportation-location-allocation problem provides the facility location (as located in Euclidean plane), allocations of demand to sources (within capacity restrictions), and the quantities to be shipped between facilities and demand points. In this formulation, supply and demand values are known and are given in an aggregate terms for the time horizon. A dynamic programming formulation was given, and optimality conditions were defined. Dynamic multiple facility problem formulations are not limited to location-allocation. Sheppard (1974) sought to extend a wide range of basic facility location models so that they include both spatial and temporal aspects of real-world problems. The author presented a variety of models which determine not only the location of multiple facilities, but also the size of the facilities and the timing of plant construction or expansion. While Sheppard’s models capture many aspects of the true location problems faced by industry or the public sector, the majority of his formulations are nonlinear, integer, and dynamic, and thus computationally intractable. Murthy (1993) modeled the allocation of n facilities to n sites as an assignment problem. In his model the assignment of the facilities to sites carry some costs which can change from period to period. The fixed relocation cost at the end of each period is included as well. Using the “Dual Ascent” procedure, the mixed integer programming model was solved for the location and relocation of the facility. Drezner (1995a) formulated the progressive p-median problem, which locates p facilities over a planning horizon of T periods, without relocation. Inputs to the progressive p-median problem include time-dependent (known) demands and times at which the facilities are to be located. The objective is to find the facilities locations which minimize total transport cost (or distance) over the time horizon. Since the general form of the problem is nonlinear, a heuristic solution procedure for finding local minima was presented. The computational complexity of most facility location problems has inspired a number of heuristic procedures for determining near-optimal solutions. In an attempt to evaluate the relative merit of such procedures, Erlenkotter (1981) compared the performance of several heuristic solution approaches to a single problem formulation. He examined a dynamic, fixed charge, capacitated, cost minimization problem with discrete time intervals, a special case of which is the static simple plant location problem. While limited in scope, Erlenkotter’s computational study suggests that combining heuristic approaches in a multiple phase solution process may prove most effective. VanRoy and Erlenkotter (1982) later studied a dynamic uncapacitated facility location problem in which goods are shipped from facilities to meet known customer demands. New facilities are allowed to be opened and initially existing facilities are allowed to be closed over the time horizon. The objective is to minimize total discounted costs, including facility location and operating costs as well as production and distribution costs for goods shipped. A branch and bound procedure with lower bounds obtained through solving LP relaxation with a heuristic dual ascent method was proposed. Driven by an application to freight carrier transportation terminals, Campbell (1990) sought simple strategies for locating and relocating facilities. Specifically, he examines the effectiveness of myopic approaches for finding near-optimal location solutions. The author developed a general continuous distribution model which includes line haul transportation and economies of scale. The model considers trade-offs between transportation, location
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and relocation costs, with the objective of overall cost minimization. Campbell developed bounds on the optimal objective value using myopic strategies which first ignore relocation costs (providing a lower bound) and then disallow relocations altogether (deriving an upper bound). Campbell showed that a myopic strategy with limited relocation is nearly optimal for locating terminals in one or two dimensions, unless relocation costs are high. Campbell thus suggested that extensive relocations may not be necessary to obtain near-optimal distribution costs. Gunawardane (1982) considered location problems within the public sector. Specifically, he examined several covering problems in which public facilities are located (and possibly relocated) over a planning horizon. Both the set covering and the maximal covering problem formulations were extended to account for a given planning horizon. Weights are decreasing over the planning period and discounting coefficients were defined for the location variables in the dynamic set covering objective function to encourage postponing facility locations until they are required. Another model formulation discourages frequent changes in locations by charging against each opening and closing a facility. Computational results were highlighted; the author reported that most LP relaxations return integer optimal solutions. All