Maximal Covering Location Problem with Price Decision for

Revenue Maximization in a Competitive Environment ∗

Frank Plastria & Lieselot VanhaverbekeMOSI - Dept. of Math. , O.R. , Stat. and Inf. Syst. for Management,

Vrije Universiteit BrusselPleinlaan 2, B-1050 Brussels, Belgium

e-mail: {Frank.Plastria, Lieselot.Vanhaverbeke}@vub.ac.be

May 13, 2008

Abstract

In this paper we extend the classical maximal covering model in a competitive envi-ronment by including a price decision. We formulate a revenue maximization model andpropose two procedures to solve it.

By a careful examination of the relationships between the maximal covering problemsfor different prices, we reveal interesting properties of the deduced revenue maximizationmodel, leading to a full enumeration solution approach. With the help of two moreproperties we develop a second, more intelligent solution procedure.

Computational experiments show promising results for a small, medium and large casestudy.

Keywords Competitive Location, Spatial Pricing, Revenue Maximization, Maximal Cov-ering Problem, Mixed Integer Programming.

1 Introduction

A new firm wants to enter a market where other players are already active. Location andprice are to be decided upon. Given that we consider a homogeneous product, we assumethat it is impossible to differentiate on that aspect. The firm’s goal is to maximize its revenuein this competitive environment.

Models for competitive location including price decisions are rather rare. The first analyticalcompetitive location study was done by Hotelling in his seminal paper ‘Stability in Competi-tion’ Hotelling H. (1929). He described the strategies of two competitors in a linear marketwith respect to price and location and he studied equilibrium questions. Several authorsreplied to Hotelling’s work and extended its assumptions in the years after the first publica-tion, for an overview see Eiselt, H.A., Laporte, G., Thisse, J.-F. (1993). Later, Hotelling’s“Principle of Minimum Differentation” was criticized by d’Aspremont et al. d’Aspremont,C., Gabszewicz, J., Thisse, J.-F. (1979). In the seventies of the 20th century, a myriad of∗This research was partially supported by the projects OZR1067 and SEJ2005-06273ECON.

1

location models emerged, both in the fields of spatial economics and industrial organizationand in the fields of operations research, regional science and geography Serra, D., ReVelle, C.(1995). At that time, considerations on the competitive aspect of facility location have beendeveloped following different approaches. Wendell and McKelvey Wendell, R., McKelvey,R. (1981) and also Hakimi Hakimi, S. (1986) looked into the game theoretical approach oflocation problems on networks with location decisions only. They found that a sub-gameperfect Nash equilibrium existed only under specified mathematical conditions. Lederer andThisse Lederer, P., Thisse, J. (1990) came to the same conclusion following a two-stage gameapproach in the more general framework of location and price decisions on a network: firstboth firms choose locations and then they simultaneously decide upon prices. A variationon the two-stage game approach was presented by Labb and Hakimi Labbe, M., Hakimi, S.(1991), with location and quantity decisions to be taken. For a tree representation of themarket, Eiselt and Laporte Eiselt, H.A., Laporte, G. (1993) showed that the existence ofequilibria depends on the distribution of demand in the geographical space if both price andlocation are to be determined (also see Eiselt, H.A. (1992)).

The approach to competitive location modeling of this paper is more in line with theideas at the basis of the PMAXCAP problem. Serra and ReVelle Serra, D., Revelle, C. (1999)formulated PMAXCAP as the maximum capture model extended with a price decision, andpresented a Competitive Price-Location Heuristic to solve the model. In what follows, we willextend the maximum covering model and propose an exact solution algorithm.

We start with the description of the problem and the formulation of the revenue maximiza-tion model in section 2. In section 3 we identify four properties affecting the full enumerationprocedure and show that with two more properties we can improve this solution procedure.Then the two procedures are described in detail in section 4. Computational experience ona small, medium and large case study is presented in section 5. Finally, some remarks andideas for improvement are given.

