The Capacitated Multi-Source Weber Problem with Facility Fixed Cost: Constructive and GRASP-Based Heuristics* Martino Luisa, Said Salhib, and Gábor Nagyb a
Othman Yeop Abdullah Graduate School of Business, Universiti Utara Malaysia, 06010 UUM Sintok, Kedah Darul Aman, Malaysia. Email:
[email protected] b The Centre for Logistics & Heuristic Optimisation, Kent Business School, University of Kent at Canterbury, Canterbury CT2 7PE, UK. Email: {S.Salhi, G.Nagy}@kent.ac.uk
Abstract A new variant of the capacitated multi-source Weber problem that incorporates facility fixed costs for opening facilities is presented. An iterative procedure that considers at each iteration the facility fixed cost as part of the stopping criterion within the use of the Weiszfeld’s equations is proposed and incorporated within the well known locate-allocate method of Cooper. A scheme to reduce the risk of getting stuck at poor local minima is also presented and shown to be very efficient. A guided constructive heuristic and a greedy randomized adaptive search procedure (GRASP) are presented. Both use restricted regions and the new local search while the latter also incorporates a learning scheme in the construction of the restricted candidate list. Three types of fixed costs, namely a constant, a zone-based and a Voronoi-based one, are considered. The four known data sets that are commonly used for the uncapacitated multi-source Weber problem, are adapted accordingly to cater for the capacity and the facility fixed costs. Interesting empirical results, based on our extensive computational experiments, are produced along with some highlights of potential future research directions.
Keywords: location on the plane, capacitated location, heuristics, GRASP, facility fixed cost. *This research has been partially supported by the Ministry of Science and Innovation of Spain under the research project ECO2011-24927, in part financed by the European Regional Development Fund (ERDF), and the Fundacion Seneca under the research project 15254/PI/10, and also by the Algerian Ministry of Education (Sciences Fondamentales), under research project PNR 8/U160/64.
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1. Introduction The location-allocation problem on the plane (also known as the multi-source Weber problem) with capacity restrictions and facility fixed costs is investigated. In this study, the number of facilities to locate can be either specified or unknown, and each facility has a capacity limit and a corresponding fixed cost. Also, a set of n customers at known locations with their respective demands has to be fully served. The aim is to determine the allocation of these customers to these facilities without violating the capacity of any of the facilities while minimising the total sum of transportation and fixed costs associated with the opening of these facilities. In this study, we consider planar location with Euclidean distances. The classical (i.e. uncapacitated) and the capacitated multi-source Weber problems both assume that the number of facilities to be located is known in advance, whereas in practice, determining the number of facilities is one of the main factors that is usually addressed at a strategic level as the establishment (opening) of a facility requires a massive investment. In this paper we investigate this capacitated multi-source Weber problem in the presence of fixed costs which we refer to for short as CMSWPF. To the best of our knowledge, this problem is new but also useful for practical applications. The closest studies to the proposed investigation include the interesting work by Brimberg et al. (2004) and that of Brimberg and Salhi (2005) where the facilities were considered uncapacitated. The paper is organised as follows. In the next section, a brief literature review is given. Then, a mathematical model for the CMSWPF is presented, followed by a discussion of the various types of fixed cost functions. Sections 5 and 6 present our solution methods, namely the constructive and the GRASP heuristics respectively. Our computational results are provided in Section 7. Finally, the last section summarises our conclusions and highlights some of those research avenues that we believe to be worthwhile exploring in the future.
2. Literature Review Our review is in two parts. First, we look at the capacitated multi-source Weber problem (CMSWP) without fixed costs and then at the uncapacitated multi-source Weber problem (MSWP) with fixed costs. We do not look at the MSWP without fixed costs, but refer the reader to Brimberg et al. (2008).
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Cooper (1972) observes that once the facilities are located, the CMSWP reduces to the classical Transportation Problem (TP). Exact and heuristic methods are presented. The latter is based on alternately solving the location-allocation problem and the TP until there is no epsilon () improvement found in the total cost. This technique, known as ATL for short, is enhanced further by Cooper (1975) and is then adapted to solve the fixed charge problem in Cooper (1976). Sherali and Shetty (1977) solve the rectilinear distance CMSWP using a convergent cutting plane algorithm originally derived from a bilinear programming problem by substituting decision variables by the difference of two non-negative variables. The problem was revisited 16 years later by Sherali and Tuncbilek (1992) who put forward a branch and bound algorithm to compute strong upper bounds. Sherali et al. (1994) study the rectilinear distance CMSWP as a mixed integer nonlinear programming formulation using a reformulation-linearization technique. Gong et al. (1997) propose a hybrid evolutionary method based on a genetic algorithm and Lagrange relaxation. Sherali et al. (2002) develop a branch and bound approach based on a partitioning of the allocation space for the capacitated Euclidean and lp distance MSWP. Aras et al. (2007a) propose three heuristic methods to tackle the CMSWP with Euclidean, squared Euclidean, and lp distances. Aras et al. (2007b) tackle the CMSWP with rectilinear, Euclidean, squared Euclidean, and lp distances by implementing simulated annealing, threshold accepting, and genetic algorithms. Zainuddin and Salhi (2007) deal with the Euclidean CMSWP by proposing a perturbation-based heuristic. Aras et al. (2008) adopt their earlier approaches to solve the CMSWP with rectilinear distance. Luis et al. (2009) solve the CMSWP by introducing the concept of region-rejection into their constructive heuristic. Mohammadi et al. (2010) design two genetic algorithms (GAs) one for the location problem and the other for the allocation of customers to those facilities. Very recently, Luis et al. (2011) present a novel guided reactive GRASP that combines adaptive learning with the concept of region-rejection.