of the dynamic deterministic problems discussed above seek an optimal or nearoptimal solution to a single objective function. Schilling (1980) considered alternate approaches to solving facility location problems, inspired by the public sector need to locate Emergency Medical Services facilities. Specifically, he considers a multi-objective maximal cover problem formulation and seeks a set of good solutions from which the decision maker can select one for implementation. Berman and LeBlanc (1984) proposed location-relocation of mobile facilities on a stochastic network. A multi-objective approach is also examined by Min (1988), who considered expanding and relocating public libraries in the Columbus metropolitan area. Another unique approach to locating facilities over time was proposed by Daskin et al. (1992). The authors acknowledge that the difficulty in solving dynamic facility location problems arises from the uncertainty surrounding future conditions. Even establishing an appropriate time horizon length is a non-trivial problem which is ignored in most formulations. They argue that the best way to manage uncertainty is to postpone decision making as long as possible, collecting information and improving forecasts as time advances. Since the first period decisions are the ones to be implemented immediately, the authors claim that the goal of dynamic location planning should not be to determine locations and/or relocations for the entire horizon, but to find an optimal or near-optimal first period solution for the problem over an infinite horizon. In this paper, the single facility location problem with multiple relocations and continuous weight functions is investigated. The procedure proposed here can be applied to general distance and weight functions. The set of possible relocation times, however, are assumed to be pre-defined and thus the problem is to choose the best relocation times in order to minimize the total cost. An algorithm which finds the optimal locations, relocation times and the total cost, for all three types of distance measurements and various weight functions, is developed. Locations are found using constant weights, and relocations times are the solution to a Dynamic Programming or Binary Integer Programming (BIP) model. The time horizon can be finite or infinite. This problem will be considered with or without relocation cost. Note that we assume that every point in the plane is a candidate for the location of facilities. This is a common assumption of continuous location models. In some applications a pre-specified set of possible locations is available.
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2 The single facility location problem The model used for the location of the facility is the Weber problem (Drezner et al. 2002). The objective cost function to be minimized is: F (X) =
m
(1)
Wi d(X, pi )
i=1
where Wi is the weight associated with demand point i, and d(X, pi ) is the distance between demand point i, located at pi = (ai , bi ) and the facility located at X = (x, y). We consider the following distance measures: • Rectilinear distance d(X, pi ) = |x − ai | + |y − bi |.
(2)
d(X, pi ) = (x − ai )2 + (y − bi )2 .
(3)
• Squared Euclidean distance
• Euclidean distance d(X, pi ) =
(x − ai )2 + (y − bi )2 .
(4)
The optimal location (x ∗ , y ∗ ) is easily obtained for each distance measure (Francis et al. 1992; Love et al. 1988). Now assume that the weights change in time. The weight Wi is a function of time wi (t), where t is the time in the range of [0, T ]. Consider the problem of finding the optimal location of the facility over the time horizon. We investigate the problem that incorporates possible relocations of facility, at a given cost, during [0, T ]. The problem is, therefore, to find the optimal relocation time(s) and locations.
3 Time dependent weights It is important to note that the weights, wi (t), within the period of [0, T ] cannot be negative. For the linear weights the following holds (Drezner and Wesolowsky 1991): wi (t) = ui + vi t,
wi (t) ≥ 0, t ∈ [0, T ]
⇒
ui ≥ 0,
vi ≥ −
ui . T
The objective function for the location problem with time dependent weights is: T m wi (t)d(X, pi ) dt F (X) = 0
(5)
(6)
i=1
which is equivalent to: F (X) =
m
T
wi (t)dt.
(7)
i = 1, . . . , m.
(8)
d(X, pi ) 0
i=1
Integrating the weights results in constant weights:
T
Wi =
wi (t)dt, o
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Fig. 1 New facility locations and relocations time Fig. 2 New facility locations and relocation time
The location problem is converted to the Weber location problem that can be easily solved (Drezner et al. 2002; Francis et al. 1992; Love et al. 1988). A more interesting problem arises when the location of the new facility is allowed to change several times during the time horizon: say, n changes are allowed during the time span [0, T ]. The variables to be determined are the time breaks B = (b1 , . . . , bn ) at which the changes take place and the associated optimal solution. Define b0 = 0 and bn+1 = T . Then, we have n time breaks. Of course, T can be infinite and this case will be addressed later.