2 Problem Description

We start from the discrete maximal covering model Church, R., ReVelle,C. (1974), allowingthe new firm to locate stores at a finite number of locations (set J), knowing the fixed locationsof the existing players (set C). The customers (set I) are represented by points, mostlyas result from an aggregation process done beforehand. The standard maximal coveringlocation problem Church, R., ReVelle,C. (1974) maximizes the demand covered [within agiven service distance (dmax)]. [For inessential goods, we take into account a maximum servicedistance. Essential goods will always be purchased, no matter how far the customer musttravel to acquire the good. We will study the latter situation; for the sake of completenesswe mention the maximum travel distance between square brackets, when applicable to theformer situation.] To extend this original model to a revenue maximization model, we needto take into account the price that is charged at the new firm’s outlets.

We integrate the price (p) into the objective function of the maximal covering model as adecision variable. It is clear that we need to define an attraction function in terms of totalpricing, i.e. mill price charged at the outlets and the traditional cost - borne by customers- for transportation between the customers and the outlets. We define dij as the distancebetween customer i and facility j ∈ J∪C and t as the transportation cost per unit of distance.

2

Customers are covered by a specific outlet [when the outlet is located close enough (withinthe given service distance) and] when the total price is the lowest, compared to other existingplayers. A covering variable (xi) is introduced for each customer i, taking the value of 1 whencustomer i is covered by at least one of the outlets of the new firm, or remains 0 if not. Eachcustomer represents a certain amount of demand (qi).

We also assume that the price must cover the costs of the firm: operating with loss is notallowed. That is why we define a minimum price (pmin) that may not be undercut. All outletsof all firms have the same cost structure. This assumption allows us to abstract from thevariable cost, as used in Serra, D., Revelle, C. (1999). Thus, in our case revenue maximizationcorresponds to profit maximization, given the identical cost structure of all outlets.

We obtain the following objective function maximizing the revenue, i.e. the demand coveredmultiplied by the price at the new firm’s outlets:

Π∗ = max∑i∈I

qixip

The total price customer i would have to pay at location j, is the sum of the mill price pcharged by the new player and the transportation cost over the distance travelled by customeri to outlet j. For technical reasons, we assume that customers show novelty behaviour Plastria,F. (2001): in case of a tie the customer patronizes the new player’s outlet. (Co-location isnot explicitly forbidden. A tie may appear when two players choose the same location, orwhen, by accident, the total price for the existing player and new player are equal.) Thus,customer i will patronize a facility at j ∈ J if its total price is the lowest in the market; inother words, if the sum of the price charged by the existing players plus his transportationcost to the existing player’s closest outlet is strictly higher than the total price at the newoutlet, i.e. in case:

p+ tdij ≤ minc∈C

(pc + tdic)

We define pij as the minimum of the total prices of all the existing player outlets forcustomer i, minus the transportation cost for the distance from customer i to new outlet j:

pij = minc∈C

(pc + tdic)− tdij (1)

We refer to pij as the critical price: if the newly set price p undercuts (or is equal to,by novelty orientation) the existing minimum total price pij , customer i will patronize a newoutlet at j. For each customer i we define a patronizing set Ni(p) containing all facilities forwhich the critical price pij is at least p as:

Ni(p) = {j ∈ J : pij ≥ p [and dij ≤ dmax]} (2)

Denote by yj the location variable which is 1 if a new outlet is opened at location j and 0otherwise. At least one facility in the set Ni(p) must be opened for customer i to be covered by

3

the new firm, under price setting p, which is expressed by the well-known covering constraint(see Church, R., ReVelle,C. (1974)): ∑

j∈Ni(p)

yj ≥ xi

The new firm is limited to a number B of new facilities to open:∑j∈J

yj = B (3)

This constraint (3) is easily modified to the budget constraint∑

j∈J fjyj ≤ B, by imple-menting fixed costs fj for opening an outlet at site j, in case this cost aspect is not firm-typical,but defined by the quality of the location.

We obtain the following formulation of the revenue maximization model:

max∑

i∈I qixip (4)s.t.

∑j∈Ni(p) yj ≥ xi ∀i ∈ I (5)∑

j∈J yj = B (6)0 ≤ xi ≤ 1 ∀i ∈ I (7)yj ∈ {0, 1} ∀j ∈ J (8)p ≥ pmin (9)

Observe that this problem is a special case of the heuristically solved problem “PMAXCAP”in Serra, D., Revelle, C. (1999): we do not take into account demand elasticity.

