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Production/inspection flow
A production line in an MMS can produce either a specific product or multiple types of the same product family. In this regard, inspections of this manufacturing line can be done per item or batch/lot. Therefore, four possible disciplines for the production/inspection flow exist (see Figure 2.3b). Figure 2.3b also shows the percentage of papers which consider these different disciplines. Although the Mixed production/batch inspection discipline is closer to the real condition of MMSs, the modeling and solving complexity of that is greater.
Inspection strategy
Two different inspection strategies have been employed by the researchers in the literature: 100% or none inspection and Sampling inspection. In the first strategy some of the QCs are not inspected, and if it is decided to inspect a QC, all the items are inspected. Sampling inspection means if it is decided to inspect a QC, a sample of items is inspected.
Figure 2.3c illustrates the percentage of papers which consider these two different strategies. Although using the Sampling inspection is more practical in high rate production systems, it is difficult to be considered because the other parameters of this strategy (e.g., sample size) need to be determined simultaneously in the optimization framework and this great number of decision variables increases the solution complexity of the problem.
Inspection errors
There are two types of errors that may happen during a part quality inspection activity:
i. Error type I: occurs when a conforming item is classified wrongly as a nonconforming one.
ii. Error type II: happens when a nonconforming item is classified wrongly as a conforming one.
The Error-Free assumption for the inspection activity is unrealistic, but a considerable number of works (i.e., 37%) considered the inspection activities free of any error (see Figure 2.3d).
Failure rate and type
A failure rate of a manufacturing stage is the proportion of defects to all the produced items. In the literature, a certain constant failure rate for each stage has been assumed in some works, whereas others have assumed either a plausible range of a failure rate or random failure under a specific distribution. In addition, two single and multiple failure types have been considered by the authors. In fact, each QC is related to a single or multiple failure modes. In a case of a single failure type for a specific QC, the QC will not be realized properly if its related failure mode is active. Similarly, for the multiple failure types, the QC will not be realized appropriately if at least one of its related failure modes is active. In this regard, for the multiple failure types, a vector of failure rates is associated with multiple failure modes ( Mandroli et al, 2006). It is notable that each manufacturing stage may contain more than one failure mode. In conclusion, as shown in Figure 2.3e, four potential combinations exist while a few papers only considered the Multiple type/random rate assumption.
Nonconforming strategy
When inspection recognized an item as a nonconforming product, four possible actions can be done. The item can be reworked, repaired, replaced, or scrapped. The decision about the appropriate action depends on the associated cost and knowledge of whether the nonconformity is reparable/reworkable or not. In this regard, a deterministic or probabilistic level of scrapping for nonconforming parts have been assumed by the researchers. In the deterministic level, for a given type of nonconformity, the scrapping level is given as one of the three different possibilities: all, none, or some of the nonconforming items are scrapped. On the other side, others have assumed a probabilistic level which means that a nonconforming item is scrapped with a certain probability, so some of the items may have a chance to be reworked, repaired or replaced (Mohammadi, 2015). According to the above-mentioned explanations, the nonconforming strategy is divided into four different subcategories as shown in Figure 2.3f.
Methodology characteristics
According to the vast review of the PQIP literature, almost all the studies have dealt with the problem through an optimization formulation. In the following subsections, first, different kinds of the considered objective functions are elaborated, and then three necessary parts of an optimization formulation as (i) constraint, (ii) uncertainty approach, and (iii) solution approach are addressed. Table 2.2 illustrates the features of the literature regarding the optimization characteristics.
Objective functions
Minimization of the total expected cost is the most common form of objective function in the literature. Total cost generally includes different cost components as production, inspection, and failure costs. The failure cost itself consists of internal and external cost. When nonconforming products are found before shipment, it imposes an internal cost to the system. This cost is specifically related to the costs of reworking, repairing, replacing, and scrapping the nonconforming product. A manufacturer undergoes the external failure costs when a defective product has been received by customer(s). These costs may be certain compensation or the lost sales and goodwill. The inspection cost contains two fixed and variable costs. The fixed inspection cost is related to a fixed amount of capital for preparing inspection tools and the variable cost directly depends on the frequency and number of inspected items. The variable inspection cost has been often assumed as a linear function, in which, the total variable inspection cost is the number of items inspected multiplied by the variable inspection cost per item (Mohammadi, 2015). There are just two works, which have treated this cost as a non-linear function (see e.g., Britney (1972) and Ballou and Pazer, (1985)). Regarding the current literature, there is no work which concurrently considers the non-linear form of variable inspection cost, fixed inspection cost and internal failure cost (i.e., scrap, rework, repair and replace). Please see Figure 2.4a-c for the details of the literature regarding the considered internal, external, and inspection cost.
Another common form of the objective function is the expected unit cost. However, there are different ways to determine the units (see Figure 2.4d). Some papers have computed the expected unit cost as total cost divided by the number of input items (i.e. total cost/input items). The other versions were dividing the total cost by the number of outputs and dividing the total cost by the number of conforming outputs (i.e., total cost/output items or total cost/conforming output items).
There are only a few authors considering maximization formulations in their studies. The maximization objectives have mainly proposed in inspection scheduling problems besides to classical PQIP problem. To the best of our knowledge, no study has considered minimizing total manufacturing time. In addition, there is a lack of applying multi-objective models in the current PQIP literature. In this respect, Mohammadi et al. (2017) minimized total manufacturing cost and warranty cost (to capture customer satisfaction) in form of two different objective functions. However, the customer satisfaction has a non-linear behavior when he/she receives the product lot —which was neglected by Mohammadi et al. (2017). Indeed, the customer satisfaction and accordingly the utility of the product lot is higher when the proportion of the conforming items is greater in the delivered lot. They solved the presented model by employing a meta-heuristic algorithm, namely Differential Evolution (DE) algorithm.
