The stochastic p -median problem with unknown cost probability distribution

Operations Research Letters 37 (2009) 135–141

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Operations Research Letters journal homepage: www.elsevier.com/locate/orl

The stochastic p-median problem with unknown cost probability distribution Roberto Tadei a,∗ , Nicoletta Ricciardi b , Guido Perboli a a

Politecnico di Torino, Turin, Italy

b

‘‘Sapienza’’, Università di Roma, Rome, Italy

article

info

Article history: Received 20 February 2008 Accepted 5 January 2009 Available online 30 January 2009 Keywords: p-median Stochastic costs Extreme value theory Asymptotic approximations Multinomial logit model

a b s t r a c t We want to find the location of p facilities which minimize the expected total cost, when the cost for using a facility is a stochastic variable with unknown probability distribution. Using the method of the asymptotic approximations the expected optimal value of the allocation variables is shown to be a multinomial Logit model. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Given a set of customers, a set of potential facility locations, partitioned into n nonempty disjoint subsets called clusters, and stochastic costs with unknown probability distribution, the Stochastic p-Median Problem with Unknown Cost Probability Distribution (SpMP-UCPD) consists in finding p, with p ≤ n, facility locations, no more than one per cluster, which minimize the expected total cost. The stochastic costs are given by the sum of a deterministic cost associated to each cluster plus a random term, with unknown probability distribution, which represents the cost heterogeneity inside each cluster. A large number of real-life situations can be satisfactorily modeled as SpMP-UCPD. For instance, this problem may arise in marketing, where a producer wishes to bring out a mix of p products which maximizes his profit (see Section 2 for details). Further, it may apply in any context where random utility choices are to be considered (e.g. transport planning, residential choice, service location, etc.). Moreover, the SpMP-UCPD may appear as a subproblem in many approaches for several combinatorial optimization problems, like stochastic scheduling and routing. Owing to that, finding good methods for coping with this problem may help to better solve stochastic scheduling and routing problems. Various results on the p-median problem on stochastic networks are given in the literature [1–5]. When the stochastic network is a tree, Mirchandani and Oudjit [6] use a selective enumeration approach for solving the 2-median problem. The uncertainty in the p-median problem concerns mainly the weights, which are considered as random variables. In particular, there are studies where the probability that the objective function exceeds a given threshold is minimized [7–9], others which analyze either the probability to have particular optimal solutions [10–12] or the uncertainty about future weights [13,14]. A different stream of studies concerns the uncertainty of the locations. In particular, future locations [15] or the number of facilities in the future [16] are considered. In several papers dealing with uncertainty in the p-median problem, assumptions on the type of the cost probability distribution are given (either the probability distribution is known or a finite number of possible states, each occurring with nonzero probability, is assumed). Unfortunately, in many real-life situations the exact shape of this distribution is unknown. Very good surveys of the p-median problem both deterministic and stochastic can be found in [17–19]. For a more general and very recent survey on location analysis see [20]. In a very recent paper dedicated to the memory of Charles ReVelle [21] the authors identify some new fields which they claim to be particularly promising in location theory. They are:

∗

Corresponding author. E-mail addresses: [email protected] (R. Tadei), [email protected] (N. Ricciardi), [email protected] (G. Perboli).

0167-6377/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2009.01.005

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R. Tadei et al. / Operations Research Letters 37 (2009) 135–141

– different structures of customer demand – models ranging from gravity types to Logit functions (which appear to the authors to be most promising in the context of location modeling, see e.g. [22]), and – congestion along the roads and at the facilities. This paper involves both the first and the second field. In fact, it considers a random evaluation of the facilities by the customers (so a different structure of customer demand) proving that, under a quite mild and reasonable assumption on the shape of the unknown cost probability distribution and the number of the potential facility locations, the probability distribution of the minimum cost becomes a Gumbel (or double exponential) one. Moreover, using such a distribution, a multinomial Logit function for the allocation variables (i.e. the variables which allocate demand to supply) of the p-median problem is derived. We observe that the multinomial Logit function has been proved to be particularly suitable to describe the dispersion, due to the cost stochasticity, of the customer preferences among the facilities [23]. The remainder of the paper is organized as follows. In Section 2 the SpMP-UCPD is introduced as a two-stage stochastic programming model with recourse. In Section 3 the stochastic minimum cost for any customer is considered and the formulation of its, still unknown, probability distribution is given. In Section 4 by applying the method of the asymptotic approximations it is proved that the above unknown distribution becomes a Gumbel (or double exponential) distribution. In Section 5 the expected optimal value of the allocation variables is shown to be a multinomial Logit model. An integer deterministic nonlinear problem to find the optimal location of the p facilities is derived in Section 6. 2. Problem definition Let V be the set of customers and U the set of potential facility locations, partitioned into n nonempty disjoint subsets U1 , . . . , Un , called clusters. Let rijk be the stochastic cost that customer i ∈ V pays for using facility j ∈ Uk , k = 1, . . . , n. We assume rijk (θ ) = ck + θij ,