Commercially available software is not suited for solving this problem directly, since it isa typical mixed-integer non-linear problem with variables p and xi in the objective function.We propose a more exact solution procedure of intelligent enumeration. Some characteristicsof the covering problem help to narrow down the solution space and avoid full enumeration.In the next section, we discuss the relevant problem properties.

3 Characteristics of the Problem

We will intensively use the fact that for a fixed p we will always have a maximal coveringproblem (CP). By examining the relationships between the covering problems for differentvalues of p, we reveal a finite domination property with respect to price of the deducedrevenue maximization model, leading to a full enumeration solution approach. A more detailedanalysis is then used to develop a more intelligent, less demanding solution procedure.

4

Given price p, define CP (p) as the maximal covering problem at p:

max∑

i∈I qixi (10)s.t.

∑j∈Ni(p) yj ≥ xi ∀i ∈ I (11)∑

j∈J yj = B (12)0 ≤ xi ≤ 1 ∀i ∈ I (13)yj ∈ {0, 1} ∀j ∈ J (14)

and let CP ∗(p) be its optimal value.

3.1 Properties affecting full enumeration

The following important property of the covering sets allows to see that if the new firm isable to cover a certain amount of demand with price p′, then at least the same amount ofdemand will be covered with a price p smaller than p′.

Property 1 When p < p′ then ∀i ∈ I : Ni(p) ⊃ Ni(p′)

Proof By definition (2): If j ∈ Ni(p′), we have the critical price pij ≥ p′ > p and thereforej ∈ Ni(p). 2

Property 2 When p < p′ then any feasible solution of CP (p′) is feasible for CP (p).

Proof The only differences between problems CP (p) and CP (p′) are the constraints (11).By property 1 we have that

∑j∈Ni(p) yj ≥

∑j∈Ni(p′)

yj , thus whenever∑

j∈Ni(p′)yj ≥ xi

we will also have∑

j∈Ni(p) yj ≥ xi. 2

Property 3 When p < p′ then CP ∗(p) ≥ CP ∗(p′) .

Proof Problems CP (p) and CP (p′) have the same objective. By property 2 the optimalsolution to CP (p′), yielding objective value CP ∗(p′), is feasible for CP (p), from which theresult immediately follows. 2

Let P be the set of all critical prices pij :

P = {pij |(i ∈ I, j ∈ J)}

To simplify matters, we will first assume in what follows that all critical prices pij are pairwisedifferent, and consider P to be ordered increasingly. We will relax this assumption in section3.3.

Property 4 When p < p′ and [p, p′[ ∩ P = ∅, then CP ∗(p) = CP ∗(p′)

Proof Since each pij ∈ P is by assumption either < p or ≥ p′, it follows that Ni(p) = Ni(p′)for all i ∈ I, showing that problems CP (p) and CP (p′) are identical. 2

Figure 1 illustrates property 4: each dot corresponding to a price in P , the half-openintervals between p and p′ with equal CP ∗(p) and CP ∗(p′) are clearly visible when we zoomin on the window of the graphical analysis in figure 2.

5

Figure 1: Property 4.

Considering the master problem’s objective function of revenue maximization, we concludethat in this case we will always prefer the price p′ to price p. Choosing for p′ the lowest criticalprice pij ∈ P larger than p shows that in order to maximize revenue we only need to examineprices in P, i.e. of critical price type pij . In other words we find that an optimal price willbe found by full enumeration of all critical prices p ∈ P , solving each time the correspondingmaximal covering problem CP (p).

In figure 2, the resulting optimal value CP ∗(p) is plotted for all the critical prices in P .We recognize the pattern, illustrated in detail in figure 1. The overall objective functionvalues Π(p) = p.CP ∗(p) for the corresponding critical prices are shown in figure 3.

3.2 Properties affecting intelligent enumeration

Although the properties 1 to 4 yield a finite method (P has cardinality |I|.|J |), it is onlyapplicable for small sized problems. The following properties will allow to reduce the numberof prices to consider.

We first derive a simple bounding rule to predict from the results obtained for one price,that some of the following higher prices in P should not be considered. We assume thatsome feasible revenue value Π∗ has already been obtained. This means that only prices in Pyielding values Π∗(p) exceeding Π∗ remain of interest.

Property 5 Let p be a price such that Π∗(p) < Π∗, then for any price p′ with p < p′ < Π∗

CP ∗(p)

we have Π∗(p′) < Π∗.

