• Cost of traveling from node

i

to node

j - c ij

• Demand per node

d i (i

• Vehicle capacity

C

• Number of vehicles

K



N-{0})

Find

• A set of at most

K

vehicle routes of total minimum cost such that – Every route starts and ends at the depot, – Each customer is visited exactly once, – The sum of the demands in each vehicle route does not exceed the vehicle’s capacity Slide  95

ADVANCED METAHEURISTICS Mathematical formulation for CVRP: INTRODUCTION - CVRP

Poznan University of Technology .

r(S)

= lower bound on the number of trucks required to service If

Problem

.

, then we have the

Multiple Traveling Salesman

Alternatively, if the edge costs are all zero, then we have the

ADVANCED METAHEURISTICS ITYPES OF METACHEURISTIC FOR CVRP

Poznan University of Technology

Four main types of metaheuristic that have been applied to the VRP:

• Simulated Annealing (SA) • Tabu Search (TS) • Genetic Algorithm (GA) • Ant Systems (AS)

Slide  97

• • • Features: Much more involved More successful Uses a better starting solution • some parameters are adjusted in a trial phase • Richer solution neighborhoods are explored • Cooling schedule is more sophisticated

Slide  98 Poznan University of Technology

Osman’s Simulated Annealing Algorithms TYPES OF METACHEURISTIC FOR CVRP

Poznan University of Technology Algorithm: Phase 1. Descent algorithm.

Step 1. (initial solution). Generate an initial solution by means of the Clarke and Wright algorithm.

Step 2. (descent). Search the solution space using the  -interchange scheme. Implement an improvement as soon as it is identified. Stop whenever an entire neighborhood exploration yields no impovement.

Phase 2. Simulated Annealing Search Step 1.(initial solution). Use as a starting solution the incumbent obtained at he end of Phase 1, or a solution produced by the Clarke and Wright algorithm.

Preform a complete neighborhood search using  -interchange generation mechanism without, however, implementing any move.

Record D max and D min , the largest and the smallest absolute changes in the objective function and compute  , the number of feasible (potential exchanges.

Slide  99

Osman’s Simulated Annealing Algorithms TYPES OF METACHEURISTIC FOR CVRP

Algorithm :

Phase 2.

Step 2. (next solution). Explore the neighborhood of x t  -interchange . using Step 3. (temperature update). Occasional increment rule: if  =1, set  t+1 :=max {  t /2,  *},  :=0 and k:=k+1 Normal decrement rule: if  =0, set  t+1 +=  t /[(n  +n  t) D max D min ]. Set t:=t+1. If k=k 3, stop.

Otherwise, go to step 2.

Poznan University of Technology Slide  100

ADVANCED METAHEURISTICS Tabu Search (TS) TYPES OF METACHEURISTIC FOR CVRP

– Two Early Tabu Search Algorithms – Osman’s Tabu Search Algorithms – Taburoute – Taillard’s Algorithm – Xu and Kelly’s Algorithm – Rego and Roucairol’s Algorithms – Barbarosoglu and Ozgur’s Algorithm – Adaptive Memory Procedure of Rochat and Taillard – Granular Tabu Search of Toth and Vigo Slide  101 Poznan University of Technology

ADVANCED METAHEURISTICS Tabu Search (TS) TYPES OF METACHEURISTIC FOR CVRP

Poznan University of Technology Taburoute – features:  the neighbourhood structure is defined by all solutions that can be reached from current solution by removing a vertex from its current route and inserting it into another route containing on of its p nearest neighbours using GENI (Generalized Insertion for the TSP. This may result in elimination in g an existing route or in creating new one  Search process examines solutions that may be infeasible with respect to the capacity or maximum route lengh constraints  Does not use a tabu list but instead uses random tabu tags.

 Uses diversification strategy Slide  102

Slide  103 Poznan University of Technology

Evolver nad PSP-OptQuest

Evolver Genetic algorithm optimization for Microsoft Excel

Poznan University of Technology

1. The application of powerful genetic algorithm based (GA) optimization techniques, can find optimal solutions to problems which are "unsolvable" for standard linear and non linear optimizers.

2. Add-in for Microsoft Excel.

3. Requires no knowledge of programming or GA theory 4. By Palisade Corporation

Slide  104

Evolver Genetic algorithm optimization for Microsoft Excel

Poznan University of Technology Slide  105

Evolver Adjustable Cells (options) Solving Methods: grouping, order, recipe, budget, project, and schedule.

•

The “Recipe” and “Order” solving methods are the most popular and they can be used together to solve complex combinatorial problems

•

The “Recipe” method treats each variable as an ingredient in a recipe, trying to find the “best mix” by changing each variable’s value independently.

•

In contrast, the “Order” solving method swaps values between variables, shuffling the original values to find the “best order.”

Slide  106 Poznan University of Technology

Crassover and Mutation Rate

Evolver Optimization Operators (Genetic operators)

•

Linear Operators

– Designed to solve problems where the optimal solution lies on the boundary of the search space defined by the constraints. This mutation and crossover operator pair is well suited for solving linear optimization problems.

•

Boundary Mutation

– Designed to Quickly optimize variables that affect the result in a monotonic fashion and can be set to the extremes of their range without violating constraints.

Slide  107 Poznan University of Technology

Evolver Optimization Operators (Genetic operators)

Poznan University of Technology •

Cauchy Mutation

– Designed to produce small changes in variables most of the time, but can occasionally generate large changes. •

Non-uniform Mutation

– Produces smaller and smaller mutations as more trials are calculated. This allows Evolver to “fine tune” answers.

•

Arithmetic Crossover

– Creates new offspring by arithmetically combining the two parents (as opposed to swapping genes).

•

Heuristic Crossover

– Uses values produced by the parents to determine how the offspring is produced. Searches in the most promising direction and provides fine local tuning.

Slide  108

Evolver Watcher Evolver Watcher

is responsible for regulating and reporting on all Evolver activity. If you are running applications other than Excel that also use Evolver, such as custom applications, the populations they create will also appear in Evolver Watcher’s population list.

Slide  109 Poznan University of Technology

Premium Solver Platform (PSP) – OptQuest Engine Tabu Search algorithm optimization for Microsoft Excel

Poznan University of Technology

1. The application of powerful tabu search optimization techniques, can find optimal solutions to problems which are "unsolvable" for standard linear and non-linear optimizers.