4 Facility relocation Given the specific weight functions in time, it could be more economical to relocate the new facility some time in the future, such that the total location and relocation costs is minimized. It is assumed that the relocation can take place only at pre-determined points in time. The total location cost, therefore, is the sum of the location costs before and after the relocation. The total cost depends on the optimal relocation time and the facility’s optimal locations prior and after the relocation. Refer to Fig. 1; Cj k is the cost of locating the facility during period [bj , bk ). In the simplest case (without relocation cost and with finite time horizon [0, T ]), we want to find the shortest path from b0 to bn+1 . The problem can be stated in terms of a shortest path problem in an acyclic network (Andretta and Mason 1994). The following Lemmas are needed for the derivation of the optimal algorithm. Lemma 1 The objective cost function is additive. Consider Fig. 2, bj −1 , bj and bj +1 are some points in time. It is obvious that the location of the new facility can be determined independently during [bj −1 , bj ) and [bj , bj +1 ), given that the relocation takes place at bj . The objective function is: F=
bj
m
d(X j −1,j , pi ) · wi (t)dt +
bj −1 i=1
= g1 (x
j −1
,y
bj +1
bj j −1
) + g2 (x , y ) j
j
m
d(X j,j +1 , pi ) · wi (t)dt
i=1
(9)
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where X j k is the optimal facility location during the [bj , bk ). This shows that the objective cost function is equal to the sum of two functions with independent variables. This shows the additivity of the objective function. Therefore, given bj , optimal location of the new facility before and after the relocation time can be determined independently. The total location cost is thus the sum of the location costs before and after the relocation. The optimal location during each interval can be calculated after integrating the weights, using the constant weight procedure. Lemma 1 helps us to find the best relocation times and new locations. However, using Lemma 1 in different situations related to time horizon and relocation cost, may be different. The time horizon can be either finite or infinite and relocation costs can exist or not; in the existence of relocation cost, various alternatives can occur. These situations will be investigated in following sections. 4.1 Facility relocation without relocation cost When there are no relocation costs, we apply the following Lemma (Drezner and Wesolowsky 1991): ∗ ≤ FL∗ , for L = 0, 1, 2, . . . . Lemma 2 FL+1
FL∗ is the cost of optimal L time break solution. Based on Lemma 2, the cost of optimal L + 1 time break solution can only be lower than the cost of optimal L time breaks solution. Thus, we should use the all opportunities (time breaks) for relocating the facility to the best locations. In addition, Daskin et al. (1992), argue that the best way to manage uncertainty is to postpone decision making as long as possible, collecting information and improving forecasts as time advances. Since the first period decisions are the ones to be implemented immediately, the goal of dynamic location planning should not be to determine locations and/or relocations for the entire horizon, but to find an optimal or near-optimal first period solution for the problem over an infinite horizon (Daskin et al. 1992). Therefore, we set T = b1 , and then using the following weights, via solving a simple single facility location problem we can find the best location from b0 up to T (Drezner 1995b; Francis et al. 1992; Mirchandani and Francis 1990): b1 =T wi (t)dt, i = 1, . . . , m. (10) Wi = b0 =0
This procedure should be repeated after each time break. 4.2 Fixed relocation cost As we saw in Sect. 4.1, without having relocation costs, relocation should be done at all break points. However, when there are relocation costs, this is no longer true. When there are relocation costs, it is better to use a forward dynamic programming. The algorithm can be stopped as soon as the first decision is made. A finite time horizon is not necessary for the procedure thus T can be infinite. Using dynamic programming, the relocation points are the stages, the last relocation time before the current stage is a state and whether to relocate or not is the decision variable. We use a forward procedure as follows: Z(bj ) = Zj = Min{C0j , S(bj −1 ) + Cj −1,j + Zj −1 }, Z(b0 = 0) = Z0 = S0 .
j = 1, 2, . . . ,
(11)
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Z(bj ) is minimum total cost during [b0 , bj ], Cj k is the cost of locating the facility during period [bj , bk ], S(t) is the relocation cost at the relocation point t and S0 is the cost of locating the facility at the time b0 . Note that based on Lemma 1, Cj k is calculated as follows: Cj k =
m
jk
Wi × d(X j k , pi )
(12)
i=1
where, jk Wi
=
j Wi
j +1 + Wi
j +2 + Wi
+ · · · + Wik−2
+ Wik−1
bk
=
wi (t)dt.
