A hybrid heuristic algorithm for the sequencing generalized assignment problem in an assembly line

Mathematical Proceedings ofModelling the 9th Vienna International Conference on Mathematical Proceedings ofModelling the 9th Vienna International Conference on Vienna, Austria, February 21-23, 2018 Mathematical Proceedings ofModelling the 9th Vienna International Conference on Vienna, Austria, February 21-23, 2018 Mathematical Modelling Available online at www.sciencedirect.com Vienna, Austria, February 21-23, 2018 Mathematical Modelling Vienna, Austria, February 21-23, 2018 Vienna, Austria, February 21-23, 2018

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IFAC PapersOnLine 51-2 (2018) 695–700 A heuristic algorithm A hybrid hybrid heuristic algorithm for for the the A hybrid heuristic algorithm for the A hybrid heuristic algorithm for the sequencing generalized assignment problem sequencing generalized assignment A hybrid heuristic algorithm forproblem the sequencing generalized assignment problem sequencing generalized assignment problem in an assembly line in an assembly line sequencing generalized assignment problem in an assembly line in an assembly line ∗ in an∗ assembly ∗∗line

S. O. S. E. E. Moussavi Moussavi ∗∗ M. M. Mahdjoub Mahdjoub ∗∗ O. Grunder Grunder ∗∗ ∗∗ S. E. Moussavi ∗ M. Mahdjoub ∗∗ O. Grunder ∗ S. E. Moussavi ∗ M. Mahdjoub ∗∗ O. Grunder ∗ ∗ S. E. Moussavi M.Bourgogne Mahdjoub O. Grunder lab, Franche-Comt´ ee,, UTBM, ∗ Nanomedicine Nanomedicine lab, Univ. Univ. Bourgogne Franche-Comt´ UTBM, ∗ F-90010 Belfort, France Univ. Bourgogne Franche-Comt´ ee,, UTBM, ∗ Nanomedicine lab, F-90010 Belfort, France Univ. Bourgogne Franche-Comt´ UTBM, ∗∗ ∗ Nanomedicine lab, Lab, Univ. Bourgogne Franche-Comt´ ee,, UTBM, 25200 ∗∗ ELLIADD Nanomedicine lab, Univ. Bourgogne Franche-Comt´ e, UTBM, F-90010 Belfort, France ELLIADD Lab, Univ. Bourgogne Franche-Comt´ UTBM, 25200 F-90010 Belfort, France ∗∗ Montb´ e liard, France F-90010 Belfort, France ELLIADD Lab, Univ. Bourgogne Franche-Comt´ e , UTBM, 25200 ∗∗ Montb´ e liard, France ELLIADD Lab, Univ. Bourgogne Franche-Comt´ e , UTBM, 25200 ∗∗ ELLIADD Lab, Univ.Montb´ Bourgogne Franche-Comt´ e, UTBM, 25200 e liard, France {seyed-esmaeil.moussavi, morad.mahdjoub, olivier.grunder}@utbm.fr Montb´ e liard, France {seyed-esmaeil.moussavi,Montb´ morad.mahdjoub, eliard, Franceolivier.grunder}@utbm.fr {seyed-esmaeil.moussavi, {seyed-esmaeil.moussavi, morad.mahdjoub, morad.mahdjoub, olivier.grunder}@utbm.fr olivier.grunder}@utbm.fr {seyed-esmaeil.moussavi, morad.mahdjoub, olivier.grunder}@utbm.fr Abstract: The generalized assignment problem (GAP) Abstract: The generalized assignment problem (GAP) is is an an enlarged enlarged version version of of the the classical classical assignment problem in which the assignment of several tasks to an agent is possible. The only Abstract: The generalized assignment problem (GAP) is an enlarged version of the classical assignment problem in which the assignment of several tasks to an agent is possible. The only Abstract: that The can generalized assignment problem (GAP) is an enlarged version of of thethe classical restriction limit the assignment of the tasks to the agents is the capacity agent. Abstract: The generalized assignment problem (GAP) is an enlarged version of the classical assignment problem in which the assignment of several tasks to an agent is possible. The only restriction that can limit the assignment of theoftasks to tasks the agents isagent the capacity of the agent. assignment problem in which the assignment several to an is possible. The only In this study, a manpower planning is modeled as a GAP in an assembly line. The considered assignment problem in which the assignment of several tasks to an agent is possible. The only restriction that can limit the assignment of the tasks to the agents is the capacity of the agent. In this study, a can manpower planning is modeled as a GAP inagents an assembly line. The considered restriction that limit the assignment of the tasks to the is the capacity of the agent. assembly line of a number of workstations in where workstation contains restriction thataconsists can limit the assignment of the tasks to series theinagents iseach the capacity of considered the agent. In this study, manpower planning is modeled as a GAP an assembly line. The assembly line consists of a number of workstations in series where each workstation contains In this study, a parallel. manpower planning isresearch modeledseeks as a to GAP in an assembly line. The considered several jobs in The present assign the best operator to jobs In this study, manpower planning isresearch modeled as a to GAP in an line. The considered assembly line of a number workstations in series where each workstation contains several jobs inaconsists parallel. The presentof seeks assign theassembly best operator to the the jobs in in assembly line consists of a number of workstations in series where each workstation contains order to minimize the processing time in each workstation and in the whole of the system. assembly line consists of a number of workstations in series where each workstation contains several jobs in parallel. The present research seeks to assign the best operator to the jobs in order tojobs minimize the processing time in each workstation and in theoperator whole of the system. several in parallel. The present research seeks to assign the best to the jobs in The generalized assignment formulation is adapted to this case. A mixed-integer mathematical several jobs in parallel. The present research seeks to assign the best operator to the jobs in order to minimize the processing time in each workstation and in the whole of the system. The generalized assignment formulation is adapted to this case. A in mixed-integer mathematical order to minimize the processing time in each workstation and the whole of the system. model is proposed for problem and that the considered is NP-complete. The order to minimize thethis processing timeproved in each workstation andAproblem in the whole ofmathematical the system. The generalized assignment formulation is adapted to this case. mixed-integer model is proposed for this problem and proved that the considered problem is NP-complete. The The generalized assignment formulation is adapted tomixed-integer this case. A mixed-integer mathematical model solved various sizes by employing Gurobi solver. This paper presents The generalized assignment formulation is adapted tomixed-integer this case. Aproblem mixed-integer mathematical is proposed for this problem proved that the considered NP-complete. The model solved in in various sizes by and employing Gurobi solver. is This paper presents is proposed for this problem and proved that the considered problem is NP-complete. The a greedy heuristic combined with a local search for the generalized assignment problem. The model is proposed for this problem and proved that the considered problem is NP-complete. solved in various sizes by employing Gurobi mixed-integer solver. This paper presents amodel greedy heuristic combined with aemploying local search for the generalizedsolver. assignment problem. The is solved in various sizes by Gurobi mixed-integer This paper presents slight deviation from the optimal solutions and the computational times which is model is solved in various by aemploying Gurobi mixed-integer solver. paper presents a heuristic combined with local search for generalized assignment problem. The slight deviation from the sizes optimal and the the computational times This which is extremely extremely a greedy greedy heuristic combined with asolutions local search for the generalized assignment problem. The reduced, validate the efficiency of the proposed heuristic approach for solving such problems. a greedy heuristic combined with a local search for the generalized assignment problem. The slight deviation from the optimal solutions and the computational times which is extremely reduced, validate the efficiency of the proposed heuristic approach for solving such problems. slight deviation from the optimal