Proof For any such p′ we have by property 3 that CP ∗(p) ≥ CP ∗(p′), hence Π∗(p′) =p′.CP ∗(p′) < Π

∗

CP ∗(p)CP∗(p) = Π∗. 2

6

Figure 2: Full enumeration.

Therefore, to find out whether a price increase would raise the optimal revenue founduntil now, an examination of the trade-off between decrease in covered demand and increasein price is recommended. A price that cannot beat the quotient of highest revenue so farand previously analysed covered demand is not worth investigating. The pay-off of the priceincrease would not compensate for the decrease in demand.

This is clearly illustrated in figure 4. We see the line A of equation Π = CP1 ∗ p throughthe origin and the point (p1, CP1 ∗ p1), corresponding to the optimal maxcovering valueCP1 = CP ∗(p1) at price p1, and the horizontal line B of highest profit Π∗ found so far withthe full enumeration procedure. The intersection of both lines indicates the price p2 that isthe next interesting critical price to consider. Property 5 shows that we can safely skip allthe prices in between.

Let us now derive some rules which allow to predict that the optimal solution for one pricein P remains optimal for the next price in P . The following property shows how the coveringsets Ni(p) behave when p increases beyond a prj ∈ P by an amount ε > 0 sufficiently smallso that no other price in P lies strictly between prj and prj + ε.

Property 6 ∀ k in I:

Nk(prj + ε) ={Nk(prj) ∀k 6= rNk(prj) \ {j} ∀k = r

Proof This is an immediate consequence of the assumption about ε and the definition 2 ofNk(p). 2

For the next property we introduce some more notation: We denote by J∗(p) and I∗(p)the set of chosen sites and the set of covered demand points in some optimal solution to themaximal covering problem CP (p). In particular this means

7

Figure 3: Full enumeration.

• i ∈ I∗(p) if and only if J∗(p) ∩Ni(p) 6= ∅

• CP ∗(p) =∑

i∈I∗(p) qi

Property 7 Let p be the next price in P for prj. Then the equality CP ∗(p) = CP ∗(prj)holds in each of the following cases:

1. r /∈ I∗(prj)

2. r ∈ I∗(prj) and j /∈ J∗(prj)

3. r ∈ I∗(prj) and j ∈ J∗(prj) and Nr(prj) ∩ J∗(prj) \ {j} 6= ∅

Proof We consider each case in turn, and show that the optimal solution of CP (prj) definingI∗(prj) and J∗(prj) remains a feasible solution for CP (p). It will follow that CP ∗(p) ≥CP ∗(prj), yielding equality by property 3 since p > prj .

1. case: r /∈ I∗(prj)For any covered i ∈ I∗(prj) we have r 6= i, so by property 6 the covering set of i remainsunchanged: Ni(p) = Ni(prj). Therefore the same set of sites J∗(prj) will still cover alli ∈ I∗(prj) at price p.

2. case: r ∈ I∗(prj) and j /∈ J∗(prj)As above, all covered i ∈ I∗(prj), except possibly r, will remain covered at price p.Customer r is covered at price prj , but since j /∈ J∗(prj), it cannot be covered by j, soit must be covered by some other chosen site j′ ∈ J∗(prj) (j′ 6= j), i.e. j′ ∈ Nr(prj).But by property 6 we then have j′ ∈ Nr(p)∩ J∗(prj), so r also remains covered at pricep.

8

Figure 4: Property 5.

3. case: r ∈ I∗(prj) and j ∈ J∗(prj) and Nr(prj) ∩ J∗(prj) \ {j} 6= ∅As above, all covered i ∈ I∗(prj), except possibly r, will remain covered at price p.Customer r is covered at price prj . By assumption there exists some j′ ∈ Nr(prj) ∩J∗(prj) \ {j}. In particular this means that j′ 6= j and j′ ∈ Nr(prj). By property 6 wethen also have j′ ∈ Nr(p) ∩ J∗(prj), so r remains covered at price p.

2

3.3 The case of non-unique critical prices

In a large data set, it might happen that for two different pairs of (i ∈ I, j ∈ J) the values ofthe critical prices coincide. This affects property 6 and property 7.

