Dynamic programming (DP)

Dynamic programming was formalised in the early 1950s by mathematician Richard Bellman. DP is a recursive method that determines the optimum solution to an n-variable problem by decomposing it into n stages, with each stage constituting a single variable sub-problem. The computational advantages are that DP optimises single variable sub-problems (Hamdy, 2003).

A number of mathematical models have been developed to determine the optimal location of inspection stations in multistage production systems, using the dynamic programming technique. Lindsay and Bishop (1964) were the first to develop a model for determining an optimal inspection policy with the lowest total cost for serial production. The inspection is assumed to be perfect (no inspection error) and the inspection at one stage is independent of the next. It was found that DP allows the determination of a minimum cost under the added assumptions of maintaining a specified quality level, or when the cost associated with outgoing defective material is linear.

White (1965) also researched this area and showed that, with replacement of defectives, the optimal plan would be characterised by 0 or 100% inspection, and could be solved by a dynamic program. The results of the work of Lindsay and Bishop (1964) are used by Pruzan and Jackson (1967) to develop an adaptive model in which the optimal inspection policy at a location depended on the previous inspection history.

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White (1969) presented two shortest route models for determining where to allocate inspection points on a serial production line. In this paper, both repairable and non-repairable defectives are considered. Hurst (1973) was the first to propose a model that considered two types of inspection errors: acceptance of non-conforming units (type I); and rejection of conforming units (type II). The production system was assumed to be serial with only one inspection operation possible after each processing stage, and units perceived to be non-conforming removed from the production flow.

In addition, Enrick (1975) and Hsu (1984) have applied dynamic programming to find the optimal location of inspection stations in serial systems. They concluded that dynamic programming was an effective technique for determining the inspection policies sequencing for a limited number of production stages.

Peters and Williams (1984) investigated the performance of five heuristics rules of thumb in a serial production system. The results indicated that a variety of economic and operating factors influenced the applicability of each of the five inspection location heuristics examined.

An inspection planning model was developed by Gunter (1985), for an assembly process free of error. The results shown that if defective items are removed from the line, then the production volume in the model will shrink as a result of inspection, in order to meet the demand the inspections decisions should be considered the production rates.

Raghavachari and Tayi (1991) developed a model as a shortest path algorithm in serial production systems to minimise the total cost. In their model, the determination of optimal initial lot size, inspection configuration and reprocessing decisions are considered simultaneously. They concluded that the developed model is flexible and versatile enough to be applied to the manufacturing environment with different characteristics.

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Chengalur et al. (1992) extended the model of Ballou and Pazer (1985) by using a dynamic model with uncertainty in the quality of incoming raw material. They assumed 100%

screening if inspection was performed at any stage. Also any defective item delivered to the customer was assigned as penalty cost. Their conclusion was that using a dynamic procedure led to cost-saving, even when unsure about the quality of the raw materials.

Bai and Yun (1996) investigated the problem in which a product consists of many identical components. In their model, only a limited number of (automatic) inspection machines are available, and the rate of production was constrained by the rate of inspection. The inspection level was defined as the percentage of components to be inspected. An inspection cost model was developed to obtain the best location for inspection points and the optimal inspection level. They found that the proposed heuristic algorithm combined with DP provides the optimal solution when the problem is small. However, as the problem increases, the heuristic algorithm provides a solution close to the optimal solution in less time comparing to the CEM.

An unreliable serial production system with known failure probabilities at each workstation was studied by Penn and Raviv (2007). The dynamic programming technique and a branch and bound method were used, to solve the problem of determining optimal quality control station configuration within the assembly line. The contribution of the model was incorporation of holding costs. Optimal quality control stations were found to reduce the load on the bottleneck stations, as well as the work in-process on the stations that followed them.

Table 2.5 presents a summary of the classifications and characteristics of the models using DP for the previously surveyed papers.

Summary

Dynamic programming technique is used in 13 papers, accounting for 25% of the papers surveyed. It is the second most common technique used, mostly used in the period from 1964

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to 1985. In fact, the use of recursion in the DP algorithm has both advantages and disadvantages: the main advantage is usually simplicity; the major disadvantage is the rapid rise in computational requirements, as the number of decision variables to be optimised is increased. This has led to various algorithms being developed that limit the number of stages, states and decision variables combinations to be evaluated. In addition, Shiau (2002, 2007), Lee and Unnikrishnan (1998) and Rau and Chu (2005) have pointed out that the DP approach becomes impractical as the set of possible combinations grows exponentially. That is why many kinds of metaheuristic methods, such as simulated annealing, Tabu search and genetic algorithm are often used to reach a satisfactory solution, even though it may not be the optimal one.

Table 2.5: Classification the main characteristics for the studied models used DP method

Article

AOQL: average of outgoing quality level.

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In summary, many models in the area of the AOIS problem have been reviewed. These models used traditional techniques to tackle the AOIS problem. However, the capability of these techniques, in terms of computational time required to obtain good solutions increases as the number of workstations increased significantly (Van Volsem and Neirynck, 2009).