Control of the Contract of a Public Transport Service

Transcription

1 Control of the Contract of a Public Transport Service Andrea Lodi, Enrico Malaguti, Nicolás E. Stier-Moses Tommaso Bonino DEIS, University of Bologna Graduate School of Business, Columbia University SRM - Reti e Mobilità January 11th, 2012 Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 1

2 Outline 1 Problem description 2 A non-cooperative game 3 Optimized performance control 4 Case study: applying the control to a mid-size city 5 Conclusions Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 2

3 Problem Description Every day 6000 bus journeys are run in a mid-size Italian city (Bologna), serving customers. Large part of the service cost is publicly subsidized. This is quite a common situation where two players are involved in a service provision: An Agency, who designs and contracts the service; An Operator, who provides the service. Some questions arise: what is the relationship between Agency and Operator? How does the Agency control the service provided by the Operator? Can the Agency trust the information provided by the Operator about its own performance? Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 3

4 Problem Description More specifically for the bus service case (similar in all the European Union): The Agency: designs the service (timetable); designs the service contract; organizes a public auction to contract the service; is in charge of controlling the service provided. The Operator winning the auction: hires the drivers; manages the buses; is in charge of providing the service. The Operator reports his performance to the Agency. If the contractual obligations are not met, the Operator receives a fine from the Agency. Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 4

5 Problem Description The issue in this scheme is the informative asymmetry between the Operator and the Agency. The Agency needs a control procedure to check the correctness of the information received from the Operator, and the latter will be fined hefty in case the information provided is false or inaccurate. Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 5

6 Agency - Operator: a non-cooperative game The Agency-Operator relationship is modeled as a dynamic non-cooperative game. 1 Contract design: the Agency fixes the fines level. For each reported skipped journey, there is a fine f. For each non-reported skipped journey, there is fine F, if discovered. 2 The Operator decides about the level of effort is going to spend to provide the service. 3 The Operator decides about being honest or not reporting missing journeys. It will be honest about the single journey with probability t. The Agency decides the average budget per journey to allocate to the controls, which determines the probability p for a journey to be checked. Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 6

7 Agency - Operator: a non-cooperative game Third stage: Operator: honest about the single journey with probability t. Agency: checks a journey with probability p. p = 0 when the agency does not check; in general p = p(b, P), where B is the budget allocated to the control and P is the control procedure. In general, p = p(b, P) cannot be computed in closed-form. We propose an optimized performance control procedure, so p = p(b, P) is the result of an optimization model. Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 7

8 Agency - Operator: a non-cooperative game Third stage equilibrium: The operator reports skipped journeys with probability t. Its payoff (per journey) is: = f (1 q)t F (1 q)(1 t)p π 3 OP Where 1 q is the probability of skipping a journey. The optimal operator pure strategy is declaring missing journeys (t = 1) when p > f /F and not declaring (t = 0) when p < f /F. Approximating p(b A ) = kba α strategy at equilibrium is: (k > 0, 0 < α 1) the operator mixed t NE = 1 1 (1 q)(η 2 + η 3 )αk( f kf ) α 1 α Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 8

9 Optimized Performance Control Problem: given a budget of working hours, check as many journeys as possible. A profit is associated with the check of each journey. Frequency check: counts the number of buses observed at a bus stops. For each stop, a few waiting times are defined, and a profit is defined depending on the number of journeys that can be observed during that time. Controllers can move in the town by riding the buses or by walking. Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 9

10 Optimized Performance Control More formally, we are given a sparse directed graph G = (V, A), where V is the set of bus stops and A the set of (bus or walking) connections. Each arc a has a positive travel time d a. Stops s V have associated waiting times t which give a profit π t s, depending on the journey observed during t. Given K controllers, each of them working W hours, we want to route them in the town so as to maximize the profit of the collected information. Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 10

11 Optimized Performance Control A MILP model for optimized performance control: We use binary variables ys t,k to denote the check of stop s during time period t of length τ by controller k. Integer variables za k denote the number of times arc a is traversed by controller k. (working hours) max πsy t s t,k k K s S t T s d a za k + τs t ys t,k W k k K a A s S t T s Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 11

12 Optimized Performance Control (balance) (office visit) (no duplication) a δ + (s) z k a = (subtour elimination) a δ (s) a δ (Σ) a δ + (0) y t,k k K t T s z k a 1 k K s 1 s S z k a t T s y t,k s z k a t T s y t,k s s S, k K k K, Σ V, s Σ, 0 / Σ ys t,k {0, 1} s S, t T, k K za k Z + a A, k K. Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 12

13 Optimized Performance Control (balance) a δ + (s) z k a = a δ (s) z k a t T s y t,k s s S, k K Note that we work with a sparse graph, and the shortest path between nodes is not computed in a pre-processing phase. Some nodes (blue) are crossed but not checked.!" #" $" y t,k i = 1 y t,k j = 0 y t,k k = 1 Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 13