Instead of analyzing one element at the time, we need to consider for each p ∈ P the setof elements for which the critical prices are equal. For that purpose, we define the sets I(p)and Ji(p) as:

I(p) = {i ∈ I : ∃j ∈ J : pij = p}

Ji(p) = {j ∈ J : pij = p}

In property 6, we use the set Ji(p) to replace the singleton {j} and we need to comparewith the set of all i ∈ I for which the price under consideration is a critical price.

So we can formulate the following, more general property:

Property 8 Assume ε sufficiently small so that no other price in P lies strictly between pand p + ε

Nk(p+ ε) ={Nk(p) k /∈ I(p)Nk(p) \ Jk(p) k ∈ I(p)

9

The proof for this property remains the same as for property 6.

Concerning property 7, for some price p ∈ P we have to consider all the customers i ∈ I(and sites j ∈ J) for which the critical price under consideration is relevant, i.e. I(p) (andJi(p)) instead of the singleton {r} (and {j}) for which p = prj . For a problem with potentialequal critical prices in P , we should use the following property 9 instead of property 7, theproof of which is totally similar.

Property 9 Let p′ be the next price in P for p. Then the equality CP ∗(p′) = CP ∗(p) holdsin each of the following cases:

1. I(p) ∩ I∗(p) = ∅

2. I(p) ∩ I∗(p) 6= ∅ and ∀i ∈ I(p) ∩ I∗(p) : Ji(p) ∩ J∗(p) = ∅

3. I(p) ∩ I∗(p) 6= ∅ and ∀i ∈ I(p) ∩ I∗(p) : Ni(p) ∩ J∗(p) \ Ji(p) 6= ∅

4 Solution Procedure

We always start by calculating all the prices pij , retaining only those at least equal to pmin,and sort them increasingly in the list P.

4.1 Full enumeration

This method consists in going through the list P , solving CP (p) for each price p, and updatethe best found revenue Π∗ if p.CP ∗(p) is better.

This involves constructing all sets Ni(p0) (i ∈ I) for the first price p0 ∈ P , and thenupdating the sets Ni(p) using property 8 at each move to the next price p ∈ P . For each pone must solve a full maximal covering problem.

This latter step is clearly the most costly, and should better be avoided as much as possible.

4.2 Intelligent enumeration

Given the properties described in the section 3.2 and 3.3, it is safe to skip certain CP opti-mization calls in the enumeration process. For this reason we call this technique ‘intelligentenumeration’.

During the enumeration and each time we solve a maximal covering problem, therebygenerating a new optimal covering for some price, we check if a better revenue has beenfound. If no higher revenue is found, we use property 5 to predict the minimum price weneed to possibly improve the revenue, and we move directly there. If a higher revenue isfound, we attempt to use property 9 to push the price further up, without changing theoptimal covering, yielding higher and higher revenues. In both cases, and when P has notyet been fully considered, we need to solve a new maximal covering problem, repeating thesame process.

More formally, we obtain following algorithm.

InitializationWe initiate the procedure with the lowest p0(≥ pmin) ∈ P for which we construct thecovering sets Ni(p0). Set p = p0 and Π∗ = 0.

10

Repeat

Solve the maximal covering problem CP (p), yielding the value CP ∗ = CP ∗(p).

If p ∗ CP ∗ > Π∗

Then update Π∗ = p.CP ∗, p∗ = p, J∗ = J∗(p) and I∗ = I∗(p),set p to the next price in P and update the sets Ni(p) using property 8.

If p ∗ CP ∗ < Π∗

Then Repeat until p ∗ CP ∗ > Π∗ or end of PSet p to the next price in P and update the sets Ni(p) using property 8.

Else Repeat until property 9 does not hold or end of PUpdate Π∗ = p.CP ∗, p∗ = p, J∗ = J∗(p) and I∗ = I∗(p), set p to the nextprice in P and update the sets Ni(p) using property 8.

until end of P .

5 Computational Results

To test the newly developed procedure, we used three data sets: a small, a medium and anartificial larger one.

We programmed the procedure and model in AMPL (A Modeling Language for Math-ematical Programming, Version 20021038) and solved them with ILOG CPLEX 9.1, usingthe default settings, on a Intel Xeon CPU, 3.4GHz, with 2GB RAM, running Windows XPProfessional SP2.