solutions and the computational times which is extremely slight deviation from the optimal solutions andheuristic the computational times which isproblems. extremely reduced, validate the efficiency of the proposed approach for solving such reduced, validate the efficiency of the proposed heuristic approach for solving such problems. reduced, validate the efficiency of the proposed heuristic approach for solving such problems. © 2018,1.IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. signments 1. INTRODUCTION INTRODUCTION signments (one (one per per day) day) are are needed needed for for this this system system and and the the objective is to minimize the production time by assigning 1. INTRODUCTION signments (one per day) are needed for this system and the the objective is to minimize the production time by assigning 1. INTRODUCTION signments (one per day) are needed for this system and the efficient workers to the jobs such the number of 1. INTRODUCTION signments (one per day) are needed for that this system and the objective is to minimize the production time by assigning Human resource planning in a production system could be the efficient workers to the jobs such that the number of is toisminimize the production timeInby assigning Human resource planning in a production system could be objective working-days respected for every workers. the studied objective is to minimize the production time by assigning the efficient workers to the jobs such that the number of considered as a specific version of the Assignment Problem working-days is respected for every workers. In the studied Human resource planning in a production system could be the efficient workers to the jobs such that the number of considered as a specific version of the Assignment Problem Human resource planning in a production system could be assembly line, the workers are able to do all of the jobs the efficient workers to the jobs such that the number of working-days is respected for every workers. In the studied (AP). This paper deals with worker assignment assembly line,isthe workersforare ableworkers. to do all of the jobs Human resource planning in the a production system could be working-days considered as specific version ofweekly the Assignment Assignment Problem respected every Intothe studied (AP). This paper deals version with the weekly worker assignment considered as aa specific of the Problem but their skills are different. The objective is select the working-days is respected for every workers. In the studied assembly line, the workers are able to do all of the jobs to the workstations/jobs in an assembly system which is but their skills are different. The objective is to select the considered as a specific the Assignment Problem (AP). This paper deals version with the weekly worker assignment line, worker the workers arejob. ableThe to do allassignments of the jobs to the This workstations/jobs in the anofassembly system which is assembly (AP). paper deals with weekly worker assignment most efficient for daily assembly line, worker the workers are able to do allto ofselect the jobs but their their skills are different. different. The objective is the modeled as aa sequencing assignment The term most efficient for each each job. The daily assignments (AP). This paper deals with the weeklyproblem. worker assignment to the workstations/jobs in an assembly system which is but skills are The objective is to select the modeled as sequencing assignment problem. The term to the workstations/jobs in an assembly system which is are correlated because of working-day/off-day restriction. but their skills are different. The objective is to select the most efficient worker for each job. The daily assignments sequencing problem in present study could are correlated becausefor of each working-day/off-day restriction. to the workstations/jobs in an assembly system which is most modeled asassignment a sequencing sequencing assignment problem. The term efficient worker job. The dailyformulation assignments sequencing assignment problem in the the problem. present study could modeled as a assignment The term Hence, a sequencing generalized assignment is most efficient worker for each job. The daily assignments are correlated because of working-day/off-day restriction. be interpreted to aa assignment number of classical Hence, a sequencing generalized assignment formulation is modeled asassignment a sequencing problem. Theassignterm are sequencing problem in the the present study could correlated because of working-day/off-day restriction. be interpreted to mean meanproblem number of the the classical assignsequencing assignment in present study could applied to model this problem mathematically by using are correlated because of working-day/off-day restriction. Hence, a sequencing generalized assignment formulation is ments that occur sequentially. Each assignment belongs to applied to model this problem mathematically by using sequencing assignment problem in the present study could be interpreted to mean a number of the classical assignHence, a sequencing generalized assignment formulation is ments that occur sequentially. Each assignment belongs to be interpreted toand mean a number ofprevious the classical assignmixed-integer programming (MIP). Hence, ato sequencing generalized assignment formulation is applied model this problem mathematically by using aabe period of time depends on the ones. Hence, mixed-integer programming (MIP). interpreted to mean a number of the classical assignments that occur sequentially. Each assignment belongs to applied to model this problem mathematically by using period of time and depends on the assignment previous ones. Hence, ments that occur sequentially. Each belongs to applied to model this problem mathematically by using mixed-integer programming (MIP). the parameters of the problem must be after ments that occur sequentially. Each belongs to The a period period of time time and depends on the assignment previous ones. Hence, term Generalized Problem mixed-integer programming (MIP). Assignment the parameters ofand thedepends problemon must be synchronized synchronized after a of the previous ones. Hence, The term Sequencing Sequencing Generalized mixed-integer programming (MIP). Assignment Problem determining an assignment. a period of time and depends on the previous ones. Hence, the parameters of the problem must be synchronized after (SGAP) was presented for the first time by Moussavi The term Sequencing Generalized Assignment Problem determining an of assignment. the parameters the problem must be synchronized after The (SGAP) was presentedGeneralized for the first time by Moussavi term Sequencing Assignment Problem the parameters of the problem must be synchronized after et determining an assignment. al. (2017) as multiple generalized assignments which The term Sequencing Generalized Assignment Problem (SGAP) was presented for the first time by Moussavi The study at determining an assignment. et al. (2017) multiplefor generalized which was as presented the first assignments time by Moussavi The study aims aims at obtaining obtaining the the optimal optimal assignment assignment (SGAP) determining an assignment. must be carried out one after the other. They have proved (SGAP) was presented for the first time by Moussavi et al. (2017) as multiple generalized assignments which of the appropriate operators to the workstations. The must be carried out one after the other. They have proved The study aims at obtaining the optimal assignment et al. (2017) as multiple generalized assignments which of thestudy appropriate operators tothe theoptimal workstations. The that The aims atline obtaining assignment SGAP an NP-Complete optimization problem. et al.the (2017) asis multiple generalized assignments which must be carried out one after the other. They have proved considered assembly consists of the workstations which that the SGAP is an NP-Complete optimization problem. The study aims at obtaining the optimal assignment of the the appropriate appropriate operators workstations. The must