Introduction 0-1 Branch and Bound Conclusion Solving an integrated Job-Shop problem with human resource constraints PMS’10 - Tours (France) E. Pinson 2 and D. Rivreau 2 O. Guyon 1 . 2 , P. Lemaire 3 , ´ 1 ´ Ecole des Mines de Saint-´ Etienne 2 LISA - Institut de Math´ ematiques Appliqu´ ees d’Angers 3 Institut Polytechnique de Grenoble April 26-28, 2010 O. Guyon, P. Lemaire, ´ E. Pinson and D. Rivreau Job-Shop problem with human resource constraints 1 / 34
Introduction 0-1 Branch and Bound Conclusion Table of contents Introduction 1 Problem Time-indexed ILP formulation Solution methods 0-1 Branch and Bound 2 Outline Initial cuts Characteristics Conclusion 3 Experimental results Concluding remarks O. Guyon, P. Lemaire, ´ E. Pinson and D. Rivreau Job-Shop problem with human resource constraints 2 / 34
Introduction Problem 0-1 Branch and Bound Time-indexed ILP formulation Conclusion Solution methods Plan Introduction 1 Problem Time-indexed ILP formulation Solution methods 0-1 Branch and Bound 2 Outline Initial cuts Characteristics Conclusion 3 Experimental results Concluding remarks O. Guyon, P. Lemaire, ´ E. Pinson and D. Rivreau Job-Shop problem with human resource constraints 3 / 34
Introduction Problem 0-1 Branch and Bound Time-indexed ILP formulation Conclusion Solution methods Nature of the problem (1/3) Get a feasible JOB-SHOP production plan Minimize EMPLOYEE TIMETABLING labor costs O. Guyon, P. Lemaire, ´ E. Pinson and D. Rivreau Job-Shop problem with human resource constraints 4 / 34
Introduction Problem 0-1 Branch and Bound Time-indexed ILP formulation Conclusion Solution methods Nature of the problem (2/3) - Employee Timetabling problem Time horizon Timetabling horizon H = δ · π where: δ number of shifts π duration time of a shift Nota : it can modelize, for example, a three-shift system Employee Timetabling Problem for a set E of µ employees A e set of machines employee e masters T e set of shifts where employee e is available O. Guyon, P. Lemaire, ´ E. Pinson and D. Rivreau Job-Shop problem with human resource constraints 5 / 34
Introduction Problem 0-1 Branch and Bound Time-indexed ILP formulation Conclusion Solution methods Nature of the problem (3/3) - Job-Shop problem Job-Shop: Schedule a set J of n jobs on m machines ∀ j ∈ J { O ji } i =1 .. m chain of operations of job j machine m ji ∈ { 1 . . . m } processing time p ji ֒ → Notation ρ jk : processing time of j on machine k can not be interrupted requires a qualified employee to use machine m ji Feasible production plan A schedule for which all operations are completed before a given scheduling completion time Cmax ≤ H. O. Guyon, P. Lemaire, ´ E. Pinson and D. Rivreau Job-Shop problem with human resource constraints 6 / 34
Introduction Problem 0-1 Branch and Bound Time-indexed ILP formulation Conclusion Solution methods Nature of the problem (3/3) Objective Assigning at minimum cost employees to both machines and shifts in order to be able to provide a feasible production plan O. Guyon, P. Lemaire, ´ E. Pinson and D. Rivreau Job-Shop problem with human resource constraints 7 / 34
Introduction Problem 0-1 Branch and Bound Time-indexed ILP formulation Conclusion Solution methods Example (6 jobs - 4 machines - 15 employees) O. Guyon, P. Lemaire, ´ E. Pinson and D. Rivreau Job-Shop problem with human resource constraints 8 / 34
Introduction Problem 0-1 Branch and Bound Time-indexed ILP formulation Conclusion Solution methods Motivation: Extension of work Continuation of a work Guyon, Lemaire, Pinson and Rivreau. European Journal of Operational Research (March 2010) Integrated employee timetabling and production scheduling problem Methods : ֒ → Decomposition and cut generation process � Simplified production scheduling problem Motivation for this new study: ֒ → Is the decomposition and cut generation process also efficient with a harder production scheduling problem (Job-Shop)? O. Guyon, P. Lemaire, ´ E. Pinson and D. Rivreau Job-Shop problem with human resource constraints 9 / 34
