A DEA-bases Approach for Multi-objective Design of Attribute Acceptance Sampling Plans

Transcription

1 Available online at htt://ijdea.srbiau.ac.ir Int. J. Data Enveloment Analysis (ISSN X) Vol.5, No.2, Year 2017 Article ID IJDEA-00422, 12 ages Research Article International Journal of Data Enveloment Analysis Science and Research Branch (IAU) A DEA-bases Aroach for Multi-objective Design of Attribute Accetance Samling Plans S. Jafarian-Namin 1*, A. Pakzad 2, M.S. Fallah Nezhad 1 (2) (1) Faculty of Industrial Engineering, Yazd University, Yazd, Iran Deartment of Indutrial Engineering, Kosar University of Bojnord, Bojnord, Iran Received February 2, 2017, Acceted Aril 5, 2107 Abstract Accetance samling (AS), as one of the main fields of statistical quality control (SQC), involves a system of rinciles and methods to make decisions about acceting or rejecting a lot or samle. For attributes, the design of a single AS lan generally requires determination of samle size, and accetance number. Numerous aroaches have been develoed for otimally selection of design arameters in last decades. We develo a multi-objective economic-statistical design (MOESD) of the single AS lan to reach a well-balanced comromise between cost and quality features. Moreover, a simle and efficient DEA-based algorithm for solving the model is roosed. Through a simulation study, the efficiency of roosed model is illustrated. Comarisons of otimal designs obtained using MOESD to economic model with statistical constraints reveals enhanced erformance of the multiobjective model. Keywords: Accetance samling, Single samling lan, MOESD, DEA. *. Corresonding author, Samrad.jafarian@stu.yazd.ac.ir

2 S. Jafarian-Namin, et al /IJDEA Vol.5, No.2, (2017) Introduction In quality assurance systems, insection is broadened to include all asects of manufacturing in recent times. Obviously, there are conditions in which insection is troublesome, i.e. when 100% insection is too costly and/or time consuming, testing is destructive, and so on [1]. In such situations, accetance samling (AS), as one of the main fields of statistical quality control (SQC), is most likely to be useful. Accordingly, a secified samle, instead of all items, is taken to evaluate whether to accet or reject the whole lot. Base on some considerations, samles can be either single, double, multile, or sequential (for more information refer to [1]). Broadly, there are two classes of AS lans: variables and attributes. Our focus is on attributes where quality features are assessed on a go, no-go basis. Moreover, single-samling lan is investigated in which the decision is made according to one random samle taken from the lot. Generally, the design of a single AS lan requires determination of samle size, and accetance number. Two traditional methods of interest in literature are as follows: 1.Two-oint method: After determining two oints on the oerating characteristic (OC) Curve, accetable and unaccetable quality levels can be secified as well as the risks regarding the accetance or rejection decisions ([2], [3]). 2.Otimization of total cost function: In this methods, the total cost function, in which the roducer's cost and the consumer's cost are considered, is minimized to determine otimal design arameters ([4], [5], [6]). Numerous rocedures for otimally selection of design arameters have been develoed in recent years. In ure economical aroaches, only cost of samling system is minimized regardless of involving roducer's risk (α) and consumer's (β) risk. In contrast, statistical aroaches mostly focus on risks and seem to be economically ineffective. In fact, the weakness of these two aroaches is in unilateral attitude to samling concet. To overcome such undesirable weakness, Hsu [7] develoed an economic model to determine the otimal design arameters by minimizing total cost while satisfying both the roducer s and the consumer s requirements. Our roosed model can be classified as a combined model with the difference that both roerties are simultaneously otimized. In this multiobjective model, our intention is to reach a well-balanced comromise between the economic and the statistical features because of their identical imortance. As a multile criteria decision-making, some researches have reared efficient solutions by alying rocedures based on data enveloment analysis (DEA) to multi-objective design of control charts recently ([8], [9]). DEA is a owerful otimization method to assess the relative efficiency of decision making units (DMUs) with multile inuts and oututs [10]. There has been continuous develoments and real alications in this field of studies since the first original work [11]. In a mathematical model, known as CCR model, the erformance of each DMU is measured with resect to the remaining DMUs. General definition of DMU for various means, and fewer assumtions involved in its modeling are of main motivations that make DEA more aealing. By exloring the literature, there is no research on efficient secification of design arameters for a mathematical modeling of AS lans. Indeed, by defining roer DMUs, we resent multiobjective economic-statistical design of single samle AS lan. The remaining sections of this aer are thus organized as follows. Basic concets and terminology of AS lan is reviewed in section 2. Then, the mathematical 1232