be carried out one algorithms after the other. They have proved considered assembly line consiststo of the the workstations which of operators to the workstations. The Accordingly, the exact are often not able must be carried out one after the other. They have proved that the SGAP is an NP-Complete optimization problem. are in series (flowshop system) and each Accordingly, theis exact algorithms optimization are often notproblem. able to to of the operators toof the workstations. The that considered assembly line consists the workstations workstations which the SGAP an NP-Complete are in appropriate series (flowshop system) and each workstation workstation considered assembly line consists of the which solve large and medium sizes of such problems. that the SGAP is an NP-Complete optimization problem. Accordingly, the exact algorithms are often not able to is composed of several jobs. In a workstation, as there solve large and medium sizes of such problems. considered assembly line consists of the workstations which are in series (flowshop system) and each workstation Accordingly, the exact algorithms are often not able to is composed several jobs. In a and workstation, as there Accordingly, are in precedence series of(flowshop system) each workstation themedium exact algorithms often not able to solve large and sizesa of ofhybrid suchare problems. are no relationships jobs, they can in precedence series of(flowshop system) and each workstation is composed composed several jobs. In aabetween workstation, as there there this paper, we propose heuristic solve large and medium sizes such problems. are no relationships between jobs, they can In is of several jobs. In workstation, as In this paper, propose heuristic algorithm algorithm solve large and we medium sizesa ofhybrid such problems. be processed in parallel. Consequently, several workers is composed of several jobs. In a workstation, as there are processed no precedence precedence relationships between several jobs, they they can which developed in MATLAB environment. algoIn this thisis paper, we propose propose a hybrid hybrid heuristic The algorithm be in parallel. Consequently, workers are no relationships between jobs, can which is developed in MATLAB environment. The algoIn paper, we a heuristic algorithm are assigned to the same workstation. Therefore, the no precedence relationships between several jobs, they can be processed processed in parallel. parallel. Consequently, workers employed to solve case study with various In thisis paper, we propose aour hybrid heuristic algorithm which developed in MATLAB environment. The algoare assigned to the sameConsequently, workstation. Therefore, the rithm be in several workers rithm is employed to solve our case study with various which developed in MATLAB environment. The algostudied problem can be as special type of be in parallel. Consequently, workers are processed assigned to the same workstation. Therefore, the the planning horizon. The results show the which is of developed in solve MATLAB environment. The algorithm employed to our case study with various studied problem can same be regarded regarded as aa several special typethe of lengths are assigned to the workstation. Therefore, lengths of the planning horizon. The results show the is employed to solve our case study withSGAP. various the generalized problem (GAP) which is aa rithm are assigned to assignment the same workstation. Therefore, the studied problem can be regarded as a special type of efficiency of the proposed approach to solve the rithm is employed to solve our case study with various lengths of the planning horizon. The results show the the generalized assignment problem (GAP) which is studied problem can be regarded as a special type of efficiency of the proposed approach to solve the SGAP. lengths of the planning horizon. The results show the combinatorial optimization problem. studied problem can be regarded as (GAP) a special typeis of the generalized assignment problem which a lengths of the planning horizon. The results show the efficiency of the proposed approach to solve the SGAP. combinatorial optimization problem. the generalized assignment problem (GAP) which is a After introducing the generalized assignment probof the proposed approach to solve the SGAP. the generalizedoptimization assignmentproblem. problem (GAP) which is a efficiency After introducing the term term generalized assignment probcombinatorial efficiency of the proposed approach to solve the SGAP. The problem contains the heterogeneous workers with difcombinatorial optimization problem. Ross Soland many researchers have Afterby introducing the term(1975), generalized assignment probThe problem contains the heterogeneous workers with dif- lem combinatorial optimization problem. lem byintroducing Ross and and the Soland (1975), many assignment researchers probhave After term generalized ferent capacities and aathe planning period of one week. The The problem contains heterogeneous workers with difproposed different exact and approximate methods to solve After introducing the term generalized assignment problem by Ross and Soland (1975), many researchers have ferent capacities and planning period of one week. The The problem contains the heterogeneous workers with difproposed different exact and approximate methods to solve lem by Ross and Soland (1975), many researchers have restriction of working-days and on the workers is The heterogeneous workers with The differentproblem capacities and aathe planning period of of one week. problems and its various extensions different scales. lem by Ross and Soland (1975), many in researchers have proposed different exact and approximate methods to solve restriction of contains working-days and off-days off-days on the week. workers is such ferent capacities and planning period one The such problems and its various extensions in different scales. proposed different exact and approximate methods to solve also considered in this research. The production system ferent capacities and a planning period of one week. The restriction of working-days and off-days on the workers is Chu and Beasley (1997) proved that the generalized asproposed different exact and approximate methods to solve such problems and its various extensions in different scales. also considered in this research. The production system restriction of working-days and off-days on the workers is Chu problems and Beasley proved that the generalized asand (1997) its various extensions in different scales. works everyday and the workers have five working-days restriction of working-days and off-days on the workers is such also considered in this research. The production system signment problem is NP-complete and the existing exact such problems and its various extensions in different scales. Chu and Beasley (1997) proved that the generalized asworks everyday and the workers have five working-days also considered in this research. The production system signment problem is NP-complete and the existing exact Chu and Beasley (1997) proved that the generalized asand two off-days per week. Therefore, seven worker asalso considered in this research. The production system works everyday and the workers have five working-days Chu and Beasley (1997) proved that the generalized assignment problem is NP-complete and the existing exact and two off-days per week. Therefore, seven worker asworks everyday and the workers have five working-days signment problem is NP-complete and the existing exact works everyday workers have five working-days and two two off-daysand perthe week. Therefore, seven worker asas- signment problem is NP-complete and the existing exact and off-days per week. Therefore, seven worker and two ©off-days per week. Therefore, seven worker as- 1 Copyright 2018 IFAC Copyright © 2018 IFAC 1 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018, 2018 IFAC 1 Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 responsibility IFAC 1 Control. Peer review© of International Federation of Automatic Copyright ©under 2018 IFAC 1 10.1016/j.ifacol.2018.03.118