14 Separation We use separation to tackle the exponentially many subtour elimination constraints. Given a fractional solution (ȳ, z), define a max-flow problem for each k by setting the arc capacities to z a k, a A. Let f s be the value of the max flow from the office (0) to s and Σ a set of vertices, including s, corresponding to a min-cut. If we have that s > f s, t T s ȳ t,k we add the corresponding violated constraint za k ys t,k. t T s a δ (Σ) Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 14

15 Additional Constraints In order to forbid solutions which visit stops which are close to each other, we add to the model a set of clique constraints of the form: y t,k k K t T s s 1 s C greedily computed on a suited incompatibility graph. Additional constraints are considered for imposing the check of special stops. Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 15

16 Algorithm Models with about 100 nodes can be solved through Branch-and-Cut with CPLEX12. For larger graphs, we get good bounds but it is hard to produce feasible solutions. This is because when we add a subtour elimination constraint, the solver easily finds an equivalent solution by re-routing on non-checked (blue) nodes. a δ (Σ) z k a t T s y t,k s!" #" Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 16

17 Algorithm Models with about 100 nodes can be solved through Branch-and-Cut with CPLEX12. For larger graphs, we get good bounds but it is hard to produce feasible solutions. This is because when we add a subtour elimination constraint, the solver easily finds an equivalent solution by re-routing on non-checked (blue) nodes. a δ (Σ) z k a t T s y t,k s!" #"!" #" Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 17

18 Algorithm To obtain good feasible solutions we designed a MIP-heuristic that uses information from the current fractional solution: consider a reduced problem restricted to nodes s having an associated y s larger then a given threshold; define an associated complete graph G; solve the model on G for a limited amount of time; map the solution back to G. The algorithm is executed during the Branch-and-Cut with a specified frequency. Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 18

19 Case study: applying the Model to Bologna We focused on 29 high frequency lines (average time between two consecutive journeys of at most 30 minutes), serving the 81.9% of the total operator passengers and visiting 1104 stops. The graph G = (S {0}, A) is derived from the real network by considering only arcs corresponding to existing network connections (i.e., G is sparse). There are two kinds of such arcs, namely: bus connections and pedestrian connections. In total, A Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 19

20 Case study: bus connections Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 20

21 Case study: overall graph (bus and pedestrian connections) Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 21

22 Case study: Iterated Control The schedule of the controllers is optimized day-by-day through the deterministic model. In order to discount recently collected information, the profits associated with lines and stops are modified as follows: π s = π s (1 1 1 int ), where int is the interval of time (in days) since the last visit to the stop. Figure: Two consecutive working days for two controllers. Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 22

23 Case study: solving the model All algorithms were coded in C and tested on a single core of a core i5 650 PC at 3.20GHz and 8GB ram under linux. We used CPLEX 12.2 as MIP solver. When solving the whole model on the sparse graph G, separation is called only for integer solutions, and we generate at most 5 cuts per controller. When solving the reduced model (MIP-heuristic) on the complete graph G, we call the cut generating procedure every 100 nodes of Branch-and-Cut, in addition to separating all integer solutions. The MIP-heuristic is executed every 100 nodes of the Branch-and-Cut with a time limit of 200 seconds. The overall method has a time limit of 2 hours. Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 23

24 Case study: solving the model We considered 6 basic scenarios with 2, 3 and 4 controllers, each one working 3 or 6 hours. Solutions are compared with those produced by an iterated greedy randomized algorithm. controllers working hours gap nodes cuts heur gap Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 24

25 Case study: check probability!"#!$%&'()*+,-,%.%/0(( )"!!# ("!!# '"!!# &"!!# %"!!# %#*+,-.+//0.1# &#*+,-.+//0.1# '#*+,-.+//0.1# $"!!#!"!!#!# (# $!# $(# %!# %(# &!# "+1*2( p(b A ) is computed through the model. Effort b A is measured in working hours w i per controller i, so p(b A ) = p(w) with w = (w 1,..., w k ). Experiments tell us that we have p(b A ) = p(w ) with W = k i=1 w i. Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 25

26 Case study: feedback on game!"#!$%&'()*+,-,%.%/0(( )"!!# ("!!# '"!!# &"!!# %"!!# %#*+,-.+//0.1# &#*+,-.+//0.1# '#*+,-.+//0.1# $"!!#!"!!#!# (# $!# $(# %!# %(# &!# "+1*2( By considering the current fines: f = 250 and F = 5000 euro, we need to have p > 5% to make honesty the optimal pure strategy for the operator (t = 1). This can be obtained with 24 hours of optimized control. Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 26

27 Conclusions We considered the problem of an Agency who contracts a public transport service to a private Operator. The informative asymmetry between the two players was addressed from a game theoretical perspective, coming to the (not too surprising) conclusion that the Operator will not provide reliable information if no one checks the information correctness. We proposed an optimized control strategy based on the solution of a price collecting routing model. The model solution describes the working day of the controllers, who will check the service at bus stops. Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 27

28 Conclusions The output of the model was used, as a feedback to the game, to evaluate the optimal level of effort to be spent in the checks. At the service contract renewal, the game theoretical/optimization approach will support the design of the new contract. Control of the Contract of a Public Transport Service (A. Lodi, E. Malaguti, N.E. Stier-Moses, T. Bonino) 28