3 IJDEA Vol.4, No.2, (2016) S. Jafarian-Namin, et al /IJDEA Vol.5, No.2, (2017) modeling is develoed for the mentioned lan. In section 4, solution rocedures for the models are rovided. Secially, an algorithm using the DEA is develoed with some modifications to solve the roosed multi-objective model. Section 5 includes a simulation study to illustrate the solution rocedure and to erform some comarisons. Finally, section 6 covers some final conclusions that can be drawn. 2. Accetance Samling Basic Concets and Terminology Accetance samling lans use statistical techniques to decide on acceting or rejecting an incoming lot. In singlesamling lans for attributes, decision rocedure is made by randomly taking one samle of n units from the lot of N units and then insecting. If the number of defectives does not exceed a redetermined accetance number (c), the whole lot is acceted. Otherwise, the lot is reject. We will use the following notations and definitions in the rest of the aer: N whole lot size n samle size c accetance number S set of design arameters α roducer s risk β consumer s risk ATI average total insection AOQ average outgoing quality D number of nonconforming items D d detected number of defective items D n not detected number of defective items C i cost of insection er item C f cost of internal failure C o cost of an outgoing defective TC total cost It is assumed that the distribution of the number of defectives (d) in a random samle of size n is binomial (n, ), where is the fraction of defective items in the lot. Thus, the robability of accetance is: c n d n d Pa P( d c n, ) (1 ) (1) d 0d In association with decisions on the obtained samle, we confront two tyes of errors: Tye I error (α): incorrectly rejection of a lot that is in fact accetable. Tye II error (β): incorrectly accetation of a lot that is indeed unaccetable. The robability of making such errors deend resectively on two levels of lot quality which are defined as follow: Accetable Quality Level (AQL) is the worst level of quality but still accetable for the consumer. The roducer tends to design a lan which has a high chance of accetance with defective level of less than or equal to AQL. Lot Tolerance Percent Defective (LTPD) is the worst level of quality that would be unaccetable for the consumer in an individual lot. It is aimed for the consumer to accet with low robability any samling lan with LTPD level of quality. Using secified AQL and LTPD, roducer s risk and the consumer s risk can be resectively calculated as following equations: c AQL n d nd Pa P( d c n, 0 AQL) d 0 (1 0) d 0 c LTPD n d nd Pa P( d c n, 1 LTPD) d 1 (1 1) d 0 (2) (3) After rejection of a lot, it is common to 100% insect the rejected lot to remove all defectives items. This corrective action is called rectifying insection. During a long series of lots in a rocess with 1233

