DSpace at VNU: Fair ticket pricing in public transport as a constrained cost allocation game

DSpace at VNU: Fair ticket pricing in public transport as a constrained cost allocation game tài liệu, giáo án, bài giản...

DOI 10.1007/s10479-014-1698-z

Fair ticket pricing in public transport as a constrained

cost allocation game

Ralf Borndörfer · Nam-D˜ung Hoang

Published online: 21 August 2014

© Springer Science+Business Media New York 2014

Abstract Ticket pricing in public transport usually takes a welfare maximization point of

view Such an approach, however, does not consider fairness in the sense that users of a shared infrastructure should pay for the costs that they generate We propose an ansatz to determine fair ticket prices that combines concepts from cooperative game theory and linear and integer programming The ticket pricing problem is considered to be a constrained cost allocation game, which is a generalization of cost allocation games that allows to deal with constraints

on output prices and on the formation of coalitions An application to pricing railway tickets for the intercity network of the Netherlands is presented The results demonstrate that the fairness of prices can be improved substantially in this way

Keywords Constrained cost allocation games· f -Nucleolus · (f, r)-Least core ·

Fair ticket prices

Mathematics Subject Classification 90C90· 91A80 · 91-08

1 Introduction

Public transport ticket prices are well studied in the economic literature on welfare opti-mization as well as in the mathematical optiopti-mization literature on certain network design

A preliminary version of this paper appeared in the Proceedings of HPSC 2009 ( Borndörfer and Hoang

2012 ) This journal article introduces better model and algorithms.

R Borndörfer

Zuse Institute Berlin, Takustr 7, 14195 Berlin, Germany

e-mail: borndoerfer@zib.de

N.-D Hoang (B)

Faculty of Mathematics, Mechanics, and Informatics, Vietnam National University,

334 Nguyen Trai Str., Hanoi, Vietnam

e-mail: hoangnamdung@hus.edu.vn

problems, see, e.g., the literature survey inBorndörfer et al.(2008) To the best of our

knowl-edge, however, the fairness of ticket prices has not been investigated yet The point is that

typical pricing schemes are not related to infrastructure operation costs and, in this sense, favor some users who do not fully pay for the costs they incur For example, we will show in this paper’s example of the Dutch IC railway network that the current distance tariff results

in a situation where some passengers in the central Randstad region of the country pay over

25 % more than the costs they incur, and these excess payments subsidize operations else-where One can argue that this is not fair We therefore ask whether it is possible to construct ticket prices that better reflect operation costs

Ticket pricing can be seen as a cost allocation problem, seeYoung(1994) for a survey/an introduction Cost allocation problems are widespread They come up whenever it is neces-sary or desirable to divide a common cost among several users or items Some examples of

applications where cost allocations have been determined using methods from cooperative game theory are, e.g, aircraft landing fees (Littlechild and Thompson 1977), water resource planning (Straffin and Heaney 1981), water resource development (Young et al 1982), dis-tribution cost of gas and oil transportation (Engevall et al 1998), and investment in electric power (Gately 1974)

The cost allocation problems in the literature are considered as cost allocation games, which require only that the output prices must be non-negative and the total prices that the players are asked to pay have to cover the total costs exactly However, real world applications often have more requirements on the output prices and on the formation of coalitions Our

ticket pricing problem is one example as it stipulates that the ticket price p ACfor a trip from

station A to station C via station B should fulfill the monotonicity conditions

0≤ p AB, pBC ≤ p AC ≤ p AB + p BC,

where p AB and p BC are ticket prices from A to B, and B to C, respectively.

