Economic and Operations Research Situations with Interval Data
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative Game Theory. Operations Research Games. Applications to Interval Games Lecture 5: Economic and Operations Research Situations with Interval Data Sırma Zeynep Alparslan G¨k o S¨leyman Demirel University u Faculty of Arts and Sciences Department of Mathematics Isparta, Turkey email:zeynepalparslan@yahoo.com August 13-16, 2011
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011Outline Introduction Cooperative interval games Classes of cooperative interval games Economic situations with interval data Operations Research situations with interval data References
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionIntroduction This lecture is based on Cooperative interval games by Alparslan G¨k which was the PhD Dissertation thesis from Middle East o Technical University. The thesis is also published as a book by Lambert Academic Publishing (LAP) Cooperative Interval Games: Theory and Applications For more information please see: http://www.morebooks.de/store/gb/book/cooperative- interval-games/isbn/978-3-8383-3430-1
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionIntroduction This lecture is also based on the papers Big boss interval games by Alparslan G¨k, Branzei and Tijs o which was published in International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems (IJUFKS), Airport interval games and their Shapley value by Alparslan G¨k, Branzei and Tijs which was published in Operations o Research and Decisions, Bankruptcy problems with interval uncertainty by Branzei and Alparslan G¨k which was published in Economics Bulletin and o Sequencing interval situations and related games by Alparslan G¨k et al. which will appear in Central European Journal of o Operations Research (CEJOR).
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionMotivation Game theory: Mathematical theory dealing with models of conﬂict and cooperation. Many interactions with economics and with other areas such as Operations Research (OR) and social sciences. Tries to come up with fair divisions. A young ﬁeld of study: The start is considered to be the book Theory of Games and Economic Behaviour by von Neumann and Morgernstern (1944). Two parts: non-cooperative and cooperative.
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionMotivation continued... Cooperative game theory deals with coalitions which coordinate their actions and pool their winnings. The main problem: How to divide the rewards or costs among the members of the formed coalition? Generally, the situations are considered from a deterministic point of view. Basic models in which probability and stochastic theory play a role are: chance-constrained games and cooperative games with stochastic/random payoﬀs.
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionMotivation continued... Idea of interval approach: In most economic and OR situations rewards/costs are not precise. Possible to estimate the intervals to which rewards/costs belong. Why cooperative interval games are important? Useful for modeling real-life situations. Aim: generalize the classical theory to intervals and apply it to economic situations and OR situations. In this study, rewards/costs taken into account are not random variables, but just closed and bounded intervals of real numbers with no probability distribution attached.
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionInterval calculus I (R): the set of all closed and bounded intervals in R I , J ∈ I (R), I = I , I , J = J, J , |I | = I − I , α ∈ R+ addition: I + J = I + J, I + J multiplication: αI = αI , αI subtraction: deﬁned only if |I | ≥ |J| I − J = I − J, I − J weakly better than: I J if and only if I ≥ J and I ≥ J I J if and only if I ≤ J and I ≤ J better than: I J if and only if I J and I = J I J if and only if I J and I = J
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative interval gamesClassical cooperative games versus cooperative intervalgames < N, v >, N := {1, 2, ..., n}: set of players v : 2N → R: characteristic function, v (∅) = 0 v (S): worth (or value) of coalition S G N : the class of all coalitional games with player set N < N, w >, N: set of players w : 2N → I (R): characteristic function, w (∅) = [0, 0] w (S) = [w (S), w (S)]: worth (value) of S IG N :the class of all interval games with player set N Example (LLR-game): Let < N, w > be an interval game with w ({1, 3}) = w ({2, 3}) = w (N) = J [0, 0] and w (S) = [0, 0] otherwise.
