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ORBEL 28 Th. PIRONET HEC-ULg ORBEL 28 QuantOM Outlines Multi-period vehicle assignment with Multi-period Vehicle stochastic load availability Assignment An example Y. Crama and Th. Pironet Stochastic Bounds HEC-Management

  1. ORBEL 28 Th. PIRONET HEC-ULg ORBEL 28 QuantOM Outlines ” Multi-period vehicle assignment with Multi-period Vehicle stochastic load availability” Assignment An example Y. Crama and Th. Pironet Stochastic Bounds HEC-Management School of the University of Li` ege Algorithms Simulation Research Group QuantOM Instances and results Robustness contact : thierry.pironet@ulg.ac.be Conclusions

  2. ORBEL 28 ” Multi-period vehicle assignment Th. PIRONET with stochastic load availability” HEC-ULg QuantOM THE PROBLEM Outlines Multi-period Vehicle assignment Vehicle To maximize profit : select loads to be transported by trucks Assignment (FTL-PDP) References : W.B Powell An example Stochastic Multi-period Bounds Confirmed and projected loads provided over some periods Algorithms Repetitive decision process period per period over an horizon Simulation Instances and Stochastic load availability results Projected loads realize or vanish Robustness Conclusions

  3. ORBEL 28 Outlines Th. PIRONET HEC-ULg ◮ Multi-period information and decision framework QuantOM ◮ The Deterministic Vehicle Assignment Problem Outlines Multi-period ◮ An example Vehicle Assignment ◮ The Stochastic version An example ◮ Bounds : a-priori and a-posteriori information Stochastic Bounds ◮ Algorithms : single or multiple scenarios approaches Algorithms ◮ Simulation Simulation Instances and ◮ Instances and Results results Robustness ◮ Robustness Analysis Conclusions ◮ Conclusions

  4. ORBEL 28 Multi-period : Rolling horizon Th. PIRONET HEC-ULg Decision : in t and t = 1 , 2 , ..., T − H = > Policy QuantOM Outlines Multi-period t+1,...,t+RH t+RH+1,...,t+H t+H+1,...,T t Vehicle Assignment Deterministic Stochastic Tail An example Stochastic Parts : decision, deterministic, stochastic Bounds Case study ( Period = day ) : Algorithms 1. Rolling horizon H = 4 P Simulation 2. Deterministic RH = 1 P , Stochastic 3 P Instances and results Dynamism of the system : Robustness Conclusions 1. Decision and actions in t (info out) 2. Roll-over 1 period, updates (info in) t + 1 → t ′ 2.1 stochastic gets deterministic t + RH + 1 → t ′ + RH 2.2 new stochastic info in t + H + 1 → t ′ + H 3. Go to 1 with t → t + 1 = t ′

  5. ORBEL 28 Deterministic Vehicle Assignment Problem Th. PIRONET HEC-ULg - Set of Cities C 1 , ..., C N and transportation times TT ( C 1 , C 2 ) QuantOM - Set of Periods 1 , ..., T Outlines - Set of Loads j ∈ J ( DepC j , ArrC j , DepP j , ArrP j , Gain j ) Multi-period - Set of Trucks i ∈ I ( LocC i , Un / Loaded i value 0 or j ) Vehicle Assignment Actions : Carry Load j , Wait in LocC i , Unladen to DepC j An example Objective : profitable paths i.e. maximize (Gains-Costs) Stochastic subject to : Bounds Max 1 Load per Truck, max 1 Truck per Load Algorithms Flow conservation constraints Simulation Instances and Network flow structure : Polynomially Solvable results Remarks : Robustness - Full-Truck-Load (FTL), no preemption Conclusions - Unloading at the end of t = ArrP j if un / loaded i = j - ArrP j = DepP j + TT ( DepC j , ArrC j )

  6. ORBEL 28 Deterministic Vehicle Assignment Problem Th. PIRONET HEC-ULg Decisions for a truck i QuantOM ◮ Carry Lj if LocC i = DepC j (Gain) Outlines ◮ Wait in LocC i (Cost) Multi-period ◮ Move Unladen from LocC i to DepC j (Cost) Vehicle Assignment An example t-1 t t+1 t+2 t+3 t+4 Stochastic Bounds Wait Wait Wait C Algorithms U C 1 • Truck 1 a n r r l Simulation a y d L e 2 n Instances and Wait Wait results C n n C 2 • • a e e r d d Robustness r a a y l l n n L Conclusions U U 4 Carry L3 Wait Wait C 3 Carry L1 • • Wait C 4 • • Truck 2

  7. ORBEL 28 Deterministic Vehicle Assignment Problem Th. PIRONET HEC-ULg An optimal solution for a period t QuantOM Outlines Multi-period t t+1 t+2 t+3 t+4 t-1 Vehicle Assignment Wait C An example C 1 • Truck 1 a r r Stochastic y L Bounds 2 Wait C n C 2 • • Algorithms a e r d r a y Simulation l n L U 4 Instances and Carry L3 results C 3 Carry L1 • • Robustness Conclusions Wait C 4 • • Truck 2 Multi-period policy : cumulated value of actions in t