(13)
bj
Note that a problem with infinite time horizon can not necessarily be solved with a finite horizon procedure. Bastian and Volkmer (1992) proposed a perfect forward algorithm for the solution of a single facility dynamic location/relocation problem on discrete locations; but Andretta and Mason (1994) provided a numerical example to demonstrate that this problem does not always have a finite horizon. They restated the original problem in terms of a shortest path problem in an acyclic network and stated a condition (which is both necessary and sufficient) for the existence of a finite forecast horizon for obtaining an optimal initial decision. However, if the minimum of the forward equation will be on the first term, we should continue the procedure and may not be able to find the optimal first decision. We propose to use finite time horizon for making a near-optimal decision. Longer time horizons normally lead to solutions which are closer to optimal. Note that the forecasted data farther into the future are not as reliable. After determining a finite time horizon we should continue the forward dynamic programming or shortest path problem in an acyclic graph, up to end of the time horizon T ∗ (Daskin et al. 1992). We propose a Binary Integer Programming (BIP) model. Step 1. There are n time candidates for relocation. Adding b0 = 0 and bn+1 = T as relocation points yields at total of n + 2 points. The interval [0, T ] is divided to n + 1 subintervals. There are m demand points. Calculate: j
Wi =
bj +1
wi (t)dt,
i = 1, . . . , m, j = 0, . . . , n + 1.
(14)
bj jk
Step 2. Define Wi : jk
j
j +1
Wi = Wi + Wi
j +2
+ Wi
+ · · · + Wik−2 + Wik−1 =
bk
wi (t)dt
(15)
bj
where i = 1, . . . , m, j = 0, . . . , n, k = j + 1, . . . , n + 1 and j < k. Calculate for every jk demand point i the value of Wi for all values of j and k. This generates the integrated weight associated with the ith demand point for the location for the facility during the time interval of [bj , bk ). Step 3. For each interval [bj , bk ) find the optimal location of the facility (x j k , y j k ). One of jk the three distance measures is used. Using the value of Wi and the coordinates of the existing facilities, the optimal solution for rectilinear distance is the median point (Francis et al. 1992; Love et al. 1988), for squared Euclidean distance the optimal location is the center of gravity (Francis et al. 1992; Love et al. 1988) and for Euclidean distance using the iterative Weiszfeld procedure (Drezner and Wesolowsky 1991; Francis et al. 1992;
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Love et al. 1988). (The convergence of the above sequence was shown in Drezner 1996; Morris 1981; Ostresh 1981; Weiszfeld 1936). Step 4. Calculate Cj k , the location cost, if the new facility has the same location during the time interval [bj , bk ), using the following relationship: Cj k =
m
jk
Wi × d(X j k , pi )
(16)
i=1
where d(X j k , pi ) is the distance between the optimal location of the new facility and demand point i for t ∈ [bj , bk ), and X j k calculated in Step 3. Step 5. Using the cost coefficients in Step 4, the following model is used to find the optimal relocation times: Minimize F =
n n+1
C j k × Zj k
j =0 k=j +1
Subject to:
n+1
Z0k = 1,
k=1 k−1
Zj k =
j =0 n
n+1
Zkl ,
k = 1, . . . , n,
(17)
l=k+1
Zj,n+1 = 1,
j =0
Zj k = 0 or 1,
∀j, k, j < k
where, Zj k =
1 0
if current relocation take place at bj and the next one at bk , otherwise.
