An optimal batch size for a JIT manufacturing system

Transcription

1 Comuters & Industrial Engineering 4 (00) 17±136 n otimal batch size for a JIT manufacturing system Lutfar R. Khan a, *, Ruhul. Sarker b a School of Communications and Informatics, Victoria University of Technology, Footscray Camus, P.O. Box1448 MCMC, Melbourne, Vic 8001, ustralia b School of Comuter Science, University of New South Wales, DF Camus, Northcott Drive, Canberra 600, ustralia bstract This aer addresses the roblem of a manufacturing system that rocures raw materials from suliers in a lot and rocesses them to convert to nished roducts. It rooses an ordering olicy for raw materials to meet the requirements of a roduction facility. In turn, this facility must deliver nished roducts demanded by outside buyers at xed interval oints in time. In this aer, rst we estimate roduction batch sizes for a JIT delivery system andthen we incororate a JIT raw material suly system. simle algorithm is develoedto comute the batch sizes for both manufacturing andraw material urchasing olicies. Comutational exeriences of the roblem are also brie y discussed. q 00 Elsevier Science Ltd. ll rights reserved. Keywords: Inventory control; Materials rocurement; JIT; Periodic delivery; Otimum order quantity 1. Introduction desirable condition in long-term urchase agreements in a just-in-time (JIT) manufacturing environment is the frequent delivery of small quantities of items by suliers/vendors so as to minimize inventory holding costs for the buyer. Consider a manufacturing system that rocures raw materials from outside suliers and rocesses them to convert into nished roducts for retailers/customers. The manufacturer must deliver the roducts in small quantities to minimize the retailer's holding cost, andaccet the suly of small quantities of raw materials to minimize his own holding costs. In the traditional JIT environment, the sulier of raw materials is dedicated to the manufacturing rm, and normally is located close by. The manufacturing lot size is deendent on the retailer's sales volume (/market demand), unit roduct cost, set-u cost, inventory holding cost and transortation cost. The raw material urchasing lot size is deendent on raw material requirement in the manufacturing system, unit raw material cost, ordering cost and inventory holding cost. Therefore, the otimal raw material urchasing * Corresonding author. addresses: khan@matilda.vu.edu.au (L.R. Khan), ruhul@cs.adfa.edu.au (R.. Sarker) /0/$ - see front matter q 00 Elsevier Science Ltd. ll rights reserved. PII: S (0)

2 18 L.R. Khan, R.. Sarker / Comuters & Industrial Engineering 4 (00) 17±136 Nomenclature D demand rate of a roduct, units er year P roduction rate, units er year (here P. D ) Q roduction lot size annual inventory holding cost, $/unit/year set-u cost for a roduct ($/set-u) f r amount/quantity of raw material r required in roducing one unit of a roduct D r demand of raw material r for the roduct in a year, D r ˆ f r D Q r ordering quantity of raw material r r ordering cost of a raw material r r annual inventory holding cost for raw material r PR r rice of raw material r Q r otimum ordering quantity of raw material r x shiment quantity to customer at a regular interval (units/shiment) L time between successive shiments ˆ x/d T cycle time measuredin year ˆ Q /D m number of shiments during the cycle time ˆ T/L n number of shiments during roduction utime T 1 roduction utime in a cycle T roduction downtime in a cycle ˆ T T 1 IP avg average nishedgoods inventory quantity may not be equal to the raw material requirement for an otimal manufacturing batch size. To oerate the JIT manufacturing system otimally, it is necessary to otimize the activities of both raw material urchasing androduction lot sizing simultaneously, taking all the oerating arameters into consideration. Unfortunately, until recently, most JIT studies in the literature have been descritive (derohunmu, Mobolurin, & Bryson, 1995; Chaman & Carter, 1990), and most of the analytical studies do not take all the costs for both sub-systems into consideration (nsari & echel, 1987; nsari & Modarress, 1987; O'Neal, 1989). Therefore, the overall result may not be otimal. larger manufacturing batch size reduces the set-u cost comonent to the overall unit roduct cost. The roducts roduced in one batch (/one manufacturing cycle) are delivered to the retailer in m small lots (ie m retailer cycles er one manufacturing cycle) at xedtime intervals. So the inventory forms a saw tooth attern during the roduction utime and a staircase attern during roduction downtime in each manufacturing cycle. Likewise, the manufacturer receives n small lots of raw material, at regular intervals, during the roduction utime of each manufacturing cycle. The raw materials are consumed at a given rate during the roduction utime only. It is assumed that the roduction rate is greater than the demand rate. So the accumulated inventory during roduction utime is used for making delivery during roduction downtime until the inventory is exhausted(see Fig. 1). Production is then resumedandthe cycle reeated. Sarker, Karim andzad(1993, 1995a) andsarker, Karim andaque (1995b) develoeda model oerating under continuous suly at a constant rate. In their aer, they considered two cases. In case I, the ordering quantity of raw material is assumed to be equal to the raw material required for one batch of the roduction system. The raw material, which is relenished at the beginning of a roduction cycle, will