2. Add-in for Microsoft Excel.

3. Requires no knowledge of programming or TS theory 4. By Frontline Systems Inc.

Slide  110

Solver parametrs

Poznan University of Technology Slide  111

Engine e.g. OptQuest

Solver parametrs – OptQuest Engine

• • • • • • • Max Time Solution Iterations Precision (Obj Fun) Precision (Dec Var) Population Size Bounduary Freq Use same sequence of random numbers with seed • Solve Without Integer Constraints • Check for Duplicated Solutions • • Bypass Solver Raports Assume Non-Negative • Show Iteration Results Slide  112 Poznan University of Technology

OptQuest Engine vs. Evolver

Poznan University of Technology Slide  113

Case study I – optimization by Evolver 1. Fleet management problem in the road transportation company (4 old trucks; 16 months) 2. Mathematical model

Decision variables 

x ij

 1  0 truck

i

is used in the period

j

otherwise Poznan University of Technology Slide  114

Case study I – optimization by Evolver

Poznan University of Technology

2. Mathematical model

   The number of vehicles replaced per time period is limited (e.g. 1 per quarter) The vehicle withdrawn from utilization can not be used again The number of vehicles is constant in the time horizon Slide  115

Case study I – optimization by Evolver

Poznan University of Technology

3. Mathematical model

 Criteria – Total maintenance cost (PLN) min

FC

 

i j x ij c ij w ij

Cost ratio Cost Decision variables

Slide  116

Case study I – optimization by Evolver

4. Evolver Options

    Solving method – recipe Crassover Rate – 0,5 Mutation Rate – 0,1 Population Size – 100   Random Number Seed – Generated Randomly Update the Display – never  Valid Trails is Less Than – 0,1% Slide  117 Poznan University of Technology

Case study I – optimization by Evolver

Poznan University of Technology Truck Truck 1 Truck 2 Truck 3 Truck 4 New truck 1 New truck 2 New truck 3 New truck 4

5. Results

•

Basic solution 339 800 PLN

1 1 1 1 1 0 0 0 0 2 1 1 1 1 0 0 0 0 3 1 1 1 1 0 0 0 0 4 1 1 1 1 0 0 0 0 5 1 1 1 1 0 0 0 0 6 1 1 1 1 0 0 0 0 7 1 1 1 1 0 0 0 0 Quarter 8 9 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 10 11 12 13 14 15 16 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 0 1 1 1

• •

Optimization by Evolver 116 436 PLN

Slide  118 Truck Truck 1 Truck 2 Truck 3 Truck 4 New truck 1 New truck 2 New truck 3 New truck 4 Quarter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

Case study II – optimization by OptQuest

Poznan University of Technology

1.

2.

Feet composition problem in the fuel transportation/distribution company



Mathematical model

Decision variables

x ij

 for  

j

 1  asigning

x x ij ij

number    a , vehicle of to customer assigned vehicle i ,    for  

j

 1 assigning

x ij

 the number vehicle of the chamber assigned for transp chamber i , orting fuel type j  2,3..., J to customer

i

, Slide  119

Case study II – optimization by OptQuest



2. Mathematical model

– Constraints Capacity of the fuel chambers in each vehicle 

p



k i I J

  

j

1 2 

P ij

such that

–

x i

1 

p

i

x ij

Eliminating fuel mix in 1 fuel chamber 

k

 

PK pk



k j J

  2   1  0   if if

x ij



k

, for otherwise a minimum 1

i

   ,  1 Poznan University of Technology Slide  120

Case study II – optimization by OptQuest

Poznan University of Technology

2. Mathematical model

 – Constraints Satisfying demand for fuel

Area A

i



1



x i

1 

P

Area B 

i j

  1 1 

x ij



K

– Working time for vehicels/drivers 

p



LD śr



LR śr

 

LKl p



1

 

LP śr

 

LKr p



V ep



T

max Slide  121

Case study II – optimization by OptQuest

Poznan University of Technology

2. Mathematical model

 – Criteria Total distribution costs [PLN]

Min FC

1 

P

 1 

KZ p

 

LD śr



LR śr

 

LKl p



1

 

LP śr

 

KS p



[

PLN

]

Slide  122

Case study II – optimization by OptQuest

5. OptQuest Options

      Max Time – 200 s Iterations – 10 000 Precision (Obj Fun) – 0,0001 Precision (Dec Var) – 0,0001 Population Size – 75 Bounduary Freq – 0,25 Poznan University of Technology Slide  123

Case study II – optimization by OptQuest

3. OptQuest Options



Use same sequence of random numbers with seed – inactive

   

Solve Without Integer Constraints – inactive Check for Duplicated Solutions – active Bypass Solver Raports – inactive Assume Non-Negative – active



Show Iteration Results - inactive

Slide  124 Poznan University of Technology

Case study II – optimization by OptQuest 6.

Results Number of a vehicle

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Vehicles (4 – cells) Vehicles (8 – cells) Value of criterion [zł] Slide  125 Poznan University of Technology

Basic solution

YES YES YES YES YES YES YES YES YES YES NO NO NO NO NO NO 7 3 10 985

Optimal solution

YES YES NO NO NO NO NO NO NO NO NO NO NO NO NO YES 3 0 5 810

Case study III – optimization by Evolver

Poznan University of Technology

1. Traveling Salesman problem Find the best way to visit all 68 cities with the least amount of traveling. The salesman must always return back to the first city to form a complete loop.

table of distances Slide  126

Case study III – optimization by Evolver

Traveling Salesman problem – NP complete Problem !!!

Poznan University of Technology

69 towns =

1 , 82  10 94

solutions

Slide  127

Case study III – optimization by Evolver

2. Evolver Options

      

Solving method – order Crassover Rate – 0,5 Mutation Rate – 0,1 Population Size – 100 Random Number Seed – Generated Randomly Update the Display – never Valid Trails is Less Than – 0,1%

Slide  128 Poznan University of Technology

Case study III – optimization by Evolver RESULTS (Raport) Valid Trials Total Recalcs Original Value + soft constraint penalties = result Best Value Found + soft constraint penalties = result Occurred on trial Time to find this value Stopped Because Optimization Started At Optimization Finished At Total Optimization Time

Slide  129 182933 291505 25479 0 25479 7824 0 7824 169321 00:09:32 Halted by User 11:43:40 11:54:31 00:10:31 Poznan University of Technology Basic Solution 1 2 3 4 5 … 13 14 15 16 17 … 29 30 … 65 66 67 68 1 Best Solution 1 58 21 23 13 … 30 28 61 18 17 … 48 35 … 65 8 36 27 1