The literature on the MSWP with fixed costs is very scarce; to our knowledge, it consists of just two papers. Brimberg and Salhi (2005) assume that fixed costs are zone-dependent, where zones are non-overlapping convex polygons. The paper mainly deals with locating a single facility. It is shown that the optimal solution falls either inside a zone with the cheapest cost, or on a zone edge. An exact algorithm, based on the above observation, is presented. For multiple facilities, discretizing the problem is suggested. Brimberg et al. (2004) use constant fixed costs and propose a multi-phase heuristic. Firstly, the corresponding discrete location problem, namely the simple plant location problem, is solved which gives a good approximation of the number of facilities needed.
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Then, Cooper’s location-allocation method is used to relocate the facilities. Finally, a local search is carried out to see whether having a few more or a few less facilities may give a better solution, as the fixed cost of adding/removing a few facilities is balanced by the decreased/increased transportation cost.
3. Mathematical Formulation In this section we first give the input data followed by the decision variables and the formulation. Input Data
M : An upper bound on the number of facilities to be located; n
: the number of fixed demand points (or customer points);
aj = a1j , a 2j : location of customer j where aj 2 (j = 1, …, n); f(X) : the fixed cost of opening a facility at point X 2; wj : demand or weight of customer j (j = 1, …, n); b
: the fixed capacity of a facility, where b č.
Decision Variables zi = 1 if the ith facility is opened, 0 otherwise (i = 1, …, M ); xij : quantity assigned from facility i to customer j, (i = 1, …, M ; j = 1, …, n); d(X, Y) : the Euclidean distance between locations X and Y where ( X, Y) 2x2;
Xi = X i1, X i2 : coordinates of facility i where Xi 2;
Formulation This problem can be formulated as follows:
xij d X i , a j f X i zi M
Minimise
n
M
i 1 j 1
(1)
i 1
Subject to M
x i 1
ij
w j , j = 1, …, n
(2)
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n
x j 1
ij
bz i , i = 1, … M
(3)
xij 0 , i = 1, … M ; j = 1,…, n
(4)
zi = {0 ,1}, i = 1, … M
(5)
The objective function (1) is to minimize the sum of the transportation costs and the fixed costs. Constraints (2) ensure the total demand of each customer is satisfied while constraints (3) guarantee that the facility capacity are not exceeded. Constraints (4) and (5) refer to the nonnegativity and the binary constraints respectively. We note that if the fixed cost fi =0 for all sites, the problem becomes the CMSWP. Also, if n
the value of b is large enough (say b w j ) the problem reduces to the classical MSWP. As a byj 1
M
product, the number of open facilities can also be derived as zi . i 1
4. The Construction of the Facility Fixed Costs The set up or opening cost of a facility may be dependent on geographical (location) areas. In other words, some areas may have cheaper costs of establishing a facility whereas others may have exorbitant costs. In this study, we investigate three types of facility fixed cost models that we consider to be practical though others could also be explored.
4.1. A Constant Fixed Cost Function The simplest model considers that the opening cost of a facility is a constant value irrespective of its location (i.e. f(X) = f, X 2). For instance, if f is set to 0 then the problem reduces to the CMSWP. For a given value of M, (M = 1, ..., M ) this fixed cost may be removed from the objective function when solving the CMSWP and then added at the end to the total cost as a constant Mf. This idea is simple and can be used for evaluating different values of M. This flexibility in assessing several values of M is useful in practice, as it provides several scenarios to the decision makers. In brief, one may try out different values of M, and then choose the solution with the smallest overall total cost.
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4.2. A Zone-Based Fixed Cost Function In this model, following Brimberg and Salhi (2005), we divide the plane into “mutually exclusive zones”, and let the fixed cost be constant within each zone. This could be conducted in different ways; we chose to tessellate the plane into Voronoi regions (see Preparata and Shamos (1985) and Figure 1), with the customer locations as seed points. In effect, this relates the cost of locating a facility to its nearest customer. As in Brimberg and Salhi (2005), we also define the fixed cost on the Voronoi edges to be the smaller of the two costs of the adjoining regions. Such a model is applicable if the Voronoi regions represent administrative regions applying different taxes. The customer can be thought of as the “main or the capital city” of the region. We note that our solution algorithm will not need to explicitly construct the Voronoi regions as it merely calculates the fixed cost for a finite number of candidate locations during the search. (Although there are polynomial algorithms to construct the Voronoi diagram but this extra task is not necessary in this particular application.)
