# Effect of Time-Varying Factors on Optimal Combination of Quality Inspectors for Offline Inspection Station

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Model Formulation

#### 3.1. Definition of Research Problem

#### 3.2. Model Notations and Abbreviations

#### Model Notations

j | type of inspectors where, j = Low skill, Medium skill and High skill |

l | low skill l = 1, 2, 3, …, L |

m | medium skill m = 1, 2, 3, …, M |

h | high skill h = 1, 2, 3, …, H |

N | lot/batch size |

n | sample size |

IT_{j} | inspection time per unit taken by jth inspector |

V | cost of inspection ($/min) |

MI | maximum allowable quality inspector |

ST | standard time of inspection of particular product |

TVC_{T} | target of total cost for inspection station |

AOQ_{T} | target of outgoing quality for inspection station |

TIQ_{T} | target of accepted quantity for inspection station |

d_{j} | number of defective items present in sample size n inspected of jth inspector |

b_{j} | learning rate of jth inspector |

${d}_{1}^{\pm}$ | deviational variables for cost of inspectors |

${d}_{2}^{\pm}$ | deviational variables for outgoing quality |

${d}_{3}^{\pm}$ | deviational variables for accepted quantity |

OQ_{j} | average outgoing quality of jth quality inspectors |

IQ_{j} | inspected quantity by jth quality inspectors |

VC_{j} | variable cost of jth quality inspectors |

NI_{j} | number of jth type of skilled labor |

#### 3.3. Outgoing Quality

_{j}is the total quantity inspected by each inspector j and p

_{j}is the probability of separating the non-conforming (NC) products from confirming (C) products than the value of NC and C can be calculated as:

_{j}can either be sent for rework or rejected. Rework quantity (RW

_{j}) and rejected quantity (RE

_{j}) can be calculated by the following equations:

_{j}is the probability of rework-able quantity. Similarly, the conforming quantity is moved for the process of sampling inspection as a lot/batch of size N. The following equation can be used to determine the value of OQ,

_{s}is the initial value, b is the learning rate and OQ(w) is the value of outgoing quality level at wth week. Similarly, the value of OQ for any inspector j can be calculated as:

_{j}is the learning rate of inspector j. This study is investigating the human labor of J types of skill levels, thus Average Outgoing Quality (AOQ) can be calculated as:

_{l}, b

_{m}, b

_{h}indicate the learning rate of inspector with low, medium and high inspection skill respectively. Finally, the value of AOQ for j type of quality inspectors is calculated by Equation.

#### 3.4. Inspection Quantity

_{j}is the inspection time taken by the jth quality inspector to inspect one item. With the passage of time, the efficiency of each quality inspector improves and inspected quantity increases because of reduction in inspection time due to learning. To calculate the improvement in inspection time, concept of wright’s learning curve [43] is used that suggests the exponential relationship between man hour and cumulative production.

_{s}is the initial value of inspection time, b is the learning rate and IT(w) is the inspection time at wth week. Similarly, the value of IT

_{j}and IQ

_{j}for a jth inspector can be calculated as:

_{j}is the learning rate of the jth quality inspector. Total inspected quantity TIQ of the offline station will include all J types of inspectors.

#### 3.5. Inspection Cost

_{j}, the VC

_{j}can be calculated on the basis of time earned TE

_{j}for the jth quality inspector.

_{j}for jth inspector at any stage w will be calculated as:

#### 3.6. Objective Functions

_{T}.

## 4. Results and Discussion

^{®}Core™ i7-7500U CPU @ 2.70 GHz Intel, 8.00 GB of RAM. Min-max GP method was used to calculate the optimal values of decision variables that also gave the optimized results of objective functions. The obtained results are summarized in Table 3 for all three stages and analysis can be divided into two parts: analysis of decision variable and analysis of objective function.

