The problem of allocating emergency units: a TU-Games approach Vito Fragnelli Università del Piemonte Orientale [email protected] The problem of allocating emergency units: a TU-Games approach 1 1 Introduction The presentation is based on a joint work with Stefano Gagliardo Emergency management −→ hard task Various types of emergencies: • medical • environmental • urban • ... Different features ⇒ different approaches, means and tools Emergency ⇔ urgency ⇔ fast intervention • medical −→ minutes • environmental −→ hours • earthquake −→ days The problem of allocating emergency units: a TU-Games approach 2 Utility of the intervention utility6 utility6 utility6 - - - distance distance distance real theoretical approximated distance may be measured as ground distance, time distance, linear distance, etc. Ambulances in Italy → red codes statistics account only intervention in 8 minutes Don’t care! If the ambulance arrives later the intervention is anyhow completed For fire planes and helicopters it is necessary to take into account the fuel tank capacity and the distance from a place for charging water The problem of allocating emergency units: a TU-Games approach 3 Candidate locations may be viewed as interacting agents ⇓ The choice of deploying an emergency unit in a candidate location takes into account its own characteristics as covered area and call frequency, but should consider also the location of the other emergency units The problem of allocating emergency units: a TU-Games approach 4 2 State of the Art Location problems are widely studied: Drezner (1995), Drezner and Hamacher (2001), Nickel and Puerto (2005), Plastria (2001), Hale and Moberg (2003), Farahani et al. (2010) Complexity of emergency management allows and requires the synergic use of different methods and approaches • pivotal contributions: Toregas et al. (1971), Church and ReVelle (1974), Schilling et al. (1979) • statistical analysis: Daskin (1983), Fujiwara et al. (1987) • simulation: Goldberg et al. (1990), Repede and Bernardo (1994) • topography and GIS: Branas et al. (2000), Derekenaris et al. (2001) • mathematical programming: ReVelle and Hogan (1989), Ball and Lin (1993) • stochastic optimization: Goldberg et al. (1990) • ... • Game Theory (of course!) The problem of allocating emergency units: a TU-Games approach 5 Game theory allows emphasizing the interactions among the ambulances deployed on the area under analysis Location games were studied for cost sharing problems: (Granot (1982), Tamir (1992), Curiel (1997), Puerto, Garcia-Jurado and Fernandez (2001), Pal and Tardos (2003), Goemans and Skutella (2004), Leonardi and Sh¨afer (2004), Xu and Du (2006), Immorlica et al. (2008), Mallozzi (2011), Puerto et al. (2011 and 2012) Non-cooperative approaches: Topkis (1998), Mazalov and Sakaguchi (2003), Mallozzi (2007) HERE Game theory supports the location of the facilities The problem of allocating emergency units: a TU-Games approach 6 3 Preliminary Example Example 1 Two ambulances have to be deployed in the area depicted below, where each box represents a zone and the value in each is its demand 123113 An ambulance located in a zone may cover it and the two adjacent ones, if any Trivially, the units have to be located in the second and in the fifth zones Allocating the units in the two most demanding zones or in the two zones with the highest aggregate demand, i.e. the sum of the demands coming from all the zones covered by a location, some zones remain uncovered The problem of allocating emergency units: a TU-Games approach 7 4 The Games Possible aims: • efficiency: satisfying the maximum number of calls received from the area we attend to; a priori analysis, based on historical data on the average frequency of emergency calls received the results need to be validated with an a posteriori experimental study on the real situation under investigation • equity: maximizing the area covered by all the ambulances, without considering the number of calls received • intermediate: maximizing the number of people living in the covered area The problem of allocating emergency units: a TU-Games approach 8 4.1 Coverage Games Hypothesis (minimal interference) the calls originated in a given time interval from the area covered by an ambulance have a time distribution that requires the minimum number of ambulances First idea: Take into account the area that each subset of candidate locations may cover Definition 1 (Coverage games) A coverage game is the TU-game (N,v) defined by v(S)= wj, S ⊆ N j A X∈ S where AS = {j ∈ M | ∃ i ∈ S s.t. cij = 1}, i.e. the set of zones which are covered by at least one emergency unit located in S, when each zone in S hosts one emergency unit and M is the set of zones and N ⊆ M is the set of zones that may host an ambulance The problem of allocating emergency units: a TU-Games approach 9 Characterization of the Shapley Value The Shapley value is a good solution for coverage games, as it strongly accounts for marginality Shapley value allows ranking by relevance all the possible locations for the emergency units Locations with the highest Shapley values are the candidates for receiving an ambulance The Shapley value has pretty good fairness properties for this problem, coverage indifference and demand indifference Usual fairness criteria, such as monotonicity and equal treatment of equals, are not so im- portant in the situation at hand The problem of allocating emergency units: a TU-Games approach 10 Definition 2 (j-th zone sub-game) Let v be a coverage game. Given j ∈ M, the j-th zone sub-game of v is the TU-game vj defined, for each S ⊆ N, by w if j ∈ A vj(S)= j S ( 0 otherwise j Lemma 1 Let v ∈ C be any coverage game. Then, for every S ⊆ N, v(S)= j∈M v (S) P The problem of allocating emergency units: a TU-Games approach 11 Definition 3 (Coverage Indifference - CI) A solution ψ satisfies coverage indifference if j for each v ∈ C and each j-th zone sub-game v of v, for any i, l ∈ N s.t. j ∈ A{i} ∩ A{l}, j j then ψi(v )= ψl(v ) The solution ψ assigns to the agents who cover a zone the same amount in the corresponding sub-game It looks at the situation from the point of view of the users and requires to give the same importance to the units that cover a zone allowing for equally sharing the demand of the zone among them, so that any of those units has the same probability to satisfy a call coming from the considered zone Definition 4 (Demand Indifference - DI) A solution ψ satisfies demand indifference if for j each v ∈ C, i ∈ N, ψi(v)= j∈M ψi(v ) The solution ψ assigns to eachP agent the sum of the amounts he receives in each zone sub-game It looks at the situation from the point of view of the emergency service provider and gives the same importance to each demand, no matter where it comes from and whatever the required intervention is The problem of allocating emergency units: a TU-Games approach 12 DI may be viewed as a restricted additivity, as it holds only for zone sub-games CI is equivalent to wj n if j ∈ A{i} j =1 clj ψi(v )= l ,i ∈ N P 0 otherwise Theorem 1 Given a coverage game v ∈ C, the Shapley value is the unique solution which satisfies CI and DI. Moreover, wj φi(v)= n ,i ∈ N =1 clj j A l ∈X{i} P The problem of allocating emergency units: a TU-Games approach 13 Algorithm for the Shapley Value CI provides a simple algorithm starting from the n × m division matrix D: wj n if cij = 1 =1 clj dij = l ,i ∈ N, j ∈ M P 0 otherwise The Shapley value of the coverage game for player i ∈ N can be obtained simply summing up the values in the i-th row of D The computational complexity of this algorithm is polynomial in n and m Moreover, the algorithm does not require the characteristic function of the coverage game The problem of allocating emergency units: a TU-Games approach 14 Example 2 - An area is described as grid of 18 zones, each with a demand of emergency calls - 3 locations are candidated for an ambulance (bullets) - The intervention time allows the ambulances to cover 9 zones a b c 1 1.3 1.8 2.8 v (•) = 19.6 1 e g v (•) = 20.7 ← max d f w 3.0 2.1 3.6 2.4 1 v (•) = 12.2 h i j 1 2.4 3.2w 3.4 v (••) = 28.7 1 m n v (••) = 29.4 ← max kw l 1.8 2.4 1.6 1.7 v1(••) = 28.1 o p q r 1 1.4 2.3 1.9 1.5 v (•••) = 36.1 The problem of allocating emergency units: a TU-Games approach 15 The division matrix D is: a b c d e f g h i j k l m n o p q r • 1.3 1.8 2.8 2.1 1.8 1.2 0.8 1.6 • 1.8 1.2 0.8 1.6 3.4 1.2 1.6 1.7 • 0.8 1.8 1.2 1.4 2.3 1.9 The Shapley Value is φ = (13.4, 13.3, 9.4) and the ranking is •, •, • The results of this game could be affected when two different locations overlap on several zones, so that both may have a great relevance even if only one of them is actually important The problem of allocating emergency units: a TU-Games approach 16 4.2 Multicoverage Games Second idea: Take into account the area that each subset of candidate locations may cover more than once Definition 5 (Multicoverage games) The multicoverage game is the TU-game (N, v˜) s.t.