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 → 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.

v˜(S)= wj, S ⊆ N ˜ jX∈AS where A˜S = {j ∈ M | ∃ i, l ∈ S s.t. cij = clj = 1}, i.e. the set of zones which are covered by at least two emergency units located in S, when each zone in S hosts one emergency unit The overlapping can be seen as a negative feature, a waste of resources for the service The problem of allocating emergency units: a TU-Games approach 17

Example 3 Referring to Example 2, the game is: a b c 2 1.3 1.8 2.8 v (•) = 0 2 g v (•) = 0 d e f w 3.0 2.1 3.6 2.4 v2(•) = 0 h i j 2 2.4 3.2w 3.4 v (••) = 11.6 v2(••) = 2.4 kw l m n 1.8 2.4 1.6 1.7 v2(••) = 4.8 o p q r v2 . 1.4 2.3 1.9 1.5 (•••) = 14 0

The Shapley value may be used for the same motivation It can be computed “dividing” the demand of each multicovered zone among the locations that cover it and againg summing up on each row The problem of allocating emergency units: a TU-Games approach 18

Multidivision matrix: a b c d e f g h i j k l m n o p q r • 1.8 1.2 0.8 1.6 • 1.8 1.2 0.8 1.6 1.2 • 0.8 1.2 The Shapley Value is φ = (5.4, 6.6, 2.0) In this case the highest the value, the highest the average multiple coverage of zones, so the ordering has to be reverted, and it is •, •, •

For this game, an isolated location, i.e. whose covered area has empty intersection with the area of any other location, has always a null marginal contribution; so, its Shapley value is always zero and this location has always the highest ranking The problem of allocating emergency units: a TU-Games approach 19

4.3 Combining Rankings

Third idea: Take into account both orderings

Referring again to Example 2 and using the Borda count: v1 v2 total • 1 2 3 • 2 3 5 • 3 1 4 so, the ranking is •, •, • The problem of allocating emergency units: a TU-Games approach 20

5 Real-world Example

Extra-urban area of ; 117 municipalities, each coinciding with a zone 65 possible locations, those in which an emergency service is located Covered area is computed according to the distances among the centers of the municipalities; a town is covered if it may be reached in less than 18 minutes The fast computation allows for both equity and efficiency solutions Looking for efficiency, 12 scenarios has been identified W 1 Winter 00 - 08 W 2 Winter 08 - 14 via the results of the research of University of Milan as W 3 Winter 14 - 24 SAmf 1 Spring-Autumn MO - FR 00 - 07 parts of the year in which the call trend is similar and SAmf 2 Spring-Autumn MO - FR 07 - 13 SAmf 3 Spring-Autumn MO - FR 13 - 24 which may be treated in the same way; the demand SAss 1 Spring-Autumn SA - SU 00 - 07 SAss 2 Spring-Autumn SA - SU 07 - 13 of each zone is the average number of calls per hour SAss 3 Spring-Autumn SA - SU 13 - 24 S 1 Summer 00 - 07 originated from it in the corresponding scenario S 2 Summer 07 - 14 S 3 Summer 14 - 24 Looking for equity, the demand of each zone corresponds to the surface of the municipality We suppose to locate 28 units Each location may host at most one ambulance 37 20 53 15 61 59 11 32 18 28 56 41 31 26 19 12 2 8 22 30 48 39 9 57 21 16 29 5 13 3 62 64 25 43 14 47 23 63 38 35 54 46 58 55 49 6 34 17 24 44 60 51 42 27 10 50 52 1 40 33 65 36 4 45 7 The problem of allocating emergency units: a TU-Games approach 22

In the following table, the second column refers to the equity approach, while the following 12 refer to the efficiency approach

Candidate locations which an ambulance is never assigned to are omitted: Bellinzago Lom- bardo (5), (8), Cassano d’Adda (13), Ceriano Laghetto (15), Cislago (20), (25), (30), (31), (39), (47), Solaro (59), Uboldo (VA - 61)