3 L.R. Khan, R.. Sarker / Comuters & Industrial Engineering 4 (00) 17± Fig. 1. Inventory level with time for roduct (to), and raw material (bottom). be fully consumedat the endof this roduction run. In case II, it is assumedthat the ordering quantity of a raw material to be n times the quantity requiredfor one lot of a roduct, where n is an integer. Though case I can be ttedin the JIT suly system, case II is not favourable for the JIT environment. n ordering olicy, for raw materials to meet the requirements of a roduction facility under a xed quantity, eriodic delivery olicy, has been develoed by Golhar and Sarker (199), Jamal and Sarker (1993), Sarker andgolhar (1993), andsarker andparija (1994). They consideredthat the manufacturer is allowedto lace only one order for raw materials er cycle, which is similar to case I above. In this case, a xedquantity of nishedgoods (say x units) is to be delivered to the customer at the end of every L units of time ( xed interval). This delivery attern forces inventory build-u in a saw-tooth fashion during the roduction utime. The on-hand inventory deletes sharly at a regular interval during the roduction downtime until the end of the cycle time, forming a staircase attern. Recently, Sarker and Parija (1996) develoed another model for urchasing a big lot of raw material that will be used in n consecutive roduction batches. In this aer, our urose is to incororate JIT concet in conventional joint batch sizing roblems in two stages. In the rst stage, the roblem considers a JIT delivery system. This roblem is somewhat similar to Sarker and Parija (1994), which considers a xed quantity-eriodic delivery olicy. The raw material urchasing quantity is equal to the exact requirement of one manufacturing cycle. Chakravarty andmartin (1991) andgolhar andsarker (199) erceivedthat the roduction time might not be comosedof exactly m full shiment eriods. To satisfy the demand for the whole cycle time, the roduction run length may extend for a fraction of a shiment eriod in addition to m shiments. owever, when the transortation cost is indeendent of the shiment quantity, the fractional shiment may not be cost effective. The retailer may not accet a fractional shiment either. So we consider the frequency of the nishedroduct suly, m, strictly as an integer. In the secondstage, the roblem

4 130 L.R. Khan, R.. Sarker / Comuters & Industrial Engineering 4 (00) 17±136 considers a JIT system for both nished roduct delivery and raw material receiving. This roblem is similar to the rst roblem excet that the raw material requirements in one manufacturing cycle will be urchasedin n small lots. Each lot of raw material will be consumedin L units of time during the roduction utime. The motivation of this research comes from the intention of examining the bene ts of JIT systems when incororatedin joint batch sizing olicy because: ² JIT systems are successful for many ractical inventory systems, and ² joint inventory olicy is considered to be an effective olicy for multi-stage inventory systems. The cost factors considered are ordering/setting u, holding and transortation costs. We have devised the total cost equation of the system with resect to the roduction quantity, nished roduct delivery frequency and/or raw material suly frequency and then solved the roblem as an unconstrained otimization roblem. The aer is organizedas follows. Following the introduction, the mathematical formulations of two coordinated olicies are given. Then, the solution methodology is rovided, followed by results and discussions. Finally, conclusions are drawn.. Model formulation The total quantity manufactured during the roduction time, T 1, must exactly match the demand for cycle time, T. It is assumedthat T 1 must be less than or equal to T. In order to ensure no shortage of roducts, we assume that the roduction rate, P, is greater than the demand rate of the nished roducts, D. There is a demand of x units of nishedgoods at the endof every L time units (due to xed-interval batch suly) and the quantity roduced during each eriod (of L time units) is PL, where (PL x) is ositive. So the nished goods inventory build-u forms a saw-tooth attern during the roduction eriod(see Fig. 1). The on-handinventory deletes sharly at a regular interval after the roduction run till the endof the cycle time. The later one forms a staircase attern. In this aer, we consider two cases of JIT manufacturing environment. Case I: JIT delivery system, where the delivery of nished goods is in m small lots of x units at regular time intervals andthe raw material urchasing quantity is exactly equal to the requirement of raw material in a manufacturing cycle. Case II: JIT suly and delivery system, modi es the raw material suly attern of case I. In this case, the raw material requiredin one manufacturing cycle will be receivedin n small lots at regular time intervals during the roduction utime. For simlicity, we assume that the length of time interval L is equal for both nished goods delivery and raw material receiving. owever, the carrying of raw material is not ermitted during roduction downtime in any of the cases. If m ˆ 1; the cases revert to the classic inventory models, where all the nished goods are delivered in one lot in a cycle and it does not have the JIT element any more. For a very large m, they become a continuous delivery system..1. ssumtions To simlify the analysis, we make the following assumtions: ² There is only one manufacturer andonly one raw material sulier for each item. ² The roduction rate is uniform and nite.