MULTIPLE OBJECTIVE METAHEURISTICS

ADVANCED METAHEURISTICS MULTIOBJECTIVE APPROACH

Poznan University of Technology

LOCAL SEARCH BASED

MOSA PSA TS - MOTS

HYBRID

PMA MOGLS

POPULATION BASED

VEGA Slide  131

ADVANCED METAHEURISTICS LOCAL SEARCH – BASED METAHEURISTICS

Poznan University of Technology 

MOSA

  A PROTOTYPE OF A MULTIOBJECTIVE S.A. METHOD FOR A SET OF WEIGHTING VECTORS  A S.A. PROCEDURE IS PERFORMED ON THE PROBLEM SCALORIZED WITH THE WEIGHTED SUM METHOD      STARTING SOLUTION

x

IS CHOSEN A SOLUTION x’ IN SOME NEIGHBORHOOD IF x IS SELECTED AND COMPARE WITH x IF f(x’)  f(x) OR

k P

  1 

k f k

(

x

' ) 

k P

  1 

k

 YES: x’ IS ACCEPTED AS A BETTER SOLUTION

f k

(

x

)  NO: x’ IS ACCEPTED WITH SOME PROBABILITY RESULT: SET OF POTETIALLY EFFICIENT SOLUTION IN DIRECTION  AFTER PROCEDURE FOR ALL   SETS OF POTENTIALLY EFFICEINT SOLUTION ARE MERGED Slide  132

ADVANCED METAHEURISTICS LOCAL SEARCH – BASED METAHEURISTICS

Poznan University of Technology 

MOTS

 BASED ON NEIGBORHOOD PRINCIPLES       STARTING POINT IS AN INITIAL SOLUTION

x

NEW SOLUTION

x’

IS SOME NEIGBORHOOD OF

x

IS SELECTED, BUT IT IS VASED ON SELECTION USING A WEIGTED DISTANCE FROM X POINT y u IN ORDER TO OVERCOME LOCAL OPTIMA, SOME SOLUTIONS IN THE NEIGHBOURHOOD ARE THE CLARED AS ”TABU” „”TABU” STATUS DEPENDS ON THE ITARATIONS PERFORM SO FAR FOR EACH WEIGHT SINGLE OBJECTIVE TS IS PERFORED AT THE END OF THE ALGORITHM POTENTIALI EFFICENT SETS OF SOLUTIONS ARE MERGED Slide  133

ADVANCED METAHEURISTICS LOCAL SEARCH – BASED METAHEURISTICS

Poznan University of Technology 

POPULATION – BASED METAHEURISTICS

 MAITAIN A WHOLE SET OF SOLUTIONS (THE POPULATIONS)  TRY TO EVOLVE THE POPULATION TOWARDS TO THE PARETO SET  MANY DIFFERENT TECHNIQUES (GA, EVOLUTIONARY ALGORITHMS) TO EVALUATE THE FITNESS OF IDIVIDUAL SOLUTIONS IN MULTIOBJECTIVE CONTEXT Slide  134

Pareto simulated annealing

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Poznan University of Technology

Similarities to single objective simulated annealing & genetic algorithms

PSA SA GA New Concepts

Slide  136

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Poznan University of Technology

Similarities to single objective simulated annealing & genetic algorithms

PSA SA GA New Concepts

• The concept of the neighborhood • Probabilistic acceptance of new neighbourhood solutions (with a certain probability) • Dependence of the acceptance probability on a parameter (temperature) • The scheme of the temperature changes Slide  137

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Poznan University of Technology

Similarities to single objective simulated annealing & genetic algorithms

PSA SA GA New Concepts

• The use of a sample (population) of solutions; each of them exploring the search space according to SA rules • The solutions may be treated as independent agents, exchanging information about their positions. A separate weight vector is associated with each of the generating solutions. Slide  138

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Poznan University of Technology

Similarities to single objective simulated annealing & genetic algorithms

PSA SA GA New Concepts

• The use of scalarizing functions locally aggregating multiple criteria functions and scalarizing functions based probabilities for acceptance of new neighborhood solutions • Automatic modifications of weights of particular objectives in each iteration according to a certain rule. The rule for updating the generating solutions’ weight vectors aims at assuring dispersion of the solutions over all regions of the nondominated set Slide  139

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Multiple objective acceptance rules

Poznan University of Technology 

Single objective SA

 

New solution no worse than current solution – acceptance – P=1 Otherwise – acceptance P<1



Each solution x can be modified (replaced) by accepting a randomly generated solution from its neighbourhood. The new solution is acceptable with some probability



PSA uses the concept of multiple objective acceptance rules (P. Serafini – 1994)

Slide  140

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING MOA rules

Poznan University of Technology

In the multiple objective case one of the following exclusive situations may occur

 y dominates or is equal to x (new solution is not worse than the current solution – P = 1 )  y is dominated by x (new solution is worse than the current solution – P < 1 )  y is non dominated with respect to x (ambiguous situation – P = ? )

y – new solution, x – current solution

Slide  141

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING The probability of accepting solution y x )

Poznan University of Technology

(compared with based on MOA rules. The case of two maximized objectives.

y

Criterion 2 (Max)

y

Slide  142

y

Criterion 1 (Max)

y

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Poznan University of Technology

y is non dominated with respect to x



Rule could be interpreted as a local aggregation of all objectives with the weighted Tchebycheff scalarizing function with reference point at f(x)

P

– probability of accepting the new solution y

T

– temperature

Λ

–weight vector Slide  143

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING y is non dominated with respect to x Graphical illustration of the rule

Poznan University of Technology Slide  144

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Poznan University of Technology

y is non dominated with respect to x The rule may be also interpreted as a local aggregation of all objectives with a weighted linear scalarizing function

P

– probability of accepting the new solution y

T

– temperature

Λ

–weight vector Slide  145

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING y is non dominated with respect to x Graphical illustration of the rule

Poznan University of Technology Slide  146

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Management of the population of generating solutions

Poznan University of Technology 

The weights used in the acceptance rules allow to influence on the direction of search in the objective space for particular generating solutions

 

The higher the weight associated with a given objective the higher the influence of this objective on the probability of acceptance of new solutions and the higher the pressure towards improvement of that objective Controling the weight vectors the method may „push” generating solutions into desired directions in the decision space



PSA controls the weight vectors associated with particular generated solution in order to achieve a form of repulsion between the solutions



The weight vector associated with a given generating solution x is modified in order to increase the probability of moving x away from its closest neighbor x’ in the generating sample



This is obtained by increasing the weights of the objectives on which x is better than x’ and decreasing the weights of the objective on which x is worse than

x’

Slide  147

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Poznan University of Technology

Management of the population of generating solutions



The Euclidean distance between solutions in the space of normalized objectives is used



The closest naighbor has to be non-dominated with respect to x. If there is no generating solution that meets this requirement each weight is either increased or decreased with probability = 0.5



Repulsion mechanism never repulses the generating solutions from the nondominated set



During the computational process some generating solutions may get stacked in regions far away from the nondominated set.