i ∈ V , j ∈ Uk , k = 1, . . . , n,

(1)

i.e., rijk is the sum of ck , a deterministic cost equal for any facility j ∈ Uk (and for any customer i ∈ V ), and θ = [θij ], where θij is a stochastic variable representing the heterogeneity of the cost paid by customer i ∈ V for using any facility j, i.e. a deviation (positive or negative) from the deterministic costs ck . We observe that rijk plays the role of a distance in a p-median problem. In practice, as the number of customers and potential facility locations is usually very high, it is impossible to observe and measure the deviations θij . In a similar situation Francis, Lowe and Tamir [24] would replace demand or supply points by aggregate points. Here, we keep all consumers i (demand points) and facilities j (supply points) as singular points and consider θij as not observable random terms. Because of that, the probability distribution of θij is in general unknown, and any assumption about it would be arbitrary. Given a facility cluster Uk , customer i shows a dispersion of his cost evaluation for any facility j ∈ Uk , due to either his limited knowledge of the actual costs or some idiosyncrasies in his own evaluation. For this reason the random variables θij , which represent such a dispersion, must be assumed independent. Moreover, there is no reason to consider different shapes for the unknown probability distribution of θij . Then, we can assume θij as independent and identically distributed (i.i.d.) random variables with a common unknown probability distribution [25–27] given by F (x) = Pr {θij ≥ x},

i ∈ V , j ∈ U k , k = 1 , . . . , n.

(2)

In order to interpret our problem as a two-stage program with fixed recourse (see e.g. [28,29]), let be: – zijk a continuous decision variable corresponding to the fraction of the demand of customers i ∈ V to facility j ∈ Uk (0 ≤ zijk ≤ 1) – fjk the fixed cost associated with opening facility j ∈ Uk – p the desired number of facilities to be opened – ykj a binary variable which takes value 1 if facility j ∈ Uk is open, 0 otherwise. The variables zijk and ykj are the first-stage decision variables. Let us now introduce the following second-stage decision variables xkij (θ ), such that xkij (θ ) = zijk , ∀θij , i ∈ V , j ∈ Uk , k = 1, . . . , n. Using the above mentioned two-stage program with fixed recourse, the SpMP-UCPD may be formulated as follows min

n X X

" fjk ykj

+ Eθ min

k=1 j∈Uk

n XXX

# rijk

(θ )

zijk

(3)

j∈Uk i∈V k=1

subject to n X X

zijk = 1,

i∈V

(4)

k=1 j∈Uk n X X

ykj = p

(5)

k=1 j∈Uk

X j∈Uk

ykj ≤ 1,

k = 1, . . . , n

(6)

R. Tadei et al. / Operations Research Letters 37 (2009) 135–141

X

zijk ≤ |V | ykj ,

137

j ∈ Uk , k = 1, . . . , n

(7)

i∈V

xkij (θ ) = zijk , zijk

∀θij , i ∈ V , j ∈ Uk , k = 1, . . . , n

, (θ ) ≥ 0,

ykj ∈ {0, 1},

(8)

∀θij , i ∈ V , j ∈ Uk , k = 1, . . . , n

xkij

(9)

j ∈ Uk , k = 1 , . . . , n

(10)

where Eθ denotes the mathematical expectation with respect to θ . The objective function (3) expresses the minimization of the total cost (given by a first-stage plus a second-stage objective function); constraints (4) ensure that the demand of each customer is satisfied; constraint (5) establishes that the number of facilities is p; constraints (6) guarantee that an optimal solution will contain no more than one facility per cluster; constraints (7) establish that no customer can use a not opened facility; constraints (8) tie the revealed customer allocation to facilities (first-stage decision variables zijk ) to the second-stage decision variables xkij (θ ) for any occurrence of the random variables θij ; constraints (9) provide lower bounds on the zijk (and xkij (θ )) variables (it is worth noting that constraints (4) and (9) imply