The test data are available at http://homepages.vub.ac.be/~lvhaverb/testdata.html.

5.1 Small Case

First we used a small case study on which we tried the full enumeration and the intelligentenumeration. We consider the region of the Belgian capital, Brussels. We have 33 customers,6 existing players on the market and the new firm considers 10 locations as possible sites. Thedistances are calculated on the Brussels road network. The prices pc set at the competitor’soutlets c ∈ C are randomly chosen between 55 and 67. The minimum price to be respectedwas set at 25. Customers are supposed to have a transportation cost per unit of distance of0.01. For the demand, we generated random values between 20 and 39.

The full enumeration procedure involved 161 prices and the same number of CP optimiza-tion calls. In table 1, the problem is solved for different values of B. The optimal revenuevalues logically increase with an increase in B, but the average revenue per facility decreases.When we take a closer look at the results in table 1, it appears that the optimal choice forthe price is fairly constant. We recognize the overall pattern of undercutting the competi-tor’s prices. We also visualized the results on a map and found that strong undercutting ofcompetitor’s prices is needed because the spatial configuration does not contain badly servedspatial niches, allowing to charge a somewhat higher price than the common market price.

11

budget price revenue full enum intell enumB p∗ Π∗ time CP opt. calls time CP opt. calls1 33,21 763,55 8s 161 1s 122 39,79 1153,98 8s 161 1s 73 40,65 1219,55 8s 161 1s 184 40,65 1260,21 8s 161 1s 225 52,41 1362,66 8s 161 1s 22

Table 1: Small Case. Sensitivity analysis on parameter B: objective function value and thecorresponding total time (in CPU secs) and CP optimization calls needed for both procedures.

budget price revenue intell enumB p∗ Π∗ time CP opt. calls1 54,72 26815,36 85s 202 54,63 34693,33 67s 183 54,71 43222,48 19s 134 54,71 49952,06 39s 145 54,71 56408,07 19s 14

Table 2: Medium Case. Sensitivity analysis on parameter B: objective function value and thecorresponding total time (in CPU secs) and CP optimization calls needed for the intelligentenumeration procedure.

Using intelligent enumeration, the number of CP optimization calls was reduced to 22, orless (depending on the parameter B). The full enumeration procedure takes eight times longerthan the intelligent enumeration procedure, since less instances of CP (p) are solved. We mayconclude that the intelligent process is much more efficient than a full enumeration.

5.2 Medium Case

In table 2, we give an overview of the results for the medium case study, corresponding to theformer province of Brabant (before 1995). Now we consider 3.337 customers (chosen as sta-tistical sectors), 35 potential sites and 17 existing competitors. The price and transportationcost are as before in the small case. We use the population of a statistical sector as a proxyfor the demand. (More information about the medium case data can be found in Plastria,F., Vanhaverbeke, L. (2007).)

Again we see a substantial decrease in number of optimization calls between the (unper-formed) full enumeration procedure of 14.013 prices and our intelligent enumeration method.Also in this case, the new firm undercuts the competitors’ prices. The difference is smallerthough, probably due to the more populated market and larger distances.

5.3 Large Case

Finally, we also generated a single large, artificial data set. For 1000 customers, x- andy coordinates between 0 and 100 are randomly produced, as is demand (between 20 and50). We positioned 100 potential sites in an inner region of the customer space (coordinates

12

Figure 5: Large Case. Histograms for minimum distance from each customer (a) to closestcompetitor and (b) to closest potential site.

between 15 and 85), to prevent extreme side effects. The 20 competitors are spread over thecustomer space and charge random prices between 6 and 10.

For every customer we calculated the minimum distance to the closest competitor and tothe closest potential site. The histograms representing these data are shown in figure 5.

With the same data set, we varied the transportation cost from 0.5 to 2, and the budgetB from 1 to 10.

Let’s first consider a fixed transportation cost of t = 1.

The resulting cumulative distribution of prices pij is represented in figure 6. The totalnumber of prices resulting from definition 1 is 100.000. However, most of these prices arenegative, due to the large distances considered. When we define a minimum price of 0, weend up with “only” 11.422 prices to examine. Happily, thanks to the solution proceduredescribed above, we do not have to optimize the CP for all of these instances.

