Abstract
In this paper, an unreliable serial production line in which nonconforming items are sent back for rework is studied. The line consists of existing machines and optional quality control stations (QCSs). The designer of such a production line needs to decide where to install the QCSs along the line and to determine the production rate, so as to maximize the expected operational profit rate obtained at a steady state. An efficient algorithm for solving this problem is presented; several extensions of the problem are discussed. An extensive simulation study proves the applicability of the model in realistic settings and is used to derive some insights about the nature of optimal solutions.
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Abbreviations
- \(a\) :
-
Arrival rate of item to the system/production rate in a stable system
- \(c_i \) :
-
Variable production cost per item on \(M_i\)
- \({c}_v^\prime (u)\) :
-
Variable inspection cost per item on \(\text{ QC}_v \) assuming the previous installed QCS is \(\text{ QC}_u\). \({c}_v^\prime (0)\) is the inspection cost on \(\text{ QC}_v\) assuming it is the first QCS in the line
- \(C(u,v)\) :
-
Total production, inspection and penalty cost per item incurred by the segment \(M_{u+1} ,\ldots ,M_v ,\text{ QC}_v\)
- \({f}_v^\prime (u)\) :
-
Fixed cost per unit of time for installation of \(\text{ QC}_v \) assuming the previous installed QCS is \(\text{ QC}_u \)
- \(L_i \) :
-
The location of the QCS that precedes \(M_i \) in a given configuration (or 0 there is no such QCS). \(L_{N+1}\) is the location of the last QCS in the line
- \(M_i \) :
-
Machine \(i\)
- \(N\) :
-
Number of machines in the line
- \(p_i \) :
-
Success probability of the operation on machine \(M_i \)
- \(p_{u,v} \) :
-
Success probability on all the machines \(M_{u+1} ,\ldots ,M_v \)
- \(\text{ QC}_i \) :
-
Quality control station located after \(M_i \)
- r\(_{B}\) :
-
Penalty incurred by a nonconforming item delivered by the system
- r\(_{G}\) :
-
Market price of a finished product
- \(T(u,v)\) :
-
Maximum throughput of the segment \(M_{u+1} ,\ldots ,M_v ,\text{ QC}_v \)
- \(x_i\) :
-
Mean processing time of operations on machine \(M_i \)
- \({x}_v^\prime (u)\) :
-
Mean processing time of inspection task on \(\text{ QC}_v \) assuming the previous installed QCS is \(\text{ QC}_u \)
- \(Y\) :
-
The set of installed QCSs in a particular solution
- \(Y^*(a)\) :
-
The optimal QCS configuration for production rate \(a\)
- \(\rho _i\) :
-
The ratio between the rework processing time and the “first time” processing time of an operation on \(M_i \)
- \(\zeta _i\) :
-
The ratio between the rework cost and the “first time” cost of an operation on \(M_i \)
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Raviv, T. An efficient algorithm for maximizing the expected profit from a serial production line with inspection stations and rework. OR Spectrum 35, 609–638 (2013). https://doi.org/10.1007/s00291-012-0304-5
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DOI: https://doi.org/10.1007/s00291-012-0304-5