Download Gradient-based optimization of the expected profit in a serial

Gradient-based optimization of the expected
prot in a serial production line with single
sampling plans
S. De Vuyst, D. Claeys, B. Raa, E.H. Aghezzaf
Ghent University, Departement of Industrial Systems Engineering and Product Design
e-mail:
Stijn.DeVuyst;Dieter.Claeys;Birger.Raa;[email protected]
We consider a serial production line consisting of K stages with single-sampling
inspection after each stage k = 1, . . . , K . A lot of N items goes through the
stages one by one until it is rejected or reaches the end of the line. Inspection
after stage k is with regard to a continuous quality characteristic (QC) Xk of
the item, pertaining to that stage. The QCs are normally distributed with mean
µk and variance σk2 at stage k and they are independent, both between stages
and between items. That is, we assume all processes to be strictly in control. If
Xk lies between xed lower and upper specication limits the item is considered
conforming, otherwise it is defective. Single sampling inspection means that nk
items are randomly selected from the lot and inspected. If the number of defective items in the sample is larger than an acceptance number dk , the entire lot is
rejected and sold. Otherwise, the lot is sent to the next production stage. Before
selling or passing to the next stage however, defective items may be reworked
and a quality cost per item is incurred either before or after the rework.
The mean µk of the QC at stage k can be controlled and if it is, the corresponding QC standard deviation follows accordingly as a given function σk (µk )
which can be seen as a constraint of the process at stage k . We refer to this
function as the deviation response of the process. The goal is to nd an optimal vector of QC means µ = (µ1 , . . . , µK ) ∈ RK which maximises the expected
prot,
µ∗ = arg max E[Prot(µ)] .
(1)
µ
We propose to solve (1) using a gradient-search procedure and provide expressions
for this gradient. The property of quasi-convexity for this objective function is
investigated as well as the possibility of expressing (1) as a dynamic programming
problem. For a number of specic scenarios, we demonstrate the performance of
the search algorithm and the nature of the solutions found.
The expected prot per lot is obtained analytically, accounting for production
cost, selling price, inspection cost, rework cost as well as a Taguchi quality cost.
The production, inspection and rework cost of an item at stage k are all assumed
to be monotone functions of that item's QC Xk , where the case of ane functions
is considered in particular. The quality cost of an item is given by the well-known
asymmetric Taguchi loss function Q(Xk ) = (Xk − T )2 (W0 1Xk <T + W1 1Xk >T ),
which favours the production of items with quality close to a target value T .
1
The optimization of the process means in a serial production line with sampling was considered before. In [1] a multi-objective optimization of a single stage
is discussed while [2] studies the impact of inspection errors. In [3] the production
of single items (instead of lots) is considered which may require multiple rework
cycles at each stage before progressing to the next. The authors of [3] show that
for their model, the optimization (1) can be done sequentially, stage by stage,
and they obtain explicit expressions for the solution.
Our model is in fact a generalisation of the study in [4] which in turn was based
on the two-stage model without quality cost in [5]. Unlike these previous models
however, we allow the production, inspection and rework costs of an item in each
stage to depend on its QC, instead of being xed. The rework and quality cost
functions may depend on whether the item belongs to the inspection sample or
not and whether the lot is accepted or not. Also, the quality cost may be incurred
either before or after the rework. This versatility of the model allows to cover
a large array of dierent system variants with dierent policies for inspecting,
reworking and assessing the quality of the lots. These policies are not necessarily
the same for each production stage, although it is required that rejected lots leave
the production line and do not progress to the next stage. Further, as in previous
studies we consider normally distributed QCs but we abandon the assumption
that their variance remains unaltered when their mean is changed.
References
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process targeting using sampling plans. Computers & Industrial Engineering,
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[2] S.O. Duuaa and A. El-Ga'aly. Impact of inspection errors on the formula
of a multi-objective optimization process targeting model under inspection
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[3] Shokri Z. Selim and Walid K. Al-Zu'bi. Optimal means for continuous processes in series. European Journal of Operational Research, 210:618623,
2011.
[4] Chung-Ho Chen and Michael B.C. Khoo. Optimum process mean and manufacturing quantity settings for serial production system under the quality
loss and rectifying inspection plan. Computers & Industrial Engineering,
57:10801088, 2009.
[5] K.S. Al-Sultan. An algorithm for the determination of the optimum target
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