  If a generating solution is dominated by at least one other generating solution for a number od iterations it is considered not promising and replaced by a solution from the set of potentially Pareto-optimal solutions, having maximum distance to the closest generating solution The idea is to move the generating solution to a poorly explored region of the nondominated set Slide  148

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Poznan University of Technology

Updating the set of potentially Pareto – optimal solutions



At the beginning of the computational procedure the set of Pareto-opitmal solutions PP is empty



PP is updated every time when a new solution is generated

Update:

• Add

f(x)

to PP if no point in PP dominates

f(x)

• Remove from PP all points dominated by

f(x)

Slide  149

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Poznan University of Technology

Updating the set of potentially Pareto – optimal solutions

Updating could be very time consuming.

The ways to avoid this disadvantage are: • A new solution

y

obtained from

x

should be used to update PP set only when it is not dominated by

x

• New solution

y

may be added to PP set only if they differ enough from all solutions in PP set (threshold – minimum Euclidean distance) • Neglect updating PP in a number of starting iterations (solutions added to PP in early iterations had good chances to be removed) • Using the data structure called quad trees to accelerate the process of updating PP (4 and more objectives) Slide  150

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Poznan University of Technology

Using partial preference information



The information concerning the DM’s preferences may help focusing on the interesting region of the nondominated set (e.g. objective 1 is more important than objective 2, solutions having value below a certain threshold on objective 3 are not interesting)



The most natural way of taking into account partial preference information in PSA is to express it in the form of constraints in the weight space

Slide  151

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Poznan University of Technology

Basic version of PSA algorithm

Slide  152

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Poznan University of Technology

APPLICATION OF PSA

MULTIOBJECTIVE BUS DRIVER’S SCHEDULING

Slide  153

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Contents



Introduction



Problem definition and mathematical formulation



Solution procedure



Computational experiments



Conclusions

Poznan University of Technology Slide  154

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Combinatorial Problem

Poznan University of Technology

Specific Crew Scheduling Problem BUS DRIVER’S SCHEDULING PROBLEM Set of duties for bus drivers

Slide  155

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Multiple objective formulation

Poznan University of Technology

STAKEHOLDERS

Trasportation company owner – cost oriented

Bus driver – convenience oriented

Slide  156

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Poznan University of Technology

Transportation company



Publicly owned, inter-city passenger transportation company, located in Poznan, Poland



The company provides medium – haul transportation services in Western Poland (Wielkopolska); operates 7 days a week



Annual sales 35 mln zl = 10 mln Euro; 125 000 vkm/ week; 62% Night transportation jobs and 38% - Local transportation jobs



3 categories of duties

   Local transportation jobs – L (31) – 9040 km (avg. 292 km) Night transportation jobs – N (20) – 5720 km (avg. 286 km) Additional tasks – P (4) 

Fleet: 103 buses (Autosan and Jelcz) in different age and technical condition



Labor force – 98 employees, incl. 52 bus drivers

Slide  157

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Poznan University of Technology Balance the workload of drivers

Objectives

Assure fair assignment of duties Slide  158

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Optimization goals

Poznan University of Technology Number of L, N, P should be balanced Number of days-off (including Sundays and Saturdays) sholud be grouped Days-off should be grouped Slide  159

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Poznan University of Technology

Decision variables

x ij

   

L N P W

if the i-th bus driver carries out the local transportation job on j-th day if the i-th bus driver carries out the night transportation job on j-th day if the i-th bus driver carries out the additional task on j-th day if the i-th bus driver has a day-off on j-th day

Slide  160

i = 1, ..., I - BUS DRIVER INDEX j = 1, ..., J - DAY INDEX

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Mathematical formulation

Poznan University of Technology

GOAL 1 - AVG. DEVIATION - LOCAL TRANSPORTATION JOBS

Min

f

1 

i I

  1

S L



j J

  1

x Lij J



100 %

I

,

where:

S L

Slide  161 

i I J

   1 1

j x Lij IJ

 100 %

x Lij

  1 if   0 if

x ij



L

otherwise

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Mathematical formulation

Poznan University of Technology

GOAL 2 - AVG. DEVIATION - NIGHT TRANSPORTATION JOBS

i I

  1

S N



j J

  1

x Nij J



100 % Min

f

2 

I

,

where:

S N

Slide  162 

i I J

   1 1

j x Nij IJ



100 %

x Nij

 

1 if

 

0 if

x ij



N

otherwise

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Mathematical formulation

Poznan University of Technology

GOAL 3 - AVG. DEVIATION – ADDITIONAL TASKS

i I

  1

S P



j J

  1

x Pij J



100 % Min

f

3 

I

,

where:

S P



i I J

   1 1

j x Pij IJ

 100 % Slide  163

x Pij

 

1 if

 

0 if

x ij



P

otherwise

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Poznan University of Technology

Mathematical formulation GOAL 4 – POINTS AWARDING THE AGGREGATION OF DAYS-OFF

Max

f

4 

i

1

j J I

   1

x W

,

where:

x W

     

0 if {x} > 0 1 if {x} = 0 3 if (x

Wij

+x

Wij+1

) = 0 and (x

Wij+1

+x

Wij+2

) = 0 6 if (x

Wij

+x

Wij+1

) = 0 and (x

Wij+1

+x

Wij+2

) = 0 and (x

Wij+2

+x

Wij+3

) = 0 X = {(x

Wij

+x

Wij+1

) + (x

Wij+1

+x

Wij+2

) + ... + (x

Win-1

+x

Win

)} for n = J

x W ij

 

0 if

 

1 if

x ij



W

otherwise

Slide  164

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Poznan University of Technology

Mathematical formulation GOAL 5 - AVG. DEVIATION – AGGREGATED DAYS-OFF



j J

  1

x W J

Min

f

5 

i I

  1

S W

 100 % ,

I

where:

x W S W

    

i I j J

   1 1

IJ

0`if {x} > 0

x W



100 %

1if {x} = 0 3if (x

Wij

+x

Wij+1

) = 0 and (x

Wij+1

+x

Wij+2

) = 0 6if (x

Wij

+x

Wij+1

) = 0 and (x

Wij+1

+x

Wij+2

) = 0 and (x

Wij+2

+x

Wij+3

) = 0

Slide  165

X = {(x

Wij

+x

Wij+1

) + (x

Wij+1

+x

Wij+2

) + ... + (x

Win-1

+x

Win

)} for n = J

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Mathematical formulation GOAL 6 - AVG. DEVIATION – SATURDAYS

i I

  1

S S



j J

  1

x Sij J

 100 % Min

f

6 

I

,

where:

Poznan University of Technology

S S



i I J

   1 1

j x Sij



100 %

IJ

Slide  166

x Sij

  1 if   0 if

x ij



W

otherwise

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Mathematical formulation GOAL 7 - AVG. DEVIATION – SUNDAYS

i I

  1

S D



j J

  1

x Dij J

 100 % Min

f

7 

I

, Poznan University of Technology

where:

S D

Slide  167 

i I J

   1 1

j x Dij

 100 %

IJ x Dij

    1 if 0 if

x ij



W

otherwise

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Length of the scheduling period Poznan University of Technology Number of drivers required for each day Types of contracts (full time, part-time) Expected absences & preferences Slide  168 Constraints Number of drivers available on each day Labour code regulations

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Solution procedure

Poznan University of Technology

Generation of solutions

PSA Multiple objective metaheuristic procedure Generation of a sample of schedules being a good approximation of the whole set of a non-dominated solution LBS-D

Review of solution

Slide  169 Evaluation of schedules according to DM’s preferences Penetration of different regions of the sample Final selection of one solution

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Customization

Poznan University of Technology Slide  170

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Poznan University of Technology

Results of the computational experiments

•

30000 feasible solutions, including 2062 PP

•

Each solution – assignment metrix (40 drivers x 30 days); allocation of jobs to the drivers

•

Computational time - 4 minutes; PC Pentium 1GHz

•

In the experiment 16 generating solutions have been used and 1856250 steps (moves) of the procedure have been preformed

•

The exemplary solution - 4366

Slide  171

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING

Exemplary schedule

Poznan University of Technology Slide  172

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING LBS-D

Poznan University of Technology 

Interactive procedure for multiple objective mathematical programming problem



User – friendly interface



Graphical facilities



Phases of decision alternating with phases of computation



Searches for a compromise solution in the neighborhood of the selected solution (middle point)



The search process is similar to projecting light onto the solution set; it is based on the definition of the DM’s preferences (aspiration levels; reference point)

Slide  173

Slide  174 Poznan University of Technology

General Scheme LBS

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Decision phase Fixing points z * and z*

Poznan University of Technology Slide  175

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Decision phase



DM’s preferences (q, p, v)



Procedure finds starting middle points

Poznan University of Technology Slide  176

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Decision phase

An outranking neighborhood is constructed Poznan University of Technology Slide  177

Acceptance of worse values Aspiration Selection

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Graphical analysis

Poznan University of Technology Slide  178

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Decision phase

Poznan University of Technology

Neighbor 3 selected as a new middle point and new neighbors are generated Final solution

Slide  179

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING The most satisfactory schedule

Poznan University of Technology Slide  180

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Comparison of solutions

Poznan University of Technology Improvement: Slide  181 3% to 24% on particular objectives

f 2

 24% ;

f 4

 12%

ADVANCED METAHEURISTICS PARETO SIMULATED ANNEALING Conclusions



Improvement of the real life solution



Flexibility



Good quality results



Efficiency of work

Poznan University of Technology Slide  182

Multiple objective genetic local search

MOGLS

ADVANCED METAHEURISTICS MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH Contents

Poznan University of Technology 

General idea of hybrid algorithms



Single objective genetic local search algorithm



Multiple objective genetic local search algorithm

Slide  184

ADVANCED METAHEURISTICS MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH General idea

Poznan University of Technology Recombination operators Local search MOGLS Slide  185 MCDM

ADVANCED METAHEURISTICS MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH General idea

Poznan University of Technology 

The algorithm hybridizes recombination operations with local search (typical single criterion algorithm)



The idea of algorithm is more general, other local huristic methods can be used



MOGLS is a multiple objective version, contains all aspects of MCDM

Slide  186

ADVANCED METAHEURISTICS MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH Motivation for creating MOGLS

Poznan University of Technology 

Great success of other hybrid genetic meta-heuristics

HGAs Slide  187 Memetic algorithms Genetic Local Search

ADVANCED METAHEURISTICS MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH Hybrid Genetic Algorithms (HGAs)

Poznan University of Technology 

Standard genetic/evolutionary algorithms working on a reduced set of solutions



Local heuristics = recombination operator (like crossover)



The efficiency grows up, because the search space is smaller



Conclusion: the local optima can be achieved in very efficient way



HGAs may be also interpreted as a modification of multiple start local search herusitcs with random initial solutions

Slide  188

ADVANCED METAHEURISTICS MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH Single objective GLS algorithm

Poznan University of Technology 

Very good efficiency - > motivation to create multiple objective version



The algorithm stops if the current population was not changed i K subsequent iterations (further improvement is not possible)

Slide  189

ADVANCED METAHEURISTICS MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH Single objective GLS algorithm

Poznan University of Technology Slide  190

ADVANCED METAHEURISTICS MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH

Poznan University of Technology

Test of three single objective methods (for the same instances)

Slide  191

ADVANCED METAHEURISTICS MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH Test results on the graph

Poznan University of Technology GLS generated best results Slide  192

Slide  193

Application of MSLS GA, GLS

Poznan University of Technology

Vehicle Routing Problem in a Road Transportation Company – case study

Vehicle Routing Problem in a Road Transportation Company

Poznan University of Technology 

Road, freight transportation & logistic company, located in Warsaw, Poland



Activities: transportation and logistic services; forwarding; customs clearence, national and international freight transportation, maintenace / service of vehicles (MAN)



Annual sales – 106 mln zl = 35 mlnEuro; 170 employees



Fleet – 230 vehicles (tractors and trailors); capacity 20 – 30 T



Transportation routes – 100 – 5000 km; 20 – 25 transportation jobs / day



Historical data – May – July 2005; 700 transportation jobs; 200 customers; 70% jobs generated by 20 customers (10%); 650 locations – 180 very important



Analyzed case: 90 transportation jobs; 30 vehicles

Slide  194

Vehicle Routing Problem in a Road Transportation Company

Poznan University of Technology 

The decision problem defined as a single objective optimization problem

  Correlation between criteria (time, cost, profit, distance) Multiple vehicle, pick-up and delivery vehicle routing problem with time windows (m-VPDVRPwTW) 