Figure 1. An illustration of Voronoi regions for 20 points in the plane
4.3. A Continuous Fixed Cost Function A disadvantage of the previous model is that the fixed cost function is not continuous; a small change in the location (if it takes us over a Voronoi edge) may yield a large change in the establishment cost. Here, we propose a continuous function that can be seen as an extension of the previous model. First, we tessellate the plane into second-order Voronoi polygons (see Preparata and Shamos (1985)), with the customer locations as seed points. A second-order Voronoi polygon
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for seed points i and j consists of all points for which the two nearest seed points are i and j, see Figure 2 for an illustration. Within each region we define the fixed cost function based on the two nearest customers i and j. First, we associate a fixed cost fi with every customer location (just like in the previous model). Then, we define the fixed cost for any point X in the region of i and j as: f (X )
f i d X , a j f j d X , ai d X , a j d X , ai
(6)
We can show that this is a continuous function. Clearly, within each second-order Voronoi region it is continuous; we just need to show continuity on the edges. Crossing a second-order Voronoi edge from the polygon of (i, j) to that of (i, k), d(X, aj) changes to d(X, ak) – but at this point the two are equal to each other, as the edge consists of points equidistant from j and k.
Figure 2. An illustration of second-order Voronoi regions for 7 points in the plane (Numbers denote fixed points, letters denote zones; their correspondence is as follows: A:1,2; B:1,4; C:1,6; D:2,3; E:2,4; F:2,5; G:3,5; H:4,5; I:4,6; J:4,7; K:5,7; L:6,7.)
5. A Constructive Heuristic for the CMSWPF First, we describe the earlier heuristic of Luis et al. (2009) as this is used as a basis of our constructive CMSWPF algorithm. A brief outline of the main idea for solving the CMSWPF is first provided. This is followed by the new modified Cooper’s method, which is used as the local search, and the algorithm.
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5.1. A Region-Rejection Heuristic for the CMSWP (without fixed costs) Luis et al. (2009) put forward a scheme known as region rejection heuristic (RR) to solve the CMSWP. First, a customer site is randomly selected to become a facility location. Then, an area around this location is declared a “restricted region” within which no facilities may be located – this is to ensure facilities are not located too close to each other. Then, another customer is chosen at random from the set of customers not covered by a restricted region, and another restricted region is constructed around it. The process is repeated until the required number of facilities (say M) is located. The authors tested different shapes and sizes for the restricted regions. Best results were found when the shape of the region is a circle and its radius is dynamically adjusted. In brief, the rejection of a region is performed as follow. A potential site whose location is y is rejected if d ( y, X i ) ri X i (i 1,...., M ) where ri is the radius of a circle whose centre is X i . Initially ri is defined as
ri1
M
2 2 Min(a1max a1min , amax amin ) with being an adjustment factor
( 0 1 ). If b Qi b is not satisfied with Qi representing the total demand of those customers in the current circle defined by X i , the radius is updated using ri k ri( k 1) where k refers to the iteration number and
b
Qi
.
A Transportation Problem (TP) is then solved to find the allocation of customers to these M open facilities yielding M subsets of customers, not necessarily disjoint as a customer may be allocated to more than one facility due its demand that could be split. We also note that as the total capacity of n
the facilities Mb may be larger than the total customer demand
w , a dummy customer with a 0 j 1
j
transportation cost will be used; this dummy customer will only contribute to the search at the transportation problem phase and will not be considered at either the location or the allocation phases. For each subset (cluster) of customers, the location of its facility is optimally found using the Weiszfeld algorithm (Weiszfeld and Plastria, 2009). This gives a new set of facilities, for which the new customers’ allocation is determined again by solving the TP.
The location and the
allocation phases are repeated until no improvement is found. This location-allocation phase is also known as the Alternating Transportation-Location (ATL) method of Cooper (1972).
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5.2. The Basic Idea of Extending the Method for CMSWP with Fixed Costs One way of solving the CMSWPF is to evaluate for each value of M starting from 1, the corresponding total cost (transportation and fixed cost), stopping when this cost starts rising as the total cost is a unimodal function of M. The following items are introduced to incorporate the facility fixed cost while speeding up the search so as to avoid unnecessary computations.
(i)
As small values of M may be infeasible, we start the search with a lower bound,
n
j 1
namely LB w j b . (ii)
As we use a heuristic rather than an exact method, we may find a local minimum. In other words, we may end up with situations where the heuristic solution for some value of M is better than for M–1 and M+1, but M is still a poor local minimum. We mitigate this in two different ways. Firstly, we stop our search only if in two successive searches the total cost goes up. Secondly, we implement our heuristics in a multi-start fashion as to increase the chance of avoiding getting trapped in local minima.
(iii)
Once we find the initial solution for M = LB, we construct subsequent solutions by adding one facility at a time to existing solutions using an efficient implementation of the ADD heuristic originally proposed by Kuehn and Hamburger (1963). The solution is then improved in the continuous space through Cooper’s modified approach which we will describe in the following subsection.