0 ≤ zijk ≤ 1 (i ∈ V , j ∈ Uk , k = 1, . . . , n)). Finally, constraints (10) are the integrality constraints. Note that constraints (4)–(7) are the first-stage constraints while constraints (8) are the second-stage ones. As stated in the introduction, SpMP-UCPD is able to model several combinatorial optimization problems. In the following, we will give some more examples of possible applications of SpMP-UCPD. Let us consider a set of potential products to be brought out by a producer, in order to maximize his profit. A situation that arises often in practice is when the products can be characterized by a market price level. In this case, a typical marketing issue is to have products differentiated enough in their price. This can be obtained by clustering them according to their market price and imposing to produce at most one product per cluster. The problem of choosing p products, one for each cluster, such that the producer’s profit is maximized can be modeled by (3)–(10), setting ck = −pk and θij = −ηij , where pk is the market price associated to cluster Uk , and ηij represents a deviation from the common price pk . A second possible application arises in the design of transportation systems. In this case, the problem consists in delivering goods to customers, whose exact location inside given clusters is unknown. Examples of this type of problem are present, in particular, in the dispatching of cars in the automotive industry [30] and in the location of intermediate depots in Multi-Echelon Vehicle Routing Problems [31]. 3. Solving the problem Let us assume that an optimal location of the facilities ykj is already given (how to find such an optimal location will be discussed in Section 6). Then, in order to solve problem (3)–(10) we look for the optimal values of the allocation (second-stage decision) variables xkij , k only. It is obvious that the optimal solution xm il , i ∈ V of problem (3)–(10), given an optimal location yj , becomes

( xm il

1,

if rilm =

0,

otherwise.

=

min

ykj =1,j∈Uk ,k=1,...,n

rijk

(11)

Let us first consider the probability distribution of the stochastic costs rijk , i ∈ V , j ∈ Uk , k = 1, . . . , n Pr rijk ≥ x = Pr ck + θij ≥ x = Pr θij ≥ x − ck .













(12)

The minimum cost r i for a customer i is given by

 ri =

min

k=1,...,n

 

min rijk

 = min

k=1,...,n

j∈Uk

min ck + θij





j∈Uk

 = min

k=1,...,n



ck + min θij .

(13)

j∈Uk

Let us define k

θ i = min θij ,

i ∈ V , k = 1, . . . , n.

j∈Uk

(14) k

The unknown probability distribution of the extreme (in this case the minimum) value θ i of the set of random variables θij for any customer i is

n

k

o

Hk (x) = Pr θ i ≥ x .

(15)

For (14), Eq. (13) becomes r i = min

k=1,...,n

n

k

ck + θ i

o

(16)

and its probability distribution G(x) = Pr {r i ≥ x} = Pr

 min

k=1,...,n

n

k

ck + θ i

o

 ≥x .

(17)

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R. Tadei et al. / Operations Research Letters 37 (2009) 135–141

As min

k=1,...,n

n

k

ck + θ i

o

n o k ≥ x ⇐⇒ ck + θ i ≥ x,

∀k = 1, . . . , n

(18)

k

and the random variables θ i are independent, using (15) Eq. (17) gives G(x) = Pr

 min

n

k=1,...,n

ck + θ

k i

o

 ≥x =

n Y

Pr

nn

k

ck + θ i

k=1

o

n n o Y o Y n k Hk (x − ck ). ≥x = Pr θ i ≥ x − ck =

(19)

k=1

k=1 k

In order to determine the unknown distributions Hk , k = 1, . . . , n let us consider that θ i ≥ x ⇐⇒ θij ≥ x, ∀j ∈ Uk . Using (2) one gets

n

o

k

Hk (x) = Pr θ i ≥ x =

Y

Pr θij ≥ x = [F (x)]Qk





(20)

j∈Uk

where Qk = |Uk | . Substituting (20) into (19), one obtains G(x) =

n Y

Hk (x − ck ) =

k=1

n Y

[F (x − ck )]Qk

(21)

