Combinatorial Benders cuts for assembly line balancing problems with setups

TL;DR: This paper describes an exact algorithm based on Benders decomposition to solve both simple and mixed-model assembly line balancing problems with setups and tests it on a set of benchmark instances and against a mixed-integer linear programming formulation of the problem solved using a commercial optimizer.

About: This article is published in European Journal of Operational Research.The article was published on 2017-06-01 and is currently open access. It has received 57 citations till now. The article focuses on the topics: Combinatorial optimization & Exact algorithm.

Summary

1 Introduction

• Assembly lines (ALs) are special flow-based production systems.
• Early designs of assembly lines were for a single product to be produced in high volumes with the corresponding problem known as the simple assembly line balancing problem .
• The rest of this paper is concerned with Type-I problems, for which reason the authors will not explicitly spell out the type unless otherwise stated.
• The computational difficulty of solving the MMALBP with setups to optimality using a commercial software was shown in [2].
• The authors reformulate the original problem as a master assignment problem with an exponential number of feasibility constraints, following which they search for infeasible assignments using the sequencing problem as a slave problem, and forbid any such assignments through infeasibility cuts in the master problem until an optimal solution is identified.

2 A Summary of Relevant Literature on ALBPs with Setups

• Setup times considerations for assembly line balancing problems have first appeared in [4] and [21], where the SALBP was extended to incorporate sequence dependent setup times between tasks, for which the authors described a mathematical programming model of the problem and proposed several heuristics including GRASP.
• Priority rules based solution procedures were developed in [17] to solve the problem, however these procedures were not effective enough in solving large-size test problems with more than 100 tasks.
• Mixed-integer programming formulations of a similar problem were developed in [23] where the authors stated that solving the problem with standard MIP solvers is not an effective solution method.
• For the Type-II version of the problem, [25] described a mathematical model similar to that in [4], and a simulated annealing (SA) algorithm.
• A mixed-integer programming formulation for another version of the problem where setup times were considered for a two sided assembly line was presented in [20], for which the authors proposed a heuristic.

3.1 Formal problem description

• Mixed-model assembly lines are used to either produce a single model of product with different features or several models on a single assembly line.
• Each model comes with a specific set of precedence relations between its tasks which can be combined into a precedence diagram for all models.
• Each task i for model m has a processing time Tim, and this may vary between the M models assembled on the line.

3.2.1 Basic formulation of the MMALBP

• Using these variables the MMALBP can be formulated as the following binary programming model.
• Constraints (2) ensure the assignment of each task to exactly one workstation.
• Constraints (3) are used to guarantee the precedence relations between tasks.
• Capacity constraints (4) ensure that the workload of each used workstation does not exceed the cycle time.

3.2.2 An improved formulation for the MMALBP with setups

• A mixed integer linear programming formulation of the MMALBP with setups was proposed in [2] but this formulation suffered from a significant number of “big-M”type constraints.
• Processing time of a common task may be different among the different models.
• Constraint sets (11), (12) and (13) order tasks and assign setup operations between them if they are assigned to the same workstation.
• If a backward setup operation has been assigned between any tasks pair then there would not be any forward setup operation, which is modeled by constraints (22).

4 A Benders Decomposition Algorithm

• Benders Decomposition [7] is based on reformulating the original problem as a so-called master problem (MP ) that has an exponential number of cuts, which are initially relaxed and separated in an iterative fashion using a so-called slave (or sub) problem.
• Some implementations look for the possible minimal subsets of variables that induce infeasibility in SP and derive a cut from these subsets rather than adding a cut containing all the x variables [11].
• The proposed algorithm is based on the observation that MMALBPS can be formulated by using two sets of assignment (Yis and As) and one set of sequencing (wijs) variables.
• Otherwise, the slave problem returns an infeasibility for at least one of the workstations, for which the following group of feasibility cuts is added to the master problem.
• Algorithm 1 Benders Decomposition Algorithm (BDA) for the MMALBPS 1: [MP ] : Master Problem; Cutset:.

5 Computational Study

• This section presents a computational study, in three parts, to assess the performance of the proposed algorithm.
• In the first part, the authors describe the way in which the instances are generated.
• The second part analyses the effect of the symmetry-breaking constraint set (10) on the computational run time of the algorithm.
• The third part presents results to numerically compare the IP and the BDA on the instances.

5.1 Instance generation

• There is no standard set of benchmark instances with setup times available in the assembly line balancing literature.
• The higher the OS value, the algorithm requires less time to solve the problem to optimality.
• The test instances that the authors used in this current paper have OS values vary between 22 and 84.
• All models and subproblems have been solved using GUROBI 6.0.

