Proceedings of MAGO 2014, pp. 81 – 84.
82 J. Fernández, A.G. Arrondo, J.L. Redondo, and P.M. Ortigosa franchisor, the new franchisee, and the franchisees of the existing facilities can agree in both location and design for the new facility, taking the corresponding economical implications of their selection into account.
2. The tri-objective model
In the model thedemandis supposed to be fixed and concentrated atndemand points, whose locations pi andbuying power wi are known. The location fj and quality of the existing fa-cilities is also known. Following Huff [3], we consider that demand points split their buying power among all the facilities proportionally to theattractionthey feel for them. The attraction function of a customer towards a given facility depends on the distance between the customer and the facility, as well as on other characteristics of the facility which determine itsquality.
The following notation will be used throughout this paper:
Indices
i index of demand points,i= 1, . . . , n.
j index of existing facilities,j= 1, . . . , m.
Variables
x location of the new facility,x= (x1, x2).
α quality of the new facility (α >0).
Datapi location of thei-th demand point.
wi demand (or buying power) atpi. fj location of thej-th existing facility.
dij distance betweenpiandfj. αij quality offjas perceived bypi.
gi(·) a non-negative non-decreasing function.
αij/gi(dij) attraction thatpifeels forfj.
γi weight for the quality of the new facility as perceived bypi.
k number of existing facilities that are part of the franchise (the firstkof themfacilities are assumed in this category,0< k < m).
Miscellaneous
dix distance betweenpiand the new facilityx.
γiα/gi(dix) attraction thatpifeels forx.
From the previous assumptions, the total market share attracted by the franchisor is
M(x, α) = Xn
i=1
wi γiα gi(dix) +
Xk j=1
αij gi(dij) γiα
gi(dix) + Xm j=1
αij gi(dij)
.
We assume that the operating costs for the franchisor due to the new facility are fixed. In this way, the profit obtained by the franchisor is an increasing function of the market share that it captures. Thus, maximizing the profit obtained by the franchisor is equivalent to maximizing the market share that it captures. This will be the first objective of the problem.
The second objective of the problem is the maximization of the profit obtained by the fran-chisee, to be understood as the difference between the revenues obtained from the market share captured by the new facility minus its operational costs. The market share captured by
A tri-objective model for franchise expansion 83 the new facility (franchisee) is given by
m(x, α) = Xn
i=1
wi
γiα gi(dix) γiα
gi(dix) + Xm j=1
αij gi(dij) and the profit is given by the following expression,
π(x, α) =F(m(x, α))−G(x, α)
whereF(·) is a strictly increasing function which determines the expected sales (i.e., income generated) for a given market share m andG(x, α) is a function which gives the operating cost of a facility located atxwith qualityα. In our computational studies we have considered F to be linear andGto be separable, of the form G(x, α) = G1(x) +G2(α), whereG1(x) = Pn
i=1Φi(dix), withΦi(dix) =wi/((dix)φi0 +φi1), φi0, φi1 >0andG2(α) =eαα0+α1 −eα1, with α0 >0andα1given values (other possible expressions forG(x, α)can be found in [2]).
The owner of the chain should also take into account that some form of competition also exists within the franchise, as expressed by the so-calledcannibalization. When the new facility enters the market, the existing franchise’s facilities might see a decrease of their own market share. To avoid this, the minimization of the cannibalization suffered by those facilities will be considered a third objective of the problem.
The market share captured by the existing facility ℓ ∈ {1, . . . , k}, before the new facility enters the market is given by
msb(ℓ) = Xn i=1
wi αiℓ gi(diℓ) Xm j=1
αij gi(dij)
which is easily seen to be strictly greater than its market shareafterentry given by
msa(ℓ,(x, α)) = Xn i=1
wi
αiℓ gi(diℓ) γiα
gi(dix) + Xm j=1
αij
gi(dij) .