Towns in red already host an ambulance The problem of allocating emergency units: a TU-Games approach 23

W SAmf SAss S W SAmf SAss S Town E 1 2 3 1 2 3 1 2 3 1 2 3 Town E 1 2 3 1 2 3 1 2 3 1 2 3 1 X XXX XXX XXX XXX 36 X XXX XXX XXX XXX 2 XXX XXX XXX XXX 37 X X X 3 X XXX XXX XXX XXX 38 Misinto 4 X 40 Opera 6 X 41 XXX XXX XXX XXX 7 X XXX XX X XXX 42 X 9 Bresso XXX XXX XXX XXX 43 Pero X XXX XXX XXX XXX 10 44 11 Caronno Pertusella 45 X X 12 46 X XX XXX XXX XXX 14 Cassina De’ Pecchi X X X XX 48 Rho X X 16 XXX XXX XXX XXX 49 X 17 50 X XXX XXX XXX XXX 18 X 51 X XXX XXX XXX XXX 19 XX XXX XXX XXX 52 X XXX XXX XXX XXX 21 XXX XXX XXX XXX 53 Saronno 22 XXX XXX XXX XXX 54 X XXX XXX XX XXX 23 X XX XXX XX 55 XX XX XX XXX 24 X XX XXX XX XXX 56 XXX XXX XXX XXX 26 XXX XXX XXX XXX 57 XXX XXX XXX XXX 27 X XXX XXX XXX XXX 58 X XXX XXX XXX XXX 28 XXX XXX XXX XXX 60 X X 29 Gorgonzola X 62 X X X XX 32 Limbiate X X X 63 X X 33 Locate Triulzi X XX XXX X X 64 XXX XXX XXX XXX 34 Magenta X 65 X 35 X The problem of allocating emergency units: a TU-Games approach 24

6 Concluding Remarks

• the nucleolus (Schmeidler, 1969), which is based on the idea of minimizing the maximum dissatisfaction of each coalition, could be another good solution, but it is very hard to compute • the set of highest ranked ambulances for a given cardinality could be non-optimal. In fact, for a generic game (N,v),

v(S) 6= φi(v) i S X∈ • the algorithm may be used as a first step when there is a large number of candidate locations, because of its low complexity: obtain an exact solution to an approximated problem, instead of an approximated solution to an exact problem • more than one ambulance could be hosted in each location in order to satisfy the whole demand The problem of allocating emergency units: a TU-Games approach 25

References

[1] Ball MO, Lin FL (1993) A Reliability Model Applied to Emergency Service Vehicle Location. Oper Res 41(1):18-36.

[2] Branas CC, MacKenzie EJ and ReVelle CS (2000) A Trauma Resource Allocation Model for Ambulances and Hospitals. Health Serv Res 35:489-507.

[3] Church RL, ReVelle CS (1974) The Maximal Covering Location Problem. Pap Reg Sci Assoc 32:101-118.

[4] Curiel I (1997) Cooperative Game Theory and Applications. Kluwer Academic Publishers.

[5] Daskin MS (1983) A Maximum Expected Covering Location Model: Formulation, Properties and Heuristic Solution. Transp Sci 17(1):48-70.

[6] Derekenaris G, Garofalakis J, Makris C, Prentzas J, Sioutas S, Tsakalidis A (2001) Integrating GIS, GPS and GSM Technologies for the Effective Management of Ambulances. Comput Environ Urban Syst 25:267-278.

[7] Drezner Z (1995) Facility Location: A Survey of Applications and Methods. Springer Verlag, Berlin.

[8] Drezner Z, Hamacher HW (2001) Facility Location: Application and Theory. Springer Verlag, Berlin.

[9] Farahani RZ, SteadieSeifi M, Asgari N (2010) Multiple Criteria Facility Location Problems: A Survey. Appl Math Model 34:1689- 1709.

[10] Fujiwara O, Makjamroen T, Gupta KK (1987) Ambulance Deployment Analysis: a Case Study of Bangkok. Eur J of Oper Res 31(1):9-18.