Table 3 gives us some interesting information about the performance of the solution pro-cedure for the case of transportation cost equal to 1.

In the column with heading (1), we see the number of times that the CP optimizationcall resulted in an update of the incumbent optimal solution. This information is relevantfor comparison with the total number of CP optimization calls (column with heading ‘CPopt. calls’). It is clear that the procedure deals very efficiently with the time-expensive CPoptimization calls: about half of all CP optimization calls appear to result in a better solution.

The columns with heading (2) and (3) give the number of times the procedure uses theproperties 5 and 8, respectively. The effectiveness of these properties is reflected in the highnumber of steps in which they both are used.

We conclude that the number of CP optimization calls for the full enumeration procedurewould have been 11.422, and that this huge number has been reduced to dramatically fewerCP optimization calls (ranging from 159 to 405, depending on the model parameter B) thanksto the properties proven in this paper.

13

Figure 6: Large Case. Cumulative distribution of all prices pij .

The optimal prices in table 3 show a somewhat strange pattern. The first two prices aresignificantly higher than the others. When we visually represent the optimal solutions, it isclear that one or two new outlets can be located in a spatial niche. The market is not spatiallysaturated and consumers from the badly served area(s) are prepared to pay a higher pricewhen they have to travel shorter distances. When more than two outlets can be opened, itis not possible anymore to solely locate in such badly served areas. The spatial competitionforces the new firm to decide on a price that heavily undercuts the competitor’s prices.

In table 4 the results are shown for the case of a constant budget B equal to 5 and varyingtransportation costs.

In table 4 too, we find an interesting phenomenon. The transport cost that is taken intoaccount for the calculation of the delivered price shows no monotone relation with the optimalmill price. Both the extreme values of transport costs (t = 0.5 and t = 2.0) lead to a highermill price than the unit transport cost (t = 1).

14

budget price revenue intell enum (1) (2) (3)B p∗ Π∗ CP opt. calls time1 10.82 21154 159 55s 65 6945 44122 10.36 38187 195 50s 86 7256 40803 4.97 54889 212 58s 96 7362 39644 5.03 71416 247 77s 114 7349 39595 5.06 84370 291 121s 139 7376 39076 5.03 96050 288 154s 140 7350 39327 5.06 106814 358 130s 172 7369 36818 5.06 116319 392 129s 186 7342 38949 5.06 123975 392 88s 187 7270 3965

10 5.06 129699 405 74s 188 7226 4008

Table 3: Large Case. Sensitivity analysis on parameter B: value of variable p and objectivefunction value, CP optimization calls needed for intelligent enumeration procedure and thecorresponding total time (in CPU secs). (1): the number of times that the CP optimiza-tion call resulted in an update of the incumbent optimal solution (2): number of times theprocedure uses property 5 (3): number of times the procedure uses property 8.

transp. cost price revenue intell enum (1) (2) (3)t p∗ Π∗ CP opt. calls time

0.5 5.52 101771 647 203s 270 12325 85750.8 5.19 90117 455 189s 220 8294 51601.0 5.06 84370 291 121s 139 7376 39071.2 9.43 83949 333 98s 129 6481 34372.0 18.17 99140 530 99s 221 4335 3075

Table 4: Large Case. Sensitivity analysis on parameter t: value of variable p and objectivefunction value, CP optimization calls needed for intelligent enumeration procedure and thecorresponding total time (in CPU secs). (1): the number of times that the CP optimiza-tion call resulted in an update of the incumbent optimal solution (2): number of times theprocedure uses property 5 (3): number of times the procedure uses property 8.

15

6 Conclusions and Extensions

We introduced a revenue maximization model, based on the maximal covering model. Wedescribed four properties of the problem which allowed to obtain exact solution methods.The intelligent enumeration method, maximally exploiting the characteristics of the maximalcovering model, safely skips steps that would have been examined in the full enumeration pro-cedure. The experimental results indicate that the efficiency gained by this method increaseswith the problem size.

As the idea for an intelligent enumeration procedure looks promising, we see some possiblealgorithmic improvements and model extensions:

• We are currently investigating how to modify the trade-off principle of property 5 whenprices are scanned in the reverse direction (from maximum price to minimum price).We expect such a procedure to be slower than the forward scanning procedure proposedin this paper. Therefore research is oriented towards a combined approach.

