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Inspection location in capacity-constrained lines

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Abstract

We consider the effects of inspection and repair stations on the production capacity and product quality in a serial garments production line with possible inspection and repair following each operation. We construct a profit function that takes into account inspection, repair, scrap, and goodwill costs, as well as the capacity of each station. Then we discuss how the profit function can be maximized and provide properties of the optimal inspection plan. Our analysis captures the possibility of increasing production capacity by scrapping or repairing defective items before a bottleneck operation station, and hence reducing the waste of operation capacity on defective products. Our numerical results show that incorporating such capacity considerations can have substantial impact on the optimal inspection policy and that optimal inspection allocations can be identified quickly even for large problem instances.

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Acknowledgements

This research was supported by the National Science Foundation under Grants CMMI-0856600 and CMMI-1536990. We would like to thank two anonymous referees for their insightful comments that led to substantial improvements to this paper.

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Correspondence to Salih Tekin.

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Appendices

Appendices

A Notation

In this appendix, we provide the notation used throughout the paper as a reference, see Table 3.

Table 3 Notation

B Nonlinear Program for the General Model

In this appendix, we formulate the nonlinear program for the general model where any station can be the bottleneck. Decision variables are the inspection plan \(a_i\), \(i=1,\ldots ,N\), and the flow rates throughout the production line (i.e., \(\lambda _i\), \(\lambda _i^R\) for \(i=1,\ldots ,N\), and \(\lambda _{N+1}\)) given the incoming defect information \(\pi _{1,j}^O\) for all j.

$$\begin{aligned}&\max \ T_P \ \text {s.t.}\nonumber \\&\quad \lambda _1 \le \lambda ;\end{aligned}$$
(33)
$$\begin{aligned}&\displaystyle \lambda _i \le \mu _i^O, \ i=1,\ldots ,N; \end{aligned}$$
(34)
$$\begin{aligned}&\displaystyle a_i \lambda _i \le \mu _i^I,\ i=1,\ldots ,N; \end{aligned}$$
(35)
$$\begin{aligned}&\displaystyle \lambda _i^R \le \lambda _ia_ir_{i}, \ i=1,\ldots ,N; \end{aligned}$$
(36)
$$\begin{aligned}&\displaystyle \lambda _i^R \le \mu _i^R, \ i=1,\ldots ,N; \end{aligned}$$
(37)
$$\begin{aligned}&\displaystyle \lambda _{i+1} = \lambda _i^R + \lambda _i[1-a_i(r_{i} +s_{i})], \ i=1,\ldots ,N; \end{aligned}$$
(38)
$$\begin{aligned}&\text{ Equations } (1)-(3), (5),(7), (9)- (12), (14)-(20)\nonumber \\&\qquad \text { with } \lambda _i^O=\lambda _i^I=\lambda _i, \lambda _{i}^{IR} =\lambda _{i}a_ir_i, \text { and } \nu _{i} =\lambda _{i}a_is_{i}, \ i=1,\ldots ,N; \nonumber \\&\quad a_i \in \{0,1\}, i=1,\ldots , N;\nonumber \\&\quad 0 \le \lambda _{N+1}, \lambda _i, i=1,\ldots , N. \end{aligned}$$
(39)

The objective \(T_P\) is the general total profit rate function. The constraints (33)–(38) represent balance equations for the flow in the system, allowing only the repair stations to become unstable. The constraint (39) is needed because some of the model parameters depend on the decision variables.

Let the solution be given by \(\bar{a}_i, \bar{\lambda }_i\), and \(\bar{\lambda }_i^R\) for \(i=1,\ldots ,N\), as well as \(\bar{\lambda }_{N+1}\). Then the repair station \(R_i\) is stable if \(\bar{\lambda }_i \bar{a}_i r_{i} \le \mu _i^R\) and unstable if \(\bar{\lambda }_i \bar{a}_i r_{i}>\mu _i^R\) in the allocation scenario that maximizes the return. We also reject at the rate \(\nu _{0}= \lambda -\bar{\lambda }_1\) at the dummy operation node \(O_0\), and stages i with \(\bar{a}_i=1\) are assigned an inspection station.