The cannibalization suffered byℓis the difference between these market shares Can(ℓ,(x, α)) =msb(ℓ)−msa(ℓ,(x, α))
and our third objective is to minimize the sum of the cannibalizations suffered by all existing members of the chain,
minCan(x, α) = Xk ℓ=1
Can(ℓ,(x, α)).
The problem considered is
max M(x, α) max π(x, α) min Can(x, α) s.t. dix ≥dmini ∀i
α∈[αmin, αmax] x∈R⊂R2
(1)
84 J. Fernández, A.G. Arrondo, J.L. Redondo, and P.M. Ortigosa where the parametersdmini >0andαmin >0are given thresholds, which guarantee that the new facility is not located over a demand point and that it has a minimum level of quality, respectively. The parameterαmaxis the maximum value that the quality of a facility may take in practice. ByRwe denote the region of the plane where the new facility can be located.
3. Obtaining a discrete approximation of the efficient set
For a majority of multi-objective problems, including location problems, it is not easy to obtain an exact description of the efficient set or Pareto-front, since those sets typically include an infinite number of points (usually a continuum set). That is why authors usually propose to present to the decision-maker a good ‘representative set’ of non-dominated points which suitably represents the whole Pareto-front. By a good representative set we mean a discrete set of points covering the complete Pareto-front and evenly distributed over it.
There is a plethora of metaheuristic methods with that purpose in literature. Nonetheless, the most common approaches utilized in literature is the use of multi-objective evolutionary algorithms (MOEAs). This is due to their ability to find multiple efficient solutions in one single simulation run. The numerous proposed variants have been surveyed, for instance, in [1]. Among them, the algorithms NSGA-II and SPEA2 have been the reference algorithms in the multi-objective evolutionary computation community for years. However, during the last five years, the multi-objective evolutionary algorithm based on decomposition MOEA/D has proved to be superior to other state-of-the-art algorithms (including both NSGA-II and SPEA2) when applied to a wide variety of multi-objective benchmark problems [7].
4. Summary
This paper describes a new tri-objective competitive facility location and design model. The efficiency of the algorithms MOEA/D and FEMOEA [6] (a recent evolutionary algorithm which has been successfully applied to other bi-objective location problems) to generate an effective approximation of the efficient set (and its corresponding Pareto-front) is investigated.
Acknowledgments
This work has been funded by grants from the Spanish Ministry of Economy and Competitive-ness (ECO2011-24927 and TIN2012-37483), Junta de Andalucía (P10-TIC-6002, P11-TIC7176 and P12-TIC301), Fundación Séneca (15254/PI/10), in part financed by the European Regional Development Fund (ERDF). Juana López Redondo is fellow of the Spanish "Juan de la Cierva"
contract program.
References
[1] C.A.C. Coello. Evolutionary multi-objective optimization: a historical view of the field. IEEE Computational Intelligence Magazine, 1(1):28–36, 2006.
[2] J. Fernández, B. Pelegrín, F. Plastria and B. Tóth. Solving a Huff-like competitive location and design model for profit maximization in the plane. European Journal of Operational Research 179:1274-1287, 2007.
[3] D.L. Huff. Defining and estimating a trading area.Journal of Marketing, 28(3):34–38, 1964.
[4] M. Kilkenny and J.F. Thisse. Economics of location: a selective survey. Computers and Operations Research, 26(14):1369–1394, 1999.
[5] F. Plastria. Static competitive facility location: an overview of optimisation approaches. European Journal of Operational Research, 129(3):461–470, 2001.
[6] J.L. Redondo, J. Fernández, J.D. Álvarez, A.G. Arrondo, P.M. Ortigosa. Approximating the Pareto-front of a planar bi-objective competitive facility location and design problem. Computers and Operations Research, doi: 10.1016/j.cor.2014.02.013, to appear.
[7] Q. Zhang and H. Li. Multiobjective optimization problems with complicated Pareto sets, MOEA/D and NSGA-II.IEEE Transactions on Evolutionary Computation, 13(2):284–302, 2009.
Proceedings of MAGO 2014, pp. 85 – 88.