4 S. Jafarian-Namin, et al /IJDEA Vol.5, No.2, (2017) fraction defective, the Average Outgoing Quality (AOQ) is obtained to evaluate rectifying insection as below equation: ( N n) AOQ Pa N. (4) Another imortant measure in relation to rectifying insection is Average Total Insection (ATI) which is calculated as follows: ATI n ( 1 P )( N n) (5) a In a samling lan, some defective items are detected and the others are not. If we indicate detected defective items by D d and not detected ones by D n, then we have (for detailed information on AS lans, refer to [1]): D n ( 1 P )( N n (6) d a ) Dn Pa ( N n) (7) 3. Design of Single AS Plan In this section, statistically constrained cost model is firstly introduced. Then, the roosed mathematical model for single samle AS lans is develoed to consider economic and statistical roerties simultaneously. A. Cost Model with Statistical Constraints With the urose of finding the otimal design arameters of AS lan, Hsu [7] develoed a model in which statistical constraints based on the roducer's and the consumer's risk were satisfied in minimum total cost. This roosed model is given by following formulation: Min TC S) C. ATI C. D C. D s. t. ( i f d 0 LTPD a AQL a 1 n (8) where TC is the total cost, S=(n, c, k) is a ossible set of design arameters, C i denotes the insection cost er unit, C f is internal failure cost (including rework, reair, and relacement of the failed unit), and C o indicates the cost of an outgoing defective [7]. In addition, α and β are the desired bounds to limit the constraints according to the considerations of decision maker (DM). B. Multi-Objective Model In addition to the statistical roerties, designing AS lans have several costly consequences as introduced above. Simultaneously considering both roerties in a multi-objective format can hel to find otimal design arameters which give the best comromise between the objectives. The roosed model is resented in the following formulation: Min TC S) C. ATI C. D C. D Min Max s. t. ( i f d 0 LTPD a AQL a ( S) ( S) LTPD a AQL a 1 n (9) The design of AS lan requires the secification of two decision variables, i.e. n, and c. In the next section, two algorithms are resented to search the otimal solution(s) based on the models introduced above. 4. Solution Procedure In this section, solution rocedures are resented for both models roosed in revious section. These algorithms are alied to determine otimal decision arameters for single samle AS lans. A. Proosed Solution for Cost Model with Statistical Constraints It is intended to find the design arameters such that both roducer's and consumer's risk are satisfied in minimum cost. The solution algorithm for otimization is resented in following stes: (i) Set re-secified values of arameters 1234

5 IJDEA Vol.4, No.2, (2016) S. Jafarian-Namin, et al /IJDEA Vol.5, No.2, (2017) in the model, i.e. a, m, α, β, δ, C i, C f and C o. (ii) Put limits on n and c according to DM s discretion. (iii) Gather the results for each set of solutions, as S=(n, c), that satisfies the constraints. (iv) Calculate the cost function for the feasible solution combinations obtained from ste (iii). (v) Select design(s) with minimum cost function as otimal. B. Proosed Solution for Multiobjective Model As reviously mentioned, the design of AS lan needs the secification of decision variables, i.e. n, and c. By using the multi-objective model, we aim to achieve a well-balanced comromise between the economic and the statistical roerties. For various control charts, algorithms based on DEA were roosed to search for the otimal design arameters using multi-objective models (for examle see [8] and [9]). DEA is a well-known otimization method to assess the relative efficiency of a grou of DMUs with multile inuts and oututs. In its first mathematical model, known as CCR, to format can be considered as either inut-oriented or outut-oriented [10]. We aly the inutoriented CCR model. Assuming n DMUs, each with m inuts and q oututs, the efficiency of a secific DMU is calculated by solving the model outlined in below format: Max E ( S ) u Y m i 1 0 r r0 r 1 s. t. v X 1 q i i 0 q u Y ( S ) v X ( S ) 0, r rj i ij r1 i 1 j 1,..., n m u 0, r 1,..., q r v 0, i 1,..., m i (10) 1235 Where u r is the weight of outut r, v i is the weight of inut i, Y rj is the value of outut r for j th DMU, and X ij is the value of inut i for j th DMU. DMUs are denoted to feasible combinations of design variables. The measured erformance of each DMU is relatively inefficient when E 0 * <1 and relatively efficient, strictly or weakly, when E 0 * =1. For the roosed model, it should be noted that the objectives including TC and a LTPD are considered as inuts and a AOQ is robed as outut. After formulating the model for each DMU, the set of weights can be discovered as decision variables. As a result, at least one DMU is exected to be efficient. Although DEA has the otential of solving various roblems, it has not been alied for the design of AS lans so far. We emloy the roosed algorithm in [8] by some modifications such as otimizing the model for designing AS lans instead of control charts and changing the objective functions and constraints, accordingly. The solution algorithm for otimization of the roosed model is outlined in following stes: (i) Set re-secified values of arameters in the model, i.e. a, m, α, β, δ, C i, C f and C o. (ii) Put limits on n, and c according to DM s idea to restrict the solution sace beforehand. (iii) Comute objective functions for each set of design arameters, as DMU. (iv) After alying the constraints of the model, gather the feasible sets with the same samle size n into a set Q n. (v) Determine the non-dominated solution oints (NDS) in terms of statistical and cost roerties for each set of Q n. (vi) Mix all determined solutions from (v) into a set W. (vii)secify efficient design set(s) with maximum score(s) calculated using the CCR model.