C Model Construction for Sect. 6.1

In this appendix, we formulate the profit rate function \(T_P\) depending on the inspection strategy and the input parameters, \(\lambda ,R, C_i, G_i, H_i\), and \(U_i\), \(i=1,2\). Let \(\mu _1^O=\mu _1\) and \(\mu _2^O=\mu _2\) denote the processing capacities at stations 1 and 2, respectively. We consider the case with \(\mu _1^I,\)\(\mu _2^I \ge \lambda \), so that the inspection operations never constrain the system. The throughput \(\lambda _{N+1}\) of the system is denoted by \(\mu \).

The optimal inspection locations are determined based on the NLP (25)–(32). Since all defects are major, we have \(s_i=p_i\) and \(r_{i}=0\) for \(i=1,2\). Moreover, \(E_i^O=C_i\), \(E_i^I=H_i\), \(\pi _{1,1}^O=0\), \(\pi _{1,1}^I=p_1\), \(\pi _{1,2}^O=\pi _{1,2}^I=0\), \(\pi _{2,1}^O=\pi _{2,1}^I=p_1(1-a_1)\), \(\pi _{2,2}^O=0\), \(\pi _{2,2}^I=p_2\), and \(\pi _{3,j}^O=p_j(1-a_j)\) for \(i=1,2\) and \(j=1,2\). Hence the NLP (25)–(32) becomes

$$\begin{aligned}&\max \mu R-\mu \bigg (p_1G_1(1-a_1)+p_2G_2(1-a_2)\bigg )\nonumber \\&\quad -\sum _{i=1}^{2}\frac{\mu }{\prod _{n=i}^{2}(1-a_ns_n)} \bigg (C_i+a_i H_i+a_ip_iU_i\bigg ) \text { s.t. } \end{aligned}$$
(40)
$$\begin{aligned}&\displaystyle \mu \le \lambda (1-a_1p_1)(1-a_2p_2), \end{aligned}$$
(41)
$$\begin{aligned}&\displaystyle \mu \le \mu _1(1-a_1p_1)(1-a_2p_2), \end{aligned}$$
(42)
$$\begin{aligned}&\displaystyle \mu \le \mu _2(1-a_2p_2), \end{aligned}$$
(43)
$$\begin{aligned}&\displaystyle a_1, a_2 \in \{0,1\}, \text { and } \mu \ge 0. \end{aligned}$$
(44)

Note that in the above allocation NLP, all quantities are known except for the decision variables \(a_1\), \(a_2\) and the throughput \(\mu \). The goodwill cost in the objective function (40) results from the fact that goodwill costs for different defects are additive.

Since the problem size is small, we can easily solve the NLP in (40)–(44) by enumerating all four possible solutions and obtain the profit function \(T_P(a_1,a_2)\). In particular, considering the nontrivial case where the profit functions are nonnegative, we have

$$\begin{aligned}&\displaystyle T_P(0,0)=\min \{\lambda ,\mu _1,\mu _2\}(R-C_1-C_2-p_1G_1-p_2G_2), \end{aligned}$$
(45)
$$\begin{aligned}&\displaystyle T_P(1,0)=\min \{(1-p_1)\lambda ,(1-p_1)\mu _1,\mu _2\} \bigg (R-\frac{C_1+H_1+p_1U_1}{1-p_1}-C_2-p_2G_2\bigg ), \end{aligned}$$
(46)
$$\begin{aligned}&\displaystyle T_P(0,1)=(1-p_2)\min \{\lambda ,\mu _1,\mu _2\}\bigg (R-\frac{C_1}{1-p_2} -\frac{C_2+H_2+p_2U_2}{1-p_2}-p_1G_1\bigg ), \nonumber \\ \end{aligned}$$
(47)
$$\begin{aligned} T_P(1,1)= & {} (1-p_2)\min \{(1-p_1)\lambda ,(1-p_1)\mu _1,\mu _2\}\nonumber \\&\times \bigg (R-\frac{C_1+H_1+p_1U_1}{(1-p_1)(1-p_2)} -\frac{C_2+H_2+p_2U_2}{1-p_2}\bigg ). \end{aligned}$$
(48)