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative interval gamesArithmetic of interval games w1 , w2 ∈ IG N , λ ∈ R+ , for each S ∈ 2N w1 w2 if w1 (S) w2 (S) < N, w1 + w2 > is deﬁned by (w1 + w2 )(S) = w1 (S) + w2 (S). < N, λw > is deﬁned by (λw )(S) = λ · w (S). Let w1 , w2 ∈ IG N such that |w1 (S)| ≥ |w2 (S)| for each S ∈ 2N . Then < N, w1 − w2 > is deﬁned by (w1 − w2 )(S) = w1 (S) − w2 (S). Classical cooperative games associated with < N, w > Border games: < N, w > and < N, w > Length game: < N, |w | >, where |w | (S) = w (S) − w (S) for each S ∈ 2N .
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative interval gamesPreliminaries on classical cooperative games < N, v > is called a balanced game if for each balanced map λ : 2N {∅} → R+ we have λ(S)v (S) ≤ v (N). S∈2N {∅} The core (Gillies (1959)) C (v ) of v ∈ G N is deﬁned by C (v ) = x ∈ RN | xi = v (N); xi ≥ v (S), ∀S ∈ 2N . i∈N i∈S Theorem (Bondareva (1963), Shapley (1967)): Let < N, v > be an n-person game. Then, the following two assertions are equivalent: (i) C (v ) = ∅. (ii) < N, v > is a balanced game.
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative interval gamesInterval solution concepts I (R)N : set of all n-dimensional vectors with elements in I (R). The interval imputation set: I(w ) = (I1 , . . . , In ) ∈ I (R)N | Ii = w (N), Ii w (i), ∀i ∈ N . i∈N The interval core: C(w ) = (I1 , . . . , In ) ∈ I(w )| Ii w (S), ∀S ∈ 2N {∅} . i∈S Example (LLR-game) continuation: C(w ) = (I1 , I2 , I3 )| Ii = J, Ii w (S) , i∈N i∈S C(w ) = {([0, 0], [0, 0], J)} .
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesClassical cooperative games (Part I in Branzei, Dimitrovand Tijs (2008)) < N, v > is convex if and only if the supermodularity condition v (S ∪ T ) + v (S ∩ T ) ≥ v (S) + v (T ) for each S, T ∈ 2N holds. < N, v > is concave if and only if the submodularity condition v (S ∪ T ) + v (S ∩ T ) ≤ v (S) + v (T ) for each S, T ∈ 2N holds.
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesConvex and concave interval games < N, w > is supermodular if w (S) + w (T ) w (S ∪ T ) + w (S ∩ T ) for all S, T ∈ 2N . < N, w > is convex if w ∈ IG N is supermodular and |w | ∈ G N is supermodular (or convex). < N, w > is submodular if w (S) + w (T ) w (S ∪ T ) + w (S ∩ T ) for all S, T ∈ 2N . < N, w > is concave if w ∈ IG N is submodular and |w | ∈ G N is submodular (or concave).
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesSize monotonic interval games < N, w > is size monotonic if < N, |w | > is monotonic, i.e., |w | (S) ≤ |w | (T ) for all S, T ∈ 2N with S ⊂ T . SMIG N : the class of size monotonic interval games with player set N. For size monotonic games, w (T ) − w (S) is deﬁned for all S, T ∈ 2N with S ⊂ T . CIG N : the class of convex interval games with player set N. CIG N ⊂ SMIG N because < N, |w | > is supermodular implies that < N, |w | > is monotonic.
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesI-balanced interval games < N, w > is I-balanced if for each balanced map λ λS w (S) w (N). S∈2N {∅} IBIG N : class of interval balanced games with player set N. CIG N ⊂ IBIG N CIG N ⊂ (SMIG N ∩ IBIG N ) Theorem: Let w ∈ IG N . Then the following two assertions are equivalent: (i) C(w ) = ∅. (ii) The game w is I-balanced.
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesSolution concepts for cooperative interval games Π(N): set of permutations, σ : N → N, of N Pσ (i) = r ∈ N|σ −1 (r ) < σ −1 (i) : set of predecessors of i in σ The interval marginal vector mσ (w ) of w ∈ SMIG N w.r.t. σ: miσ (w ) = w (Pσ (i) ∪ {i}) − w (Pσ (i)) for each i ∈ N. Interval Weber set W : SMIG N I (R)N : W(w ) = conv {mσ (w )|σ ∈ Π(N)} .