The cooperative game theory literature has already considered games where coalition formation and payoff vectors are required to fulfill additional constraints A very general approach in (Maschler et al 1992) considers (profit allocation) games given by a pair(Π, F),

whereΠ is a topological space and F is a finite set of real and continuous functions defined

onΠ It can be shown that a general nucleolus can be computed using an algorithm that

iteratively shrinks the least core such that the general nucleolus is, in particular, non-empty

under natural conditions In the special case of so-called truncated games F is given by

the coalition excesses and coalition payoffs are constrained by lower bounds This concept

is easily extended to our constrained cost allocation setting, which explicitly allows for arbitrary linear constraints, in order to derive, in particular, the non-emptiness of the general nucleolus We show in this paper how such a computation can be performed efficiently in our constrained cost allocation setting by a cutting plane algorithm The key idea is to combine linear independence and complementary slackness arguments in order to fix crucial coalition excesses and to rule out redundant ones We show that, in this way, large-scale instances

of a ticket pricing constrained cost allocation game can be solved, substantially improving fairness

In this paper, we model ticket pricing as a constrained cost allocation game in order to

deal with pricing constraints We present an( f, r)-least core and argue that the ( f, r)-least

core of this game can be used to determine fair prices The( f, r)-least core can be computed

by solving several linear programs, whose numbers of constraints are exponential in the number of players They can be solved for large-scale instances using a constraint generation approach

The article is structured as follows Section2presents some concepts from cooperative game theory Section3considers several computational approaches in order to calculate game theoretical solutions concepts A model that treats ticket pricing as a constrained cost allocation game is presented in Sect.4 The final Sect.5is devoted to the Dutch IC railway example

2 Game theoretical setting

We present in this section the notion of constrained cost allocation game as a generalization

of the classical cost allocation game It deals with the determination of fair prices subject

to restrictions on the set of possible output prices and on the set of possible coalitions in a way that is very similar to truncated games, seeMaschler et al.(1992) Such a game can be formally defined as follows

We are given a finite set of players N = {1, 2, , n} that can form a family of possible

coalitions  ⊆ 2 N We denote the set of non-empty coalitions by+:= \{∅} and assume

N ∈ +(the grand coalition N is possible), + = {N} (there are non-empty coalitions

other than the grand coalition) For S ⊆ N, let χ S denote the incidence vector of S, i.e., χ i

S

is 1 if i ∈ S and 0 else For a set family Ω ⊆ 2 N, we denoteχ Ω := {χ S | S ∈ Ω} The

familyΩ is called full dimensional if dim span χ Ω = n.

Associated with each coalition S is a cost c (S) ≥ 0 for operating the service on its own,

and a weight f (S) ≥ 0 for averaging purposes, satisfying c(S) > 0 and f (S) > 0 for all

non-empty coalitions S ∈ +; typical weights are f (S) = |S|, f (S) = c(S), or f (S) ≡ 1.

To operate the service together in the grand coalition N , the players will be asked to pay prices in an imputation set

P := {x ∈Rn

≥0| Ax ≤ b},

that is defined by polyhedral constraints Ax ≤ b, x ≥ 0 We assume P to be non-empty and

to imply the cost recovery condition



i ∈N

x i = c(N).

Note that P is then bounded, i.e., a polytope For each imputation x = (x1, x2, , xn) ∈ P

and each non-empty coalition S ∈ +, we define the price of coalition S as x (S): =i∈S x i and the f -excess of S at x as

e f (S, x) := c (S) − x(S)

f (S) .

The f -excess represents the (weighted average) gain (or loss, if it is negative) of coalition S,

if its members accept to pay x (S) instead of operating some service on their own at cost c(S).

The f -excess measures price acceptability: the smaller e f (S, x), the less favorable is price x

for coalition S, and for e f (S, x) < 0, i.e., in case of a loss, x will be seen as unfair by the

members of S The constrained cost allocation game Γ = (N, c, P, ) is to determine a

“fair imputation” x ∈ P of the common cost c(N) among the players in N If  = 2 N

and P = {x ∈Rn

≥0| x(N) = c(N)}, then the constrained cost allocation game reduces to

a (classical) cost allocation game If P = {x ∈Rn

≥0| x(N) = c(N), x(S) ≤ u S ∀S ∈ },

where u∈R∞is a vector of (possible infinite) upper bounds on coalition prices, then it is equivalent to a truncated game

We proceed with game theoretical concepts for constrained cost allocation games

Definition 1 For a constrained cost allocation gameΓ , a weight function f , and ε ∈R, the set