  8. ORBEL 28 Stochastic Vehicle Assignment Problem Th. PIRONET HEC-ULg Stochastic framework : Stochastic Load Availability QuantOM If DepP j ∈ t + RH + 1 , .., t + H , (*) Outlines the stochastic availability of load j is represented by a Multi-period discrete distribution law : Vehicle Assignment � if x = 1 p j An example P ( q j = x ) = (1) Stochastic 1 − p j if x = 0 Bounds Algorithms Projection q j materializes (1) or not (0) when t + RH + 1 → t ′ + RH Simulation Instances and Scenario : specific outcome of q j ∀ j ∈ J if (*). results Robustness Simulation : Stochastic becomes Deterministic Conclusions 1 scenario technique Multiple scenarios : Separately or Simultaneously

  9. ORBEL 28 Specific Scenarios = > Bounds Th. PIRONET HEC-ULg Optimal policy for the stochastic problem : E ∗ QuantOM Bounds from deterministic scenarios : Outlines 1. Myopic or a-priori policy over RH : O ∗ Multi-period RH Vehicle 2. Oracle or a-posteriori policy over H : O ∗ Assignment H 3. Oracle or a-posteriori solution over T : O ∗ An example T Stochastic Expected Value Scenario = > Expected Value ’Solution’ EVS Bounds H ≥ E ∗ ≥ EVS ≥ O ∗ Maximization : O ∗ T ≥ O ∗ Algorithms RH Simulation T - E ∗ ≥ 0 VPI : Value of the Perfect Information O ∗ Instances and results Robustness Conclusions

  10. ORBEL 28 A picture : maximization Th. PIRONET HEC-ULg QuantOM O ∗ T Outlines VPI VTI Multi-period O ∗ H 100% Vehicle Assignment VAI An example Stochastic E ∗ Bounds VSS Algorithms Simulation VMPM EVS Instances and results Robustness Conclusions O ∗ RH 0% Problem : Found E ∗ the optimal policy

  11. ORBEL 28 Algorithms : single or multiple scenarios Th. PIRONET approaches HEC-ULg QuantOM Approximations of E ∗ Outlines Multi-period Single scenarios (Mean, Modal, ” Optimist” , Dedicated) Vehicle Multiple Scenarios Approaches (MSA) Assignment An example ◮ Consensus (Cs) : Stochastic 1. Solve N scenarios Bounds 2. Create a new solution with frequent decisions Algorithms ◮ Restricted Expectation (RE) : Solve scenarios i,j and Simulation cross-evaluate action i over scenarios j Instances and results 1. Insert actions of solution i in scenario j Robustness 2. Scenarios i � = j ( i , j ∈ N ) ⇒ Solutions (i in j) Conclusions 3. Cumulated value of Solutions (i in j) 4. Select the best action i ◮ Subtree : Solve ST scenarios and add non-anticipativity constraints (Linear Relaxation = Optimal in practice)

  12. ORBEL 28 Statistical validation and Biases Th. PIRONET HEC-ULg Statistical validation : QuantOM How to compare Policy 1 with Policy 2 values ?( µ 1 , µ 2 ) Outlines Outclassment = significant statistical difference of means Multi-period Vehicle ” Paired sample comparison” Assignment Hypothesis : µ 1 � = µ 2 , µ 1 > µ 2 ? An example Solve 30 scenarios by instance Stochastic Normality check, confidence level, Z-test Bounds Algorithms Warm-up and End of horizon biases : Simulation Warm-up : remove H periods Instances and results End of horizon : T long, unit per period Robustness Conclusions

  13. ORBEL 28 Instances and Results Th. PIRONET HEC-ULg 10 Trucks, 10- 15-20 -25 Cities, 150 -200 Loads, 20P ( RH = 1, QuantOM H = 4) stochasticity linked to city sizes, Subtree (30 scenarios) Outlines Info LB EVS UB Multi-period Inst./Alg. O ∗ O ∗ EVS Cs ST 30 O ∗ T RH H 5-15-25 A 222.0 0 73.6 80.0 79.2 100 Vehicle 6-15-25 A 156.1 0 78.6 90.8 89.7 100 Assignment 7-15-25 A 171.0 0 57.2 68.0 70.7 100 8-15-25 A 187.3 0 54.3 13.8 53.4 100 An example 5-15-25 B 153.1 0 57.7 61.2 81.6 100 Stochastic 6-15-25 B 165.7 0 55.8 42.8 60.3 100 7-15-25 B 194.7 0 56.5 60.4 61.0 100 Bounds 8-15-25 B 201.4 0 86.7 60.8 100.0 100 5-15-25 C 192.4 0 64.1 53.8 78.8 100 Algorithms 6-15-25 C 125.9 0 62.7 78.3 88.0 100 Simulation 7-15-25 C 179.2 0 63.9 49.6 70.4 100 8-15-25 C 192.0 0 47.0 20.0 63.5 100 Instances and 5-20-25 A 195.1 0 63.9 45.2 65.9 100 results 6-20-25 A 153.8 0 52.1 54.4 74.3 100 7-20-25 A 253.9 0 38.6 32.1 44.5 100 Robustness 8-20-25 A 225.7 0 7.3 -36.5 21.9 100 5-20-25 B 141.9 0 62.9 33.2 68.4 100 Conclusions 6-20-25 B 147.4 0 62.7 53.4 74.2 100 7-20-25 B 176.7 0 52.1 52.7 66.1 100 8-20-25 B 165.1 0 49.8 25.6 54.2 100 5-20-25 C 171.7 0 51.4 61.2 67.7 100 6-20-25 C 215.3 0 39.1 23.6 56.1 100 7-20-25 C 142.9 0 53.6 54.0 61.3 100 8-20-25 C 150.3 0 67.3 41.7 71.3 100 Average 178.4 0 56.6 46.7 67.6 100

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