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In this study, the SGAP formulation is adapted to worker assignment in a production system and modeled by applying mixed-integer linear programming. The mathematical model is solved for different sizes of the problem by employing the Gurobi mixed-integer solver. As a result, the exact solver is not able to solve the model for the medium and large scale problems. Therefore, we have implemented a heuristic algorithm for solving the SGAP by considering working and off days of the operators. The results of the exact and heuristic algorithm demonstrate the effectiveness of the heuristic method that can solve the problem in a few seconds with a slight deviation from the optimal solution.

methods are practical only for the instances where there are no additional constraints. They proposed a genetic algorithm which is able to solve the GAPs with more constraints. They also compared their adaptive genetic algorithm with various existing approaches, for instance, tabu search and simulated annealing which were adapted for the GAP by Osman (1995). Thereafter Osorio and Laguna (2003) and Diaz and Fernandez (2001) considered the generalized assignment problems as the sub-problems in their studies and they added more details and assumptions to the classical GAP. They prove that these types of problems are much more complicated than classical GAP to solve. Hence, they proposed the tabu search and logic cuts in a branch and bound algorithm for solving such problems.

The remaining of this paper is organized as follows. In section 2, the problem and its related mathematical model is presented in detail. The proposed heuristic algorithm is explained in section 3. The case study is introduced in section 4 and the computational results of the exact and heuristic methods are presented in this section. Finally, in section 5, the conclusion and the future research opportunities are presented.