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesThe interval Shapley value The interval Shapley value Φ : SMIG N → I (R)N : 1 Φ(w ) = mσ (w ), for each w ∈ SMIG N . n! σ∈Π(N) Example: N = {1, 2}, w (1) = [0, 1], w (2) = [0, 2], w (1, 2) = [4, 8]. 1 Φ(w ) = (m(12) (w ) + m(21) (w )); 2 1 Φ(w ) = ((w (1), w (1, 2) − w (1)) + (w (1, 2) − w (2), w (2))) ; 2 1 1 1 Φ(w ) = (([0, 1], [4, 7]) + ([4, 6], [0, 2])) = ([2, 3 ], [2, 4 ]). 2 2 2
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Economic situations with interval dataClassical big boss games versus big boss interval games Classical big boss games (Muto et al. (1988), Tijs (1990)): < N, v > is a big boss game with n as big boss if (i) v ∈ G N is monotonic, i.e., v (S) ≤ v (T ) if for each S, T ∈ 2N with S ⊂ T ; (ii) v (S) = 0 if n ∈ S; / (iii) v (N) − v (S) ≥ i∈NS (v (N) − v (N {i})) for all S, T with n ∈ S ⊂ N. Big boss interval games: < N, w > is a big boss interval game if < N, w > and < N, w − w > are classical (total) big boss games. BBIG N : the class of big boss interval games. Marginal contribution of each player i ∈ N to the grand coalition: Mi (w ) := w (N) − w (N {i}).
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Economic situations with interval dataProperties of big boss interval games Theorem: Let w ∈ SMIG N . Then, the following conditions are equivalent: (i) w ∈ BBIG N . (ii) < N, w > satisﬁes (a) Veto power property: w (S) = [0, 0] for each S ∈ 2N with n ∈ S. / (b) Monotonicity property: w (S) w (T ) for each S, T ∈ 2N with n ∈ S ⊂ T . (c) Union property: w (N) − w (S) (w (N) − w (N {i})) i∈NS for all S with n ∈ S ⊂ N.
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Economic situations with interval dataT -value (inspired by Tijs(1981)) the big boss interval point: B(w ) := ([0, 0], . . . , [0, 0], w (N)); the union interval point: n−1 U(w ) := (M1 (w ), . . . , Mn−1 (w ), w (N) − Mi (w )). i=1 The T -value T : BBIG N → I (R)N is deﬁned by 1 T (w ) := (U(w ) + B(w )). 2
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Economic situations with interval dataHolding situations with interval data Holding situations: one agent has a storage capacity and other agents have goods to store to generate beneﬁts. In classical cooperative game theory, holding situations are modeled by using big boss games (Tijs, Meca and L´pez (2005)). o For a holding situation with interval data one can construct a holding interval game which turns out to be a big boss interval game. Example: Player 3 is the owner of a holding house which has capacity for one container. Players 1 and 2 have each one container which they want to store. If player 1 is allowed to store his/her container, then the beneﬁt belongs to [10, 30] and if player 2 is allowed to store his/her container, then the beneﬁt belongs to [50, 70].
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Economic situations with interval dataExample continues ... The situation described corresponds to an interval game as follows: The interval game < N, w > with N = {1, 2, 3} and w (S) = [0, 0] if 3 ∈ S, w (∅) = w (3) = [0, 0], / w (1, 3) = [10, 30] and w (N) = w (2, 3) = [50, 70] is a big boss interval game with player 3 as big boss. B(w ) = ([0, 0], [0, 0], [50, 70]) and U(w ) = ([0, 0], [40, 40], [10, 30]) are the elements of the interval core. T (w ) = ([0, 0], [20, 20], [30, 50]) ∈ C(w ).