^{−}and overachieved value as d

^{+}. Min-max GP method provided optimum results of decision variables such that the set target of each objective function is attained. Even though underachieved and overachieved values are also there for different objective functions but all these deviational values do not violate the given conditions. Like in Table 3, overachieved value of the inspection quantity (d

^{+}) are 83, 138 and 28 for stage A, B and C respectively. However, it is still according to the constraints given in Section 3.6. Inspection quantity per day should not be less than the set target but presented results gave over achieved value, which is a positive side of the results. Similarly in Table 3, underachieved values of variable cost (d

^{−}) are 48, 17 and 189 for stage A, B and C respectively. Since the constraint of the proposed model is to retain this variable cost low as much as possible so, these underachieved values also fulfill the already mentioned constraints.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Authors | Inspection | Learning Behavior | Study Objective | ||||
---|---|---|---|---|---|---|---|

Strategy | Error | Cost | Time | Skill | |||

Jaber and Guiffrida [42] | √ | √ | Proposed quality learning curve | ||||

Finkelshtein, Herer, Raz and Ben-Gal [16] | Sampling | √ | √ | Optimal ID policy | |||

Duffuaa and Khan [50] | 100% | √ | √ | Repeat inspection plan to measures the performance | |||

Anily and Grosfeld-Nir [11] | Both | √ | Optimal inspection policy | ||||

Elshafei, et al. [51] | 100% | √ | √ | Optimal inspection sequence for repeat inspection plan | |||

Wang [17] | Sampling | √ | √ | Optimal ID policy | |||

Duffuaa and Khan [52] | Both | √ | √ | Optimal inspection cycles | |||

Wang and Hung [18] | Sampling | √ | √ | Optimal ID policy | |||

Jaber and Guiffrida [44] | √ | Proposed QLC by relaxing assumptions | |||||

Wang and Meng [12] | Both | √ | √ | Optimal inspection policy | |||

Colledani and Tolio [53] | Sampling | √ | Analytical method of evaluation | ||||

Tzimerman and Herer [6] | Sampling | √ | √ | Optimal inspection policy | |||

Bendavid and Herer [19] | Sampling | √ | √ | Optimal ID policy | |||

Vaghefi and Sarhangian [54] | Sampling | √ | √ | Optimal inspection policy | |||

Wang, Sheu, Chen and Horng [20] | Sampling | √ | √ | Optimal ID policy | |||

Yu, Yu and Wu [28] | Both | √ | √ | Optimal inspection policy | |||

Khan, et al. [55] | 100% | √ | Economic order quantity with learning in production | ||||

Jaber and Khan [45] | √ | Proposed QLC by relaxing assumptions | |||||

Yang [56] | Both | √ | Optimization of K- stage inspection system | ||||

Khan, et al. [57] | 100% | √ | √ | Economic order quantity | |||

Tsai and Wang [21] | Sampling | √ | √ | Optimal IDR Policy | |||

Yu and Yu [29] | Both | √ | √ | Optimal inspection policy | |||

Khan, et al. [58] | 100% | √ | √ | Effect on human factors on cost of supply chain | |||

Avinadav and Sarne [59] | Both | √ | √ | Selection of inspection systems | |||

Avinadav and Perlman [13] | Sampling | √ | √ | Optimal inspection interval | |||

Duffuaa and El-Ga’aly [36] | 100% | √ | Maximization of income, profit, product uniformity | ||||

Duffuaa and El-Ga’aly [37] | Sampling | √ | Maximization of income, profit, product uniformity | ||||

Bouslah, et al. [60] | Sampling | √ | Joint production control and economic sampling plan | ||||

Khan, et al. [61] | 100% | √ | √ | √ | √ | Integrated supply chain model | |

Liu and Liu [62] | Sampling | √ | Resubmitted sampling scheme | ||||

Aslam, et al. [63] | Sampling | √ | Mixed acceptance sampling plan | ||||

Yang and Cho [64] | 100% | √ | Optimal inspection cycles | ||||

Mohammadi, et al. [65] | Sampling | √ | √ | Effective robust inspection planning | |||