On longest edge division in simplicial branch and bound
∗Juan F. R. Herrera1, Leocadio G. Casado1, and Eligius M. T. Hendrix2
1Informatics Department, Universidad de Almería (ceiA3), Spain, juanfrh@ual.es, leo@ual.es 2Department of Computer Architecture, Universidad de Málaga, Spain, Eligius.Hendrix@wur.nl
Abstract Simplicial partitions are suitable to divide a bounded area in branch and bound. In the iterative refinement process, a popular strategy is to divide simplices by their longest edge, thus avoiding needle-shaped simplices. A range of possibilities arises when the number of longest edges in a simplex is greater than one. The behaviour of the search is different depending on the selected longest edge. In this work, we investigate the importance of the rule to select an edge.
Keywords: longest edge bisection, branching rule, branch and bound, simplex
1. Introduction
Global Optimization pursuits the search of the global optima. Several methods can be used to find the solution. Within deterministic methods, the branch and bound method (B&B) guar-antees to find a global minimum point up to a guaranteed accuracyǫ. This method iteratively divides the search space discards subsets that are proven not to contain anǫglobal solution.
Generally, five rules define the method:
Branching rule: determines how to divide a subproblem into subproblems.
Bounding rule: defines how to obtain upper and/or lower bounds of the subproblem’s solution.
Selection rule: chooses a subproblem among all subproblems stored in a working set.
Rejection rule: discards subproblems which are proven not to contain a global solution.
Termination rule: defines when the given accuracy has been reached. Once a subprob-lem meets this criterion, it is not further divided. Otherwise, it is stored in the working set.
Every B&B rule plays an important role in the efficiency of the algorithm. Careless decisions in one of the rules may lead to inefficient algorithms. This work focuses in the efficiency of the branching rule using longest edge bisection within simplicial B&B optimization methods.
For some problems like mixture design, the search space is a regular simplex. Here, we fo-cus on box-constrained problems, where the search space is ann-dimensional hyper-rectangle that can be partitioned into a set of non-overlappingn-simplices. Ann-simplex is a convex hull ofn+ 1affinely independent vertices.
A recent study shows how the number of generated sub-simplices varies when different heuristics are applied in the iterative bisection of a regular n-simplex [1]. In that study, the complete binary tree is built by bisecting the heuristically-selected longest edge of a sub-simplex until the width, determined by the length of their longest edge, is smaller or equal to a given accuracyǫ. A large reduction in the number of generated sub-simplices can be achieved when heuristics, different from bisecting the first longest edge in terms of vertex indexation (the default method), were used.
∗This work has been funded by grants from the Spanish Ministry (TIN2008-01117 and TIN2012-37483) and Junta de Andalucía (P11-TIC-7176), in part financed by the European Regional Development Fund (ERDF). J.F.R. Herrera is a fellow of the Spanish FPU program.
86 Juan F. R. Herrera, Leocadio G. Casado, and Eligius M. T. Hendrix In this context, we specifically study whether the reduction factor of the search tree is still large when longest-edge selection heuristics are applied to a simplicial B&B optimization method on box-constrained problems, where the initial search region is not a regular simplex and the termination criterion is based on the bounding rule.
Section 2 briefly explains the main features of the used simplicial B&B algorithm. Section 3 describes the studied division heuristics and Section 4 concludes.
2. Simplicial branch and bound method
In this section we cover both the initial space and the rules that define the simplicial B&B method to solve multidimensional global optimization problems.
Initial space
Most B&B methods use hyper-rectangular partitions. However, other types of partitions may be more suitable for some optimization problems. For the use of simplicial partitions, the feasible region is partitioned into simplices. The most preferable initial covering is face-to-face vertex triangulation. It involves partitioning the feasible region into a finite number of n-dimensional simplices with vertices that are also the vertices of the feasible region. A standard method [5] is triangulation into n!simplices. All simplices share the diagonal of the feasible region and have the same hyper-volume. Figure 1 depicts a hypercube of dimension three partitioned into six irregular simplices.
Figure 1: Division of a hypercube into six irregular simplices