This is a Binary Integer Programming (BIP) model with n constraints and n(n + 1)/2 variables. The first constraint ensures that the relocation decision starts at time b0 = 0. The second set enforces the consideration of the next relocation time exactly after the last relocation time. The last constraint guarantees that the decision making will continue to the end of planning horizon, T ∗ . The BIP model can be solved using optimization software such as LINGO and CPLEX. The solution to this BIP model yields the times for relocation. The facility location for each time interval is found in Step 3. The total cost of this policy is F . 4.3 Constraints on budget and the number of relocations The maximum number of possible relocations is: n n+1
Zj k .
(18)
j =0 k=j +1
If we wish to limit this number to L, then the following constraint is added (16): n n+1 j =0 k=j +1
Zj k ≤ L.
(19)
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This constraint can also be in the form of an equality. We must make sure, however, that L ≤ n, otherwise the constraint cannot be tight and is not constraining the solution. Now assume that Sj = S(bj ) is the relocation cost at time bj . The objective function should be modified as follows: Min F =
n n+1
C j k × Zj k +
j =0 k=j +1
n n+1
S j × Zj k .
(20)
j =1 k=j +1
It is also possible to have a combination of a fixed relocation cost and predetermined number of relocations. In such a case the objective function will be revised as explained above and the relocation constraint (19) will be added to the model. 4.4 Location dependent relocation cost Another situation closer to real world applications is the case in which relocation cost depends on the location of the facility; i.e. the facility relocation cost at the time bj is Sj (X = x, y) instead of Sj . Here X = (x, y) is the location of facility. For instance, one of major important part of the cost for locating facility is the price of land. Price of land for locating a facility will differ from one location to another. In this model we solve the problem similarly to the procedure described in Sect. 4.2. The only difference is that the cost of the best location during [bj , bk ) will be the minimum of the function: Cj k = Sj (X) +
m
jk
Wi × d(X j k , pi ).
(21)
i=1
This means that the second term of the objective function (20) in our BIP model is not independent of the first term. If Sj (X = x, y) is differentiable, the minimum of Cj k could be calculated directly. When squared Euclidean distances are used, Cj k is differentiable and there is no need to use Hyper Approximation Procedure (HAP) for the calculation of the optimal Cj k (Drezner and Wesolowsky 1991; Francis et al. 1992; Love et al. 1988). 4.5 Relocation cost depend on the location of facility before and after relocation Sometimes, relocation cost at the time bj depends on the location of facility before and after bj . In practical cases there are situations in which the cost of relocating to a closer location is lower than to a farther location. For example in mobile facilities, the relocation cost will be a function of distance between the location of facility before and after relocation time. In this case the objective function is: Cj k = u(bj ) × d(X hj , X j k ) +
m
jk
Wi × d(X j k , pi )
(22)
i=1
where bh is the last relocation point before bj , d(X hj , X j k ) is the distance between the location of facility before bj (X hj ) and after bj (X j k ). In addition, u(bj ) is the unit distance cost for relocating a facility from its location to another new location, at the time bj . In this case Lemma 1 is not satisfied and the objective function is not additive; thus we can not use the previous techniques. For a finite time horizon we propose to perform complete enumeration and find all of feasible paths from b0 to bn+1 (Fig. 1). When n is large we must resort to
Ann Oper Res Fig. 3 Traffic rate function for five regions
heuristic algorithms. For each feasible solution we should calculate the objective function and select the minimum. The number of these feasible paths can be calculated with respect to the number of relocations (0, 1, 2, . . .) as follows: n n n (23) 1+ + + ··· + = 2n . 1 2 n With a longer time horizon we need more data and of course the solution will be closer to optimality for making the first decision. For solving the objective function of each feasible path, note that if d(X j k , pi ) and d(X hj , X j k ) both are rectilinear distance we can solve an equivalent LP substituting the absolute function with new variables and constraints. If d(X j k , pi ) and d(X hj , X j k ) both are squared Euclidean distance, the objective function is differentiable and the problem easily can be solved (Drezner and Wesolowsky 1991; Francis et al. 1992; Love et al. 1988). If the type of distances related to d(X j k , pi ) and d(X hj , X j k ) is different or both are Euclidean we can use the Hyper Approximation Procedure (HAP) (Drezner 1996; Drezner and Wesolowsky 1991; Francis et al. 1992; Love et al. 1988; Morris 1981; Ostresh 1981; Weiszfeld 1936).