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Operations Research situations with interval dataAirport situations with interval data In airport situations, the costs of the coalitions are considered (Driessen (1988)): One runway and m types of planes (P1 , . . . , Pm pieces of the runway: P1 for type 1, P1 and P2 for type 2, etc.). Tj [0, 0]: the interval cost of piece Pj . Nj : the set of players who own a plane of type j. nj : the number of (owners of) planes of type j. < N, d > is given by N = ∪m Nj : the set of all users of the runway; j=1 d(∅) = [0, 0], d(S) = j Ti i=1 if S ∩ Nj = ∅, S ∩ Nk = ∅ for all j + 1 ≤ k ≤ m. S needs the pieces P1 , . . . , Pj of the runway. The interval cost of the used pieces of the runway is j Ti . i=1
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Operations Research situations with interval dataAirport situations with interval data m ∗ Formally, d = k=1 Tk u∪m Nr , where r =k ∗ 1, K ∩ S = ∅ uK (S) := 0, otherwise is the dual unanimity game. Interval Baker-Thompson allocation for a player i of type j: j m γi := ( nr )−1 Tk . k=1 r =k Proposition: Interval Baker-Thompson allocation agrees with the interval Shapley value Φ(d).
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Operations Research situations with interval dataAirport situations with interval data Proposition: Let < N, d > be an airport interval game. Then, < N, d > is concave. Proof: It is well known that non-negative multiples of classical dual unanimity games are concave (or submodular). By formal deﬁnition of d the classical games d = m T k uk,m and k=1 ∗ m ∗ |d| = k=1 |Tk | uk,m are concave because T k ≥ 0 and |Tk | ≥ 0 for each k, implying that < N, d > is concave. Proposition: Let (N, (Tk )k=1,...,m ) be an airport situation with interval data and < N, d > be the related airport interval game. Then, the interval Baker-Thompson rule applied to this airport situation provides an allocation which belongs to C(d).
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Operations Research situations with interval dataExample: < N, d > airport interval game interval costs: T1 = [4, 6], T2 = [1, 8], d(∅) = [0, 0], d(1) = [4, 6], d(2) = d(1, 2) = [4, 6] + [1, 8] = [5, 14], ∗ ∗ d = [4, 6]u{1,2} + [1, 8]u{2} , Φ(d) = ( 1 ([4, 6] + [0, 0]), 2 ([1, 8] + [5, 14])) = ([2, 3], [3, 11]), 2 1 1 1 γ = ( 2 [4, 6], 2 [4, 6] + [1, 8]) = ([2, 3], [3, 11]) ∈ C(d).
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Operations Research situations with interval dataSequencing situations with interval data Sequencing situations with one queue of players, each with one job, in front of a machine order. Each player must have his/her job processed on this machine, and for each player there is a cost according to the time he/she spent in the system (Curiel, Pederzoli and Tijs (1989)). A one-machine sequencing interval situation is described as a 4-tuple (N, σ0 , α, p), σ0 : a permutation deﬁning the initial order of the jobs α = ([αi , αi ])i∈N ∈ I (R+ )N , p = ([p i , p i ])i∈N ∈ I (R+ )N : vectors of intervals with αi , αi representing the minimal and maximal unitary cost of the job of i, respectively, p i , p i being the minimal and maximal processing time of the job of i, respectively.
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Operations Research situations with interval dataSequencing situations with interval data To handle such sequencing situations, we propose to use either the approach based on urgency indices or the approach based on relaxation indices. This requires to be able to compute α p p either ui = p i , αi (for each i ∈ N) or ri = αi , αii (for each p i i i i ∈ N), and such intervals should be pair-wise disjoint. Interval calculus: Let I , J ∈ I (R+ ). We deﬁne · : I (R+ ) × I (R+ ) → I (R+ ) by I · J := [I J, I J]. Let Q := (I , J) ∈ I (R+ ) × I (R+ {0}) | I J ≤ I J . I I I We deﬁne ÷ : Q → I (R+ ) by J := [ J , J ] for all (I , J) ∈ Q.