C ε, f (Γ ) :=x ∈ P | e f (S, x) ≥ ε, ∀S ∈ +\{N}

is called the(ε, f )-core of Γ In particular, C0, f (Γ ) is the f -core of Γ The f -least core of

the gameΓ , denoted LC f (Γ ), is the intersection of all non-empty (ε, f )-cores Equivalently,

letε f (Γ ) be the largest ε such that C ε, f (Γ ) is non-empty, i.e.,

ε f (Γ ) = max

x∈P S∈min+\{N} e f (S, x) = max

s.t x (S) + εf (S) ≤ c(S), ∀S ∈ +\{N} (2)

thenLC f (Γ ) = C ε f (Γ ), f (Γ ) In other words, the f -least core is the set of all imputations

x ∈ P that maximize the minimum f -excess over all coalitions in +\{N} The number

ε f (Γ ) is called the f -least core radius of Γ We call the LP (1 3) the f -least core (radius) problem LCP( Γ ) associated with Γ and (2) the coalition constraints.

Obviously there holds the following lemma

Lemma 1 The f -least core of a constrained cost allocation game Γ is non-empty.

Proof The f -least core problem LCP( Γ ) is feasible Indeed, since P = ∅ and +\ {N} is

finite, we can choose some x ∈ P and find ε sufficiently small such that

x (S) + εf (S) ≤ c(S), ∀S ∈ +\{N}.

LCP(Γ ) is also bounded, because

ε ≤ c (S)

f (S) , ∀S ∈ +\{N}.

Therefore, LCP(Γ ) has an optimal solution Let ε∗be the optimal objective value Then

ε∗is the f -least core radius of Γ and the f -least core of Γ is the non-empty set {x ∈

Rn | (x, ε∗) is an optimal solution of LCP(Γ )}.

The f -least core of a constrained cost allocation game Γ may contain, in general, more than

one point However, if the coalition family is full dimensional, uniqueness can be enforced

by imposing a lexicographic order on the f -excesses as follows For each x∈Rn, letθ f (x)

be the vector inR|+|−1whose components are the f -excesses e f (S, x) of S ∈ +\{N},

arranged in increasing order, i.e.,

θ i

f (x) ≤ θ j

f (x), ∀1 ≤ i < j ≤ |+| − 1.

For x , y ∈Rn,θ f (x) is called lexicographically greater than θ f (y), denoted θ f (x)  θ f (y),

if there exists an index i0such that

θ i0

f (x) > θ i0

f (y) and θ i

f (x) = θ i

f (y), ∀i < i0.

In this case, we say that x is a more acceptable price than y We write θ f (x)  θ f (y) if

θ f (x)  θ f (y) or θ f (x) = θ f (y).

Definition 2 The f -nucleolus of a constrained cost allocation game Γ = (N, c, P, ) is

the set

N f (Γ ) := {x ∈ P | θ f (x)θ f (y), ∀y ∈ P}

of all price vectors in P that maximize θ f with respect to the lexicographic ordering

Algorithm 1 computes the f -nucleolus of a constrained cost allocation game Γ = (N, c, , P) It considers a finite sequence of “shrinking subgames” Γk := (N, c, P k,  k+)

ofΓ , k = 1, , k∗, i.e., k+∗ +k∗ −1· · ·+1 = +, P

k∗ ⊆ P k∗ −1⊆ · · · ⊆ P1= P,

and computes their associated f -least core radii which satisfy ε1≤ ε2· · · ≤ ε k∗

The

algo-rithm determines in iteration k a set of coalitions B k with optimal f -excess ε kand fixes their

prices by adding the constraints x (S) = c(S)−ε k f (S), S ∈ B k , to P k The new constraint set

P k+1 also fixes the prices (and excesses) of all coalitions S with linearly dependent incidence

vectors, i.e.,χS ∈ span χ {N}∪k

i=1B i; these are removed from the coalition set+k to produce the next coalition set+k+1 (N is not removed in order to make +k a valid coalition family) This makes the procedure computationally efficient It stops when all prices have been fixed;

P k∗ +1then describes the f -nucleolus of Γ The algorithm thereby maximizes gradually the

attractiveness of a cooperation for all coalitions by improving their prices Algorithm1is a generalization and improvement of the one inHallefjord et al.(1995), which considers the nucleolus of classical cost allocation games

Algorithm 1 Computing the f -nucleolus of Γ = (N, c, P, )

1: k := 1, 1+:= +, P1:= P, F1:= {N}.