The most important contribution of this paper is the consideration of the availability of the workers and possibility of job rotation in the worker assignment problem. In this way, Alfares (2002) and Elshafei and Alfares (2008) studied the availability of the workers and they considered the restrictions on the working-days and off-days in their workforce scheduling model. Alfares tried to minimize the number of worker regarding the successive working-days and successive off-day for the workers in his research. Then Elshafei and Alfares developed a dynamic model for labor assignment by aiming to minimize the labor costs.

2. MATHEMATICAL MODEL FOR THE GENERALIZED ASSIGNMENT PROBLEM The considered assignment problem is composed of a series of GAPs where each GAP corresponds to a period of the planning horizon. In this study, the elements of the matrix of assignment present the processing time of the workers in different jobs. The production system is composed of a serial of the workstations in which each workstation contains a number of jobs in parallel. The objective of this assignment is to minimize the production time. The production time is from the starting time of the first workstation to the ending time of the last workstation (the sum of the processing time of workstations). The workstations are in series and each workstation is started once the precedent workstation is ended. For most of the workstations, there is more than one operator that work in parallel. The processing time in such workstations is equal to the maximum operating time of the workers who work there. By considering these conditions, the aim is to assign the most efficient workers to the workstations to minimize the production time. Thus for each day, we have an optimal assignment and its related production time. Note that, the optimal daily assignment can not be repeated for all days of the planning horizon because the workers do not work every days. In fact, they have a predefined number of working-days and off-days during the planning horizon. It means, there is a number of available workers that is more than the number of jobs. In each day, the number of working-workers is equal to the number of jobs and the rest of the workers are off-workers in this day. Therefore, the worker assignment and the production time vary from one day to another. For this sequential assignment, a mixed-integer mathematical model is proposed. In the next sections, the model is presented by its parameters, variables, objective function, and constraints.

In more recent studies, various heuristic and metaheuristic approaches have been proposed for the generalized assignment problem. In this way, Cohen et al. (2006) developed a heuristic approximation approach to adapt knapsack’s solving methods for the GAP. Tasgetiren et al. (2009) presented a continuous optimization algorithm based on differential evolution for the GAP and they hybridized their algorithm by employing variable neighborhood search to improve the quality of the solutions. Moccia et al. (2009) proposed a column generation algorithm to compute a lower bound by linear relaxation and find a feasible integer solution for the dynamic GAP. Sadykov et al. (2015) also proposed a column generation for the GAP. A combinatorial solution approach to solve the GAP was developed by Woodcock and Wilson (2010). They combined the exact branch and bound method with the meta-heuristic Tabu Search algorithm. Posta et al. (2012) presented an exact algorithm based on the decomposition of the GAP formulation by fixing certain variables. In recent years, the literature has been more focused on different extensions of the GAP. In this way, for the Generalized Quadratic Assignment Problem (GQAP), as an extension of the GAP, McKendall and Li (2016) proposed a meta-heuristic tabu search algorithm. They applied GQAP formulation to solve a location problem and assign the machines to different locations. Location/Allocation consideration was studied in the GAP by Ghoniem et al. (2016). They developed a heuristic algorithm based largescale neighborhood search for this problem. Ghoniem et al. (2016) in another research presented an exact algorithm based on the branch and bound algorithm for their extended version of the GAP. In this year, Sethanan and Pitakaso (2016) developed a hybrid algorithm composed of three local search techniques which are added to the differential evolution (DE) to solve the classical GAP.

2.1 Parameters U D : Upper bound on the number of working-days during the planning. 2

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n 

LD : Lower bound on the number of working-days. m : Total number of workers for the assignment. I = {1, ..., m}

j=1

Xijp ≤ 1

697

∀ i ∈ I, p ∈ P ;

The maximum and minimum number of working-days (for each operator during the planning horizon) which are considered in this study as the predefined conditions in the workforce scheduling. These assumptions are imposed on the model by the following sets of the constraints:

n : Total number of jobs. J = {1, ..., n}

s : Number of workstations. K = {1, ..., s}

Nk : Number of jobs in workstation k ∈ K. (N1 = 4 : There are 4 parallel jobs in workstation 1) t : Number of periods in the planning horizon. P = {1, ..., t}

l  n 

Xijp ≤ U D

∀ i ∈ I;

W Sk : The jobs which are carried out in workstation k.

l  n 

Xijp ≥ LD

∀ i ∈ I;

p=1 j=1

Yij : Operating time of worker i in job j. (Decision matrix)

(W Sk = {j ∈ J |

k−1  l=0

Nl < j ≤

k 

p=1 j=1

Nl : where N0 = 0}). Scheduling constraints The scheduling constraints of the model contain the precedence of the production operations and the procedure of the production time calculation. The first set of the scheduling constraints presents the working time of the operators during a production cycle.  Wikp = Xijp ∗ Yij ∀ i ∈ I, k ∈ K, p ∈ P ;

l=0

2.2 Variables In the model, there is a set of decision variables named Xijp and some other variables as: Xijp : Binary, where Xijp = 1 if operator i is assigned to job j in period p, and Xijp = 0 otherwise.

j∈W Sk

CM AXp : Production time (for producing one product) in period p.

These constraints imply the working time of worker i if he is assigned to workstation k in period p; Note that the Wikp will be zero if worker i is not assigned to the jobs of workstation k in period p. The second set of the scheduling constraints signify the processing time of each workstation in each day. While there is more than one operator working in a workstation, the processing time of the workstation is the maximum working time of the operators. This assumption is modeled as the following:

Z : Average of the production times.