5 An example The management of the highway police department is looking for the best location to place a new police station. The planning horizon is set to 10 years. Management has agreed to possibly relocate the police station at the end of year t = [0, 1, 2, 4, 5.5, 7, 8, 10], if the changes in the traffic rate in the region justify the move. The police station is providing the services for five regions. The distance from the new police station is measured based on Euclidean norm, from the center of these five regions located at: p1 = (1, 1), p2 = (0, 3), p3 = (3, 9), p4 = (4, 4) and p5 = (1, 1). The traffic rates for these regions, wi , i = 1, . . . , 5, are changing in time. The functions of the traffic rates in time are: w1 (t) = t 2 − 10t + 26, w2 (t) = 2.6t + 2, w3 (t) = −2.8t + 30, w4 (t) = 28e−1.5t , w5 (t) = −0.1t 3 + t 2 + 10. The first function w1 (t) represents a forecast of declining demand in the near future but anticipating that it will pick up after 5 years. The next two functions w2 (t) and w3 (t) are
Ann Oper Res Table 1 Weights of the existing facility in each time interval
t1 –t2
t2 –t3
t3 –t4
t4 –t5
t5 –t6
t6 –t7
t7 –t8
W1
21.3
13.3
10.66
1.875
4.125
7.3
34.66
j W2 j W3 j W4 j W5
3.3
5.9
19.6
21.525
27.375
21.5
50.8
28.6
25.8
43.2
25.05
18.75
9
9.6
14.5
3.327
0.883
0.042
0.002
0
0
10.31
11.96
32.67
32.65
36.72
23.96
35.07
j
Table 2 Cumulative weights of existing facilities
tj –tk
jk
W1
jk
W2
jk
W3
jk
W4
jk
W5
1_2
21.33
3.3
28.6
14.5
10.31
1_3
34.67
9.2
54.4
17.74
22.27
1_4
45.3
28.8
97.6
18.62
54.93
1_5
47.21
50.33
122.65
18.66
87.59
1_6
51.33
77.7
141.4
18.66
124.31
1_7
58.67
99.2
150.4
18.66
148.26
1_8
93.3
150
160
18.66
183.33
2_3
13.33
5.9
25.8
3.24
11.96
2_4
24
25.5
69
4.12
44.63
2_5
25.87
47.03
94.05
4.16
77.28
2_6
30
74.4
112.8
4.16
114
2_7
37.33
95.9
121.8
4.16
137.96
2_8
72
146.7
131.4
4.16
173.03
3_4
10.66
19.6
43.2
0.88
32.67
3_5
12.54
41.13
68.25
0.92
65.32
3_6
16.67
68.5
87
0.92
102.04
3_7
24
90
96
0.92
126
3_8
58.67
140.8
105.6
0.92
161.07
4_5
1.87
21.53
25.05
.045
32.65
4_6
6
48.9
43.8
0.049
69.37
4_7
13.33
70.4
52.8
0.05
93.33
4_8
48
121.2
62.4
0.05
128.4
5_6
4.13
27.37
18.75
.002
36.72
5_7
11.46
48.87
27.75
.002
60.68
5_8
46.12
99.67
37.35
.002
95.75
6_7
7.33
21.5
9
0
23.96
6_8
42
72.3
18.6
0
59.03
7_8
34.66
50.8
9.6
0
35.07
typical for forecasting demand by linear regression. The fourth function w4 (t) represents an exponentially declining demand that is not expected to pick up again. The last function represents a forecast of demand slowly increasing and is predicted to decline towards the end of the period. Figure 3 shows these five functions.
