Quality. Mokhtari. for improving. characteristic, to an out- select the. in-control. of quality. in a. classical. .iust.ac.ir/

Transcription

1 International Journal Industrial Engineering & Production Research (16 Setember 16, Volume 7, Number htt://ijiepr..iust.ac.ir/ Simultaneous Otimization Production and Deterioration Process Quality in a Ali Salmasnia, Hossein Fallah Ghadi & Hadi Mokhtari Ali Salmasnia, Deartment Industrial Engineering, Faculty Engineering and echnology, University Qom Hossein Fallah Ghadi, Deartment Industrial Engineering, Faculty Engineering and echnology, University Qom Hadi Mokhtari, Deartment Industrial Engineering, Faculty Engineering, University Kashan, Kashan KEYWORDS Production lanning, deterioration rocess, roduction run length, mean quality characteristic, drift, Loss function ABSRAC Achieving otimal roduction cycle time for imroving manufacturing rocesses is one the common roblems in roduction lanning. During recent years, different aroaches have been develoed for solving this roblem, but most them assume that mean quality characteristicc is constant over roduction run length and sets it on customer s target value. However, the rocess mean may drift from an in-control to an out- -control at a random oint in time. his study aims to select the roduction cycle time and the initial setting mean quality characteristic, so that the exected total cost, consisting quality loss and maintenance costs as well as ordering and holding costs, already considered in the classic models is minimized. o investigate the effect mean rocess setting, a comutational analysis on a real world examlee is erformed. Resultss show the sueriority the roosed aroach comared to the classical economic roduction quantity model. 16 IUS Publication, IJIEPR. Vol. 7, No. 3, Reserved All Rights 1. Introduction1 In many industries, lanning for quantity roduction and its scheduling have been consistently associated with restrictions other manufacturing sectors. hus, other sectors olicies may affect roduction lanning. For examle, maintenancee olicy, quality control rograms, safety and health rograms. Corresonding author: Hadi Mokhtari mokhtari_ie@kashanu. ac.ir Received 13 December 15; revised May 16; acceted December 16 Although roduction lanning and quality control are two fundamental and interrelated factors in many industrial rocesses, they have been ten investigated searately in the literature. In other words, the classical models for determining the economic roduction quantity (EPQ assume that roduction rocess is erfect, which means quality defect never haens. he classical EPQ model has been studied by many researchers. he model basically includes the setu, holding and urchasem or manufacturing cost the roduct. However, the effect an imerfect rocess on the EPQ model has not been studied extensively. International Journal Industrial Engineering & Production Research, Setember 16, Vol. 7, No. 3