6 S. Jafarian-Namin, et al /IJDEA Vol.5, No.2, (2017) Simulation Results In order to find the otimal design arameters of single AS lans for the roosed models, the re-defined coefficients are considered according to [7]. These values are listed in Table 1 by considering some other values for cost, lot size and fraction defective arameters. Moreover, we consider 1 n 300 increases by 1, and 0 c 10 increases by 1 to limit the solution sace. The results for the models and the comarisons are rovided in the following. Note that all calculations have been facilitated under coded rograms in the MATLAB (version R2016b) environment. For the cost model with statistical constraints, after setting re-secified values of arameters according to Table 1, DM s discretions are ut on design arameters to limit the solution sace. Then, the feasible combinations, as S=(n,c), are gathered using the constraints of the model. Next, the cost function is calculated for each feasible solution. The design with minimum cost function is chosen as otimal. For the multi-objective model, the solution sace is firstly confined under the limits ut on design arameters. Furthermore, objective functions are comuted for entire ossible combinations, i.e =6300 combinations. Then, the constraints are alied to determine feasible solutions. Next, the NDS oints in terms of statistical and cost roerties for each set of Q n are chosen. Finally, the relative efficiency score of each DMU is calculated to secify a design set with maximum score as the most efficient one. From the results resented in Tables 2, some imortant oints can be mentioned as follows: All combinations have received efficiency score 1 using their own models. Although re-secified values of arameters are changed, a few combinations are introduced through whole Table 2. In fact, (131, 5) and (300, 10) for the first model and (285, 14) for the second model are reeated as otimal design arameters. Besides, the results of MOESD are more robust to altered arameters. Comaring the most efficient units obtained using multi-objective model shows significant difference against the first model. In a secial case, the efficient units resectively from the first and second row in Table 2 are comared. It is observed that a AOQ has imroved about 5.05%, and a LTPD has a slight decrease about 1.13%. However, TC is increased about 2.96%. In site of the increase in cost, statistical erformances are imroved noticeably using multi-objective model. For more investigation, we considered these re-secified values of arameters in the model: C i =, C f =2, C o =10, =, and N=1000. Figure 1, deicted in Minitab statistical software 17, shows three curves including oerating characteristic (OC), ATI, and AOQ for comaring the results of two different models. Generally, the ower of a samling lan in distinguishing good and bad lots gets closer to the ideal state, called 100% insection, when samle size increases [1]. According to the OC curve in Figure 1, the greater sloe of samling lan obtained using multi-objective model is an indication of its greater discriminatory ower. Thus, the increase in cost for this model can be justified. On the other hand, it is desired to design a samling lan so that the OC curve gives high robability of accetance at the AQL and, also, low robability of accetance at the LTPD [1]. In Table 3 for AQL=2%, robability of accetance using the multi-objective model is higher than that of in the cost model. However for LTPD=7%, there is no significant difference between them. Another imortant measure, shown in Figure 1, is ATI. Clearly, larger ATI is obtained from multi-objective model for 1236

7 IJDEA Vol.4, No.2, (2016) S. Jafarian-Namin, et al /IJDEA Vol.5, No.2, (2017) fraction defectives almost lower than 3. After that, there is no significant difference between to models. As indicated in [1], AOQ is the quality of lot that results from the alication of rectifying insection. It is exected to be lower than incoming fraction defective () over a long term. Considering AOQ curve in Figure 1 and its results from Table 3, the erformance of lan obtained using the multi-objective model is obviously better with lower AOQ values for AQL and LTPD oints. These results can totally endorse the enhanced erformance of the multiobjective model and disclose the insufficiency of the cost model with statistical constraints in such sace. 6. Conclusion and Future Researches This research aimed at to design a single AS lan in which samle size, and accetance number must be determined. Moreover, it was tried to reach a wellbalanced comromise between cost (economical) and the roducer's and consumer's quality and risk (statistical) features that never had been considered simultaneously. For these reasons, a multi-objective model was develoed and a simle and efficient otimization algorithm using DEA was emloyed to solve it. Through a simulation study, the roosed model erformed better than the existing cost model with statistical constraints. As mentioned earlier, samles can be either single, double, multile, or sequential. We investigated the most common (single) samling lan. The other lans can be used as future researches. Table1. Inut values of arameters for erforming different simulations Parameters Magnitudes AQL ( 0 ) 0.02 LTPD ( 1 ) 0.07 Failure rate (), 0 Producer s risk (α) 0.05 Consumer s risk (β) 0 C i 0, 1.00 C f 2.00, 3.00 C o 10.00, Lot size (N) 1000,