Ann Oper Res Table 3 Calculation of facility location and its cost in each time interval
tj –tk
(x j k , y j k )
Cj k
1_2
(3.6730, 4.1023)
293.2603
1_3
(3.3285, 4.3206)
552.1851
1_4
(3.2463, 4.4727)
1030.0497
1_5
(3.3148, 4.2868)
1394.4947
1_6
(3.2229, 3.9056)
1802.7860
1_7
(2.8208, 3.6524)
2014.5586
1_8
(2.0720, 3.0607)
2468.7318
2_3
(2.9908, 4.9438)
256.3221
2_4
(3.0891, 4.8847)
733.4052
2_5
(3.2031, 4.3918)
1099.8467
2_6
(3.2090, 3.8996)
1480.5302
2_7
(3.0446, 3.6020)
1732.8257
2_8
(2.1261, 3.0325)
2194.2234
3_4
(3.1540, 4.8568)
476.9398
3_5
(3.2946, 4.2351)
842.2666
3_6
(3.3085, 3.7058)
1220.1237
3_7
(3.1246, 3.4164)
1437.4945
3_8
(1.9902, 2.9242)
1924.2919
4_5
(3.6664, 3.6081)
361.3012
4_6
(3.6220, 3.1730)
733.0883
4_7
(3.2098, 3.0095)
982.7101
4_8
(1.6538, 2.6874)
1413.5560
5_6
(3.5161, 2.8644)
369.7872
5_7
(2.7575, 2.7809)
615.7689
5_8
(1.3070, 2.5230)
1024.6501
6_7
(1.8581, 2.5860)
241.7077
6_8
(1.0559, 2.3526)
635.0770
7_8
(0.9570, 2.1910)
389.2160
j
Step 1. Calculate the weight of each facility for each time interval [tj , tj +1 ), Wi . The results are depicted in Table 1. jk Step 2. Calculate the cumulative weight of the existing facilities, Wi . The results are depicted in Table 2. Step 3. For each interval find the optimal facility location using the formulas presented for the Euclidean distances. The results are presented in Table 3. Step 4. Calculate the location cost associated with placing the facility in its optimal location in the relevant time interval. The results are presented in Table 3. Step 5. Using the cost coefficients, the BIP model is formed and solved using LINGO. The model and its output are presented in Fig. 4. Based on the results, the optimal solution, depicted in Table 4, includes four locations for the police station and three relocation times.
Ann Oper Res
Fig. 4 LINGO input and output for the optimal solution
Ann Oper Res Table 4 Optimal solution
Time interval
Location
Cost
t1 = 0 to t2 = 1
(3.6730, 4.1023)
293.2603
t2 = 1 to t3 = 2
(2.9908, 4.9438)
256.3221
t3 = 2 to t7 = 8
(3.1246, 3.4164)
1437.4945
t7 = 8 to t8 = 10
(0.9570, 2.1910)
389.2160
Total
2376.2900
6 Conclusions Single facility location problems can be extended to include time dependent weights. An optimal algorithm has been presented to identify the location of the new facility during a time interval. It is assumed that the new facility can be relocated several times. The set of possible relocation times are given in advance. The algorithm generates the optimal relocation times, and the optimal location of the facility between two consecutive relocations, such that the total location cost is minimized. The important characteristic of the approach presented here is that the cost functions are continuous and the algorithm is not restricted to any specific distance norm. In addition, infinite time horizon and types of relocation cost are considered. For future research, one can consider the continuous planning horizon. This means that the relocation times can happen anywhere within the planning period (Drezner and Wesolowsky 1991). Another extension to this model is to consider multi-facility location and relocation problems.