k=1

which is the distribution of the minimum cost r i for a customer i as a function of three terms: – the number of facilities in each cluster k, Qk – the deterministic cost, ck – the distribution of the random variables θij , F (x). Unfortunately, the distribution (21) cannot be used as it is because it implies the knowledge of the probability distribution F (x), which is still unknown. In order to solve this problem, the method of the asymptotic approximations derived from the extreme value theory [32] will be used. 4. The asymptotic approximation of the probability distribution of the minimum cost The method of the asymptotic approximations is based on the following observation: if under mild assumptions on the probability distribution F (x) of the random terms θij , the distribution of the stochastic variables rijk (and then of their minimum r i ) tends to a specific functional form as the system becomes large (i.e. when |Uk |, k = 1, . . . , n become large. The concept of ‘‘large’’ depends on the kind of systems we are dealing with and it ranges from some decades, like in marketing, to thousands of units, like in scheduling and routing), then further specific knowledge of the probability distribution F (x) is not needed. The asymptotic theory of extreme values seems particularly appropriate for this kind of problem, as it deals with the asymptotic behavior of maxima and minima of sequences of random variables. Because our problem actually deals with the minimum of a sequence of random variables, some results from that theory can then be used. The only very mild assumption we take for the probability distribution F (x) is that it is asymptotically exponential in its left tail, i.e. there is a constant β > 0 such that 1 − F (x + y)

lim

y→−∞

1 − F (y)

= eβ x , ∀ x ∈ Z .

(22)

Loosely speaking, this assumption states that F (x) acts as an exponential function in its left tail. This property is widely used in the extreme value theory and defines the so-called domain of attraction of the double exponential distribution [32]. Moreover, many probability distributions show such a behavior, among them the following widely used distributions: Gumbel, Logistic, Gamma, and Generalized Exponential. It will be proved that under assumption (22) F (x) tends to a specific functional form as the number of potential facility locations in each cluster becomes large. Consider the following simple but very important remark. If an arbitrary constant is added to the random variables θij (and then to their k

minima θ i , k = 1, . . . , n), this leaves the solution of problem (3)–(10) unchanged. As a specific choice, let us use the constant aQk root of the equation 1 − F (aQk ) = 1/Qk . k

(23)

k

Replacing θ i with θ i − aQk in (19) one has from (21) G(x | Qk ) =

n Y 

F (x − ck + aQk )

Qk

.

k=1

Let us assume that Qk , k = 1, . . . , n are large enough to use limQk →∞ G(x | Qk ) as an approximation of G(x).

(24)

R. Tadei et al. / Operations Research Letters 37 (2009) 135–141

139

In order to obtain an explicit form of the probability distribution G(x) consider the following. Theorem 1. Under assumption (22) G(x) = lim G(x | Qk ) = exp −Aeβ x ,




(25)

Qk →∞

where A=

n X

e−β ck .

(26)

k=1

Proof. By (24) one has n Y 

G(x) = lim G(x | Qk ) = lim Qk →∞

Qk →∞

F x − ck + aQk


Qk

.

(27)

k=1

As limQk →∞ ak = −∞, from assumption (22) one obtains lim

1 − F (x − ck + aQk )

= eβ(x−ck ) .

1 − F (aQk )

Qk →∞

(28)

By (23) and (28) one has lim F (x − ck + aQk ) = lim

Qk →∞



Qk →∞

1−

eβ(x−ck )

 (29)

Qk

and lim



Qk →∞

Qk

F (x − ck + aQk )

 = lim

Qk →∞

1−

eβ(x−ck )

Qk

Qk


= exp −eβ(x−ck ) .

(30)

Substituting (30) into (27) one gets G(x) =

n Y

exp −eβ(x−ck ) = exp −Aeβ x . 