8 S. Jafarian-Namin, et al /IJDEA Vol.5, No.2, (2017) Table 2. Efficient designs of different models under various values of arameters LTPD AQL C i C f C o N Model n c TC a a Efficiency Multi-objective Multi-objective Multi-objective Multi-objective Multi-objective Multi-objective Multi-objective Multi-objective Multi-objective Multi-objective Const. Multi-objective Multi-objective Multi-objective Multi-objective Multi-objective Multi-objective

9 IJDEA Vol.4, No.2, (2016) S. Jafarian-Namin, et al /IJDEA Vol.5, No.2, (2017) Table 2. (Continued) C i C f C o N Model n c TC LTPD a AOQ a Efficiency Multi-objective Multi-objective Const Multi-objective Multi-objective Multi-objective Multi-objective Multi-objective Multi-objective Multi-objective Multi-objective Multi-objective Multi-objective Multi-objective Multi-objective Multi-objective Multi-objective

10 S. Jafarian-Namin, et al /IJDEA Vol.5, No.2, (2017) Table 3.Comarison of different models for C i =, C f =2, C o =10, =, and N=1000 Model n c AQL% LTPD% P a AOQ ATI Const Multi-objective Figure 1.a) OC curve, b) ATI curve, and c) AOQ curve for efficient designs obtained using different models when C i =, C f =2, C o =10, =, and N=

11 IJDEA Vol.4, No.2, (2016) S. Jafarian-Namin, et al /IJDEA Vol.5, No.2, (2017) References [1] Montgomery, D.C. (2008). Introduction to Statistical Quality Control. New York: Wiley. [2] Pearn, W.L., Wu. C.W. (2006). Critical accetance values and samle sizes of a variables samling lan for very low fraction of nonconforming. Omega, 34(1), [3]Aslam, M., Niaki, S.T.A.., Rasool, M., Fallahnezhad, M.S. (2012). Decision rule of reetitive accetance samling lans assuring ercentile life. Scientia Iranica, 19(3), [4] Ferrell, W. G., Chhoker, Jr. A. (2002). Design of economically otimal accetance samling lans with insection error. Comuters and Oerations Research, 29(1), Chart. International Journal of Data Enveloment Analysis, 2(2), [9] Jafarian-Namin, S., Hasanzadeh, H., Mobin, M. (2016). Multi-objective economic-statistical design of the new t- Chart based on Process Caability Index. Proceedings of the 2016 International Conference on Industrial Engineering and Oerations Management, Detroit, Michigan, USA, [10] Charnes, A., Cooer, W.W. Rhodes, E. (1978). Measuring the efficiency of decision making units, Euroean Journal of Oerational Research, 2(6), [11] Emrouznejad, A., Yang, G. L. (2017). A survey and analysis of the first 40 years of scholarly literature in DEA: Socio-Economic Planning Sciences, 61, 4-8. [5] Niaki, S.T.A., Fallahnezhad, M.S. (2009). Designing an otimum accetance lan using bayesian inference and stochastic dynamic rogramming. Scientia Iranica, 16(1), [6] Fallahnezhad, M.S., Niaki, S.T.A., VahdatZad, M.A. (2012). A new accetance samling design using bayesian modeling and backwards induction. International Journal of Engineering, Transactions C: Asects, 25(1), [7] Hsu, J.-T. (2009). Economic Design of Single Samle Accetance Samling Plans. Journal of Hungkuang University, [8] Jafarian-Namin, S., Amiri, A., Najafi, E. (2014). Efficient Selection of Design Parameters in Multi-Objective Economic- Statistical Model of Attribute C Control 1241

12 S. Jafarian-Namin, et al /IJDEA Vol.5, No.2, (2017)