Assumptions

This section provides the assumptions for the Extended Problem as follows:

 The system has N manufacturing stages arranged serially and processes P types of products with K different quality characteristics.

 Different quality characteristics may be processed in a same manufacturing stage.

 Nonconformities are generated only at the manufacturing processes and other activities such as movement, setup and inspection activities do not make nonconformity.

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 Each manufacturing stage has a failure rate of producing nonconforming items.

 Two types of conformity (CI) and monitoring (MI) inspections are considered, while considering MI for a manufacturing stage decreases the failure rate of that stage.

 CI subjects to both errors type I and II.

 Two inspection strategies may be taken at each manufacturing stage as no inspection and full inspection.

 The frequency of MI is fixed.

 Detected nonconforming items are directly scrapped and no rework or repair operation is considered.

 A unit scrap cost is imposed to the system in case of detecting a nonconforming item. The scrap cost depends on both the number of manufacturing stage and the quality characteristics.

 Different machines with specific features (i.e., time, cost, capability, etc.) exist to operate the items and machine can operate a set of quality characteristics.

 Only one machine is allocated for operating each quality characteristics.

 Different inspection tools with specific features (i.e., errors, detection rate, time, cost, etc.) exist to inspect the items and these tools can inspect a set of quality characteristics.

 Only one inspection tool is allocated for inspecting each quality characteristic.

 In-process items must wait in a queue to receive services (i.e., machinery or inspection). Therefore, Machines and Inspection tools are modeled as a M/M/1 queuing system.

 The production system reaches a steady state but machines and inspection tools are subject to disruption and breakdown.

 In case of disruption, the processing and inspection rates are degraded to 0.

 Machines and inspection tools are disrupted with a random rate and are retrieved again with a random rate.

 A capacity constraint is assumed for both machines and inspection tools.

3.3.2.2. Notations

Before the mathematical model is presented, necessary notations are first provided in this section.

Sets:

𝑜, 𝑜^′ ∈ {1,2, … , 𝑂 + 1} Set of operations 𝑝 ∈ {1,2, … , 𝑃} Set of products 𝑚 ∈ {1,2, … , 𝑀} Set of Machines 𝑖 ∈ {1,2, … , 𝐼} Set of inspection tools 𝑘 ∈ {1,2, … , 𝐾} Set of quality characteristics

62 Parameters:

𝑓𝑟_{𝑜𝑘𝑝𝑚}¹ Failure rate of operation o for quality characteristic k for product p on machine m with monitoring inspection.

𝑓𝑟_{𝑜𝑘𝑝𝑚}² Failure rate of operation o for quality characteristic k for product p on machine m without monitoring inspection.

𝑑_𝑜𝑘^𝑝𝑖 Detection rate of conformity inspection assigned to operation o for quality characteristic k in product p using inspection tool i.

𝛼_𝑜𝑘^𝑝𝑖 Type I error of conformity inspection assigned to operation o for quality characteristic k in product p using inspection tool i.

𝛽_𝑜𝑘^𝑝𝑖 Type II error of conformity inspection assigned to operation o for quality characteristic k in product p using inspection tool I (𝛽_𝑜𝑘^𝑝𝑖 = 1 − 𝑑_𝑜𝑘^𝑝𝑖).

𝐺_𝑘^𝑃 Relative importance of quality characteristic k in product p.

𝑛_𝑇^𝑝 Total number of raw parts of product p fed to the production process.

𝑝𝑐_𝑜𝑝^𝑚 Unit production cost per time for operation o in product p on machine m.

𝑝𝑡_𝑜𝑝^𝑚 Unit production time of operation o in product p on machine m.

𝑠𝑐_𝑜^𝑝 Scrap cost of nonconforming items detected between operations o and o+1 in product p.

𝑛𝑐_𝑘^𝑝 Cost of nonconforming items in the market due to quality characteristic k in product p.