Duffuaa and El-Ga’aly [35] | Sampling | √ | √ | Maximization of income, profit, product uniformity | |||

Sarkar and Saren [14] | Sampling | √ | √ | Product inspection policy | |||

Ramzan and Kang [7] | Both | √ | √ | √ | MOO model to determine inspectors of different skills | ||

Jaber [38] | √ | √ | √ | A review of studies linking quality with learning | |||

Kang, Ramzan, Sarkar and Imran [8] | Both | √ | √ | √ | MOO model to determine inspectors for different products | ||

This paper | Both | √ | √ | √ | √ | √ | MOO model to determine optimal quality inspectors |

Notation | At Stage A | At Stage B | At Stage C |
---|---|---|---|

ST (mins) | 0.96 | 0.96 | 0.96 |

OQ_{T} | 0.07 | 0.06 | 0.05 |

OQ | 0.12 | 0.10 | 0.08 |

OQ_{m} | 0.07 | 0.05 | 0.04 |

OQ_{h} | 0.04 | 0.03 | 0.02 |

TC_{T} ($) | 3900 | 4690 | 5470 |

C_{l} ($) | 222 | 247 | 263 |

C_{m} ($) | 374 | 440 | 483 |

C_{h} ($) | 516 | 645 | 692 |

TIQ_{T} (Units) | 3125 | 3750 | 4375 |

IQ_{l} (Units) | 185 | 206 | 219 |

IQ_{m} (Units) | 311 | 366 | 403 |

IQ_{h} (Units) | 430 | 537 | 577 |

V ($/min) | 1.25 | 1.25 | 1.25 |

MI (Units) | 12 | 12 | 12 |

Decision Variables | Values | Objectives | Target | Deviation Variables | ||
---|---|---|---|---|---|---|

Set | Achieved | d^{+} | d^{−} | |||

Stage A | ||||||

Low skill | 3 | Inspection cost ($) | 3900 | 3852 | 0 | 48 |

Medium skill | 3 | Outgoing quality | 0.07 | 0.07 | 0 | 0 |

High skill | 4 | Inspected quantity (Units) | 3125 | 3208 | 83 | 0 |

Stage B | ||||||

Low skill | 3 | Inspection cost ($) | 4688 | 4671 | 0 | 17 |

Medium skill | 6 | Outgoing quality | 0.06 | 0.06 | 0 | 0 |

High skill | 2 | Inspected quantity (Units) | 3750 | 3888 | 138 | 0 |

Stage C | ||||||

Low skill | 4 | Inspection cost ($) | 5469 | 5280 | 0 | 189 |

Medium skill | 5 | Outgoing quality | 0.05 | 0.05 | 0 | 0 |

High skill | 2 | Inspected quantity (Units) | 4375 | 4403 | 28 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Babar Ramzan, M.; Mohsin Qureshi, S.; Irshad Mari, S.; Saad Memon, M.; Mittal, M.; Imran, M.; Waqas Iqbal, M.
Effect of Time-Varying Factors on Optimal Combination of Quality Inspectors for Offline Inspection Station. *Mathematics* **2019**, *7*, 51.
https://doi.org/10.3390/math7010051

**AMA Style**

Babar Ramzan M, Mohsin Qureshi S, Irshad Mari S, Saad Memon M, Mittal M, Imran M, Waqas Iqbal M.
Effect of Time-Varying Factors on Optimal Combination of Quality Inspectors for Offline Inspection Station. *Mathematics*. 2019; 7(1):51.
https://doi.org/10.3390/math7010051

**Chicago/Turabian Style**

Babar Ramzan, Muhammad, Shehreyar Mohsin Qureshi, Sonia Irshad Mari, Muhammad Saad Memon, Mandeep Mittal, Muhammad Imran, and Muhammad Waqas Iqbal.
2019. "Effect of Time-Varying Factors on Optimal Combination of Quality Inspectors for Offline Inspection Station" *Mathematics* 7, no. 1: 51.
https://doi.org/10.3390/math7010051