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Operations Research situations with interval dataSequencing situations with interval data Example (a): Consider the two-agent situation with p1 = [1, 4], p2 = [6, 8], α1 = [5, 25], α2 = [10, 30]. We can compute 4 5 u1 = 5, 25 , u2 = 3 , 15 and use them to reorder the jobs as the 4 intervals are disjoint. Example (b): Consider the two-agent situation with p1 = [1, 3], p2 = [4, 6], α1 = [5, 6], α2 = [11, 12]. Here, we can compute r1 = 5 , 1 , r2 = 11 , 1 , but we cannot reorder the jobs 1 2 4 2 as the intervals are not disjoint. Example (c): Consider the two-agent situation with p1 = [1, 3], p2 = [5, 8], α1 = [5, 6], α2 = [10, 30]. Now, r1 is deﬁned but r2 is undeﬁned. On the other hand, u1 is undeﬁned and u2 is deﬁned, so no comparison is possible; consequently, the reordering cannot take place.
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Operations Research situations with interval dataSequencing situations with interval data Let i, j ∈ N. We deﬁne the interval gain of the switch of jobs i and j by αj pi − αi pj , if jobs i and j switch Gij := [0,0], otherwise. The sequencing interval game: w := Gij u[i,j] . i,j∈N:i<j Gij ∈ I (R) for all switching jobs i, j ∈ N and u[i,j] is the unanimity game deﬁned as: 1, if {i, i + 1, ..., j − 1, j} ⊂ S u[i,j] (S) := 0, otherwise.
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Operations Research situations with interval dataSequencing situations with interval data The interval equal gain splitting rule is deﬁned by 1 IEGSi (N, σ0 , α, p) = 2 Gij + 12 Gij , for each j∈N:i<j j∈N:i>j i ∈ N. Proposition: Let < N, w > be a sequencing interval game. Then, 1 i) IEGS(N, σ0 , α, p) = 2 (m(1,2...,n) (w ) + m(n,n−1,...,1) (w )). ii) IEGS(N, σ0 , α, p) ∈ C(w ). Proposition: Let < N, w > be a sequencing interval game. Then, < N, w > is convex. Example: Consider the interval situation with N = {1, 2}, σ0 = {1, 2}, p = (2, 3) and α = ([2, 4], [12, 21]). The urgency indices are u1 = [1, 2] and u2 = [4, 7], so that the two jobs may be switched. We have: G12 = [18, 30], IEGS(N, σ0 , α, p) = ([9, 15], [9, 15]) ∈ C(w ).
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Operations Research situations with interval dataBankruptcy situations with interval data In a classical bankruptcy situation, a certain amount of money E has to be divided among some people, N = {1, . . . , n}, who have individual claims di , i ∈ N on the estate, and the total claim is weakly larger than the estate. The corresponding bankruptcy game vE ,d : vE ,d (S) = (E − i∈NS di )+ for each S ∈ 2N , where x+ = max {0, x} (Aumann and Maschler (1985)). A bankruptcy interval situation with a ﬁxed set of claimants N = {1, 2, . . . , n} is a pair (E , d) ∈ I (R) × I (R)N , where E = [E , E ] [0, 0] is the estate to be divided and d is the vector of interval claims with the i-th coordinate di = [d i , d i ] (i ∈ N), such that [0, 0] d1 d2 . . . dn and E < n di. i=1 BRI N : the family of bankruptcy interval situations with set of claimants N.
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Operations Research situations with interval dataBankruptcy situations with interval data We deﬁne a subclass of BRI N , denoted by SBRI N , consisting of all bankruptcy interval situations such that |d(N S)| ≤ |E | for each S ∈ 2N with d(N S) ≤ E .(∗) We call a bankruptcy interval situation in SBRI N a strong bankruptcy interval situation. With each (E , d) ∈ SBRI N we associate a cooperative interval game < N, wE ,d >, deﬁned by wE ,d (S) := [vE ,d (S), vE ,d (S)] for each S ⊂ N. Note that (∗) implies vE ,d (S) ≤ vE ,d (S) for each S ∈ 2N . SBRIG N : the family of all bankruptcy interval games wE ,d with (E , d) ∈ SBRI N .