5 L.R. Khan, R.. Sarker / Comuters & Industrial Engineering 4 (00) 17± ² There are no shortages. ² The delivery of the roduct is in a xed quantity at a regular interval. ² The raw material suly is available in a xedquantity whenever required. ² The roducer is resonsible to transort the roduct to the retailers' location... Case I: JIT delivery system The total cost function, for case I, can be exressedas follows: TC 1 ˆ Dr Q r 1 D Q r r P r 1 D Q 1 IP avg 1 where D r Q r ˆ ordering cost of raw materials r D P Q r r ˆ inventory holding cost of raw materials D Q ˆ set-u cost of finishedroducts IP avg ˆ inventory holding cost of finished roducts The rst two terms in Eq. (1) reresent the inventory cost for raw material andthe next two terms reresent the inventory costs for nishedgoods. Since the raw materials are carriedfor only T 1 time units, the holding cost in second term, r, is rescaledby the factor T 1 =T ˆ D =P to comute the raw material inventory carrying cost. lso, unlike the assumtion of a one-to-one conversion of raw materials to nished goods a raw material may be transformed to a new roduct through the manufacturing rocess at a different conversion rate, f r ˆ D r =D ˆ Q r =Q : Therefore, substituting D =Q for D r =Q r ; and f r Q for Q r ; Eq. (1) leads to TC 1 ˆ D Q r 1 D f r Q P r 1 D Q 1 IP avg in which the total inventory costs for both raw materials and nishedgoods are exressedin terms of manufacturing batch size, Q : The average nishedgoods inventory er cycle can be exressedas follows: IP avg ˆ Q 1 D P m 1 x The exression for IP avg is obtainable by calculating the relevant area of Fig. 1 and dividing it by ml. The area consists of n triangles of area PL =; (n n = rectangles of area (PL Lx), anda number of rectangles of area Lx. 3

6 13 L.R. Khan, R.. Sarker / Comuters & Industrial Engineering 4 (00) 17±136 The details of the derivation of this exression can be found for integer m in Sarker and Uddin (1995) andfor continuous m in Sarker andparija (1994). For simlicity, m has been assumedto be an integer in our study. Using the relationshi in Eq. (3), the total cost in Eq. (1) may be written as TC 1 ˆ D Q r 1 1 D P f r Q r 1 Q 1 D P m 1 x 4 fter simli cation, Eq. (4) becomes TC 1 ˆ D Q r 1 1 Q D P f r r D P 1 x m 1 x 5 It is requiredto minimize the function TC 1 to determine Q where Q ˆ mx:.3. Case II: JIT Suly and delivery system The total cost function, for case II, can be exressedas follows: TC ˆ D Q 1 Q 1 D P 1 D r Q r 1 D f r Q r r P r m 1 x 6 where D Q ˆ set-u cost of finishedroducts Q 1 D P m 1 x ˆ inventory holding cost of finished roducts D r Q r r ˆ ordering cost of raw materials D P f r Q r r ˆ inventory holding cost of raw materials It is requiredto minimize the function TC to determine Q where Q ˆ mx; Q ˆ nq r and Q r ˆ PL: ere Q ˆ mx and Q ˆ nq r indicate that mx must be equal to nq r : In other words, Q r is equal to the ratio of m and n multiliedby x that is (m=n)x. In this aer, we assume that the time intervals for raw material receiving will be L time units. So we can write Q r ˆ PL; that means the raw material urchasing quantity is known and it is indeendent of Q : Substituting the relationshi Q r ˆ PL in Eq. (6) andthen simlifying, we get the total cost equation as follows: TC ˆ D Q 1 Q 1 D P x m 1 f rd PL r 1 D P f r PL r 1 x 7