(31)

k=1

A few words of comment on Theorem 1 are worthwhile. The main assumption is that F (x) is an exponential function in its left tail. The second important assumption, which can be considered true in many applications of our concern, is that Qk , k = 1, . . . , n are quite large, in order to justify the asymptotic approximation for G(x) (of course, the larger are the clusters Uk the more correct is the use of an asymptotic approximation for G(x)). The two above assumptions allow to calculate G(x) as the asymptotic form of the probability distribution of the extreme value (the minimum cost in our problem) of a large set of i.i.d. random variables, which is consistent with the objective of using those facilities which minimize the expected total cost. It is interesting to observe that consequently G(x) becomes a Gumble (or double exponential) distribution [33], which will imply for the allocation variables xkij of our problem a multinomial Logit form, as it will be shown in Section 5. 5. Finding the expected optimal value of the allocation variables The expected value of the minimum cost r i for a customer i becomes

e ri = E ( r i ) =

Z

+∞

xdG(x) = − −∞

Z

+∞

x exp −Aeβ x Aeβ x β dx.




(32)

−∞

Substituting for t = Aeβ x one gets

e ri = −1/β

Z

= −1/β

Z

+∞

log(t /A)e−t dt 0

+∞

e

−t

log tdt + 1/β log A

0

+∞

Z

e−t dt 0

= γ /β + 1/β log A = 1/β(γ + log A) R +∞ −t where γ = − 0 e log t dt ' 0.5772 is the Euler constant.

(33)

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R. Tadei et al. / Operations Research Letters 37 (2009) 135–141

By (33) and (26), the expected value of the minimum total cost e R becomes

X

e R =

e ri =

X

i∈V

1/β(γ + log A)

i∈V

= |V | 1/β γ + log

n X

! e

−β ck

.

(34)

k=1

We are now interested in finding the expected value e xkij∗ of the optimal solution xkij∗ of the SpMP-UCPD (3)–(10). The following theorem holds Theorem 2. For any given solution ykj of problem (3)–(10), the expected value e xkij∗ of the optimal solution xkij∗ is e−β ck

e xkij∗ =

,

n

P

i ∈ V,

k = 1, . . . , n/∃j ∈ Uk , ykj = 1.

(35)

e−β cl

l=1,l/∃j∈Ul ,ylj =1

Proof. From the Total Probability Theorem [34] and assumptions (22), one obtains

e xkij∗ =

+∞

Z

Pr {x < ck ≤ x + dx} Pr {cs > x, ∀s 6= k}

−∞ Z +∞

=

β eβ(x−ck ) exp(−eβ x A)dx

−∞ +∞

Z

= e−β ck

e−At dt = 0

=

e−β ck A

e−β ck n P

e−β cl

l =1

where t = eβ x .



Let us observe that the formulation of e xkij∗ in (35) represents a multinomial Logit model. This model is widely used in choice theory and particularly suitable to describe the dispersion of customer preferences (due in our case to the cost stochasticity) among different alternatives (the facilities). Moreover, it is interesting to note that in the literature [23] the multinomial Logit model is usually derived by assuming for the probability distribution of the random terms the following Gumbel (or double exponential) distribution F (x) = exp −eβ x .




(36)

Here, vice versa, we have decided, as F (x) may not be observable, not to hypothesize any specific form for it, but only its asymptotic exponential behavior (see (22)), and assume (which is often true) that the number of alternatives in any cluster is large enough to allow the use of the asymptotic approximations. As a result we get the distribution (25) for G(x), which is actually a Gumbel distribution, but obtained a posteriori by Theorem 1. 6. Finding the optimal solution of the location variables Until now, we have assumed that an optimal location of the facilities ykj in problem (3)–(10) is given. Now, we drop this assumption and look for such an optimal location. As (35) is still valid, problem (3)–(10) has to be solved in ykj , only. It is easy to see that, using (34), the following integer deterministic nonlinear problem holds min n o ykj

n X X

fjk ykj

+ |V | 1/β γ + log

k=1 j∈Uk

n X

! ykj e−β ck

(37)

k=1

subject to n X X

ykj = p

(38)

k=1 j∈Uk

X

ykj ≤ 1,

k = 1, . . . , n

(39)

j ∈ Uk , k = 1, . . . , n.

(40)

j∈Uk

ykj ∈ {0, 1},

R. Tadei et al. / Operations Research Letters 37 (2009) 135–141

141

By solving problem (37)–(40), a p-median optimal solution ykj ∗ and the corresponding expected value of the minimum total cost can be obtained. It is easy to prove, e.g. by reduction from the Dominating Set problem, that problem (37)–(40) is NP-complete. Because of that, for solving real-life instances of this problem in a reasonable computational time, efficient heuristics are required. A particular efficient and effective one, developed for a similar problem, can be found in [35]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

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