2 76 A. Salmasnia, H. Fallah Ghadi & H. Mokhtari In some industrial rocesses, rocess mean may drift continuously over time in either ositive or negative directions. In other words, Production rocess starts to roduce in the in-control state, and after a eriod time, the rocess will eventually drift from in-control state to out-oan assignable case such as corrosion, fatigue or cumulative wear. In this situation, roduction ercentage control state due to occurrence non-conforming items increases extremely. Hence, in many industries, selection the otimum rocess mean setting has been a focal oint research in the area quality control engineering. his issue arises because the rocess mean affects the defect rate, material cost, scra or rework cost and ossible losses due to the deviation roduct erformance from the roducer s target value. If the rocess mean is set too low, then the manufacturing cost might be reduced, but the manufacturer may exerience the high rejection cost associated with non- the rocess mean is set tooo high, then the roortion non-conforming roducts may become low and the manufacturer tends to save rejection costs associated with non-conforming roducts [ 1]. conforming roducts. On the other hand, if However, a higher rocess mean tyically results in a higher cost associated with an increased amount contents. herefore, setting the otimal rocess target mean is an imortant asect in the design any manufacturing rocess. Furthermore, a short roduction run length may result in greater set-u costs and restoration costs, and ossibly fewer non-conforming items. A longer roduction run will result in lower setu costs and restoration costs, and ossibly more non-conforming items []. herefore, there is a need to determine an otimal roduction run length, such that the exected cost er unit time is minimized. Consequently, setting the mean quality characteristic under consideration and determining the roduction run length are two imortant factors any roduction system. Effective integration these two factors will give an industry a cometitive advantage. Production rocesses generally deteriorate due to consumtion or ageing, and the rocess mean may drift or shift from an in-control to an out-- control state. As such, without any restoration action, roduction rocesses will roduce more non-conforming items. on the other hand, if restoration time is longer than normal range, we cannot roduce in a normal situation because long eriod run length. Simultaneous Otimization Production and Quality he effect an imerfect roduction rocess was initially studied by Rosenblatt and Lee [3]. he elased time to the rocess shift was assumed to be exonentially distributed, and the otimal roduction run was found to be shorter than that the classical EPQ model. Recently, Kim and Hong [4] have develoed an EPQ model to find an otimal roduction run in a deteriorating rocess. Brandolese [5] integrated roduction and maintenance lanning to minimize the total roduction cost. Gibra [6] rovided an economic model combining resetting costs and enalty costs for non-conforminequation that may be roducts and derived an solved grahically to find the otimal resetting time. Arcelus [7] dealt with the roblem obtaining the otimum roduction run when the quality characteristic the roduct is subject to a systematic linear shift in both the mean and variance. Rahim and Lashkari [8], Rahim and Raouf [9], and several others extended Gibra s work and studied the otimal determination the roduction run where the rocess mean is subject to linear drift. In the above studies, it was assumed that the drift the rocess mean starts right from the beginning the roduction cycle. his assumtion, however, is not aroriate for most industrial rocesses where the deterioration starts at a random oint in time resulting from the occurrence an assignable cause. Ganeshan [1] introduced the aguchi quality loss function into inventory modeling develoment. he urose was to determine the otimal levels inventory and roduction order quantity in order to minimize inventory and quality related costs. Hou [11] considered an economic roduction quantity model with imerfect roduction rocesses, in which the setu cost and rocess quality are functions caital exenditure. Rahim [1] and Rahim and Ben-Daya [13] consideredd the effect EPQ on the economicc design -control chart. hey develoed an integrated model for the inventory and quality control roblems for a class deteriorating rocesses where the in-control eriod follows a general robability distribution with increasing failure rate. he rocess was insected using a samling frequency that increases with the age the system. hese insections rovided information on the state the rocess. In fact, the quality the roduct was monitored by a -controll chart. Recently, Ben- Daya [14] develoed a model integrating the otimization the design the control chart, the economic roduction quantity, and maintenance requirements. Moreover, we can International Journal Industrial Engineering & Production Research, Setember 16, Vol. 7, No. 3