5.3. The modified Cooper’s method with fixed cost We describe this procedure for the case of M facilities where for each facility and its customer set the Weiszfeld recursive equations, as given in Equation (7), and the incorporation of fixed cost are considered at each iteration to assess whether or not to accept the new continuous location. The output is the M independent set of allocations, each subset I i consisting of ni fixed points where i 1,..., M and
M
n i 1
i
n . Note that we used ‘’ instead of ‘=’ as some customers may have
their demand split between different facilities during the allocation stage. The iterative procedure which we describe below, is based on the Weiszfeld equations
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wij a1j
X
1( k ) i
d(X jI i
d(X jI i
( k 1) i i j ( k 1) i
wij a 2j
,aj )
and X
w
2( k ) i
,aj )
d(X jI i
d(X jI i
( k 1) i i j ( k 1) i
,aj )
(7)
w
,aj )
where the superscript k is the iteration number within the Weiszfeld iterative procedure;
I i is the set of customers that are served either fully or partially by facility i (i = 1, 2, …, M); k
k
X ik ( X i1 , X i2 ) denotes the new location of the ith facility at iteration k (i = 1, 2, …, M);
wij ( wij w j ) corresponds to all or a fraction of the jth customer demand that is served by facility i (i=1, 2,…, M);
The Iterative Procedure (IP) (i)
At iteration k when using Equations (7), compute the allocation cost for this ith facility
and
add
k
k
the
fixed
cost
that
relates
to
the
location
defined
by X ik ( X i1 , X i2 ) . (ii)
Repeat the above step (i) until there is either (a) less than improvement in the total cost (we set this threshold to 0.01) (b) or when the total cost starts rising instead of decreasing due to a new larger fixed cost in the vicinity.
(iii)
In the case of (a) we retain the last location whereas in (b) we choose the location found before last.
It is also worth noting that in each of these M subsets the capacity is not violated so the total allocation cost is computed as the sum of distances from the new facility to those customers in the subset. This is because the capacity of all the facilities is considered the same, and also the customers’ set remains unchanged. Once all the M locations are determined, a TP is then solved and a new customer allocation is found. The above iterative procedure is performed within each subset. The entire process of `locateallocate’ is repeated until there is no improvement in the total cost. The main steps of Cooper’s modified method are summarised below.
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Cooper’s modified method (1) Choose an initial facility configuration E {X1 , ... , X M } (2) Solve the TP using these M open facilities and record the ni customers served by facility i (i 1,..., M ) with their corresponding demands.
(3) For each open facility i 1,..., M apply the procedure (IP) starting with X i1 X i to find the new X i ( X 1i , X i2 ) . Let E {X1 , ... , X M } be the new facility configuration. (4) If there is no change in the total cost or in the set E, set E* E and stop, else go to (2).
5.4. The proposed Constructive CMSWPF Algorithm The following additional notations are used,
X i : the selected location for the i th facility, RRi : the restricted region around facility i, CT (.) : the total transportation cost of using all the facilities in the set (.), C ( M ) : the overall total cost which includes both the fixed costs and the transportation costs .
The algorithm is given below. Step 1. (Initialisation) Find the solution to the CMSWPF for M LB by using Cooper’s modified method of Subsection 5.3. Record C ( M ) and define E {X1 , ... , X M } as the set of chosen facilities and set R
M
RRi i 1
Step 2. (Generation of the subset of potential sites) Randomly choose a subset of fixed points S { j {1,..., n}: a j R} . Step 3. (Selection of the new fixed point) (a) For each jS, evaluate the incremental cost saving of adding a facility at a fixed point j as j CT ( E ) [CT ( E { j}) f (a j )] . (b) Select the new fixed point as i ArgMax jS ( j ) , set E E {i} and let M M 1 .
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Step 4. (Local Search Phase- Improvement in the continuous space) Use Cooper’s modified method of Subsection 5.3 using the above M locations, record C ( M ) and define E {X1 , ..., X M } .
Step 5. (Construction of the forbidden regions) For each of the open M facilities, construct ( RRi )i 1,...,M and set R
M
RRi . i 1
Step 6. (Testing for local minimum and stopping criterion within a given run) If M = LB+1, return to Step 2. If C(M–1) < C(M–2) or C(M) < C(M–1), return to Step 2. Let M* = M–2, record C(M*) as the cost of the best solution found for this run. Step 7. (Stopping criterion of the whole procedure) Repeat Steps 2 to 6 for K times, report the best overall solution and stop. We would like to note that in step 3, the function i is an approximation for the saving in the total cost as the transportation costs are based on using the new but fixed point i (not a continuous point). To find the value of this saving accurately, we would need to apply the modified Cooper’s method for each potential site, which can be time-consuming.