Introduction Problem 0-1 Branch and Bound Time-indexed ILP formulation Conclusion Solution methods Motivation: Case treated in literature Case treated in literature Artigues, Gendreau, Rousseau and Vergnaud [AGRV09]. Computers and Operations Research (2009) Aim: To experiment hybrid CP-ILP methods on an integrated job-shop scheduling and employee timetabling problem Methods : ֒ → CP with a global additional constraint corresponding to the LP-relaxation of the employee timetabling problem Our study : specific case of mapping activities - machines ֒ → 8 of the 11 instances of [AGRV09] can be used O. Guyon, P. Lemaire, ´ E. Pinson and D. Rivreau Job-Shop problem with human resource constraints 10 / 34
Introduction Problem 0-1 Branch and Bound Time-indexed ILP formulation Conclusion Solution methods Plan Introduction 1 Problem Time-indexed ILP formulation Solution methods 0-1 Branch and Bound 2 Outline Initial cuts Characteristics Conclusion 3 Experimental results Concluding remarks O. Guyon, P. Lemaire, ´ E. Pinson and D. Rivreau Job-Shop problem with human resource constraints 11 / 34
Introduction Problem 0-1 Branch and Bound Time-indexed ILP formulation Conclusion Solution methods ILP model (1/4) Binary decision variables x eks = 1 iif employee e is assigned to the pair (machine k ; shift s ) y ikt = 1 iif job i starts its processing on the machine k at instant t O. Guyon, P. Lemaire, ´ E. Pinson and D. Rivreau Job-Shop problem with human resource constraints 12 / 34
Introduction Problem 0-1 Branch and Bound Time-indexed ILP formulation Conclusion Solution methods ILP model (2/4) Objective function [ P ] min Θ = � � � s ∈ T e c eks · x eks e ∈ E k ∈ A e Employee Timetabling Problem specific constraints � σ � s =0 x eks = 0 e = 1 , . . . , µ k / ∈ A e � � ∈T e x eks = 0 e = 1 , . . . , µ s / k ∈ A e � k ∈ A e ( x eks + x ek ( s +1) + x ek ( s +2) ) ≤ 1 e = 1 , . . . , µ s = 0 , . . . , σ − 3 x eks ∈ { 0 , 1 } e = 1 , . . . , µ k = 1 , . . . , m s = 0 , . . . , σ − 1 O. Guyon, P. Lemaire, ´ E. Pinson and D. Rivreau Job-Shop problem with human resource constraints 13 / 34
Introduction Problem 0-1 Branch and Bound Time-indexed ILP formulation Conclusion Solution methods ILP model (3/4) Job-Shop scheduling problem specific constraints � d ik − ρ ik t · y ikt + ρ ik ≤ C max i = 1 , . . . , n k = m im t =0 � d ik − ρ ik y ikt = 1 i = 1 , . . . , n k = 1 , . . . , m t = r ik � r ik t =0 y ikt + � C max t = d ik − ρ ik +1 y ikt = 0 i = 1 , . . . , n k = 1 , . . . , m � t u = r ik + ρ ik y ilu − � t − ρ ik u = r ik y iku ≤ 0 i = 1 , . . . , n j = 1 , . . . , m − 1 k = m ij l = m i ( j +1) t = ρ ik + p ik , . . . , d il − ρ il � min( d ik − ρ ik , t ) � n u =max( r ik , t − ρ ik +1) y iku ≤ 1 k = 1 , . . . , m t = 0 , . . . , C max i =1 y ikt ∈ { 0 , 1 } i = 1 , . . . , n k = 1 , . . . , m t = 0 , . . . , C max O. Guyon, P. Lemaire, ´ E. Pinson and D. Rivreau Job-Shop problem with human resource constraints 14 / 34
Introduction Problem 0-1 Branch and Bound Time-indexed ILP formulation Conclusion Solution methods ILP model (4/4) Coupling constraints e ∈ E x eks − � n � min( d ik − ρ ik , t ) � u =max( r ik , t − ρ ik +1) y iku ≥ 0 k = 1 , . . . , m t = 0 , . . . , C max i =1 s = ⌊ t /π ⌋ O. Guyon, P. Lemaire, ´ E. Pinson and D. Rivreau Job-Shop problem with human resource constraints 15 / 34
Introduction Problem 0-1 Branch and Bound Time-indexed ILP formulation Conclusion Solution methods Plan Introduction 1 Problem Time-indexed ILP formulation Solution methods 0-1 Branch and Bound 2 Outline Initial cuts Characteristics Conclusion 3 Experimental results Concluding remarks O. Guyon, P. Lemaire, ´ E. Pinson and D. Rivreau Job-Shop problem with human resource constraints 16 / 34
Introduction Problem 0-1 Branch and Bound Time-indexed ILP formulation Conclusion Solution methods Solution methods Three exact methods MIP Decomposition and cut generation process 0-1 Branch and Bound based on the work (or not) for each pair (machine, shift) O. Guyon, P. Lemaire, ´ E. Pinson and D. Rivreau Job-Shop problem with human resource constraints 17 / 34
Introduction Outline 0-1 Branch and Bound Initial cuts Conclusion Characteristics Plan Introduction 1 Problem Time-indexed ILP formulation Solution methods 0-1 Branch and Bound 2 Outline Initial cuts Characteristics Conclusion 3 Experimental results Concluding remarks O. Guyon, P. Lemaire, ´ E. Pinson and D. Rivreau Job-Shop problem with human resource constraints 18 / 34
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