3 Simultaneous Otimization Production and Quality A. Salmasnia, H. Fallah Ghadi & H. Mokhtari 77 find many studies on quality characteristics in. Problem Descrition ractical environments in literaturee such as [15- In this aer, we consider a ractical 18]. manufacturing situation, where a roduction In the roduction and quality analysis literature, the economic roduction quantity has been rocess starts in good running condition, in which conforming items are roduced. After running studied under various conditions. Most the some time, due to the nature the rocess and models discussed are based on the assumtions that the roduction rocess never fails, and it always roduces items accetable quality. In machine wastage, At a random oint in time,τ, a assignable cause occurs that leads to the rocess starting to drift either into a ositive or negative other words, the traditional economic roduction direction following the drift function, W (t,. quantity model assumes that the roduct is Afterwards, non-conformi ing items are roduced erfect for a manufacturing rocess. However, a as can be seen in Figure 1. more realistic situation is one in which the quality We consider a rocess with knownn and fixed is not always accetable because the condition variance that always startss in the in-control state. the roduction rocess may deteriorate with time. After a random running time, the rocess shifts Hence, the occurrence a defective roduct for from in-control to out--control state. In such a these models has been neglected. situation, the mean value the quality he role the roduction run length and rocess characteristic drifts according to a linear function, mean setting in controlling quantity and quality while the variance is assumed to remain roduct is more evident and imortant than unchanged. ever. In other words, roduction quantity and According to deviation from target value, the cost roduct quality are two interrelated roblems. In rocess mean drift due to drift function is site the mentioned fact, these two roblems calculated by aguchi s quadratic loss function. are ten investigated searately. herefore, Quality loss in in-control state (Figure, due to develoing models that cature interdeendency secification limits and rocess mean that between these two main asects in the modern roduces conforming items, is slight. But when roduction rocess is an undeniablee necessity. rocess goes in out-f-control state (Figure 3, o fill research gas and overcome the above- the items that are rejected by costumer, is more the quality loss, due to roducing not conforming mentioned drawbacks, this study integrates concets quality control and inventory control significant. in a unified model. It aims to determine On the other hand, we should consider roduct simultaneously the otimal initial setting the cost with quality cost to determine otimum rocess mean and otimal roduction cycle roduction run and cycle time. By otimization length for a rocess which deteriorates over time quality, relacement, and holding costs, we in a way that the exected total cost, consisting can obtain otimum initial mean, roduction run, the holding, set u, and quality control costs, is and roduction quantity. herefore, we can minimized. Furthermore, in contrast to the most achieve an otimum annual exected cost by existing models which assume the otimization initial mean setting and deterioration starts at the beginning the roduction run length. rocess, in this aer, it is suosed that the he role the roduction run length and rocess deterioration starts at a random oint in time. mean setting in controlling quantity and quality he structure this article is as follows: he roduct is more evident and imortant than roblem descrition is resented in Section. In ever. Production quantity and roduct quality Section 3, the roosed mathematical model is the roduction rocess are interrelated roblems. exressed in detail. In Section 4, a numerical raditionally, these two roblems are dealt with examle to illustrate imortant asects the searately. here is a need for develoing models roosed integrated model is resented. In that cature the interdeendence between these Section 5, solution aroach to solve the two main asects any modern roduction suggested mathematical rogramming is rocess leading to their joint otimization. resented. In Section 6, the sensitivity analysis is However, regardless roduction lanning, given to determine the most effective arameters determining the volume roduction during the on the exected total cost er roduction cycle. cycle and resetting the rocess mean cause more Finally, the concluding remarks are given in cost for roducer. In this situation, we need to Section 7. identify the initial mean setting, µ, and the length the cycle time,, due to otimization roduction arameter such as Q. he rocess International Journal Industrial Engineering & Production Research, Setember 16, Vol. 7, No. 3

4 78 A. Salmasnia, H. Fallah Ghadi & H. Mokhtari mean resetting to its initial setting is usually done at a certain resetting cost. We should ay attention to, because the rocess mean resetting is done due to rocess run cycle. Simultaneous Otimization Production and Quality herefore, the goal is to find an initial mean setting, µ, and a cycle length,, which otimize roduction quantity. -1. Notation We use the following notation for aer to solve resented model. Notation K A R P D h f ( he density function the occurrence time the assignable cause f ( e,, Descrition ab. 1. Notations he initial mean quality characteristic the roduct when the rocess begins in an in-control state having variance σ he otimal initial rocess mean Cycle length P he otimal cycle length: D Production Run Otimal Production Run Losss arameter: ( K A / Quality loss index (Reject cost Distance the mean from the secification limits n 1 ( USL Z.5 Z.5 he cost each installation or relacement ye I error, When the rocess is in in-control state Drift rate er time unit Production rate er time unit Demand rate er time unit Inventory holding cost er unit er time unit he elased time until the occurrence the assignable cause. It is a random variable and is assumed to be exonentially distributed with a mean 1/λ; A random art that is normally distributed with mean,variance International Journal Industrial Engineering & Production Research, Setember 16, Vol. 7, No. 3