P Tkp = M AXi∈I,j∈k Wikd

2.3 Model formulation

The constraint is linearized as: P Tkp ≥ Wikd K, p ∈ P

Skp , P Tkp , Fkp : They present respectively the starting, processing and ending time of workstation k in period p. Wikp : Working time of worker i in workstation k in period p. Where Wikp = Yij if worker i is assigned to job j in the period p and job j is in workstation k; Wikp = 0 otherwise.

Objective function The model is aimed to minimize the average of the production time on the planning periods and defined as the following: M inimize Z =

Sk = Fk−1

CM AXp /t;

Assignment constraints The first set of constraints is the classical assignment constraint and implies that each job in each period is fulfilled by one and only one operator.

i=1

Xijp = 1

∀k ≥2

where

S1 = 0;

Finally, the last constraint of this model presents the makespan or total production time for producing one product that is the ending time of the last workstation. The makespan varies from a period to another because the assignments are not the same for all the periods.

p=1

m 

∀ i ∈ I, k ∈

The next set of the constraints is the classical formula for the precedence of the operations in a serial system like a production line. In this study, the workstations are in series and each one can be started once the last one would be finished and the first workstation is started at T = 0.

The mixed-integer mathematical model which is presented in this study, contains a minimization objective function together with two different kinds of constraints as the assignment and the scheduling constraints.

t 

∀ k ∈ K, p ∈ P ;

CM AXp = P Tkp

∀p∈P

where

k = s;

The objective of the model is to minimize the average of the makespans on the planning horizon. The presented model is applicable on various types of the production systems as the parallel, serial and combinatorial systems. The presented problem is NP-hard and the exact methods are not able to solve its related MIP model for the large scales. In the next section, we propose a heuristic algorithm which is able to solve the considered assignment problem in a few seconds with a very slight deviation from the optimum solution.

∀ j ∈ J, p ∈ P ;

The second set of constraints implies that each worker during a day can not be assigned to more than one job. Thus the worker has whether one job (if he is in his working-day) or without the job (if he is in his off-day). This condition is defined as follows: 3

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considered as the decision parameters. For the classical assignment problem, the first step of the algorithm is to select the best score in each row. This score presents the minimum operating time for the corresponding job, and the column number of the selected score presents the most appropriate worker for the job. In this way, a primal solution is found which is the best possible solution. Most of the time, the obtained solution is infeasible because of the overlapping, where a worker is selected for more than one job or a job is assigned to more than one worker. For the generalized assignment problem where more than one worker is needed in a workstation, obtaining the primal solution is more complicated. As an example, if a workstation is composed of three jobs (operators), three best scores must be selected from its related row. Note that, in this case the rows of the assignment matrix represent the workstations. Once the best possible solution is obtained, its feasibility must be examined. For the feasibility evaluation, the classical assignment constraints are applied. In this way, two set of constraints must be respected; 1. Each operator must be assigned to one and only one job, 2. Each job must be fulfilled by one and only one operator. If the primal solution is feasible, the algorithm is finished and the obtained solution is the optimum solution for the assignment problem. But if there are the operators assigned to the more than one job, in at least one job, he must be replaced by another available operator. 3.2 Moving towards feasibility Once the primal solution does not respect all the constraints, the operators who cause the violation must be replaced by others. As an example, if an operator is assigned to the two jobs at the same time, he must be replaced in one of them by another available operator. At first, we have to choose a job for the operator reassignment. Then, for the chosen job, an available operator must be selected to take place the previously assigned operator. For this purpose, there are various possibilities which are compared and the best one is selected.

Fig. 1. The procedure of the heuristic algorithm 3. HEURISTIC ALGORITHM FOR GENERALIZED ASSIGNMENT PROBLEM As mentioned in section 2, the generalized assignment problem is NP-hard and solving such problems by mathematical modelization and exact algorithms could be very complicated for the large scales. In this paper, we present a hybrid heuristic algorithm for the sequencing generalized assignments. The algorithm is a greedy heuristic combined with a local search. Its starts from the best possible solution and moves to the feasibility. Four principal steps of the algorithm are as follows: 1. Choosing the best possible solution (without caring about feasibility), 2. Going towards the feasibility, 3. Applying two-exchange neighborhood for improving the solution and finally, 4. Developing the algorithm for the sequential assignment. Figure 1 shows the principal steps of the algorithm in summary. It is explained in more detail in the next sections.

In this way, a penalty matrix with two rows is created. The number of columns of the matrix is equal to the number of available operators. Each element of this matrix indicates the replacing penalty. For instance, the element in the first row and second column implies the penalty of replacing the violating operator at his first job by the second available operator. The element with the minimum penalty is selected and the replacement is made. After this replacement, the assignment is re-evaluated in terms of its feasibility. If the assignment is now feasible, hence, it is considered as the output of this step and the input of the step three as is explained in the next section. If there are still the operators to which two or more jobs are assigned, another replacing penalty matrix is created and the above procedure is repeated until a feasible solution is obtained.

