Non-linear programming (NLP)

Non-linear programming is the process of solving a system of equalities and inequalities, collectively termed „constraints‟, over a set of unknown real variables. The objective function

12

is maximised or minimised, where some of the constraints or the objective function are non-linear (Hillier and Lieberman, 2010).

Numerous models in the existing literature review have been formulated the inspection allocation problem using the NLP technique. Ballou and Pazer (1982) developed a computer program to perform a „what-if‟ simulation analysis of the serial systems with inspection errors. In a series of experiments, the authors found that inspection error rates have a major impact on cost. Type I errors (rejection of conforming units) were found to have a greater impact than type II errors (acceptance of non-conforming units). This contradicts the description in section 2.1.3. They interpreted that many real world systems continue to put pressure on the inspector to avoid type II errors which may be detected further down the line while failing to properly audit type I errors which may exist among the items discarded at the inspection station.

The original model of Ballou and Pazer (1982) was extended by the same authors in (1985), to analyse the relative merits of enhanced inspection and process improvement. The framework for their work assumes a multistage serial production system, with the possibility of end point and intermediate inspection. The inspection configuration chosen is one which minimises cost per good unit delivered to the customer and, accordingly, incorporates various cost trade-offs.

Tayi and Ballou (1988) noted that the traditional inspection procedures which incur costs are only used to identify and remove defective units. They proposed a model that considered both inspection and reprocessing activities. However, they assumed that the inspection configuration is given and fixed, and obtained a simple formula for determining the optimal initial lot size and the reprocessing batch size, which minimised the total system costs.

13

Barad (1990) described a break-even approach for performing inspection in a multistage production process. This paper was assumed that when inspection does take place, 100% of the processed product at that stage is inspected. One of the variables used to decide whether to inspect is the quality level at some points in the manufacturing process. Barad suggested allocating most of the inspecting resources to stages with a relatively high proportion of non-conforming product.

Jewkes (1995) noted that previous models of optimal inspection allocation in their review did not consider the case when a repair was carried out on-line. This paper was modelled inspection policies for a single stage manufacturing system as queues with two phases of service (processing and inspection), in which items can be inspected or repaired as necessary.

Several examples were given to illustrate the process of finding the optimal effort.

Narahari and Khan (1996) proposed an approximate analytical technique based on mean value analysis for a non-serial manufacturing system. The aim was to predict the mean cycle time and throughput rate of such models under different inspection strategies. The proposed method has been validated using a simulation technique.

Lee and Unnikrishnan (1998) combined the inspection allocation problem with multiple inspection stations, in a scenario controlled by the inspection time constraint. Owing to the complexity of the problem, they developed three heuristic methods. The optimal solution was determined by the inspection plan that minimises the total cost per conforming item which exits the system. They found that the results of the heuristic methods were close to the optimised solution.

Shiau (2002) noted that the inspection error still needs to be considered for solving the allocation problem in a multistage manufacturing system. This paper has introduced an inspection error model and as a result of the complexity of the problem, two heuristic

14

methods were developed. The results show that solutions obtained by the heuristic methods were close to the optimal solution.

Kogan and Raz (2002) applied optimal control theory to determine the best AOIS in a serial system, to minimise the sum of the inspection cost. They assumed that the defect detection rate and the cost are linearly proportional with inspection. They obtained the optimal sequencing of inspection activities at a point in time, as well as optimal timing of switching between inspection points.

Emmons and Rabinowitz (2002) dealt with an inspection system for detecting malfunctioning operations in a multistage production system. The authors proposed a heuristic procedure for inspection assignment and scheduling that enables the prediction of the system performance under any inspection capacity. The main contribution of their paper was a theoretical foundation for the further development of models and solution procedures for more realistic problems. They demonstrated that the solutions obtained by the heuristic method are close to the optimal solution.

Shiau (2003a) noted that the inspection error needs to be considered even when applying the same inspection station to monitor various workstations that have different manufacturing capabilities. Based on the limited inspection resource constraint, Shiau developed a unit cost model and introduced two heuristic solution methods. It was found that the heuristic methods produce solutions near to the optimal solution, with less processing time comparing with CEM. Shiau (2003b) extended his previous model to include external costs, and proposed a heuristic method to solve the allocation problem. The results obtained were very similar to the original work.

Hadjinicola and Soteriou (2003) developed a mathematical model for a multistage production system. The authors noted that previous literature did not adequately examine the impact of

15

changes in the yield of each production stage on the total cost, from defects observed at each production stage. They developed a generalised model concerned with allocating a limited inspection budget to the different production stages. The aim was to improve the yield of the production stages and to minimise the cost. They concluded that the optimal budget allocation, leading to reduce in the annual expected cost, resulted from defects observed at all production stages.

Work on allocating inspection stations has been done by Rau and Chu (2005), which considered non-serial production systems. Owing to their complexity, a heuristic solution method was developed and proved to have much less calculation time, even when the number of workstations increases. The results of work Rau and Chu (2005) were used by Rau et al.

(2005) to develop a mathematical model to find an optimal solution for allocating inspection stations in non-serial production systems. They used similar assumptions and considerations for the treatment of detected non-conforming items, as in the original model. To approach the complexity of the problem, a heuristic method was developed and the results obtained were very similar to the previous work.

Summary

The non-linear programming technique is the most popular method in the literature review, and was used by 15 papers, or 29% of the total. This is because of the nature of the inspection allocation problem in which some of the decision variables can only have integer values; for example, whether or not to inspect at the workstation. Moeini and Afshar (2009) explained that, for an actual manufacturing system, the computations required by NLP will escalate considerably as the number of workstations increases. However, its capability is limited in terms of solving a large-scale problem. In addition, they described that, within the past decade, many researchers have shifted their focus on optimisation problems from traditional optimisation techniques, based on linear and non-linear programming, to the implementation

16

of metaheuristic methods. Table 2.3 presents a summary of the classifications and characteristics of the studied models using NLP for the previously surveyed papers.

Table 2.3: Classification the main characteristics for the studied models used NLP method

Article

Lee (98) Serial Inspection time

Shiau (03a) Serial Limited inspection stations

7 I and II Yes _ Yes Yes NLP

Shiau (03b) Serial Limited inspection a general mixed and pure integer linear programming (ILP) problem (Hamdy, 2003). B&B is a general algorithm for finding optimal solutions to various optimisation problems, especially in discrete and combinatorial optimisation. Dorigo and Stutzle (2004) defined combinatorial optimisation problems as: involve finding values for discrete variables such that the optimal