Conteporary Enneern Scences, Vol. 9, 2016, no. 7, 315-324 HIKARI Ltd, www.-hkar.co http://dx.do.or/10.12988/ces.2016.617 A Mult Due Date Batch Scheduln Model on Dynac Flow Shop to Mnze Total Producton Cost Tenku Nuranun * Departent of Industral Enneern, Unverstas Isla Neer Sultan Syarf Kas (UIN Suska) 28293, Pekanbaru, Rau Indonesa * Correspondn author Ahad Fudhol Solar Enery Research Insttute, Unverst Kebansaan Malaysa 43600 Ban Selanor, Malaysa Msra Hartat Departent of Industral Enneern, Unverstas Isla Neer Sultan Syarf Kas (UIN Suska) 28293, Pekanbaru, Rau Indonesa Rado Yendra Departent of Matheatcs, Faculty of Scence and Technoloy Unverstas Isla Neer Sultan Syarf Kas (UIN Suska) 28293, Pekanbaru, Rau Indonesa Isu Kusuanto Departent of Industral Enneern, Unverstas Isla Neer Sultan Syarf Kas (UIN Suska) 28293, Pekanbaru, Rau Indonesa Copyrht 2016 Tenku Nuranun et al. Ths artcle s dstrbuted under the Creatve Coons Attrbuton Lcense, whch perts unrestrcted use, dstrbuton, and reproducton n any edu, provded the ornal work s properly cted.
316 Tenku Nuranun et al. Abstract Ths research solved a batch scheduln proble on dynac flow shop envronent, n whch new orders are consdered to nsert n the current work. Decson akn s ade by coparn the total cost of the nserton schedule wth the ornal schedule. Another feature of ths odel s ts ablty to accoodate ult product orders that are delvered on dfferent due dates n whch revsed due date s allowed for each order. The proposed odel s known as Mx Inteer Non Lnear Proran (MINLP) wth a convex objectve functon. An alorth s developed to fnd the best solutons of batch sze and sequence that nze total cost. A nuercal experence s perfored to show how the alorth works. Keywords: Batch scheduln, dynac flow shop, ult due dates, ult products, total cost 1 Introducton Batch scheduln has proven as an effectve way to ncrease the anufacturn perforance. Many lteratures have dealt that cobnaton of batch sze, and sequences are relable n nzn cost or te lenth producton. Nowadays, uncertan envronent poses a copany to the response all possbltes of chann rapdly. Dsturbances ht be caused by varous factors such as dsrupted achne, arrval of new orders, cancellaton of current orders, updatn due dates, and others. Scheduln research has ostly focused on nventn a odel or alorth to solve deternstc probles. In recent decades, further research has been developed to solve dynac and uncertan envronents [1-5]. In ost real producton envronents, jobs arrve to the syste randoly, and the job's arrval and release dates are not known n advance. A sple approach of ths condton s nsertn the new jobs nto the scheduled sequence. Several works have been done to arrane the latest order to the executed schedule by desnn the nserton technque [6, 7]. Ths stuaton akes a devaton fro ornal schedule and potentally ncreases the total of producton cost. Coon resource that s used for dfferent jobs also ncreases the coplexty of proble and becoes ore dffcult to fure t. In fact, copletn all jobs on ther due dates has becoe a coplex ssue n a way of ncreasn servce level to the custoer. Further, n dynac stuaton, both WIP and new orders need to eet due date. Each job has ts release date that s sorely portant to be accoplshed. Very lttle work s reported on the cost effect of new order's arrval [8, 9]. Earler copleton te arses consequence to the holdn cost, n contrast, later copleton te effects at the penalty cost. Hence, trade off aon varous costs are needed n akn decson, whether contnun the current schedule or adjustn the ornal schedule wth all dynac stuatons. Ths paper s aed to develop a ore flexble odel and technque to accoodate new orders wth acceptable cost consequence. The odel has consdered setup cost, WIP cost, nventory cost, and lost sale cost. All jobs are
A ult due date batch scheduln odel 317 processed on the sae routn but ht be dfferent n specfcaton (ult te). Each job has ts due date (ult due date) and possble to revse t. An adaptve jobnserton based heurstc s presented to deterne batch sze and schedule the resultn batches wth the objectve to nze total producton cost. 2 Total producton cost There are four eleents of producton cost that are consdered n ths odel, they are set up cost, work-n-process (WIP) cost, nventory cost, and lost sale cost. Suppose that there are n jobs of te to be process on stae, each stae conssts of snle achne. Jobs are batched nto N batches wth batch sze Q ( = 1, 2,..., N). All jobs that are processed n the sae batch only need a snle setup before processn. The setup te on a achne s denoted as s, and setup cost of a achne s denoted as C1,, then setup cost s forulated by ultplyn setup te wth setup cost. Thus, total setup cost can be wrtten as follows: M TC N s C (1) 1, 1, 1 Interedate processn s defned as te lenth of a batch to be processed. If F s the fnshn te of a batch on achne, and B1 s the startn te of a batch on achne 1, then WIP s calculated as a devaton between the fnshn te and the startn te. Mnzn WIP cost wll also nze the te lenth producton sultaneously. If Interedate cost of te s denoted as C2, then forulaton of total WIP cost can be seen as follows: N TC2, r, F B1 Q C2, (2) 1 1 Ths odel has consdered about real te condton where each te has specfc due date. If the shop capacty s tht, t s dffcult to eet all due dates but earler fnshed oods are pertted. Ths condton ay ncur an nventory cost untl the products are released. If holdn cost of te s denoted as C3,, then we