3.1 Selecting the best primal solution

3.3 Two-exchange neighborhood

In the assignment matrix, the rows represent the jobs and the columns represent the workers. The elements of the matrix present the decision parameters. In this study, the operating times of the operators at different jobs are

The solution obtained through two previous steps, is a feasible solution such that each operator is assigned to one and only one job and each job is carried out by one and only one operator. In this step, a local search is developed 4

Proceedings of the 9th MATHMOD Vienna, Austria, February 21-23, 2018

S.E. Moussavi et al. / IFAC PapersOnLine 51-2 (2018) 695–700

699

workstation have to carry out their jobs in less than eight minutes. The processing time of the mentioned workstation is equal to the maximum operating time of the four operators working in this workstation in parallel. Thus in each workstation, there is a critical operator who determines the processing time of the workstation. The objective is to find the best job assignment to reduce total production time (the sum of the processing time of the three studied workstations) for one product.

by aiming to improve the obtained feasible solution. For this purpose, all the two-exchange possibilities are examined. the two-exchange neighborhood in this algorithm represents another solution where two workers change their jobs or two jobs change their assigned workers. The best exchange is selected and the objective function (production time) is evaluated. If the best two-exchange improves the solution, then the mentioned exchange is applied and this procedure will be repeated until no improvement is made. The output of this step is the best solution found by the algorithm for the classical assignment problem.

For solving the mentioned generalized assignment problem, we have proposed a mixed-integer mathematical model as explained in section 2. This model is solved for a planning horizon of one week by employing an exact solving software named Gurobi. As a result, the optimum assignment (workforce schedule) and its related objective value (production time) are obtained. The assignment of the operators during the first period is presented in table 1. This table shows the structure of the studied production system and the problem considered in this research. As shown, the best operator is assigned to each of the jobs so that the processing time of the corresponding workstation is minimized. Among four operators which are assigned to the workstation number one (W S1), the operator number nine (O9) or the job number one (J1) correspond to the maximum operating time. The operating time of this operator/job determines the processing time of the workstation. There are three workstations in series, thus the total production time is equal to the sum of the processing times of the workstations.

3.4 Developing the algorithm for the multiple assignment The above three steps of the proposed algorithm generate the best operator assignment for one period of time. The studied problem in this research considers a series of assignments where each assignment is related to a period (day) and the assignment is developed for several periods. As mentioned before, the best assignment can not be repeated for all of the periods because of the limitations which are defined by the number of the working-days for the operators. Therefore, the availability of the operators must be considered in the algorithm. For this purpose, after each assignment, the availability variable (the number of working-days which remains) for each worker is synchronized. As an example, if operator number one has worked two days during the three first days and the maximum number of working-days for him is predefined to be five days, the availability variable for him at the beginning of the fourth day is three. According to the proposed algorithm, the difference between the availability of different workers must not exceed one score (Ai − Aq ≤ 1 ∀ i, q ∈ I and i = q). In this way, after each assignment (period), for the operators with the minimum availability, their related score in the assignment matrix are replaced by a big score to avoid the same assignment for the next period. This procedure is followed until the assignment is made for all the periods.

The generalized assignment is an NP-hard problem and hence the computational time to solve such problems is highly increased by raising the size of the assignment matrix. Therefore, a heuristic algorithm for the GAP is presented that is able to solve the large size sequential and generalized assignments in a few seconds. Both of two solving methods have been employed to solve the considered problem on the studied assembly line. The results of the solving methods for the different numbers of periods are compared in table 2.(PT: Production Time, CT: Computational Time to solve the problem)

4. NUMERICAL APPLICATION

As shown in the table, by increasing the length of the planning horizon, the computational time for solving the problem by the exact method is exponentially grown. The proposed heuristic algorithm is able to solve this sequential assignment with a very small deviation from the optimal solution, less than 1%, where the computational time is highly decreased. Note that the planning horizon for the job assignment in the considered case study in real-time is one week (seven periods or days). The last column of table 2, where P = 7, shows that the exact method solves this instance in about five hours (17000 seconds) whereas the heuristic approach spends just ten seconds to attain a solution quite near to the optimal solution. From the CT columns of the table, it can be concluded that the heuristic algorithm is very efficient in the computational time mainly where the number of planning period is raised.

In the present study, a production system has been considered for evaluating the proposed mathematical model and heuristic algorithm. For this system, a sequential daily assignment is needed to manage the operators during a week. The studied production system is a truck assembly line consisting of the workstations, machines, operators, ... The considered part of this assembly line contains three workstations in which there are respectively four, five and five jobs in parallel. The system works seven days per week whereas the operators have five working-days and two offdays per week. Therefore, for these three workstations, fourteen (4 + 5 + 5) operators are needed per day where the system needs twenty (14 ∗ 7/5 = 19, 6 ∼ 20) available operators for the weekly manpower planning. The production cycle time in the system is eight minutes and each operator, depending on his capacities, has different operating times at different jobs which are between seven and eight minutes. It means the more efficient operators in each job, have the operating times near to seven minutes.