6. A Guided Hybrid GRASP for the CMSWPF We introduce a GRASP method that nonetheless retains the concept of region-rejection while using the local search of Subsetcion5.3. 6.1. A Guided Hybrid GRASP for the CMSWP (without fixed costs) The greedy randomized adaptive search procedure (GRASP for short) is a multi-start heuristic technique consisting of constructive and local search phases to tackle hard combinatorial optimization problems (see Resende and Ribeiro (2010)). In the first phase of GRASP, known as the construction phase, a feasible initial solution is built one at a time. The construction of these feasible solutions is based on the creation of a restricted candidate list (RCL) made up of good attributes including those of the current best solution. The choice of this list is a crucial issue in the success in using GRASP. This can be based on the size of the set (say the best |RCL| elements are
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selected), or alternatively, it may be attribute-based (i.e., those candidates whose solution quality is better than a certain threshold are chosen). From this set RCL, one element is chosen one at a time either randomly or following a certain selection rule (usually pseudo-randomly) until the full solution is completed. The second phase or the improvement phase is a standard local search applied to explore the neighbourhood of the constructed solution in order to find a better solution. The two phases are reiterated a certain number of times either independently or using some learning scheme and the best overall local optimum is then selected as the final result. Luis et al. (2011) proposed a hybrid heuristic that combines a guided reactive GRASP with region-rejection when constructing RCL. Both size-based and attribute-based aspects are taken into account when creating RCL while considering only those fixed points that lie outside the restricted regions. The local search used was based on Weiszfeld’s equations as given in (7). This procedure iteratively improves the location of a facility in the continuous space. Four different types of GRASP were explored and analysed.
The variant that combined region-rejection with a
dynamically adjusted radius and a learning process within GRASP was found to be the best. Here, we use this variant.
6.2. A Guided Hybrid GRASP for the CMSWP (with fixed costs) Our GRASP algorithm has a similar structure to our constructive heuristic given in the previous section, retaining the ADD greedy procedure and the concept of restricted regions. The differences are only in Steps 1 and 3. In Step 1, we modify the recently developed GRASP method of Luis et al. (2011) to cater for fixed cost when finding the solution for the M facility problem. In Step 3 (b), instead of adding the location from S with the maximum incremental cost saving using i ArgMax jS ( j ) , we add a location pseudo-randomly from RCL as follows.
First,
we
construct
RCL S
by
including
all
elements
jS
such
that
j MinkS (k ) [ MaxkS (k ) MinkS (k )] with ]0,1] . Note that if 1 this reduces to
selecting the facility with the maximum incremental saving (greedy) and if 0 , RCL will contain all the elements of S (fully random). In addition, we also impose that at least K 0 elements of the best solutions with respect to j are part of RCL. The latter condition will guarantee that we have at least r elements to choose from. In other words, we have
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RCL {( j S such that j MinkS (k ) [MaxkS (k ) MinkS (k )]} and | RCL | K0 .
(8)
The size of the candidate list depends on the two parameters namely (0 1) and
K0 (1 K0 | S |) . We then choose pseudo-randomly an element from RCL to be the next location. This is performed as follows: We randomly generate a random number U (0,1) and select i
i RCL such that P(i 1) P(i ) where P(i ) p( j ) represents the cumulative probability *
*
*
j 1
distribution with p( j )
j
kRCL
k
denoting the probability distribution of the j th facility. For
simplicity and for ease of experimental repeatability we sort the elements of RCL in decreasing order of j using i 1 for the one with Max jRCL ( j ) and i | RCL | for the one with Min jRCL ( j ) . The local search in the second phase of GRASP is performed using our modified Cooper’s method as described in the Subsection 5.3. Note that this local search works on the continuous space as we are solving a global optimisation problem instead of a combinatorial optimisation one. This may sound a bit different to the classical local searches that operate on the discrete space such as moving/closing/adding facilities. Once the solution for the M facilities is found, the process continues in the same way as in our constructive heuristic (same steps 5-7).
7. Computational Experiments The proposed methods were coded in C++, compiled with Salford FTN95 compiler, and run on a PC with an Intel 1.5 GHz Pentium M processor and 1.3 GB RAM. We tested our approaches on three classes of instances. To evaluate the performance of our proposed methods, we adapted the four well known data sets from Brimberg et al. (2000) with the addition of facility fixed costs and capacity. These data sets vary from 50 to 1060 customers and use the Euclidean distance measure. As the above data sets do not have a capacity of the facilities, we created the capacity values ourselves; these are given in Table 1. Every facility is set to have the same capacity irrespective of location. We tested our proposed methods using three types of fixed costs. Type I is a constantbased fixed cost where the fixed cost is set to be a constant number. In Type II and Type III, the
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fixed costs at the fixed points were generated from discrete uniform distributions in the range of [2, 8] for 50 fixed points, [50, 400] for 287 fixed points, [1000, 10000] for 654 fixed points, and [10000, 100000] for 1060 fixed points, see Luis (2008) for more details. For each data set, we present three instances for the 50 fixed points and five instances for the other data sets. The instances are available from the authors upon request. We used the following parameter setting. In step 2, we set | S | Max{20, Min(3M ,50)} . This provides a compromise between an excessive use of TPs (note that each addition of a potential facility requires solving a TP) and an appropriate level of solution quality for selecting a smaller
subset of potential sites. In step 7 we set K max 100,5LB, 3 n LB . In Equation (8), we allow the parameter α to vary dynamically as a linear function with a learning process and we let
r Max(5,
M 2
) . This setting was found, after extensive testing, to be promising in Luis et al.