5 Simultaneous Otimization Production and Quality A. Salmasnia, H. Fallah Ghadi & H. Mokhtari 79 Notation y (t, (t, W (t, EC Descrition he random variable denoting the quality characteristic the roduct at time t y (t, (t, t he rocess mean at time t, (t, W (t, t he drift function arget value the quality characteristic under consideration Exected total cost er roduction cycle -. Assumtion We consider the following assumtions in the roosed model: 1. Loss cost ( L(t, follows a non-linear and quadratic function. he shortage cost er unit time is infinite. 4. he variance quality characteristic is assumed a fixed value and does not shift over time. 5. Every year is assumed to be 5 days. 6. Every day is assumed to be 8 hours. 5. At the end each roduction run (, the resetting action is done with installation cost. 6. he elased time until the occurrence the assignable cause is a random variable and follows an exonential distribution with a mean 1/λ. 7. Production run ( is the same t in the classic EPQ model ( =tt. 8. Drift function is deendent on time t and the occurrence time assignable cause, i.e., w(, t (t. 3. he Proosed Mathematical Model In this section, the exected total cost (EC, consisting quality loss, holding, and setu costs, is resented. It is notable that the quality loss cost deends on the considered loss function. he exected quality loss deends on the elased time until the occurrence the assignable cause; hence, it can be obtained as follows: E[L(t] E (L(t t P( t E(L(t t P(t 1 where shows the occurrence time an assignable cause. In other words, the rocess roduces conforming items before, while the rocesss starts drifting after. Now, we can calculate quality loss deending on t and as in equation (. L (, t K ( y ( t, Where y( t, is a random variable that shows value the quality characteristic the roduct at time t. International Journal Industrial Engineering & Production Research, Setember 16, Vol. 7, No. 3

6 8 A. Salmasnia, H. Fallah Ghadi & H. Mokhtari t y (t, w( t, t According to equations ( and (3, the quality loss is calculated as follows: Simultaneous Otimization Production and Quality 3 Lt (, K ( K ( w( t, 4 Consequently, the exected quality loss within in-control and out--control states for a certain time can be calculated as in equation (5. KE[( ] t E [ Lt (, ] (5 KE[( w( t, ] t We know that E[x ] Var[x] ( E [x], where Var [ ] denotes variance the quantity in the brackets. herefore, equation (5 can be rewritten as follows: KVar[( ] KE ( [( ] t ELt [ (, ] (6 KVar[( w( t, ] KE ( [( w( t, ] t According to equations (6 and (1, the exected quality losss in a certain time can be calculated as follows: t E[L(t] [ KVar[( ] K( E[( ] ] e (7 [ KVar [( w( t, ] K( E[( w( t, ] ] (1 e t If E[( ], Var[( ] and (, t KVar[( w( t, ] K( E[( w( t, ] ] (1 e t are considered, equation (7 can be simlified as follows: E[L(t] K ( ( e t (,t (8 Now, the exected quality loss for a roduction cycle is calculated by integrating E [L(t] in the interval [, ] International Journal Industrial Engineering & Production Research, Setember 16, Vol. 7, No. 3

7 Simultaneous Otimization Production and Quality A. Salmasnia, H. Fallah Ghadi & H. Mokhtari 81 L(, t E[L(t]dtt K ( ( e (,tdt (9 L(, K ( ( (1 e (,tdt (1 Since drift function is consideredd as wt (, (t and follows an exonential distribution with arameter, the exected quality loss in the time t and the exected quality losss for a roduction cycle can be rewritten as in equations (11 and (1, resectively. E[L(t] K ( ( e t K ( ( ( t ( 1 e t (11 K L(, ( ( (1e K e ( (1 e (1 3 (1 K ( t ( [1e ]dt According to the classical EPQ model, we know that: Q Q P d,,, Q P, P D, d ( D P D P D R hp ( D herefore, according to Figure 3, the setu and holding costs are and, resectively. P With resect to equations (1-(1 and the above exlanations, the exected total cost by considering both roduction and quality in a deterioration rocess is: D R hp ( D DK 1 EC P ( (1 e 1 3 e ( 1 e (1 ( t ( [1e ] dt (13 By otimization EC, will be obtained and by Q P, the otimumm roduction quantity is attained. 4. Solution Aroach Since the decision variables roblem are continuous, and the article swarm otimization (PSO has good erformance in otimizing such variables, we use PSO algorithm for otimizing the exected total cost. PSO is a robust stochastic otimization techniquee based on the movement and intelligence swarms. PSO is based on the metahor social interaction and communicationn (e.g., fish schooling and bird flocking. his technique uses collaboration among a oulation simle search agents, called articles, to find otima in search sace. PSO algorithm has effective erformance in otimizing difficult roblems in a variety fields. In PSO, the otential solutions (articles fly through the roblem sace by following the current otimum articles. All the articles are evaluated by the fitness function to be otimized and have velocities that direct the flying the articles. A general flowchart solution aroach is deicted in Figure 4. International Journal Industrial Engineering & Production Research, Setember 16, Vol. 7, No. 3