5. CONCLUSION In this study, a manpower planning is developed in a combined production system consisting of the jobs in parallel and in series. The planning is performed by adapting a

According to the structure and the restrictions of the studied production system, all four operators in the first 5

Proceedings of the 9th MATHMOD 700 Vienna, Austria, February 21-23, 2018

S.E. Moussavi et al. / IFAC PapersOnLine 51-2 (2018) 695–700

Table 1. Manpower planning and production time for the first period WS1 Operator O9 O16 O13 O14

Job J1 J2 J3 J4

Time 7.17 7.06 7.09 7.08

Job J5 J6 J7 J8 J9

WS time: 7.17

WS2 Operator Time O18 7.05 O11 7.01 O6 7.01 O1 7.04 O5 7.06 WS time: 7.06

Job J10 J11 J12 J13 J14

WS3 Operator Time O2 7.07 O10 7.01 O17 7.06 O19 7.01 O7 7.00 WS time: 7.07

PT: 21.30

Table 2. Gurobi vs Heuristic for solving generalized assignment problem P =1 PT

P =2 CT

PT

P =3 CT

PT

P =4 CT

PT

P =5 CT

PT

P =6 CT

PT

P =7 CT

CT

Gurobi

21.30

< 1s

42.69

< 1s

64.28

40s

85.57

36s

107.00

196s

128.55

12 ∗ 10 s

149.98

17 ∗ 103 s

Heuristic

21.32

< 1s

42.87

< 1s

64.87

< 1s

86.15

< 1s

107.67

< 3s

129.55

< 5s

151.25

< 10s

Gap

0.09%

0.42%

0.76%

0.67%

specific type of the assignment problem to the considered production system. According to the characteristics of the system, a sequencing generalized assignment problem is attributed to our optimization problem. In the research, two extra considerations are added to the classical generalized assignment. The first one is the restriction which is imposed on the working-days and off-days of the operators. The second consideration is the dynamics of the assignment which can be changed from a period to another. Because of these additional concerns, the problem is much more complicated than the classical GAP which is an NP-complete problem. For this problem, a mixedinteger mathematical model is presented. This model is solved by employing the Gurobi mixed-integer solver for the various sizes of the problem. The solver is only able to solve the small-scale problems. Therefore, a heuristic algorithm is proposed for solving such problems in the medium and large sizes. This algorithm presents the solutions with a very slight deviation from the optimal ones. The numerical results demonstrate the great reduction in the computational time in comparison with the Gurobi solver. Developing and generalization of this algorithm and comparing that with the meta-heuristic approaches can be the interesting subjects of the future researches.

0.62%

0.77%

3

PT

0.84%

Dıaz, J.A. and Fern´andez, E., 2001. A tabu search heuristic for the generalized assignment problem. European Journal of Operational Research, 132(1), pp.22-38. Chu, P.C. and Beasley, J.E. 1997. A genetic algorithm for the generalised assignment problem. Computers & Operations Research, 24(1), 17-23. Sethanan, K. and Pitakaso, R., 2016. Improved differential evolution algorithms for solving generalized assignment problem. Expert Systems with Applications, 45, 450-459. Ghoniem, A., Flamand, T. and Haouari, M., 2016. Exact solution methods for a generalized assignment problem with location/allocation considerations. INFORMS Journal on Computing, 28(3), pp.589-602. Ghoniem, A., Flamand, T. and Haouari, M., 2016. Optimization-based very large-scale neighborhood search for generalized assignment problems with location/allocation considerations. INFORMS Journal on Computing, 28(3), pp.575-588. McKendall, A. and Li, C., 2017. A tabu search heuristic for a generalized quadratic assignment problem. Journal of Industrial and Production Engineering, 34(3), 221-231. Posta, M., Ferland, J.A. and Michelon, P., 2012. An exact method with variable fixing for solving the generalized assignment problem. Computational Optimization and Applications, 52(3), pp.629-644. Moccia, L., Cordeau, J.F., Monaco, M.F. and Sammarra, M., 2009. A column generation heuristic for a dynamic generalized assignment problem. Computers & Operations Research, 36(9), pp.2670-2681. Tasgetiren, M.F., Suganthan, P.N., Chua, T.J. and AlHajri, A., 2009, May. Differential evolution algorithms for the generalized assignment problem. In Evolutionary Computation, 2009. CEC’09. IEEE Congress on (pp. 2606-2613). IEEE. Moussavi, S.E., Mahdjoub, M. and Grunder, O., 2017. Productivity improvement through a sequencing generalised assignment in an assembly line system. International Journal of Production Research, 55(24) pp.7509-7523. Alfares, H.K., 2002. Optimum workforce scheduling under the (14, 21) days-off timetable. Advances in Decision Sciences, 6(3), pp.191-199. Sadykov, R., Vanderbeck, F., Pessoa, A. and Uchoa, E., 2015. Column generation based heuristic for the generalized assignment problem. XLVII Simp´ osio Brasileiro de Pesquisa Operacional, Porto de Galinhas, Brazil.

REFERENCES Cohen, R., Katzir, L. and Raz, D., 2006. An efficient approximation for the generalized assignment problem. Information Processing Letters, 100(4), pp.162-166. Woodcock, A.J. and Wilson, J.M., 2010. A hybrid tabu search/branch and bound approach to solving the generalized assignment problem. European journal of operational research, 207(2), pp.566-578. Ross, G.T. and Soland, R.M., 1975. A branch and bound algorithm for the generalized assignment problem. Mathematical programming, 8(1), pp.91-103. Osman, I.H. 1995. Heuristics for the generalised assignment problem: simulated annealing and tabu search approaches. Operations-Research-Spektrum, 17(4), 211225. Osorio, M.A. and Laguna, M. 2003. Logic cuts for multilevel generalized assignment problems. European Journal of Operational Research, 151(1), 238-246. Elshafei, M. and Alfares, H.K. 2008. A dynamic programming algorithm for days-off scheduling with sequence dependent labor costs. Journal of Scheduling, 11(2), 85. 6