(2011). Tables 1 to 3 summarise the obtained results using the constructive heuristic and GRASP for the three types of fixed cost functions namely constant, zone-based and continuous. ‘Bold’ numbers represent the least costs found. Based on these results, GRASP gives better or equal solutions for all the instances when compared to the constructive heuristic results, which is in line with the findings of Luis et al. (2011) for the case of no fixed costs. In the case of the constant fixed cost, both methods produce similar solutions, see Table 1. For example, when n = 50, all the instances show the same total costs for both methods. This is because the fixed cost is constant and therefore for a fixed value of M, this can be removed from the objective function and added once the M facilities are found. The same process is then repeated for M+1, etc. In fact, both methods produce the same final facility configurations. For the case of the zone-based and the continuous fixed cost, the constructive heuristic is found to be inferior when compared to the results found by GRASP in most instances. For example, in the zone-based case (see Table 2) with n = 287 and b = 1264, GRASP finds M = 12 and a total cost of 8131.69 whereas the constructive heuristic yields a cost of 8413.99, yielding a deviation of about 3.5%. In the continuous case (see Table 3) with n = 654 and b = 66, GRASP produces a total cost of 149183.10 with M = 15 but the constructive heuristic gives a total cost of 155544.30 with M = 16, a deviation of approximately 4.3%. It can be observed that GRASP
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outperforms the constructive heuristic in almost all the instances. However, it is worth noting that the constructive heuristic is relatively faster than GRASP. We can observe that using two, rather than one, increasing moves has shown to be a worthwhile look ahead strategy that helps to avoid poor local minima. For example, let us look at the case of the zone-based fixed cost with n = 654 customers and b = 131, this is given in Table 4 and is also illustrated in Figure 3. We can see that had we stopped at M = 8 because C(8) < C(9), we would have paid a penalty of 34%. This behaviour may obviously not have been presented if an exact method was used instead. One way to reduce the risk of such behaviour even more would be to have three increasing moves as our stopping criterion. Another way would be to re-solve the problems in the neighbourhood of a suspected minimum using a more powerful local search.
8. Conclusions and Future Research This paper studies a new variant of the classical multi-source Weber problem in the presence of capacity restrictions and fixed costs for opening facilities. Two approaches, namely a constructive and GRASP-based heuristics, both using the concept of region-rejection and a new local search are adopted to tackle this difficult location problem on the plane. This local search is an iterative procedure that works on the continuous space contrarily to those standard local searches that are based on discrete moves. A Cooper’s modified method that uses the well known locateallocate mechanism is given. A promising scheme that is based on a look ahead strategy is proposed to reduce the risk of obtaining poor local minima.
Three types of facility fixed costs are
investigated: a constant fixed cost, a Voronoi zone–based and a continuous function based on second-order Voronoi polygons. The proposed schemes were evaluated using some adaptations of well-known instances taken from the MSWP literature.
GRASP has shown empirically to
outperform its counterpart the constructive heuristic by producing results that could be used for future benchmarking.
Research avenues that we consider to be worth investigating in the future include other forms of fixed costs. For instance, we could consider a fixed cost function that contains terms inversely proportional to its distance to the customers as it is usually cheaper to locate facilities further away
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for inhabited areas, the cost of a facility could also be dependent on its throughput. Note that in this study, the capacity at the facilities is considered constant. This could be extended to having various capacities instead but note that such flexibility could become very cumbersome and complex to incorporate. This difficulty arises due to the fact that feasibility could easily be violated from one iteration to the next when applying the Weiszfeld recursive equations. In other words, the capacity of the facility chosen at that particular iteration may not be large enough to cater for all customers in the subset. Some guidance on how to implement these recursive equations while maintaining feasibility, or the development of new ones to maintain feasibility and guarantee convergence is a challenging aspect that is worth the investigation. It may also be interesting to design and analyse other powerful meta-heuristics to tackle the current continuous location problem and its related versions as mentioned above given that these techniques proved to be effective in tackling other hard combinatorial and global optimisation problems.
References Aras, N., Altınel, I. K. and Orbay, M., 2007a. New Heuristic Methods for the Capacitated Multi-Facility Weber Problem. Naval Research Logistics 54, 21-32. 2. Aras, N., Yumusak, S. and Altınel, I. K., 2007b. Solving the Capacitated Multi-Facility Weber Problem by Simulated Annealing, Threshold Accepting and Genetic Algorithms. In Doerner, K. F., Gendreau, M., Greistorfer, P., Gutjahr, W. J., Hartl, R. F. and Reimann, M., editors. Metaheuristics: Progress in Complex Systems Optimization, Springer, 91-112. 3. Aras, N., Orbay, M. and Altinel, I. K., 2008. Efficient Heuristics for the Rectilinear Distance Capacitated Multi-Facility Weber Problem. Journal of the Operational Research Society 59, 64-79. 4. Brimberg, J., Hansen, P., Mladenović, N. and Taillard, E. D., 2000. Improvements and Comparison of Heuristics for Solving the Uncapacitated Multisource Weber Problem. Operations Research 48, 444-460. 5. Brimberg, J., Mladenović, N. and Salhi, S., 2004. The Multi-Source Weber Problem with Constant Opening Cost. Journal of the Operational Research Society 55, 640-646. 6. Brimberg, J. and Salhi, S., 2005. A Continuous Location-Allocation Problem with ZoneDependent Fixed Cost. Annals of Operations Research 136, 99-115. 7. Brimberg, J., Hansen, P., Mladenović, N. and Salhi, S., 2008. A Survey of Solution Methods for the Continuous Location-Allocation Problem. International Journal of Operations Research 5, 1-12. 8. Cooper, L., 1964. Heuristic Methods for Location-Allocation Problems. SIAM Review 6, 37-53. 9. Cooper, L., 1972. The Transportation-Location Problem. Operations Research 20, 94-108. 10. Cooper, L., 1975. The Fixed Charge Problem – I: A New Heuristic Method. Computers and Mathematics with Applications 1, 89-95. 11. Cooper, L. 1976. An Efficient Heuristic Algorithm for the Transportation-Location Problem. Journal of Regional Science 16, 309-3015. 1.