8 8 A. Salmasnia, H. Fallah Ghadi & H. Mokhtari Start Initialize articles with random osition Evaluate the fitness each article ( Get the article best (best Obtain the Global best (gbest Calculate article velocity No If the stoing criterion met? Yes Sto Fig. 4. Solution aroach 5. Alication the resented model he roosed model in this aer can be alied to various engineering roblems, such as automotive industry, rocess industry, etc. A case study is used to illustrate the imortant asects Simultaneous Otimization Production and Quality the roosed integrated model. Our case study is a factory sring-making in the city Qom in Iran. Quality characteristic roduct is the sring-constant with the scale Newton er centimeter. Based on the quality control olicies, it is assumed that the target value the quality characteristic is 1, and the accetable region is in the interval [9, 11]. he rocess outut follows a normal distribution with a known and constant standard deviation σ = 1. he drift rate is considered to be θ =.555 er hour. he rocess mean drifts at a random oint in time that is exonentially distributed at a rate once every 6..5 hours oeration (λ= =.15. he relacement cost is estimated to be R= $3. he target mean the roduct is assumed to be 1. Confidence level is considered 95% (ye I error, when the rocess is in in-control state is. 5%. he roduction rate is 35 units er hours. he demand rate is 3 units er hour. It is assumed that the number working days year is days and working hours are 8 hours for a day. herefore, each year is equivalent to 16 hours. he inventory holding cost er unit er hour is h= =$.45($7er year. he quality loss index is A=$.5 er hour ($4 er year. According to the above assumtions and able 1: Distance the mean from the secification limits is =.39, and consequently the loss arameter is K=$.163 er hour ($6. 8 er year. Results otimization by PSO algorithm is shown in able. Method Proosed model EPQ ab.. Comarison the roosed model with the EPQ model Q EC $ $ Comarison resultss two methods indicate that the EPQ model, because ignoring the roduction non-conforming items, sets the roduction run length too larger than the roosed model. Furthermore, the resented mathematical model in contrast to the EPQ model, by considering drift function, does not set the initial value quality characteristic on its target value. Finally, the attained results confirm that the roosed method outerforms the EPQ model with resect to EC. 6. Sensitivity Analysis Normalized data in Figures (5-(8 show the effects the major arameters consisting A,, P, and D on otimization result. Sensitivity analysis is done by varying one arameter at each time, while the other arameters remain in their initial values. Effect the quality loss index A on,, EC is investigated in Figure 5. By ncreasing the quality loss index, the loss arameter K A / also increases, and International Journal Industrial Engineering & Production Research, Setember 16, Vol. 7, No. 3

9 Simultaneous Otimization Production and Quality A. Salmasnia, H. Fallah Ghadi & H. Mokhtari consequently EC grows, while decrease. and A μ EC Fig. 5. Effect quality loss index Effect variation on the drift rate is investigated in Figure 6. By increasing in the drift rate, roduction rocess will deteriorate in a short time. o reduce the number the roduced non- conforming items in a cycle, the roduction run length should be decreased. his issue leads to ncreasing set u cost and EC. As can be seen in Figure 7, increasing the roduction rate, as exected, results in reduction roduction run length and growing exected total cost and initial value rocess mean Ɵ μ EC Fig. 6. Effect International Journal Industrial Engineering & Production Research, Setember 16, Vol. 7, No. 3

10 84 A. Salmasnia, H. Fallah Ghadi & H. Mokhtari Simultaneous Otimization Production and Quality P μ EC Fig. 7. Effect Production rate According to the results illustrated in Figure 8,,, and EC are deely influenced as the demand rate varies. It can be observed that with an increase in the demand rate, the value roduction run length increases dramatically, while the exected total cost and the initial value rocess mean decrease. D μ EC 7. Conclusion his aer resented an integrated model for simultaneous otimization roduction and quality in a deterioration rocess with finding the otimal initial setting the rocess mean, and the otimal roduction cycle length. In contrast to the EPQ model, this study considered that the rocess mean may drift from an in-control state to an out--control state at a random oint in time. he quality loss due to deviation quality characteristic from the target value was described by a quadratic function. PSO algorithm was used for otimizing the exected total cost. o investigate the effect mean rocess setting, a comutational analysiss on a real world examle was erformed. Resultss showed the sueriority the roosed aroach comared to the EPQ Fig. 8. Effect Demand rate model. A sensitivity analysis has been roosed to find the most effective arameters on decision variables and EC. As for future research, we suggest to extend this aer in two directions: first, integration three concets roduction lanning, maintenance olicy, and rocess monitoring; second, simultaneous monitoring mean and variance quality characteristic under consideration. Referenses [1] M. A. AL-Fawzan and M. A. Rahim Otimal Control a Deteriorating Process with a Quadratic Loss Function. Quality and Reliability Engineering International, Vol. 17, (1, International Journal Industrial Engineering & Production Research, Setember 16, Vol. 7, No. 3