Not many reports about the solution procedures for this problem

 J. Desrosiers and others – small instance – application of Branch and Bound Slide  195

Mathematical Formulation of the Problem Criterion



Maximal profit in the time horizon [PLN]

Poznan University of Technology Max

z

 

Max

Z

 

r

 1          

W R

 

w

 1

i w i r dod

 

wr

 

c r dod

  

b B

  1 

wkm c br

 

b B

  1   

i r dod



c r dod

  

b B

  1 (

c wkm br

(

c wkm br

 

l br l br r



R

   1 

l br t r dod z b B

  1 

r c b prac c b prac



c b prac



t b prac



t b prac

)

t b prac

 )

b B

  1    

b B

  1

c b prac c b prac

 

t b prac t b prac

  (

T

  

t r

    )       Slide  196

Mathematical Formulation of the Problem Constarints

Poznan University of Technology 

Each order must be either completely fulfilled (by 1 or more vehicles) or rejected



Each vehicle types and loads must match



Capacity dimensions of the vehicle should exceed weight/dimensions of the load



Loading and unloading must be carried out in concrete time windows



Working time for drivers is defined by the labor code

Slide  197

Mathematical experiment Visualization of the optimal solution (GLS – 20 iterations)

Poznan University of Technology Slide  198

The computational efficiency MSLS vs. GLS vs. GA;Time of solution

Poznan University of Technology

PC Pentium IV/2,8 GHz; 30 vehicles and 90 orders

Slide  199

The computational efficiency MSLS vs. GLS vs. GA;Time of solution

Poznan University of Technology

PC Pentium III 750 MHz; 30 vehicles and 90 orders

Slide  200

Intuitive vs. Computer planning

Poznan University of Technology Parameter Intuitive planning Forwarder 1 Forwarder 2 Time of planning [min] Profit [PLN] ~ 230 40000 ~ 190 PC; program VR MSLS ~ 40 GA ~ 40 GLS ~ 40 36800 38500 38500 42200 Slide  201

Conclusion

Poznan University of Technology 

The real-life VRP is characterized by high computational complexity



GLS is the most efficient metaheuristic algorithm (compared with MSLS & GA)



For practical reasons it is advised to use computers with high computational power to solve VRP



Practical results – computer system VR reduces labor intensity by 80% and improves profits by 5.5%

Slide  202

ADVANCED METAHEURISTICS MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH MOGLS

Poznan University of Technology 

Goal



set generate good approximations of the nondominated



Finding the whole nondominated set = finding the optima of all weighted Tchebycheff and all weighted linear scalarizing functions



In fact the goal is a simultaneous optimalization of all Tchebycheff and all weighted linear scalarizing functions



„Optimization” is understood as a tendency of the algorithm to improve values of all scalarizing functions (with normalized weight vectors)

Slide  203

ADVANCED METAHEURISTICS MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH MOGLS

Poznan University of Technology 

MOGLS implements the idea of simultanous optimization of all weighted Tchebycheff, all weighted linear or all composite scalarizing functions with normalized weight vectors by random choice of the scalarizing function optimized in each iteration



MOGLS tries to improve the value of a randomly selected scalarizing function in each iteration



Single iteration consists of a single recombination of a pair of solutions and application of a local heuristic that takes into account the value of the current scalarizing function

Slide  204

ADVANCED METAHEURISTICS MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH

Poznan University of Technology • To draw at random the scalarizing funcion, a normalized weight vetor is drawn at random by the algorithm The algorithm assures that weight vectors are drawn with uniform probability distribution

p(

Λ)

Rand() returns a value from <0,1> Slide  205

ADVANCED METAHEURISTICS MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH General scheme of the MOGLS

Poznan University of Technology Slide  206

ADVANCED METAHEURISTICS MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH Selection of solutions for recombination

Poznan University of Technology 

In single objective GLS the method combines features of two good solutions



MOGLS combines features of solutions that are already good on the current scalarizing function



In each iteration MOGLS algorithm constructs a temporary elite population (TEP) composed of K different solutions being the best on the current scalarizing function among all known solutions.



Two different solutions are drawn for recombination from (TEP)



The idea of recombining good solutions is motivated by „global convexity”

– In single objective optimization this means – good solutions are similar – In multiple objective optimization – good solutions on a given scalarizing function being close in the objective space are similar Slide  207

ADVANCED METAHEURISTICS MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH Generating the initial set of solutions

Poznan University of Technology 

Construction by applying iteratively the local heuristic to random starting solutions



Local heuristic optimizes the scalarizing functions with randomly generated weight vectors



The number of the initial solutions S is the additional parameter of the method



The method allows to stop generating the initial solutions when the avarage quality of K best solutions in the set of initial solutions over all scalarizing functions is the same as the avarage quality of solutions generated by the local heuristic used for optimization of these functions

Slide  208

ADVANCED METAHEURISTICS MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH Example of generated solutions for TSP

Poznan University of Technology Slide  209

ADVANCED METAHEURISTICS MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH Management of the current set of solutions

Poznan University of Technology 

The idea of storing all solutions in CS is very time and memory consuming for MOGLS



CS is organized as a queue of size KxS (K- number of best solutions, S – number of initial solutions)



In each iteration the newly generated solution is added to the beginning of the queue (if the conditions are met); if it is better than the worst solution in TEP and different form all solutions in TEP



If the size of queue is bigger than KxS the last solution is removed

Slide  210

ADVANCED METAHEURISTICS MULTIPLE OBJECTIVE GENETIC LOCAL SEARCH

Poznan University of Technology

Updating the reference point



The reference point is an important parameter in case of weighted Tchebycheff and composite scalarizing functions



In MOGLS the ideal point (best known values of the objective functions) is used as reference point



The reference point changes in the run of the procedure



The first approximation is obtained by applying local heuristic to optimization of each objective individually.