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12. Gong, D., Gen, M., Yamazaki, G. and Xu, W., 1997. Hybrid Evolutionary Method for Capacitated Location-Allocation Problem. Computers and Industrial Engineering 33, 577-580. 13. Kuehn, A. A. and Hamburger, M. J., 1963. A Heuristic Program for Locating Warehouses. Management Science 9, 643-666. 14. Luis, M., 2008. Meta-heuristics for the Capacitated Multi-Source Weber Problem, Ph.D. Thesis, University of Kent, UK. 15. Luis, M., Salhi, S. and Nagy, G., 2009. Region-Rejection Based Heuristics for the Capacitated Multi-Source Weber Problem, Computers and Operations Research 36, 2007-2017. 16. Luis, M., Salhi, S. and Nagy, G., 2011. A Guided Reactive GRASP for the Capacitated MultiSource Weber Problem, Computers and Operations Research 38, 1014-1024. 17. Mohammadi, N., Malek, M. R., and Alesheikh, A. A., 2010. A New GA Based Solution for Capacitated Multi Source Weber Problem, International Journal of Computational Intelligence Systems 3, 514-521. 18. Preparata, F. and Shamos, M. I., 1985. Computational Geometry. Springer, New York. 19. Resende, M. G. C. and Ribeiro, C. G., 2010. Greedy Randomized Adaptive Search Procedures: Advances, Hybridizations and Applications. In Gendreau, M. and Potvin, J-Y., Editors. Handbook of Metaheuristics (2nd Edition), Springer, NY, p. 283-319. 20. Rosing, K. E., 1992. The Optimal Location of Steam Generators in Large Heavy Oil Fields. American Journal of Mathematical and Management Sciences 12, 19-42. 21. Sherali, H. D. and Shetty, C. M., 1977. The Rectilinear Distance Location-Allocation Problem. AIIE Transaction 9, 136-143. 22. Sherali, H. D. and Tuncbilek, C. H., 1992. A Squared-Euclidean Distance Location-Allocation Problem. Naval Research Logistics 39, 447-469. 23. Sherali, H. D., Ramachandran, S. and Kim, S., 1994. A Localization and Reformulation Discrete Programming Approach for the Rectilinear Distance Location-Allocation Problem. Discrete Applied Mathematics 49, 357-378. 24. Sherali, H. D., Al-Loughani, I. and Subramanian, S., 2002. Global Optimization Procedures for the Capacitated Euclidean and lp Distance Multifacility Location-Allocation Problem. Operations Research 50, 433-448. 25. Weiszfeld, E. and Plastria, F., 2009. On the Point for Which the Sum of the Distances to n Given Points is Minimum. Annals of Operations Research 167, 7-41. 26. Zainuddin, Z. M. and Salhi, S., 2007. A Perturbation-Based Heuristic for the Capacitated Multisource Weber Problem. European Journal of Operational Research 179, 1194-1207.