𝑓𝑚_𝑜𝑘^𝑝𝑚𝑖 Fixed cost of an MI between operations o and o+1 for quality characteristic k in product p on machine m using inspection tool i.

𝑓𝑐_𝑜𝑘^𝑝𝑚𝑖 Fixed cost of an CI between operations o and o+1 for characteristic k in product p on machine m using inspection tool i.

𝑓𝑝_𝑚 Fixed cost of utilizing machine m.

𝑓𝑎_𝑜𝑝^𝑚 Fixed cost of performing operation o in product p using machine m.

𝑐𝑝_𝑖 Fixed cost of utilizing inspection tool i.

𝑐𝑎_𝑜𝑘^𝑝𝑖 Fixed cost of conformity inspection of quality characteristic k in operation o in product p using inspection tool i.

𝑣𝑚_𝑜𝑘^𝑝𝑚𝑖 Unit variable cost of MI per time between operations o and o+1 for quality characteristic k in product p on machine m using inspection tool i.

𝑣𝑐_𝑜𝑘^𝑝𝑚𝑖 Unit variable cost of CI per time between operations o and o+1 for quality characteristic k in product p on machine m using inspection tool i.

𝑚𝑡_𝑜𝑘^𝑝𝑖 Unit time of MI between operations o and o+1 for quality characteristic k in product p on machine m using inspection tool i.

𝑐𝑡_𝑜𝑘^𝑝𝑖 Unit time of CI between operations o and o+1 for quality characteristic k in product p on machine m using inspection tool i.

𝜇𝑝_𝑜𝑝^𝑚 Production rate of machine m for performing operation o in product p (𝜇𝑝_𝑜𝑝^𝑚 = 1 𝑝𝑡⁄ _𝑜𝑝^𝑚).

𝜇𝑐_𝑜𝑘^𝑝𝑖 CI rate (part/time) of inspection tool i for quality characteristic k of operation o in product p (𝜇𝑐_𝑜𝑘^𝑝𝑖 = 1 𝑐𝑡⁄ _𝑜𝑘^𝑝𝑖).

𝜇𝑚_𝑜𝑘^𝑝𝑖 MI rate (part/time) of inspection tool i for quality characteristic k of operation o in product p (𝜇𝑚_𝑜𝑘^𝑝𝑖 = 1 𝑚𝑡⁄ _𝑜𝑘^𝑝𝑖).

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𝑓𝑝_𝑜𝑝^𝑚 Breakdowns rate of machine m for performing operation o in product p.

𝑟𝑝_𝑜𝑝^𝑚 Retrieve time rate of machine m for performing operation o in product p.

𝑓𝑐_𝑜𝑘^𝑝𝑖 Breakdowns rate of inspection tool i for performing CI of quality characteristic k of operation o in product p.

𝑟𝑐_𝑜𝑘^𝑝𝑖 Retrieve time rate of inspection tool i for performing CI of quality characteristic k of operation o in product p.

𝑓𝑚_𝑜𝑘^𝑝𝑖 Breakdowns rate of inspection tool i for performing MI of quality characteristic k of operation o in product p.

𝑟𝑚_𝑜𝑘^𝑝𝑖 Retrieve time rate of inspection tool i for performing MI of quality characteristic k of operation o in product p.

𝑓𝑠_𝑜^𝑝 Fixed space cost per part of performing inspection between operations o and o+1 in product p.

𝜁_𝑜′𝑜

𝑝 Is 1 if two operations 𝑜^′ and o are dependent in product p and 0 otherwise.

𝜓_𝑜𝑘^𝑝 Is 1 if quality characteristic k belongs to operation o in product p and 0 otherwise.

𝑚𝑓_𝑘^𝑝 Monitoring frequency for quality characteristic k of operation o in product p.

𝑐𝑓_𝑘^𝑝 Conformity frequency for quality characteristic k of operation o in product p.

ℳ A big number.

Decision Variables:

𝑁𝑃_𝑜𝑘^𝑝 Number of nonconforming items due to characteristic k from operation o in product p.

