A review on solving approaches

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The MMJIT problem, showing nonlinear OF and binary variables, has no polynomial solutions as far as we know. However, a heuristic solution approach can be effective. To get to a good solution, one among dynamic programming, integer programming, linear programming, mixed integer programming or nonlinear integer programming (NLP) techniques can be used. However, those methodologies usually require a long time to find a solution, so are infrequently used in real production systems1. Just a few studies used other methods such as statistical analysis or the Toyota formula2. The most renowned heuristics are the Miltenburg’s3 and the cited Goal Chasing Method (GCM I) developed in Toyota by Y. Monden. Given the products quantities to be processed and the associated processing times, GCM I computes an “average consumption rate” for the workstation. Then, the processing sequence is defined choosing each successive product according to its processing time, so that the cumulated consumption rate “chases” its average value. A detailed description of the algorithm can be found in4. GCM I was subsequently refined by its own author, resulting in the GCM II and the Goal Coordination Method heuristics5.

The most known meta-heuristics to solve the MMJIT6, 7, 8 are:

Simulated Annealing;
Tabu Search;
Genetic Algorithms;
Scalar methods;
Interactive methods;
Fuzzy methods;
Decision aids methods;
Dynamic Programming.

Table 6.1 Alternative configurations of most common MMJIT models
Parameter / major alternatives		Alternatives
Model structure		Mathematical programming		Simulation			Markov Chains			Other
Decision variables		Kanban number		Order interval			Safety Stock level			Other
Performance measures		Kanban number			Utilization ratio				Leveling effectiveness
Objective function	Minimize cost	Setup cost		Inventory holding cost			Operating cost			Stock-out cost
	Minimize inventory
	Maximize throughput
Setting	Layout	Flow-shop			Job-shop				Assembly tree
	Period number	Multi-period					Single-period
	Item number	Multi-item					Single-item
	Stage number	Multi-stage					Single-stage
	Machine number	Multiple machines					Single machine
	Resources capacity	Capacitated					Non-capacitated
Kanban type		One-card					Two-card
Assumptions	Container size	Defined					Ignored (container size equals one item)
	Stochasticity	Random set-up times	Random demand			Random lead times		Random processing times			Determinism
	Production cycles	Manufacturing system					Continuous production
	Material handling	Zero withdrawal times					Non-zero withdrawal times
	Shortages	Ignored					Computed as lost sales [43]
	System reliability	Dynamic demand	Breakdowns possibility			Imbalance between stages		Reworks			Scraps

In some experiments9 Tabu Search and Simulated Annealing resulted to be more effective than GCM; however, the computational complexity of these meta-heuristics – and the consequent slowness of execution – makes them quite useless in practical cases, as the same authors admitted.

Another meta-heuristic based on an optimization approach with Pareto-efficiency frontier – the “multi objective particle swarm” (MOPS) – to solve the MMJIT with setups was proposed through a test case of 20 different products production on 40 time buckets10.

In11, the authors compared a Bounded Dynamic Programming (BPD) procedure with GCM and with an Ant Colony (AC) approach, using as OF the minimization of the total inventory cost. They found that BDP is effective (1,03% as the average relative deviation from optimum) but not efficient, requiring roughly the triple of the time needed by the AC approach. Meanwhile, GCM was able to find the optimum (13% as the average relative deviation from optimum) on less than one third of the scenarios in which the AC was successful.

A broad literature survey on MMJIT with setups can be found in12 while a comprehensive review of the different approaches to determine both kanban number and the optimal sequence to smooth production rates is present in13.

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