7 3. Solution methodology In this section, the solution rocedures for both the roblems stated in Section are resented. Ef cient algorithms may be aliedto solve the above roblems using an otimization technique. To solve a similar roblem, Moinzadeh andggarwal (1990) roosedan algorithm for discrete otimization, Park andyun (1984) develoeda heuristic method, andgolhar andsarker (199) used one-directional search rocedure. In this aer, we roose a heuristic method which is simle, easy to aly andcomutationally ef cient. In this method, an initial solution for Q is determined by relaxing the integrality requirement for m. The corresonding value of m is comuted. If the comuted m is not integer, then we search for an integer m in the neighbouring oints, which gives minimum TC Problem I In order to nd an otimal manufacturing quantity, it is necessary to differentiate TC 1 with resect to Q (Eq. (5)). But the function is non-differentiable as it contains an integer variable m. owever, it can be shown that TC is a convex function of Q for a given m. s we assumedin an earlier section, the integer variable m can be relacedby Q =x: So the new cost equation is TC 1 ˆ D Q r 1 1 Q D P f r r D P 1 1 x It can be shown that TC 1 is a convex function. Now, differentiating TC 1 with resect to Q andthen equating to zero, we get s Q D 1 r ˆ 9 KK where L.R. Khan, R.. Sarker / Comuters & Industrial Engineering 4 (00) 17± KK ˆ D P f r r D P 1 Now the calculated m ˆ Q =x may not be an integer as we relaced m by a continuous variable in Eq. (8). In the case of non-integer m, we take the neighboring integer value corresonding to which the total cost is minimum. The algorithm to solve the batch sizing roblem for roblem I is as follows: lgorithm I: nding batch size Ste 0. Initialize andstore D ; P; ; r ; and r : Ste 1. Comute the number of batch size Q using Eq. (9). Ste. Comute m ˆ Q =x : If m is an integer, then sto. Ste 3. Comute TC 1 using Eq. (5) for m ˆ m and m : Choose the m ˆ m that gives minimum TC 1. Sto. 8

8 134 L.R. Khan, R.. Sarker / Comuters & Industrial Engineering 4 (00) 17±136 Fig.. Total cost curve. 3.. Problem II Substituting the integer variable m by Q =x; we can rewrite Eq. (7) as follows: TC ˆ D Q 1 Q 1 D P Q 1 D r PL r 1 D f r PL P r 1 x 10 It can also be shown that TC is a convex function. Now, differentiating TC with resect to Q andthen equating to zero, we get v u Q ˆ u t D 1 D P Now the calculated m ˆ Q =x may not be an integer as we relaced m by a continuous variable in Eq. (10). In case of non-integer m, we take the neighbouring integer value corresonding to which the total cost is minimum. The algorithm to solve the batch sizing roblem for roblem II is as follows: lgorithm II: nding batch size Ste 0. Initialize andstore D ; P; ; r ; and r : Ste 1. Comute the number of batch size Q using Eq. (11). Ste. Comute m ˆ Q =x : If m is an integer, then sto. Ste 3. Comute TC using Eq. (7) for m ˆ m and m : Choose the m ˆ m that gives minimum TC. Sto Results and discussions The total cost functions with integer m are lottedin Fig.. s an assumtion made m ˆ Q =x in an earlier section, the total cost is comutedfor each airedm (integer) and Q data. The data used, to

9 L.R. Khan, R.. Sarker / Comuters & Industrial Engineering 4 (00) 17± Table 1 Otimum TC, m and Q ( indicates otimum TC) Calculatedvalues With uer integer (m) With lower Integer (m) Solution Q m m TC m TC m Q generate these curves, are similar to those in Jamal andsarker (1993) andsarker andparija (1994): D ˆ 400 units/year, P ˆ 3600 units/year, ˆ $300/set-u, r ˆ $00/order, ˆ $/unit/year, r ˆ $1/unit/year, f r ˆ 1 and x ˆ 100 units/shiment. The behavior of the cost function, in Fig., re ects a smooth andconvex attern. If we restrict m to an integer for the roblems statedin Golhar andsarker (199), Jamal and Sarker (1993), andsarker andparija (1994), then their cost equations become similar to our Eq. (5). With the data mentioned earlier, the Otimum TC, m and Q are shown in Table 1. The solution is Q ˆ 1300; m ˆ 13 andtc ˆ $ s er the solutions of Golhar andsarker (199), Q ˆ 134 units/batch which is incidentally similar to our Q solution with Eq. (9), shown in the rst column of Table 1. The otimum Q is 1346 as determinedby Jamal andsarker (1993) andsarker andparija (1994). The exerimental analysis of the total cost against raw material ordering cost for both case 1 and case shows that case is more sensitive than case 1 to the ordering cost when other arameters are xed. From the analysis, it can be concluded that case is more attractive for very low ordering cost. That means if the ordering cost is very low, which is a condition for JIT suly, case can achieve more savings, which is consistent with the reasons for using JIT systems. 5. Conclusions In a JIT manufacturing environment, a sulier is exected to deliver goods frequently in small lots. Ideally, a sulier to the JIT buyer is exected to synchronize his roduction caacity with the buyer's demand so that the inventory in the suly ieline is reduced and eventually eliminated. n inventory model is develoed for erfect matching. The total cost function is non-differentiable with resect to manufacturing batch size. owever, it can be shown that the total cost function is convex for a given m. simle algorithm is develoed to comute otimal manufacturing batch size and the associatedraw material urchasing lot size. The quality of solution is discussedandsensitivity analysis is rovided. In roblem II, it is assumedthat Q ˆ nq r ; where Q r ˆ PL: To generalize the roblem, the relation Q r ˆ PL needs to be relaxed. That is a toic for further research. cknowledgements The authors are thankful to the anonymous referees for their comments.

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