Aghezzaf, El-Houssaine

Claeys, Dieter

De Vuyst, Stijn

Raa, Birger

English

Ghent University Academic Bibliography

Gradient-based optimization of the expectedproﬁt in a serial production line with singlesampling plansS. De Vuyst, D. Claeys, B. Raa, E.H. AghezzafGhent University, Departement of Industrial Systems Engineering and Product Designe-mail: Stijn.DeVuyst;Dieter.Claeys;Birger.Raa;ElHoussaine.Aghezzaf@ugent.beWe consider a serial production line consisting ofK stages with single-samplinginspection after each stage k = 1, . . . ,K. A lot of N items goes through thestages one by one until it is rejected or reaches the end of the line. Inspectionafter stage k is with regard to a continuous quality characteristic (QC) Xk ofthe item, pertaining to that stage. The QCs are normally distributed with meanµk and variance σ2k at stage k and they are independent, both between stagesand between items. That is, we assume all processes to be strictly in control. IfXk lies between ﬁxed lower and upper speciﬁcation limits the item is consideredconforming, otherwise it is defective. Single sampling inspection means that nkitems are randomly selected from the lot and inspected. If the number of defec-tive items in the sample is larger than an acceptance number dk, the entire lot isrejected and sold. Otherwise, the lot is sent to the next production stage. Beforeselling or passing to the next stage however, defective items may be reworkedand 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 corre-sponding 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 thisfunction as the deviation response of the process. The goal is to ﬁnd an opti-mal vector of QC means µ = (µ1, . . . , µK) ∈ RK which maximises the expectedproﬁt,µ∗ = argmaxµE[Proﬁt(µ)] . (1)We propose to solve (1) using a gradient-search procedure and provide expressionsfor this gradient. The property of quasi-convexity for this objective function isinvestigated as well as the possibility of expressing (1) as a dynamic programmingproblem. For a number of speciﬁc scenarios, we demonstrate the performance ofthe search algorithm and the nature of the solutions found.The expected proﬁt per lot is obtained analytically, accounting for productioncost, 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 assumedto be monotone functions of that item's QC Xk, where the case of aﬃne functionsis considered in particular. The quality cost of an item is given by the well-knownasymmetric Taguchi loss function Q(Xk) = (Xk − T )2(W01Xk<T +W11Xk>T ),which favours the production of items with quality close to a target value T .1The optimization of the process means in a serial production line with sam-pling was considered before. In [1] a multi-objective optimization of a single stageis discussed while [2] studies the impact of inspection errors. In [3] the productionof single items (instead of lots) is considered which may require multiple reworkcycles at each stage before progressing to the next. The authors of [3] show thatfor 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 basedon the two-stage model without quality cost in [5]. Unlike these previous modelshowever, we allow the production, inspection and rework costs of an item in eachstage to depend on its QC, instead of being ﬁxed. The rework and quality costfunctions may depend on whether the item belongs to the inspection sample ornot and whether the lot is accepted or not. Also, the quality cost may be incurredeither before or after the rework. This versatility of the model allows to covera large array of diﬀerent system variants with diﬀerent policies for inspecting,reworking and assessing the quality of the lots. These policies are not necessarilythe same for each production stage, although it is required that rejected lots leavethe production line and do not progress to the next stage. Further, as in previousstudies we consider normally distributed QCs but we abandon the assumptionthat their variance remains unaltered when their mean is changed.References[1] S.O. Duﬀuaa and A. El-Ga'aly. A multi-objective optimization model forprocess targeting using sampling plans. Computers & Industrial Engineering,64:309317, 2013.[2] S.O. Duﬀuaa and A. El-Ga'aly. Impact of inspection errors on the formulaof a multi-objective optimization process targeting model under inspectionsampling plan. Computers & Industrial Engineering, 80:254260, 2015.[3] Shokri Z. Selim and Walid K. Al-Zu'bi. Optimal means for continuous pro-cesses in series. European Journal of Operational Research, 210:618623,2011.[4] Chung-Ho Chen and Michael B.C. Khoo. Optimum process mean and man-ufacturing quantity settings for serial production system under the qualityloss and rectifying inspection plan. Computers & Industrial Engineering,57:10801088, 2009.[5] K.S. Al-Sultan. An algorithm for the determination of the optimum targetvalues for two machines in series with quality sampling plans. InternationalJournal of Production Research, 32:3745, 1994.

Gradient-based optimization of the expected profit in a serial production line with single sampling paths

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Gradient-based optimization of the expected profit in a serial production line with single sampling paths

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