2: Solve the f -least core problem (LP k ) associated with the subgame Γ k = (N, c, P k , +k )

max

s.t x (S) + εf (S) ≤ c(S), ∀S ∈ +k \ {N} (5)

x ∈ P k (6) Let(x k , ε k ) and (λ k , μ k ) be primal and dual optimal solutions of (LP k ), where λ kcorresponds to the constraints ( 5 ) andμ kto constraints ( 6 ) Define

Π k := suppλ k = {S ∈  k+\ {N} | λ k > 0}.

3: ChooseB k ⊆ Π ksuch that

F k+1:= F k ∪ B k = {N} ∪

k



i=1

B i

gives rise to a maximal linearly independent setχ F k+1 Define

 k+1+ := {S ∈ +| χ S /∈ span χ F k+1} ∪ {N},

P k+1:= {x ∈ P | x(S) + ε i f (S) = c(S), ∀S ∈ B i , i = 1, , k}.

4: If|F k+1 | < dim span χ then k ← k + 1 and goto 2, else set k∗:= k and stop.

There holds the following result

Theorem 1 Algorithm1 terminates after k∗ steps, k∗ ≤ dim spanχ  − 1, with the f

-nucleolus N f (Γ ) = Pk∗ +1of Γ The f -nucleolus is, in particular, non-empty If the coalition

family  is full dimensional, the f -nucleolus contains a unique point.

Proof The proof proceeds by induction over k We prove the following claims:

1 Γk = (N, c, P k, +

k ), k = 1, , k∗, is a well defined constrained cost allocation game.

2 χS /∈ span χ F k , S ∈  k+\ {N}, k = 1, , k∗+ 1.

3 k≤ |F k | ≤ dim span χ , k = 1, , k∗+ 1

4 ∀k = 1, , k∗: LC f (Γk) ⊆ LC f (Γi ), ∀i : 1 ≤ i ≤ k.

Due to Lemma1, claim 1 implies that the algorithm is well-defined, claim 2 is technical, claim 3 proves that it terminates, and claim 4 helps to show that it produces the correct output

In the base case k = 1, claim 1 holds since N ∈ += {N} and P = ∅, claim 2 as

χ +

1\{N} ∩ span χF1 = χ +\{N} ∩ span 1 = ∅,

claim 3 as|F1| = |{N}| = 1, and claim 4 as k = i = 1.

Now consider the induction step k → k + 1 By the induction hypothesis, Γ k is well defined Then Lemma1implies that(LPk) has optimal primal and dual solutions (x k , ε k )

and(λ k , μ k ); in particular, ε k is the f -least core radius of Γk and x k belongs to the f -least

coreLC f (Γk) of Γk By duality, we have thatλ k≥ 0 and



S ∈ k+\{N}

λ k

S f (S) = 1.

From this it follows thatλ k = 0 and Π k = ∅ As Π k ⊆  k+\ {N}, claim 2 implies

{S ∈ Π k | χ S ∈ span χ F k } = ∅.

Hence we can chooseB k = ∅ and then |F k+1 | > | F k | ≥ k, i.e., | F k+1 | ≥ k + 1 As

F k+1 ⊆ +, we have

k+ 1 ≤ |F k+1 | ≤ dim span χ += dim span χ ,

i.e., claim 3 holds Claim 2 follows directly from the definition of+k+1

We next show claim 1 thatΓk+1is well defined by checking the conditions on k+1+ and