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Operations Research situations with interval dataBankruptcy situations with interval data We notice that wE ,d ∈ SBRIG N is supermodular because vE ,d and vE ,d ∈ G N are convex. The following example illustrates that wE ,d ∈ SBRIG N is supermodular but not necessarily convex. Example: Let (E , d) be a two-person bankruptcy situation. We suppose that the claims of the players are closed intervals d1 = [70, 70] and d2 = [80, 80], respectively, and the estate is E = [100, 140]. Then, the corresponding game < N, wE ,d > is given by wE ,d (∅) = [0, 0], wE ,d (1) = [20, 60], wE ,d (2) = [30, 70] and wE ,d (1, 2) = [100, 140]. This game is supermodular, but is not convex because |wE ,d | ∈ G N is not convex.
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Operations Research situations with interval dataMinimum cost spanning tree situations with interval data There are also interesting results for interval extension of minimim spanning tree situations. For further details please see the paper Connection situations under uncertainty and cost monotonic solutions by Moretti et al. which was published in Computers and Operations Research.
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [1]Alparslan G¨k S.Z., Cooperative interval games, PhD o Dissertation Thesis, Institute of Applied Mathematics, Middle East Technical University, Ankara-Turkey (2009). [2]Alparslan G¨k S.Z., Cooperative Interval Games: Theory and o Applications, Lambert Academic Publishing (LAP), Germany (2010) ISBN:978-3-8383-3430-1. [3]Alparslan G¨k S.Z., Branzei R., Fragnelli V. and Tijs S., o Sequencing interval situations and related games, to appear in Central European Journal of Operations Research (CEJOR). [4]Alparslan G¨k S.Z., Branzei R. and Tijs S., Airport interval o games and their Shapley value, Operations Research and Decisions, Issue 2 (2009).
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [5]Alparslan G¨k S.Z., Branzei R. and Tijs S., Big boss interval o games, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems (IJUFKS), Vol. 19, no:1 (2011) pp.135-149. [6]Aumann R. and Maschler M., Game theoretic analysis of a bankruptcy problem from the Talmud, Journal of Economic Theory 36 (1985) 195-213. [7]Bondareva O.N., Certain applications of the methods of linear programming to the theory of cooperative games, Problemly Kibernetiki 10 (1963) 119-139 (in Russian). [8] Branzei R. and Alparslan G¨k S.Z., Bankruptcy problems with o interval uncertainty, Economics Bulletin, Vol. 3, no. 56 (2008) pp. 1-10.
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [9]Branzei R., Dimitrov D. and Tijs S., Models in Cooperative Game Theory, Springer, Game Theory and Mathematical Methods (2008). [10]Curiel I., Pederzoli G. and Tijs S., Sequencing games, European Journal of Operational Research 40 (1989) 344-351. [11]Driessen T., Cooperative Games, Solutions and Applications, Kluwer Academic Publishers (1988). [12]Gillies D. B., Solutions to general non-zero-sum games. In: Tucker, A.W. and Luce, R.D. (Eds.), Contributions to the theory of games IV, Annals of Mathematical Studies 40. Princeton University Press, Princeton (1959) pp. 47-85. [13] Moretti S., Alparslan G¨k S.Z., Branzei R. and Tijs S., o Connection situations under uncertainty and cost monotonic solutions, Computers and Operations Research, Vol.38, Issue 11 (2011) pp.1638-1645.
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [14] Muto S., Nakayama M., Potters J. and Tijs S., On big boss games, The Economic Studies Quarterly Vol.39, No. 4 (1988) 303-321. [15]Shapley L.S., On balanced sets and cores, Naval Research Logistics Quarterly 14 (1967) 453-460. [16] Tijs S., Bounds for the core and the τ -value, In: Moeschlin O., Pallaschke D. (eds.), Game Theory and Mathematical Economics, North Holland, Amsterdam (1981) pp. 123-132. [17] Tijs S., Big boss games, clan games and information market games. In:Ichiishi T., Neyman A., Tauman Y. (eds.), Game Theory and Applications. Academic Press, San Diego (1990) pp.410-412.
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6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [18]Tijs S., Meca A. and L´pez M.A., Beneﬁt sharing in holding o situations, European Journal of Operational Research 162 (1) (2005) 251-269. [19] von Neumann, J. and Morgernstern, O., Theory of Games and Economic Behaviour, Princeton: Princeton University Press (1944).
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