𝑌𝐶_𝑜𝑘^𝑝 1 if operation o in product p needs CI for characteristic k; and 0, otherwise.

𝑌𝑀_𝑜𝑘^𝑝 1 if operation o in product p needs MI for characteristic k; and 0, otherwise.

𝑋𝐶_𝑜′𝑜

𝑘𝑝𝑖 1 if CI of operation 𝑜^′ for characteristic k in product p is performed between operations o and o+1 using inspection tool i (𝑜^′ ≤ 𝑜); and 0, otherwise.

𝑋𝑀_𝑜′𝑜

𝑘𝑝𝑖 1 if MI of operation 𝑜^′ for characteristic k in product p is performed between operations o and o+1 using inspection tool i (𝑜^′ ≤ 𝑜); and 0, otherwise.

𝑁_𝑜^𝑝 Number of in-process parts entering operation o in product p.

𝑁𝑀_𝑜𝑘^𝑝𝑖 Number of MIs performed using inspection tool i between operations o and o+1 for quality characteristic k in product p.

𝑁𝐶_𝑜𝑘^𝑝𝑖 Number of CIs performed using inspection tool i between operations o and o+1 for quality characteristic k in product p.

𝑁𝑆_𝑜^𝑝 Is 1 if there is an inspection station between operations o and o+1 in product p.

𝒮_𝑜𝑘^𝑝 Number of scrapped part between operations o and o+1 due to quality

64 characteristic k in product p.

𝑆_𝑜^𝑝 Total number of scrapped parts between operations o and o+1 for product p.

𝑍_𝑚 Number of machine m that must be purchased/utilized.

𝑍_𝑖 Number of inspection tool i that must be purchased/utilized.

𝑈_𝑜𝑝^𝑚 Is 1 if operation o in product p is performed on machine m.

𝑊𝑃_𝑜𝑝^𝑚 Waiting time of parts for performing operation o of product p on machine m.

𝑊𝐶_𝑜𝑘^𝑝𝑖 Waiting time during a CI of quality characteristic k of operation o in product p using inspection tool i.

𝑊𝑀_𝑜𝑘^𝑝𝑖 Waiting time during a MI of quality characteristic k of operation o in product p using inspection tool i.

𝑂𝐹𝑉_𝜏^{𝐷−𝐸𝑃} Deterministic value of the 𝜏th objective function for the Extended Problem (𝜏 = 1,2,3).

Auxiliary variables:

𝔸_𝑜′𝑜

𝑘𝑝𝑖 Linear form of 𝑋𝐶_𝑜′𝑜

𝑘𝑝𝑖× 𝑁_𝑜𝑝′. 𝔹_𝑜′𝑜

𝑘𝑝𝑖 Linear form of 𝑋𝑀_𝑜′𝑜

𝑘𝑝𝑖× 𝑁_𝑜𝑝′. 𝔻_𝑜′𝑜

𝑘𝑝𝑖 Linear form of 𝑋𝐶_𝑜′𝑜

𝑘𝑝𝑖× 𝑁𝑃_𝑜′𝑘 𝑝 . 𝔼_𝑜′𝑘

𝑝 Linear form of 𝑁𝑃_𝑜′𝑘

𝑝 × 𝑌𝐶_𝑜′𝑘 𝑝 . 𝔽_𝑜′𝑘

𝑝 Linear form of 𝑁𝑃_𝑜′𝑘

𝑝 × 𝑌𝑀_𝑜′𝑘 𝑝 . 𝕃_𝑜^𝑝 Linear form of 𝑁_𝑜^𝑝× 𝑁𝑆_𝑜^𝑝. 𝕌_𝑜′𝑘

𝑝 Linear form of 𝑁_𝑜𝑝′ × 𝑌𝐶_𝑜′𝑘 𝑝 . 𝕍_𝑜′𝑘

𝑝 Linear form of 𝑁_𝑜𝑝′ × 𝑌𝑀_𝑜′𝑘 𝑝 .