P k+1 If the termination criterion in step 4 is not fulfilled, i.e,|F k+1 | < dim span χ =

dim spanχ +, then the set{S ∈ +| χ S /∈ span χ F k+1} =  k+1+ \ {N} is non-empty and

hence+k+1 = {N}; N ∈  k++1 by definition By complementary slackness, the optimal solution(x k , ε k ) of (LPk) satisfies

x k (S) + ε k f (S) = c(S), ∀S ∈ Πk ,

i.e., x k ∈ P k+1 This proves claim 1 It also provesLC f (Γk) ⊆ Pk+1and henceε k+1≥ ε k

By the induction hypothesis, in order to show claim 4, we only have to prove that

LC f (Γk+1) ⊆ LC f (Γk) Let x be a vector in LC f (Γk+1) Since x k , x ∈ Pk+1, we have that

x k (S) + ε i f (S) = c(S) = x(S) + ε i f (S), ∀S ∈ B i , i = 1, , k,

i.e.,

x k (S) = x(S), ∀S ∈ B i , i = 1, , k.

Since x k (N) = c(N) = x(N), it follows that

x k (S) = x(S), ∀S : χS ∈ spanF k+1,

i.e.,

e f (S, x k ) = e f (S, x), ∀S ∈ +\ +

k+1

From this and since(x k , ε k ) is an optimal solution of (LPk), we have

e f (S, x) = e f (S, x k ) ≥ ε k , ∀S ∈ (+\ +k+1 ) ∩ (+k \ {N}) = +k \  k+1+ .

On the other hand, since x∈LC f (Γk+1 ), there holds

min

S∈+

k+1\{N} e f (S, x) = ε k+1 ≥ ε k

Hence (x, ε k ) is a feasible and therefore an optimal solution of (LPk) for every x ∈

LC f (Γk+1), i.e., LC f (Γk+1) ⊆ LC f (Γk).

k∗≤ dim span χ  − 1 follows from claim 3 by setting k = k∗+ 1 We now prove that

N f (Γ ) = Pk∗ +1 For every y ∈ P k∗ +1, since x k∗ ∈ P k∗ +1, similar to the proof of claim 4, there holds

x k∗(S) = y(S), ∀S : χS ∈ spanF k∗ +1.

SinceF k∗ +1contains dim spanχ linearly independent vectors, we have

{S ∈  | χ S∈ spanF k∗ +1} = .

Therefore

x k∗(S) = y(S), ∀S ∈ 

andθ f (x k∗

) = θ f (y) Hence LC f (Γk∗) = Pk∗ +1and if we can prove thatθ f (x k∗

)  θ f (z)

for every z ∈ P\P k∗ +1 then there holdsN f (Γ ) = Pk∗ +1 Indeed, let z be a vector in

P \P k∗ +1 The proof of claim 1 shows thatLC f (Γk) ⊆ Pk+1 for k = 1, , k∗ Therefore

P \P k∗ +1= P1\P k∗ +1=

k∗



k=1

P k \P k+1⊆

k∗



k=1

P k\LC f (Γk).

Hence, z ∈ P k\LC f (Γk) for some k, 1 ≤ k ≤ k∗ Since z ∈ P k and z∈LC f (Γk )

min

S ∈+k \{N} e f (S, z) = min

S ∈+k \{N}

c (S) − z(S)

On the other hand, due to claim 4, we have x k∗

∈ P k∗ +1=LC f (Γk∗) ⊆ LC f (Γk) and hence

min

S∈+

k \{N} e f (S, x k∗

S∈+

k \{N}

c (S) − x k∗

(S)

Since x k∗, z ∈ Pk, similar to the proof of claim 4, we have that

e f (S, x k∗

From (7)–(9) it follows thatθ f (x k∗

)  θ f (z).

Finally, if is full dimensional, then there holds

|F k∗ +1| = dim span χ = n.