5.2 Analysis on the effect of the symmetry-breaking constraint set

• To numerically confirm whether this is the case, some experiments are conducted by running the MP with and without the constraint set (10) on a subset of the test instances.
• The feasible solutions given in the third and seventh columns of Table 3 are the objective values of the best solution found by MP after one hour.
• The gap values given in the fourth and eighth columns of Table 3 are the percentage differences between the best solutions found by MP and the lower bound value calculated by the solver for the problem.
• As can be seen from Table 3, the constraint set (10) has a significant effect on reducing the CPU time as the problem size gets larger.
• For some cases the MP cannot identify an optimal solution without the constraint set (10), however the incumbent solution found by the MP is the same as the optimum solution.

5.3 Performance evaluation of the proposed Benders decomposition algo-

• This part of the computational analysis concerns the performance evaluation of the BDA and the IP on the test bed of instances listed in Table 2 in terms of solution time.
• Additionally, the tables contains the average solution times for SP and MP , and the number of added feasibility cuts (NFC) for each instance.
• Here the authors conclude that the BDA is superior to the IP in terms of solution time, since the average solution computational time for instances 39–46 is 349.30 seconds for the latter and 0.61 seconds for the former.
• For the other four instances numbered 19, 38, 56 and 57, the algorithm was not able to find optimal solutions within one hour.
• On the other hand, the IP found optimal solutions for the single and two model instances with up to 21 tasks, and for three model instances with up to 19 tasks.

6 Conclusions

• The authors described a Benders decomposition algorithm for single and mixed-model Type-I assembly line balancing problems with setups.
• The model contains the assignment subproblem of the assembly lines and the sequencing subproblem related to the sequence dependent setup times between tasks.
• By exploiting this structure the authors devised a Benders decomposition algorithm, which solves the assignment subproblem as a master problem and the sequencing subproblem as a slave problem in order to generate combinatorial Benders cuts.
• The performance of the proposed algorithm was tested on a set of literature-based benchmark instances and the results are compared against a mixed-integer linear programming formulation of the problem solved using an off-the-shelf optimizer.
• The results confirm the superior performance of the proposed algorithm in terms of computational time.

Figures

Table 1: Model notation

Table 2: Main characteristic of the test problems

Table 6: Computational results for three model problems

Table B.1: Forms of converting absolute value constraints into linear constraints

Table B.2: Auxiliary variables

Table 5: Computational results for two model problems

Table 4: Computational results for single model problems

Table A.1: Model notation and their definitions

Table 3: Analysis on the Effect of the Symmetry-Breaking Constraint Set

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Combinatorial benders cuts for assembly line balancing problems with setups" ?

This paper adopts such an approach and describes an exact algorithm based on Benders decomposition to solve both simple and mixed-model assembly line balancing problems with setups. 

Q2. What is the purpose of the paper?

The authors reformulate the original problem as a master assignment problem with an exponential number of feasibility constraints, following which the authors search for infeasible assignments using the sequencing problem as a slave problem, and forbid any such assignments through infeasibility cuts in the master problem until an optimal solution is identified. 

Q3. What is the definition of an assembly line?

The design of such systems gives rise to the assembly line balancing problem (ALBP), which consists of assigning assembly tasks to a number of workstations in order to optimize a given objective. 

Q4. What is the type of assembly line balancing problem?

Type-I (resp. TypeII) problems deal with the assignment of tasks to workstations with the aim of minimizing the number of workstations (resp. the cycle time) for a predetermined cycle time (resp. predetermined number of workstations) by respecting the precedence relations of the tasks involved in the assembly. 

Q5. What is the main idea of the paper?

The mixed-model version of the assembly line balancing problem with setups was studied in [18], and the variant with sequence dependent setup times between tasks was studied in [2], and hybrid meta-heuristic algorithms, including a combination of ant colony optimization and genetic algorithm and a multiple colony hybrid bees algorithm were described in [1] and [3]. 

Q6. What was the heuristic for the assembly line balancing problem?

A mixed-integer programming formulation for another version of the problem where setup times were considered for a two sided assembly line was presented in [20], for which the authors proposed a heuristic. 

Q7. What is the purpose of the article?

Early designs of assembly lines were for a single product to be produced in high volumes with the corresponding problem known as the simple assembly line balancing problem (SALBP). 

Q8. What is the difference between a single model and a mixed model?

Single-model assembly lines did not prove efficient for products with a high variety, required by a consumer-centric market, necessitating a high degree of flexibility in the manufacturing system [29]. 

Q9. What was the proposed formulation of the SALBP?

Another mathematical formulation of the SALBP was proposed in [14], along with a combination of aparticle swarm optimization algorithm and variable neighbourhood search.