• In future work, we plan to analyse the impact on the efficiency of the algorithm whenwe use the linear relaxation of the model CP (p). We expect the forecasts of the relaxedmodel solutions to be weaker than the ones obtained from the currently used MIPprograms. This would result in a larger number of CP optimization calls, but the linearprogram should be easier and thus faster to solve. For large data sets this might lowerthe total solution time.

• On a more economic note: we have assumed that the demand is defined in terms offrequency, rather than in terms of volume. This assumption plays a conceptual rolesince customers, located at the same place, might bundle their demand and in thisway decrease their collective transportation cost, which would increase the attractionfunction value. We have used frequency as a criterion for demand in the model, therebyovercoming this phenomenon which is all together seldom adopted by clients in commonretail context on small scale. If we look at a more international context, behaviour ofthat kind is more often observed. The model can be adapted to take into account thismore complicated version of customer behaviour.

• The price setting in a retail outlet does not allow for the very precise figures we proposehere. We are interested to see the impact of rounded prices on the solution of the model.Another option we consider is the use of a selective price choice heuristic. Imagine theretailer to be interested in certain ranges of price, what would be the effect on theoptimal model solution?

• The competitive environment offers the possibility to go into the question of equilibria.The reaction of the existing competitors possibly consists of a price decrease. Locationdecisions are considered to be long term, but prices are much more volatile. Would themarket enter a price war, or would a cooperative equilibrium gradually arise? Furtherresearch on cooperative and non-cooperative equilibria is planned.

• We considered the stipulation of one and the same price for all the newly opened fa-cilities. This single price assumption allowed us to exploit the unidimensionality of theprice-search to improve the performance of the solution procedure. It would be inter-esting to consider different prices for the different facilities in order to obtain a higherrevenue, however this leads to 2 difficulties:

16

– It will be necessary to define rules for the assignment of customers to facilities: acustomer might belong to more than one market area and more specific informationabout the facility choice is then essential to be able to calculate the overall revenue.

– It is not clear how to handle the resulting multidimensionality in price.

References

Church, R., ReVelle,C. 1974. The Maximal Covering Location Problem. Papers inregional science, Vol. 32(1), 101–118.

d’Aspremont, C., Gabszewicz, J., Thisse, J.-F. 1979. On Hotelling’s “Stability inCompetition”. Econometrica, Vol. 47(5), 1145–1150.

Eiselt, H.A. 1992. Hotelling’s Duopoly on a Tree. Annals of operations research, Vol.40(1), 195–207.

Eiselt, H.A., Laporte, G. 1993. The Existence of Equilibria in the 3-facility HotellingModel in a Tree. Transportation science, Vol. 27(1), 39–43.

Eiselt, H.A., Laporte, G., Thisse, J.-F. 1993. Competitive Location Models: A Frame-work and Bibliography. Transportation science, Vol. 27(1), 44–54.

Hakimi, S. 1986. p-median Theorems for Competitive Locations. Annals of operationsresearch, Vol. 6(4), 75–98.

Hotelling H. 1929. Stability in Competition. Economic journal, Vol. 39(153), p. 41–57.

Labbe, M., Hakimi, S. 1991. Market and Location Equilibrium for Two Competitors.Operations research, Vol. 39(5), 749–756.

Lederer, P., Thisse, J. 1990. Competitive Location on Networks under Delivered Pricing.Operations research letters, Vol. 9(3), 147–153.

Plastria, F. 2001. Static Competitive Facility Location: An Overview of OptimisationApproaches. European journal of operational research, Vol. 129(3), 461–470.

Plastria, F., Vanhaverbeke, L. 2007. Aggregation without Loss of Optimality in Com-petitive Location Models. Networks and spatial economics, Vol. 7(1), 3–18.

Serra, D., ReVelle, C. 1995. Facility location. Springer. Chap. Competitive Location inDiscrete Space, pages 337–356.

Serra, D., Revelle, C. 1999. Competitive Location and Pricing on Networks. Geographicalanalysis, Vol. 31, 109–129.

Wendell, R., McKelvey, R. 1981. New Perspectives in Competitive Location Theory.European journal of operations research, Vol. 6(2), 174–182.

17