Hence, sinceχ F k∗+1is independent, P k∗ +1contains at most one vector On the other hand,

since x k∗ belongs to P k∗ +1, we have P k∗ +1 = {x k∗

} That means the f -nucleolus of Γ

contains a unique point, namely, x k∗

For the f -nucleolus concept, the coalitions consisting of one player have the same priority

as the other coalitions The role of each individual player is, however, more important The social critic HL Mencken once quipped that, “a wealthy man is one who earns $100 a year more than his wife’s sister’s husband” That means personal objectives are quite local It does not rely on some absolute measure but is relative to what other people have It is very hard

to convince someone that his price is fair, while somebody else has to pay just a fraction of his price for one unit Game theoretical fairness (or coalitional fairness) means that the price should reflect the position of each coalition and its cost by considering all possible groupings Individual fairness tends to equality, i.e., each player has to pay the same amount of money for one unit The( f, r)-least core, which is defined below, is a compromise between coalitional

fairness and individual satisfaction

Let r∈RN

>0be a vector that satisfies



i∈N

r i = c(N).

Vector r is called a reference price-vector of Γ For example for the ticket pricing problem

we can choose r as the distance-price vector The distance-price of a passenger is the product

of the traveling distance and some base-price for a passenger for a distance unit The ratiox i

r i

in this case is nothing else than the ratio between the price that player i is asked to pay for a distance unit and the base-price Each individual player i prefers a small ratio x i

r i A price x i

with a big ratiox i

r i will be seen as unfair by player i , since in this case there exists a player j

with much smaller ratio x r j

j Our goal is to find a price vector in the f -least core of Γ that

is as “near” as possible to r It means that from the point of view of the cooperative game theory our price is fair since it belongs to the f -least core and hence the minimal weighted

benefit of the coalitions in+\{N} is as large as possible On the other hand, from the point

of view of each individual player, the increment of the price of each player in comparison to its reference price is as small as possible Define

and

R : Λ →R>0 , R(N) = c(N) and R({i}) = ri , ∀i ∈ N. (11)

Function R is called a reference price-function of Γ We have then a new constrained cost

allocation game Δ := (N, R, LC f (Γ ), Λ) This is indeed a constrained cost allocation

game sinceLC f (Γ ) is non-empty due to Lemma1and for each x ∈ LC f (Γ ) there holds

x (N) = c(N) = R(N) For each price vector x and player i ∈ R, the R-excess of the

coalition{i} at x is

e R({i}, x) = R ({i}) − xi

R ({i}) = 1 −

x i

r i

Due to Theorem1, the R-nucleolus of Δ is well-defined and contains a unique vector It

maximizesθR ,Δ (x) in LC f (Γ ) with respect to the lexicographic ordering, where θR ,Δ (x)

is the R-excess vector of Δ at x, i.e., the vector in RN whose components are the R-excesses e R({i}, x), i ∈ N, arranged in increasing order Let us define ϑR ,Δ (x) as the vector

inRNwhose components are the ratiosx i

r i , i ∈ N, arranged in decreasing order Then,

equiva-lently, the R-nucleolus of Δ minimizes ϑR ,Δ (x) in LC f (Γ ) with respect to the lexicographic

ordering That means, by using the R-nucleolus of Δ as the price, the ratios x i

r i , i ∈ N, are

kept as small as possible

Definition 3 Given are a constrained cost allocation gameΓ = (N, c, P, ), a weight

func-tion f : +→R>0 , and a reference price-vector r∈RN

>0ofΓ The set Λ and the function R

are defined in (10) and (11), respectively The R-nucleolus of Δ = (N, R, LC f (Γ ), Λ) is

called the( f, r)-least core of Γ , denoted by LC f ,r (Γ ).

Due to Theorem1, there holds the following corollary

Corollary 1 Given a constrained cost allocation game Γ = (N, c, P, ), a weight function

f : + →R>0 , and a reference price-vector r ∈RN

>0 of Γ The ( f, r)-least core of Γ is

well-defined and contains a unique vector.