11 Follow his Article at he Following Site Mokhtari H, Fallah Ghadi H, Salmasnia A. Simultaneous Otimization Production and Quality in a Deterioration Process. IJIEPR. 16; 7 (3 :75-85 URL: htt://ijier.iust.ac.ir/article en.html Simultaneous Otimization Production and Quality A. Salmasnia, H. Fallah Ghadi & H. Mokhtari 85 [] Hall RI, Eillon S Controlling roduction [1] Rahim MA Joint determination rocesses which are subject to linear trend. Oerational Research Quarterly, Vol. 14, ( roduction quantity, insection schedule and control chart design. IIE rans, Vol. 6, (1994, [3] Porteus, E. L Otimal Lot Sizing, Process Quality Imrovement and Set-U Cost Reduction. Oerations Research, Vol. 34, (1986, [4] Kim CH, Hong Y An otimal roduction run length in deteriorating roduction rocesses. International Journal roduction Economics, Vol. 58, (1999, [5] Brandolese M, Franci M, Pozzetti A Production and maintenance integrated lanning. International Journal Production Research, Vol. 34, (1996, [6] Gibra IN Otimal control rocesses subject to linear trends. Journal Industrial Engineering, Vol. 18, (1967, [7] Arcelus FJ, Banerjee PK, Chandra R Otimal roduction run for a normally distributed quality characteristic exhibiting non-negative shifts in rocess mean and variance. IIE ransactions, Vol. 14, (198, [8] Rahim MA, Lashkari RS Otimal decision rules for determining the length roduction run. Comuters and Industrial Engineering, Vol. 9, (1985, [9] Rahim MA, Raouf A Otimal roduction run for a rocess having multilevel tool wear. International Journal Systems Science, Vol. 1, (1988, [1] Ganeshan, R., Kulkarni, S., and Boone,., Production economics and rocess quality: a aguchi ersective. International Journal Production Economics, Vol. 71, (1, [11] Hou, K., An EPQ model with setu cost and rocess quality as function caital exenditure. Alied mathematical Modelling, Vol. 31, (7, [13] Rahim MA and Ben-Daya model for the joint determination Production run, Insection schedules and M A Generalized economicc control chart design. Int J Prod Res, Vol. 36, (1996, [14] Ben-Daya M Integrated Production Maintenance and Quality Model for Imerfect Processes. IIE rans, Vol. 31, (1999, [15] Sholuwe, M Ofori-Boadu, A. Waller, L. Bock-Hyeng, C Quality Imrovement Practices Award-Winning Residential Builders and Housing Develoers, International Journal Industrial Engineering & Production Research, Vol. 3, (1, [16] Ameli, M. Mirzazadeh, A. Shirazi M. Entroic Economic Order Quantity Model for Items with Imerfect quality Considering Constant Rate Deterioration under Fuzzy Inflationary Conditions, International Journal Industrial Engineering & Production Research, Vol. 4, (13, [17] Yahyatabar Arabi, A., Eshraghnia Jahromi, A., Shabannataj, M Develoing a new model for availability otimization alied to a series-arallel system, International Journal Industrial Engineering & Production Research, Vol. 4, (13, [18] Zadbood, A., Noghondarian, K., Zadbood, Z Multiresonse surface otimization via Harmony search algorithm, International Journal Industrial Engineering & Production Research, Vol. 4, (13, International Journal Industrial Engineering & Production Research, Setember 16, Vol. 7, No. 3