[Paper Review] Solving Assembly Line Balancing Problems by Combining IP and CP

Alexander Bockmayr, Nicolai Pisaruk|ArXiv.org|Jun 1, 2001

Assembly Line Balancing Optimization14 references22 citations

TL;DR

This paper proposes a hybrid algorithm combining integer programming (IP) and constraint programming (CP) to solve simple assembly line balancing problems (SALBP), leveraging branch-and-cut with valid inequalities and CP-based propagation to significantly reduce search space. The key contribution is a cooperative framework that alternates between IP cut generation and CP domain reduction, achieving up to 85% reduction in variable domains and solving previously intractable instances within hours, outperforming standalone IP or CP.

ABSTRACT

Assembly line balancing problems consist in partitioning the work necessary to assemble a number of products among different stations of an assembly line. We present a hybrid approach for solving such problems, which combines constraint programming and integer programming.

Motivation & Objective

• To address the NP-hard simple assembly line balancing problem (SALBP) with precedence constraints and station capacity limits.
• To integrate integer programming (IP) and constraint programming (CP) for improved solution efficiency and pruning of the search tree.
• To develop a cooperative framework where IP generates cuts and CP performs domain reduction, enhancing problem size reduction.
• To evaluate the effectiveness of hybridization on benchmark instances, particularly for hard or large-scale problems.
• To provide a scalable modeling platform for real-world assembly line problems with complex side constraints.

Proposed method

• Formulates SALBP as an integer program with binary variables xij indicating task j assigned to station i, subject to assignment, capacity, and precedence constraints.
• Applies a branch-and-cut approach using lifted cover and (1,d)-configuration inequalities to strengthen the IP relaxation and cut off fractional solutions.
• Develops a CP model using domain variables startj for task j, with constraints enforcing precedence and station capacity via domain filtering.
• Implements a two-phase cooperation: first, CP propagation reduces variable domains; second, IP solves relaxed subproblems with domain-restricted variables.
• Uses LP relaxation solutions from IP to guide CP labeling: for each task, identify active stations via LP values and propagate via CP to fix domains.
• Introduces 'rounding off' of LP solutions by building a CP model on support sets and running limited labeling to find feasible solutions.

Experimental results

Research questions

• RQ1 Can combining IP and CP significantly reduce the search space in SALBP through cooperative pruning?
• RQ2 How effective is the integration of branch-and-cut cuts with CP propagation in reducing variable domains and solving hard instances?
• RQ3 Does the hybrid approach outperform standalone IP or CP on benchmark SALBP instances with tight precedence constraints?
• RQ4 To what extent can the hybrid solver handle real-world side constraints beyond standard SALBP?
• RQ5 Can the framework be extended to solve unsolved instances from standard benchmarks like Warnecke (73-22) within practical time limits?

Key findings

• The hybrid approach reduced the total number of variable domains (sum of |Sj|) by up to 85% on instances like Sawyer30 (47-7), from 210 to 33.
• For the hard instance Warnecke (73-22), the hybrid solver completed in 1 hour 36 minutes, while standalone IP or CP failed to solve it within one hour.
• The use of all cut types (standard and generated) in LP subproblems reduced domain sizes more effectively than using only standard cuts.
• Problem reduction via CP propagation alone reduced domain sizes significantly, with Lutz3 (118-14) dropping from 1246 to 285 initial assignments.
• The hybrid solver achieved a 90% reduction in search space on Tonge70 (251-14), reducing domain sum from 980 to 394 after CP and standard cuts.
• The cooperation between IP and CP proved essential: neither solver alone could solve the most challenging instances within practical time limits.