3 Computational aspects

The f -nucleolus and the ( f, r)-least core belong to the f -least core Finding a vector in

the f -least core of a constrained cost allocation game is NP-hard in general.Faigle et al

(2000) show that computing a vector in the f -least core of min-cost spanning tree games,

which is a special cost allocation game, is NP-hard The biggest challenge is that there is exponential number of possible coalitions This problem can be overcome, however, by using

a constraint generation approach (Hallefjord et al 1995)

In this section, to apply constraint generation approaches, we only consider the so called constrained combinatorial cost allocation games This class is large enough as the cost

func-tion is often given by a minimizafunc-tion problem A constrained combinatorial cost allocafunc-tion game Γ = (N, c, P, ) is a constrained cost allocation game where the cost function c is

given by an optimization problem of the following form

∀S ∈ +: c(S) := min

ξ c ξ

s t Bξ ≥ CχS

D ξ ≥ d

ξ j ∈ Z j , j = 1, 2, , k, (12) whereχS is the incidence vector of S, Z jis the set of either real, or integer, or binary numbers,

and B, C, and D are matrices of suitable dimensions We assume that the weight function f satisfies f = α + β| · | + γ c with α, β, γ ≥ 0 and α + β + γ > 0 Define

Q:=(x, ε) ∈Rn+1 | x(S) + εf (S) ≤ c(S), ∀S ∈ \{∅, N}.

In order to construct the above mentioned constraint generation approach, we firstly consider

the separation problem of Q: Given is a vector ( ¯x, ¯ε), we ask whether ( ¯x, ¯ε) belongs to Q If

the answer is “no”, then find a valid cut that cuts off( ¯x, ¯ε) from Q To do this, we only have

to consider the following optimization problem

max

If the optimal value of (13) is non-positive, then( ¯x, ¯ε) ∈ Q Otherwise, let T be an optimal

solution of (13), then

x (T ) + εf (T ) ≤ c(T )

is a valid cut of Q which cuts off ( ¯x, ¯ε) Due to (12), ifγ ¯ε ≤ 1, then we can rewrite (13) as follows

max

(z,ξ) α¯ε +

i ∈N ( ¯xi + β ¯ε)z i + (γ ¯ε − 1)cξ

s t Bξ ≥ Cz

D ξ ≥ d

ξj ∈ Z j , j = 1, 2, , k,

1≤

i∈N

z i ≤ |N| − 1,

where the variable z represents the incidence vector of the set S in (13)

If = 2 N , then the constraint z ∈ χ  is nothing else than z ∈ {0, 1} N

Based on the above separation problem, we can calculate the( f, r)-least core using the

constraint generation approach presented in Algorithm2

Algorithm 2 Computing the( f, r)-least core of Γ = (N, c, P, )

Given a (small) subsetΩ of  satisfying Ω  N and Ω\{∅, N} = ∅.

1: Compute the f -least core radius ε ΩofΓ Ω := (N, c|Ω , P, Ω) The f -least core of Γ Ωis the following set

LC f (Γ Ω ) =x ∈ X (ΓΩ ) | x(S) ≤ c(S) − ε Ω f (S), ∀S ∈ Ω+\{N},

whereΩ+:= Ω\{∅}.

2: Compute the ( f, r)-least core of Γ Ω , i.e., the R-nucleolus of the constrained cost allocation

game(N, R, LC f (Γ Ω ), Λ), using Algorithm 1 and obtain a vector x Ω.

3: Consider the separation problem

max

S∈+\{N}



If the optimal value is positive, then find a set T in +\{N} that satisfies

x Ω (T ) + ε Ω f (T ) − c(T ) > 0,

setΩ := Ω ∪ {T } and go to 1.

4:{xΩ } is the ( f, r)-least core of Γ

Remark 1 In Algorithm2, sinceε Ω is the f -least core radius of Γ Ω, one can easily prove thatγ ε Ω ≤ 1 and hence we can rewrite (15) as (14) with¯x = x Ωand¯ε = ε Ω

Remark 2 In practice, we do not add only one but several violated coalitions T in each

step Computational results show that our constraint generation approach works well in practice For the IC-ticket-price example in Sect.5, instead of 285it only need to consider

781 coalitions

In the case that there exists a violated coalition T , finding the optimal solution of the

sep-aration problem, i.e., the most violated coalitions, is not necessary and very time-consuming Instead we should be able to quickly find sufficiently good solutions for the separation prob-lem In the following, we consider several heuristics for the separation probprob-lem Heuristics