[11] Goemans MX, Skutella M (2004) Cooperative Facility Location Games. J Algorithms 50:194-214.

[12] Goldberg J, Dietrich R, Chen JM, Mitwasi MG (1990) A Simulation Model for Evaluating a Set of Emergency Vehicle Base Locations: Development, Validation and Usage. Socio-Econ Plan Sci 24:124-141. The problem of allocating emergency units: a TU-Games approach 26

[13] Granot D (1982) The Role of Cost Allocation in Locational Models. Oper Res 35:234-248.

[14] Hale TS, Moberg CR (2003) Location Science Research: A Survey. Ann of Oper Res 123:21-35.

[15] Immorlica N, Mahdian M, Vahab SM (2008) Limitations of Cross-Monotonic Cost-Sharing Schemes. ACM Trans on Algorithms, 4(2) Article 24.

[16] Leonardi S, Sh¨afer G (2004) Cross-Monotonic Cost Sharing Methods for Connected Facilities Location Games. Theor Comput Sci 326:431-442.

[17] Mallozzi L (2007) Noncooperative Facility Location Games. Oper Res Lett 35:151-154.

[18] Mallozzi L (2011) Cooperative Games in Facility Location Situations with Regional Fixed Costs. Optim Lett 5:173-181.

[19] Mazalov V, Sakaguchi M (2003) Location Game on the Plane. Int Game Theory Rev 5(1):13-25.

[20] Nickel S, Puerto J (2005) Location Theory: A Unified Approach. Springer Verlag, Berlin.

[21] Pal M, Tardos E (2003) Group Strategyproof Mechanisms Via Primal-Dual Algorithms. Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 584-593.

[22] Plastria F (2001) Static Competitive Facility Location: An Overview of Optimisation Approaches. Eur J of Oper Res 129:461- 470.

[23] Puerto J, Garcia-Jurado I, Fernandez FR (2001) On the Core of a Class of Location Games. Math Methods of Oper Res 54:373-385.

[24] Puerto J, Tamir A, Perea F (2011) A Cooperative Location Game Based on the 1-center Location Problem. Eur J of Oper Res 214:317-330. The problem of allocating emergency units: a TU-Games approach 27

[25] Puerto J, Tamir A, Perea F (2012) Cooperative Location Games Based on the Minimum Spanning Steiner Subgraph Problem. Discrete Appl Math 160:970-979.

[26] Repede JF, Bernardo JJ (1994) Developing and Validating a Decision Support System for Locating Emergency Medical Vehicles in Louisville, Kentucky. Eur J Oper Res 75:567-581.

[27] ReVelle CS, Hogan K (1989) The Maximum Reliability Location Problem and α-reliable p-center Problem: Derivatives of the Probabilistic Location Set Covering Problem. Ann Oper Res 18(1):155-173.

[28] Schilling DA, Elzinga DJ, Cohon JL, Church RL, ReVelle CS (1979) The TEAM-FLEET Models for Simultaneous Facility and Equipment Sitting. Transp Sci 13:163-175.

[29] Schmeidler D (1969) The Nucleolous of a Characteristic Function Game. SIAM J of Appl Math 17:1163-1170.

[30] Tamir A (1992) On the Core of Cost Allocation Games Defined on Locational Problems. Transp Sci 27:81-86.

[31] Topkis D (1998) Supermodularity and Complementarity. Princeton University Press.

[32] Toregas C, Swain R, ReVelle C, Bergman L (1971) The Location of Emergency Service Facilities 19(6):1363-1373

[33] Xu D, Du D (2006) The k-level Facility Location Game. Oper Res Lett 34:421-426. The problem of allocating emergency units: a TU-Games approach 28

Contents

1 Introduction 1

2 State of the Art 4

3 Preliminary Example 6

4 The Games 7 4.1 CoverageGames ...... 8 4.2 MulticoverageGames ...... 16 4.3 CombiningRankings ...... 19

5 Real-world Example 20

6 Concluding Remarks 24