Normalization of objectives, updating the set of PP, using partial preference information Analogy with PSA

Slide  211

Slide  212 Poznan University of Technology

MOGLS pseudocode

Memetic algorithm and Pareto memetic algorithm

ADVANCED METAHEURISTICS MEMETIC ALGORITHM AND PARETO MEMETIC ALGORITHM Contents



Memetic



Memetic algorithm



Pareto Memetic algorithm

Poznan University of Technology Slide  214

ADVANCED METAHEURISTICS MEMETIC ALGORITHM AND PARETO MEMETIC ALGORITHM

Memetics - Genetics

Meme

Poznan University of Technology “the basic unit of cultural transmission, or imitation” Richard Dawkin , ethologist Slide  215 “an element of culture that may be considered to be passed on by non-genetic means” English Oxford Dictionary

ADVANCED METAHEURISTICS MEMETIC ALGORITHM AND PARETO MEMETIC ALGORITHM

Memetics - Genetics Poznan University of Technology 

Mem is defined per analogy to gen



Evolution is not only based on genetics



Term Memetic algorithm was first used by Moscato in 1989 in the sense of population-based hybrid genetic algorithm with some learing procedures

Slide  216

ADVANCED METAHEURISTICS MEMETIC ALGORITHM AND PARETO MEMETIC ALGORITHM Memetic algorithm

Inspiration Poznan University of Technology

Darwinian natural evolution Dawkins’ conception

Techniques

of a meme

Slide  217 Search algorithm (LS) Evolutionary algorithm (GA)

ADVANCED METAHEURISTICS MEMETIC ALGORITHM AND PARETO MEMETIC ALGORITHM General scheme of the memetic algorithm

Poznan University of Technology 

Initiation: generating an initial population



Iteration (until termination conditions are reached)

 Improvement of current solutions (by local optimalization methods)  Developing of new generation (solutions) by evolutionary algorithm For improvements MA can use any local optimalization method like local search, tabu search or another one Slide  218

ADVANCED METAHEURISTICS MEMETIC ALGORITHM AND PARETO MEMETIC ALGORITHM General scheme of the memetic algorithm

General initial population Select individuals for nest generations Poznan University of Technology Crossover Mutation Local search Population complete?

Enough generations found?

Slide  219

ADVANCED METAHEURISTICS MEMETIC ALGORITHM AND PARETO MEMETIC ALGORITHM Pseudocode of the memetic algorithm

Poznan University of Technology Slide  220

ADVANCED METAHEURISTICS MEMETIC ALGORITHM AND PARETO MEMETIC ALGORITHM Pareto memetic algorithm

Poznan University of Technology 

Author: A. Jaszkiewicz, Poznan University of Technology



Modification of MOGLS (Multiple objective genetic local search)

Slide  221

ADVANCED METAHEURISTICS MEMETIC ALGORITHM AND PARETO MEMETIC ALGORITHM Two stages of the algorithm

Poznan University of Technology

Stage 1:



Initiation



Generation of the first approximation of the ideal point



Generation of the initial set of solutions Stage 2



Probabilistic choice of two solutions



Recombination and improvement

Slide  222

ADVANCED METAHEURISTICS MEMETIC ALGORITHM AND PARETO MEMETIC ALGORITHM Initiation

Poznan University of Technology 

At the beginning a set of Pareto-optimal solutions is empty.

PP:= Ø



The current set of solutions is empty, too.

CS:= Ø

Slide  223

ADVANCED METAHEURISTICS MEMETIC ALGORITHM AND PARETO MEMETIC ALGORITHM

Poznan University of Technology

Generation of the first approximation of the ideal point



Random creation of a possible solution x



Optimalization x to x’ by local heuristic algorithm



Adding x’ to CS



Updating set PP with x’

Slide  224

ADVANCED METAHEURISTICS MEMETIC ALGORITHM AND PARETO MEMETIC ALGORITHM Generation of the initial set of solutions

Poznan University of Technology

• Randoming a weight vector Λ



Random creation of a possible solution x



Optimalization of the scalarizing function (z,.. Λ ) x to x’ by local search



Adding x’ to CS

• Updating set PP with x’

This phase is iterated until stopping condition is met Slide  225

ADVANCED METAHEURISTICS MEMETIC ALGORITHM AND PARETO MEMETIC ALGORITHM Probabilistic choice of two solutions



Randoming a weight vector Λ



Drawing randomly a sample of solutions from CS

Poznan University of Technology Slide  226

ADVANCED METAHEURISTICS MEMETIC ALGORITHM AND PARETO MEMETIC ALGORITHM Recombination and improvement

Poznan University of Technology 

Recombination of the best and second best solution on s(z,..., Λ) – x

1



Optimization of s(z,.. Λ ) x

1

search to x

1

’ by local



Adding x

1

’ to CS and updating PP if x

1

’ is better than the second best solution in a sample

Slide  227

Slide  228 Poznan University of Technology

Pseudo code of PMA

Slide  229

Application of PMA

Poznan University of Technology

Vehicle Assignment Problem in the Bus Transportation Company – case study

Introduction (I)

Poznan University of Technology 

The essence of the vehicle assignment problem (VAP) in a bus transportation company

 transportation companies utilise vehicles (buses) to transport passengers on given routes according to a given timetable  general problem in such a situation is: How to assign particular buses to given routes?



Many formulations of the VAP are known, for example

    linear programming formulations which can be solved with an application of simplex method, network algorithms or assignment method (Cook 1985; Lotfi et al. 1989) linear, integer programming formulations (Löbel 1998, Rushmeier et al. 1997), sometimes transformed into a non-linear, continuous form (Beaujon et al. 1991) formulations based on the queuing theory (Green et al. 1995, Whitt 1992) formulations considering the homogeneous (Beaujon et al. 1991) or a non homogeneous fleet (Ziarati et al. 1999) Slide  230

Introduction (II)

Poznan University of Technology    formulations which combine the VAP with other fleet management problems, such as: fleet sizing (Beaujon et al. 1991) or fleet scheduling (Löbel 1998) formulations referring to specific transportation environments, such as: urban transportation (Löbel 1998), rail transportation (Ziarati et al. 1999) or air transportation (Rushmeier et al. 1997) formulations with a single objective function (all mentioned above) or, sometimes, with a multicriteria objective function (Zeleny 1982) 

Proposed problem formulation

  is expressed in terms of multicriteria, non-linear, integer, mathematical programming determines the optimal assignment of non-homogeneous fleet of buses to a given set of routes in an international passenger transportation company one week time horizon is assumed for the problem analysis  Slide  231

Computational Experiment Decision situation (I)

Poznan University of Technology 

A Polish, passenger transportation company operating on the 17 routes between 34 Polish and 47 European cities is analysed



All the routes are characterised by the following parameters:

     length

S i

between

1818

and

4048

kilometres average number of passengers travelling weekly on particular routes

P i

between

2

and

796

average income per one passenger (ticket price)

p pas i

and

721

PLN* between

188

average load index

w i

fixed cost

k ij

per route

i

between

0.25

and bus

j

and

0.46

between

3 530

and

14 809

PLN / ride

* PLN – Polish New – Polish currency. 1 PLN = 0.24 USD in December 2001

Slide  232

Computational Experiment Decision situation (II)