References Andretta, G., & Mason, F. M. (1994). A perfect forward procedure for a single facility dynamic location/relocation problem. Operations Research Letters, 15, 81–83. Ballou, R. H. (1968). Dynamic warehouse location analysis. Journal of Marketing Research, 5, 271–276. Bastian, M., & Volkmer, M. (1992). A perfect forward procedure for a single facility dynamic location/relocation problem. Operations Research Letters, 12, 11–16. Berman, O., & LeBlanc, B. (1984). Location-relocation of mobile facilities on a stochastic network. Transportation Science, 8, 315–330. Campbell, J. F. (1990). Locating transportation terminals to serve an expanding demand. Transportation Research, 24B(3), 173–192. Chand, S. (1988). Decision/Forecast horizon results for a single facility dynamic location/relocation problem. Operations Research Letters, 7(5), 247–251. Daskin, M. S., Hopp, W. J., & Medina, B. (1992). Forecast horizon and dynamic facility location planning. Annals of Operations Research, 40, 125–151. Drezner, Z. (1995a). Dynamic facility location: the progressive p-median problem. Location Science, 3(1), 1–7. Drezner, Z. (1995b). Facility location: a survey of applications and methods. New York: Springer. Drezner, Z. (1996). A note on acceleration the Weiszfeld procedure. Location Science, 3, 275–279. Drezner, Z., & Wesolowsky, G. O. (1991). Facility location when demand is time dependent. Naval Research Logistics, 38, 763–771. Drezner, Z., Klamroth, K., Schobel, A., & Wesolowsky, G. (2002). In Z. Drezner & Hamacher (Eds.), The Weber problem in facility location: applications and theory. Berlin: Springer. Erlenkotter, D. (1981). A comparative study of approaches to dynamic location problems. European Journal of Operational Research, 6, 133–143. Francis, R. L., McGinnis, L. F., & White, J. A. (1992). Facility layout and location: an analytical approach (2nd edn.). Englewood Cliffs: Prentice Hall.
Ann Oper Res Gunawardane, G. (1982). Dynamic version of set covering type public facility location problems. European Journal of Operational Research, 10, 190–195. Hormozi, A. M., & Khumawala, B. M. (1996). An improved algorithm for solving a multi-period facility location problem. IIE Transactions, 28, 105–114. Love, R. F., Morris, J. G., & Wesolowsky, G. O. (1988). Facilities location: models and methods. New York: North-Holland. Min, H. (1988). Dynamic expansion and relocation of capacitated public facilities: a multi-objective approach. Computers and Operations Research, 15(3), 243–252. Mirchandani, P. B., & Francis, R. L. (1990). Discrete location theory. New York: Wiley. Morris, J. G. (1981). Convergence of the Weiszfeld algorithm for Weber problems using a generalized distance function. Operations Research, 26, 37–48. Murthy, I. (1993). Solving the multiperiod assignment problem with start-up costs using dual ascent. Naval Research Logistics, 40, 325–344. Ostresh Jr., L. M. (1981). On the convergence of a class of iterative methods for solving the Weber location problem. Operations Research, 26, 597–609. Schilling, D. A. (1980). Dynamic modeling for public-sector facilities: a multi-criteria approach. Decision Science, 11, 714–724. Scott, A. J. (1971). Dynamic location-allocation systems: some basic planning strategies. Environment and Planning, 3, 73–82. Sheppard, E. S. (1974). A conceptual framework for dynamic location-allocation analysis. Environment and Planning A, 6, 547–564. Sweeney, D. J., & Tatham, R. L. (1976). An improved long-run model for multiple warehouse location. Management Science, 22(7), 748–758. Tapiero, C. S. (1971). Transportation location-allocation problems over time. Journal of Regional Science, 11(3), 377–384. VanRoy, T. J., & Erlenkotter, D. (1982). A dual-based procedure for dynamic facility location. Management Science, 28(10), 1091–1105. Weiszfeld, E. (1936). Sur le point pour lequel la somme des distances de n points donnes est minimum. Tohoku Mathematical Journal, 43, 355–386. Wesolowsky, G. O. (1973). Dynamic facility location. Management Science, 19(11), 1241–1248. Wesolowsky, G. O., & Truscott, W. G. (1976). The multi-period location-allocation problem with relocation of facilities. Management Science, 22(1), 57–65.
