Methodology

According to the vast literature on inspection planning problems, most of researchers have solved the problem through an optimization formulation. Almost all

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objective functions have been to minimize the total cost including the costs of inspection, scrap, rework, warranty, and so on. In this section, different components of the cost objective function are discussed first; next, the three fundamental parts of an optimization formulation as (i) objective function; (ii) constraint; and (iii) solution approach, are discussed. The category of methodology has been illustrated in detail in Figure 2.2.

Cost components

In almost all researches, the authors have considered specific components for the cost objective function including production, inspection, and failure costs. The failure cost itself contains internal and external cost. An internal failure cost is incurred when nonconforming items are detected before reaching the customers.

This cost specially reflects the costs associated with reworking, replacing, or scrapping a nonconforming item. External failure costs are all costs that manufacturer faces with when a nonconforming product is sold and delivered to the customers. These costs may be a certain penalty or compensation, as well as the lost sales and costs for restoring the reputation of the product. The external failure costs have not been taken into account in every research.

On the other hand, the inspection cost involves two fixed and variables costs.

The fixed inspection cost corresponds to fixed amount of capital for providing inspection tools and the variable cost directly depends on the frequency and number of inspected items. The variable inspection cost has been often considered 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. Some other researchers have treated this cost as a quasi-concave function (Britney, 1972).

Objective functions

The most common form of objective functions in the literature is minimizing total expected cost to optimize the inspection plans. Another common treatment is expected unit cost, instead of the total expected cost. However there are different ways to determine the units. Some papers have computed the expected unit cost as total cost divided by the number of input items; i.e., (𝑡𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡) (𝑖𝑛𝑝𝑢𝑡 𝑖𝑡𝑒𝑚𝑠)⁄ . Another version is dividing total cost by number of output; i.e., (𝑡𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡) (𝑜𝑢𝑡𝑝𝑢𝑡 𝑖𝑡𝑒𝑚𝑠)⁄ . Another idea for the second form is dividing total cost by the number of conforming outputs; (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 inspection planning problem. To the best of our knowledge, there is no study that has considered minimizing total manufacturing time as well as maximizing customer satisfaction.

16 Constraints

The constraints in typical inspection planning problem are mostly related to the type of production structure, the type of nonconformance, the type of inspection.

The authors have derived other constraints such as: (i) an upper bound for inspection time; (ii) limited number of inspection stations; (iii) limited number of rework and the times that an inspection can be repeated; (iv) limited budget for manufacturing and inspection actions; (v) a limited places that an inspection can be performed, and (vi) a lower bound requirement on throughput or production capacity (Mandroli et al., 2006). It is noteworthy that constraints (i) and (ii) can be categorized as a special form of constraints (vi) and (iv), respectively.

Other constraints in inspection plans could be the dependency between different quality characteristics that need to be inspected. For example, two quality characteristics must be inspected at the same time or vice versa. Besides to quality characteristics dependency, operations that realize the characteristics might be dependent and there is no possibility to stop a specific operation to inspect a characteristic and we have to wait once the second operation is terminated. For more information regarding to operation dependency, interested readers are referred to the work done by Mirdamadi (2014). There may be other constraints applicable in domain of inspection planning problems that have not gained lot of attention such as limited capacity of operating machines and inspection tools to treat the items.

Solution approaches

The authors have proposed a wide variety of approaches for solving small and large size instances. In small size instances, approaches such as dynamic programming (DP), integer programming (IP) and nonlinear programming (NLP) have been utilized. Among these, DP has gained more popularity due to multistage structure of production systems following by NLP and IP methods in lower popularity.

An important limitation of these approaches is their incapability of solving medium and large size problems due to requirement of high computational time and memory. This limitation led to arise of heuristic and metaheuristics algorithms in this domain such as Simulated Annealing (SA) and Genetic Algorithms (GAs), while they provide near optimal solutions in considerable low computational time. Another optimization approach includes using simulation to solve the problem.

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Figure 2.1. Criteria of production system characteristics

Production System Characteristics

Production

Structure Production & Inspection

Flow Inspection

Type Inspection Strategy Inspection Error Failure Type & Rate Nonconforming Strategy

Nonserial

Convergent

Serial Single Prod./single Insp. Mixed Prod./single Insp.

Single Prod./batch Insp. Mixed Prod./batch Insp. Conformity Monitoring SamplingInspection

Full Inspection

No Inspection Error-Free

Error Type II

Error Type I Constant rate/single type Constant rate/multiple type

Random rate/single type Random rate/multiple type No Scrap Scrapping all

Scrapping some Probabilistic

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Figure 2.2. Criteria of methodology

Methodology

Internal Failure External

Failure Inspection Cost

Objective

Function Constraint Solution Approach

Scrap

Replace

Rework Defect Dependent Fixed

Defect Independent Linear Nonlinear Production Total Cost / Conforming Output

Total Cost / Output

Total Cost / Input Number of Repeated Inspection

Number of Inspection Station

Inspection Time & Place Budget Integer Programming

Dynamic Programming Nonlinear Programming Heuristics & Metaheuristics Simulation

Variable Cost

Cost Component

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In the following, the literature is first reviewed based on two main categories described in Section 2.1 as separate and simultaneous optimizations; next, the surveyed papers are analyzed to drag the gaps.