Poznan University of Technology 

Analysed company utilises a fleet of 30 buses (Hyundai, Neoplan, Scania, Volvo) characterised by:

 vehicle-kilometre cost

k wkm ij

between

1.49

and

2.01

PLN / kilometre  number of seats (capacity)

c j

between

31

and

57

 comfort level

f j

between

3

and

9

points (comfort level ranges from 1 to10 points) Slide  233

Mathematical Formulation of the Problem Input data – model parameters

Poznan University of Technology        

S i

– length of route

i

[kilometres]

P i

– average number of passengers travelling weekly on route [persons]

i p pas i

– average income per one passenger travelling on route price) [monetary units]

i

(ticket

w i

– average load index of a bus on route

i

[-], expressed as a quotient of an average number of tickets sold for a particular ride on route

i

an average number of passengers in a bus during this ride,

w i

 and

{0,1} k ij

– fixed cost per route

i

and bus

j

, including drivers’ salaries, highway fares, tolls, insurance and licence fees etc. [monetary units / ride]

k wkm ij

– variable (vehicle-kilometre) cost per bus

j

and route fuel and maintenance cost [monetary unit / kilometre]

i

, including

c j

– capacity of bus

j

– number of seats [-]

f j

– travelling comfort level of bus

j

[-], expressed in points according to the following characteristics of bus

j

: seats’ comfort (size, softness), air conditioning, toilet, video etc.,

f j



{1, 2, 3, 4, ..., f max = 10}

Slide  234

Mathematical Formulation of the Problem

Decision variables and Criteria Poznan University of Technology • The integer decision variable 

ij



{0, 1, 2, 3, ...}

, denominates a number of rides carried out weekly by a vehicle

j

on route

i

Criterion

1. Total weekly profit 2. Capacity utilisation – –

Z WL

3. Total number of weekly lost (rejected) customers (passengers) –

SK

4. Comfort of travel for passengers –

WK

Unit

[monetary units] - - [points]

Dp

max min min

Consequence

The maximal number of passengers should be transported with minimal costs Average capacity utilisation should be close to 80% (assumed optimal level), percentage of empty rides should be minimal Assures that all demand will be satisfied,- high customers’ satisfaction max High quality service Slide  235

Mathematical Formulation of the Problem

Criteria Poznan University of Technology

Criterion Formula

1. Total weekly profit –

Z

2. Capacity utilisation –

WL

3. Total number of weekly lost customers –

SK

4. Comfort of travel for passengers –

WK

Slide  236

Max Z Min WL

 

i I

  1

WL opt

   

W i W i

      

S P i i

 

i

 1

j J I

  1    

j J

  1 

SK i



ij



k wkm p pas i ij

min 

P oi

,

c j



cj

     

ij

 

j J

  1 

ij



k ij

   

i

 1

j J I

  1 

ij

 

P oi

  

P i j J

  1 

ij

  

w i



i SK i Min SK



i I

  1

SK i

 max    0 ,

P i



j J

  1   min

P oi

,

w i c w i j

 

ij

    

Max WK



i

 1

j J I

  1 min 

P oi

,

c j

  

ij



f j i

 1

j J I

  1 min 

P oi

,

c j

  

ij



f

max

Mathematical Formulation of the Problem Constraints

Poznan University of Technology 

The presented model takes into consideration the following constraints:

 real riding time by bus

j

on route

i

should be consistent with the timetable  weekly working time of bus

j

should not be grater than its maximal weekly working time, including maintenance (repair and service) times Slide  237

Mathematical Formulation of the Problem Output data – the results

Poznan University of Technology 

As a result DM obtains the most satisfactory solution of the problem from the company’s and its customers’ point of view:

  bus assignment expected values of considered criteria 

ij

=             1 0 ...

1 5 0 ...

0 Route

i

0 3 ...

0 0 0 ...

4 ...

...

...

...

...

...

...

...

2 1 ...

1 0 1 ...

0 2 0 ...

0 1 0 ...

0 0 0 ...

1 0        1 ...

  0   Slide  238

Computational Experiment Stage one - results of PMA

Poznan University of Technology 

A sample of Pareto - optimal solutions generated after 60 000 iterations (recombination and local improvements) is composed of 2 985 different solutions (possible assignments of buses)



The range of considered criteria: Capacity Profit - Z

[PLN]

utilisation - WL Number of lost passengers – SK

[-] [-]

Min -2 669 000 0.11 0

Comfort of travel - WK

[-]

0.85 Max 1 802 570 0.74 181 0.90 Slide  239

Computational Experiment Stage two - settings of LBS method

Poznan University of Technology Slide  240

Computational Experiment Stage two - results of LBS method

Poznan University of Technology 

DM is interested in solution A2960 which outranks the present middle point on criterion 1 (by 300 000 PLN) and is indifferent on the other criteria

 Slide

Solution A2960 becomes a new middle point

 241

– its neighbourhood consists of 38 solutions including solution A2959 which has been selected by DM as the most satisfactory, compromise solution

Computational Experiment Stage two – „the best” assignment of buses (solution A2959)

Poznan University of Technology

1 1

0

10 3 11 1 12 2 13 1 14 1 15 0 20 0 21 0 22 0 23 0 24 0 25 1 30 1

2

0 1 0 0 0 0 0

3

0 0 0 1 0 0 0

4

0 0 0 0 0 1 0

5

0 1 0 0 0 0 0

6

0 0 0 0 1 0 0

7

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1

8

Routes i

9 10 11 12 13 14 15 16 17

0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 … 0 0 1 0 0 0 ... 0 0 0 0 0 0 ... 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0

Slide  242

Computational Experiment Compromise solution vs. other Pareto - optimal solutions

Poznan University of Technology

Objective

Other Pareto – optimal solutions Min Max The most satisfactory assignment (solution A2959)

Profit – Z Capacity utilisation - WL Number of lost passengers – SK Comfort of travel - WK

- 2 669 000 1 802 670 0.11 0.74 0 0.85 181 0.90 1 752 140 0.15 22 0.87 Slide  243

Conclusions

Poznan University of Technology 

The presented methodology lets DM to define the most satisfactory assignment of buses to particular routes



The methodology can by applied in a long-distance passenger transportation companies utilising a non-homogeneous fleet of buses



The methodology leads to the profitability analysis of particular routes. Based on the analysis of criterion 1 certain, non-profitable routes can be eliminated from the existing portfolio of the transportation services. It also allows to define the minimal ticket price for each route to assure its acceptable profitability and maintain this service in the portfolio



The methodology of solving VAP combined with an appropriate database let us create the modern DSS for such a problem in the future

Slide  244