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Table 1. Results for the Case of a Constant Fixed Cost Function Number of Customers (n) 50
287
654
1060
Capacity (b)
Fixed Cost (fi)
10 5 3 1264 632 422 316 211 131 66 33 22 17 212 106 53 36 27
5 3 2 350 250 200 100 50 10000 8000 5000 3000 1000 100000 80000 50000 30000 10000
Smallest # facilities (LB) 5 10 17 5 10 15 20 30 5 10 20 30 40 5 10 20 30 40
Constructive
GRASP
Least Cost using LB
M*
Least Cost using M*
CPU (secs)
Least Cost using LB
M*
Least Cost using M*
CPU (secs)
101.52 78.17 59.81 12235.46 10935.16 10014.11 7745.38 5688.03 371969.69 244716.80 207357.31 168830.49 91359.35 2371544.64 2088424.42 1856628.50 1563406.53 1362247.52
7 12 19 10 14 18 27 38 11 15 24 34 45 8 11 20 30 40
89.47 71.76 59.68 10285.31 9107.84 8738.95 6377.04 4676.11 210132.86 201612.79 181535.31 151644.56 83643.39 2224272.48 2067868.60 1856628.50 1563406.53 1362247.52
3 5 7 115 154 180 587 1333 300 485 1050 2070 4790 355 583 1024 2163 3638
101.52 78.17 59.81 12235.38 10935.00 9995.19 7745.36 5688.03 371969.69 244716.80 207357.31 168830.49 91359.35 2371544.64 2088424.42 1856628.50 1563406.53 1362086.12
7 12 19 10 14 21 28 38 11 15 25 34 45 8 11 20 30 40
89.47 71.76 59.68 10204.84 9061.22 8520.12 6131.37 4580.47 210132.86 201552.52 180527.07 151644.56 83643.39 2224272.48 2067868.60 1856628.50 1563406.53 1362086.12
8 29 93 179 383 1517 4291 11185 563 1381 7264 22204 43276 1134 1105 3526 10345 22661
Table 2. Results for the Case of the Zone-based Fixed Cost Function Number Smallest # of Capacity facilities Customers (b) (LB) (n) 10 5 50 5 10 3 17 1264 5 632 10 287 422 15 316 20 211 30 131 5 66 10 654 33 20 22 30 17 40 212 5 106 10 1060 53 20 36 30 27 40
Constructive
GRASP
Least Cost using LB
M*
Least Cost using M*
CPU (secs)
Least Cost using LB
M*
93.71 90.19 99.45 11735.46 10196.04 9392.15 9315.88 10859.66 347987.58 220716.80 192381.16 212859.47 215268.27 2221544.65 1675180.02 1690831.25 1912378.86 2041853.24
9 12 18 12 13 18 23 31 12 15 23 32 40 9 13 21 32 41
82.05 83.76 98.94 8413.99 8373.43 8692.08 9161.27 10679.37 153443.79 148612.37 176031.92 199178.43 215268.27 1635855.70 1611733.66 1662551.25 1821326.68 2012700.96
4 5 5 189 132 190 292 295 491 516 852 1269 1277 489 1205 1696 5280 6039
93.71 91.10 99.32 11735.46 10195.80 9162.67 9283.84 10771.38 346971.99 220716.80 192381.16 212859.47 209655.23 2221544.65 1675106.66 1569320.06 1897549.25 2041853.24
9 12 19 12 16 20 23 31 14 15 22 32 40 11 12 20 32 41
Least Cost using M*
CPU (secs)
16 81.92 29 83.00 94 98.50 281 8131.69 650 8220.61 8396.55 1226 8990.75 1508 10643.24 2355 147199.56 1319 139552.52 1370 173048.47 3382 197831.42 9934 209655.23 10328 1558082.23 2858 1571835.08 1767 1569320.06 3553 1765815.94 23390 2012700.96 35732
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Table 3. Results for the Case of a Continuous Fixed Cost Function
Number Smallest of Capacity # facilities Customers (b) (LB) (n) 10 5 50 5 10 3 17 1264 5 632 10 287 422 15 316 20 211 30 131 5 66 10 654 33 20 22 30 17 40 212 5 106 10 1060 53 20 36 30 27 40
Constructive
GRASP
Least Cost using LB
M*
Least Cost using M*
CPU (secs)
Least Cost using LB
M*
Least Cost using M*
CPU (secs)
95.05 91.68 97.44 11738.38 10310.82 9542.66 9373.61 10911.57 347582.96 221651.96 209650.61 224259.94 240465.69 2154044.58 1673323.58 1683940.67 1964836.78 2184908.54
8 12 17 13 14 16 23 32 12 16 22 31 40 11 12 20 30 40
81.09 81.77 97.44 8439.36 8591.78 8972.53 9203.13 10699.59 154280.76 155544.30 185436.68 218516.54 240465.69 1600478.55 1613478.77 1683940.67 1964836.78 2184908.54
4 4 5 201 155 104 310 422 499 592 606 866 1349 647 868 521 2162 3646
95.05 91.68 97.33 11738.38 10306.11 9307.46 9348.14 10801.66 346332.01 221651.96 209650.64 224657.43 239132.80 2154044.58 1673323.58 1672401.91 1964836.78 2184908.54
8 13 17 12 13 18 26 32 14 15 23 34 40 11 12 20 30 40
81.09 81.52 97.33 8180.17 8516.38 8571.86 8907.67 10607.88 149929.33 149183.10 182340.27 214481.92 239132.80 1569610.77 1606542.46 1672401.91 1964836.78 2184908.54
13 32 62 294 290 752 2976 3318 1341 1388 4526 16466 10401 2866 1798 3591 10417 22787
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Table 4. An illustration of a poor local minimum: zone-based fixed cost for n = 654 and b = 131
M 5 6 7 8 9 10 11 12 13 14 15 16
Least Total Cost 346971.99 248542.66 221940.87 197533.39 198977.91 162695.53 159132.86 153499.32 149598.38 147199.56 148057.77 149269.27
360000 330000
Total Cost
300000 270000 240000 210000 180000 150000 120000 5
6
7
8
9
10
11
12
13
14
15
16
Number of Facilities (M )
Figure 3. A graphical illustration of a poor local minimum
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