For approaching to the real situation of MMSs, where they face to the conflict objectives for optimizing their systems, developing the multi-objective mathematical models which simultaneously optimize several conflict objectives is an excellent research direction for future in this field. For instance, when the quality of the system is maximized through employing more inspection stations, the production cycle time is increased. Hence, developing a new bi-objective model to establish a trade-off between these two conflict objectives seems interesting and practical.
Constraints
The constraints in the typical PQIP problem are mostly associated with the different types of production structure and nonconforming strategy. Moreover, other constraints have been imposed to the optimization formulation. For example, some of them are: limitations on the inspection time, number of inspection and rework stations, number of inspection repetition, the budget for production and inspection activities, required place for an inspection station, and minimum throughput or production capacity. Figure 2.4e illustrates the percentage of papers which considered different constraints.
Other constraints in the developed optimization formulation could be the dependency between different QCs that require to be inspected. For instance, two QCs must be inspected in parallel or vice versa. In addition to QCs dependency, in some situations, there are dependent production stages, and there is no possibility for stop a particular stage to inspect a QC and you need to wait for the following operation(s) to be completed (Mirdamadi, 2014; Mohammadi, 2015). There are the other applicable and realistic constraints in the domain of PQIP problem that have not considered yet such as the limited capacity of operating machines, waiting time for inspection and capable inspection tools to treat the items.
Optimization under uncertainty
The PQIP problem inherently contains different sources of uncertainty. Accordingly, the necessity for consideration of associate uncertain parameters and obtaining a robust solution has been implied by most of the researchers, and it should be extended more. One of the main sources of uncertainty in this problem refers to the condition of production stages for processing items in conformity with specification region. This uncertain condition results in the uncertainty of the failure production rate parameter. In this regard, many works considered the random failure rate in a probabilistic manner (see Table 2.1). They assumed a specific probability distribution (e.g., Bernoulli distribution) for this parameter. By the above-mentioned source of uncertainty, the proportion of nonconforming items which is repairable, reworkable or need to be scrapped or replaced, is uncertain. This uncertain data has been also treated as a probabilistic nonconforming strategy in the literature (See Table 2.1). Regarding the uncertainty about inspection tools and inspection operators, Error type I and Error type II have been considered probabilistically by the majority of researchers (see Table 2.1). Beside these uncertainties which are rooted in the internal reasons, there are the other uncertainties that are related to the uncertain external condition such as price fluctuations (Azadeh et al., 2015a) and demand amount. Azadeh et al. (2015a) described the cost components (i.e., inspection, rework and penalty of defects shipment) by applying the fuzzy numbers. They proposed a fuzzy model for the problem and converted it into an equivalent auxiliary crisp model by employing the Jimenez ’s definition of expected value. Mohammadi et al. (2017) considered alteration ranges for the production and inspection times, errors type I and II of the inspection activities, dispersion and misadjustment of the production processes, and developed a global robust model based on optimizing the expected value and variance of the objective function in the form of the Taguchi method. Regarding these recent works, it would be an excellent research direction to propose an approach (e.g., a robust possibilistic programming approach) to make use of the advantages of the fuzzy and robust approaches for considering uncertainties related to the external reasons which are being usually implied by experts (subjective data).
Table of contents :
Chapter 1: Introduction
1.0. Chapter purpose and outline
1.1. Introduction
1.2. PQIP problem
1.3. The relation of PQIP with maintenance and production issues
1.4. Conclusion
Chapter 2: Literature review
2.0. Chapter purpose and outline
2.1. Introduction
2.2. MMS characteristics
2.2.1. Production structure
2.2.2. Production/inspection flow
2.2.3. Inspection strategy
2.2.4. Inspection errors
2.2.5. Failure rate and type
2.2.6. Nonconforming strategy
2.3. Methodology characteristics
2.3.1. Objective functions
2.3.2. Constraints
2.3.3. Optimization under uncertainty
2.3.4. Solution approaches
2.4. Integrated optimization
2.5. Conclusion
Chapter 3: Mathematical formulation
3.0. Chapter purpose and outline
3.1. Main problem description
3.1.1. Assumptions
3.1.2. Indices
3.1.3. Parameters
3.1.4. Variables
3.1.5. Mixed-Integer Linear Mathematical Programming Model
3.2. Extended problem description
3.2.1. Assumptions
3.2.2. Indices
3.2.3 Parameters
3.2.4. Variables
3.2.5. Bi-Objective, Mixed-Integer, and Non-Linear Mathematical
3.3. Conclusion
Chapter 4: Methodology
4.0. Chapter purpose and outline
4.1. Stage 1: Piecewise linear approximation
4.2. Stage 2: Robust possibilistic programming model
4.3. Stage 3: Single objective counterpart model
4.4. Conclusion
Chapter 5: Experimental Results
5.0. Chapter purpose and outline
5.1. Numerical example
5.1.1. Main model
5.1.2. Extended model
5.2. Case study
5.2.1. Main model
5.2.2. Extended model
5.3. Conclusion
Chapter 6: Conclusion and Future Research Direction
6.0. Chapter purpose and outline
6.1. Conclusion
6.1.1. Main model
6.1.2. Extended model
6.2. Future research directions
6.2.1. MMS characteristics
6.2.2. Methodology characteristics