Next, we provide the total profit functions for the input and capacity constrained systems. We start with the case where the system is constrained by the arrival rate, so that \(\lambda < \min \{\mu _1,\mu _2\}\). In this case, we cannot see the side benefit of having inspection before the bottleneck on the optimal decision and our inspection allocation decision agrees with traditional wisdom in that we choose to inspect whenever the expected inspection cost for a unit is less than the expected cost of not inspecting (see, e.g., Bai and Yun 1996; Chen 1998; Hurst 1973; Lindsay and Bishop 1964; Raz and Kaspi 1991). More specifically, rearranging the profit functions (45)–(48), we obtain

$$\begin{aligned} \begin{aligned} \frac{T_P(1,0)-T_P(0,0)}{\lambda }&=p_1(C_2+p_2G_2+G_1)-(H_1+p_1R+p_1U_1), \\ \frac{T_P(0,1)-T_P(0,0)}{\lambda }&=p_2(p_1G_1+G_2)-(H_2+p_2R+p_2U_2), \\ \frac{T_P(1,1)-T_P(0,0)}{\lambda }&= (p_1G_1+p_2G_2+p_1C_2) \\&-\,[H_1+p_1U_1+(H_2+p_2U_2)(1-p_1) \\&+\,(p_1+p_2-p_1p_2)R], \end{aligned} \end{aligned}$$
(49)

where \((H_i+p_iR+p_iU_i)\), \(i=1,2\), can be considered as the expected cost of inspection for defect i only per unit, and \([H_1+p_1U_1+(H_2+p_2U_2)(1-p_1)+(p_1+p_2-p_1p_2)R]\) as the expected cost of inspection when inspecting both defects simultaneously. Similarly, \(p_1(C_2+p_2G_2+G_1)\), \(p_2(p_1G_1+G_2)\), \((p_1G_1+p_2G_2+p_1C_2)\) can be viewed as the expected cost of not inspecting at locations 1, 2, and 1 and 2 together, respectively.

Next we derive the profit functions for the interesting case where the production line is constrained by the capacity of the second operation station having processing rate \(\mu _2\), assuming that \(\mu _1>\lambda >\mu _2/(1-p_1)\). The first condition ensures that \(O_1\) is not a bottleneck, and the second condition ensures that we have in all cases enough input for \(O_2\) to be a bottleneck. Hence, as a result of Theorem 1, we reject any incoming job with probability \([\lambda -\mu _2/(1-p_1a_1)]/\lambda \) to stabilize the system. In this case, we can observe the effects of throughput considerations on the inspection allocation decision. Note that by inspecting for defect 1 after the first operation station and scrapping defective items, we not only remove the defective items but also increase the capacity of the production line from \(\mu _2\) to \(\mu _2/(1-p_1)\). Thus, in the capacity constrained case, inspecting after the first operation has gained an advantage with magnitude depending on the value of \(p_1\). We will see that even under the same defect distribution and cost parameters, inspecting after operation station 1 for defect 1 becomes more beneficial as compared to the first case. Similar to the first case, we can compare all three different inspection decisions to the no inspection case using the profit functions (45)–(48) to obtain

$$\begin{aligned} \frac{T_P(1,0)-T_P(0,0)}{\mu _2}= & {} p_1G_1-\frac{(p_1C_1+H_1+p_1U_1)}{1-p_1},\nonumber \\ \frac{T_P(0,1)-T_P(0,0)}{\mu _2}= & {} p_2(p_1G_1+G_2)-(H_2+p_2U_2+p_2R), \nonumber \\ \frac{T_P(1,1)-T_P(0,0)}{\mu _2}= & {} (p_1G_1+p_2G_2)\nonumber \\&-\left[ \frac{(p_1C_1+H_1+p_1U_1)}{1-p_1} +(H_2+p_2U_2+p_2R)\right] .\nonumber \\ \end{aligned}$$
(50)

The profit function (\(T_P\)) formulation and the pairwise profit comparisons obtained in this appendix can be used to identify the optimal inspection decisions in the examples of Sect. 6.1.

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Tekin, S., Andradóttir, S. Inspection location in capacity-constrained lines. Cent Eur J Oper Res 28, 905–937 (2020). https://doi.org/10.1007/s10100-018-00604-x

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