can calculate total fnshed oods nventory cost as follows: N (3) TC 3, 1 1 r d F M Q C 3, In ths forulaton, a job s not allowed to delver after ts due date. If jobs are possble to be fnshed on ther due date, then they wll not be processed on the shop floor. It eans the copany wll lose ther opportunty to delver the products and caused the lost sale cost as penalty. In case the arrval of a new job, odel wll copare nventory cost f the products are fnshed earler, and lost sale cost f the products are refused to be process. Due date can be rearraned wth the custoer based on the feasblty of the odel. It clearly asssts the decson aker n dealn wth the custoer edately. The result of ths consderaton s yes or no decson. It wll nvolve a Bnner varable, X, n whch the varable value s 1 f the new order s accepted or the varable value s 0 f the new order s rejected. If lost sale cost of te s denoted as C4,, then total lost sale cost s forulated as follows:
318 Tenku Nuranun et al. TC 4, 1 X C 4, (4) achne M1 L 11 Earlness M2 L 12 s 2 te In Process Batch B 1,1 F 1,2 d F. 1 Producton cost 3 Proble forulatons The varables that are nvolved n ths odel forulaton can be denoted as below: B : Startn te for batch, whch s sequenced n poston on achne F : Fnshn te for batch, whch s sequenced n poston on achne Q : Batch sze whch s sequenced n poston N : The nuber of batches r : Bnner varable for te wthn batch whch s sequenced n poston X : Bnner varable for arrval of new order TC : Total Producton Cost The paraeters that are used n ths proposed odel can be defned as follows: M : The nuber of achnes : The nuber of te n : The nuber of te to be requested on d d : Due date of te t : The processn te for te processed on achne s : The setup te requred before any batch s processed on achne A : Avalablty of achne C1, : Setup cost for achne C2, : Interedate cost for te C3, : Holdn Cost of te There are n jobs of te ( = 1, 2,... ) that are requested to be processed on a staes flow shop, each stae consst of a snle achne, whch s each te wll be delvered on ts due date, d1, d2,..., d. All tes have the sae routn but dfferent n specfcaton, e.. colour, sze, etc., All varant tes wll be processed n batches,
A ult due date batch scheduln odel 319 and setup te s requred before any batch enters a achne. The proble s deternn batch szes and sequences that nze total producton cost sultaneously. Ths odel has accoodated arrval of new orders, so that a varable to check avalablty of achne s needed. The functon of ths varable wll be clearly explaned n proposed alorth. The odel s developed under several assuptons: Holdn cost of raw ateral s nelble. Setup te and setup cost for all batches that are processed on the sae achne s fxed. The arrval of a new job s not pertted to nterrupt the current work. None of the batches that s processed on ore than one achne, and none of achnes that run the process for ore than one batch. The objectve of the developed odel s to nze total producton cost. It conssts of several cost eleents, and we need the trade-off between the cost varables as the result. Thus, the forulaton of total producton cost s: TC N M 1 1 1 s C N r 1, N 1 1 r F B 1 d F, M Q C3, 1 Q C X 2, C 4, Where, B1,1 A1 s1 (6) FN, M d (14) F1,1 B1,1 r1, t,1 Q1 (7) r, 0,1 (15) B 1 ; 1,, N ; 1,, F 1,1 s1 ; 2, N (8) r 1 ; 1, N, 1, t Q ; 2, N F, 1 B1 r,,1, B 1 ax F, 1, 1 A ; 2, M (10) 1,, 1 F t Q ; 2, M F1, B1, r1,, 1,,, 1 (9) r, Q n ; 1,, N B ax 1, F 1, s (12) 1, nteer ; 2,, N ; 2,, M F B t Q ; 2,, N ; 2,, M r 1, 1 N 1, f r, 0 ; 1,, X 1 0, else (16) (17) (18) (11) N (19) (13) Q (20) Constrant (6) and (7) show the bennn and fnshn te of the frst batch on achne 1. Constrant (8) and (9) show the bennn and fnshn te of batch that s sequenced on achne 1. Constrant (10) and (11) show the bennn and fnshn te of the frst batch on achne. Constrant (12) and (13) shows the bennn and fnshn te of batch that s sequenced on achne. Constrant (14) to ake sure that fnshn te of all products s earler or precsely on ther due dates. Constrant (15) s a Bnner varable to deterne the partcular te that (5)
320 Tenku Nuranun et al. s placed n batch where s sequenced on poston. Constrant (17) verfes the total of jobs that are processed n a batch equal wth the order nuber. Constrant (18) s a Bnner varable to deterne the new order, whether accepted or rejected t. Constrant (19) shows that the nuber of batch s equal or ore than the nuber of te. Constrant (20) s avalable to ake sure that the nuber of a batch s ore than 1 and nteer. 4 Proble soluton Ths paper has consdered about the arrval of new orders n the flow shop. An alorth s proposed to et the soluton of ths proble. There are two alorths developed for ths proble. Frstly, we wll ntroduce an alorth to nsert a new order to the exstn schedule. Secondly, we wll show an alorth to splt jobs nto batches. In eneral, we can explan the dea of the dynac flow shop alorth as below: Check the avalablty of achnes to perfor the new order by re-count the nuber of jobs and reann te needed for unfnshed works and deterne the ost possble pont of te perod to ben the arrval order. roup the new work that s dentcal wth the current works and calculate the total nuber of revsed order n a roup. Re-schedule. Further, these are the detals of the proposed alorth to solve the proble: a. Insertn New Order Alorth Step 1. Identfy the current works that are processn on the shop floor. Step 2. Set the fnshn te of current works on all achnes as the avalablty of achne (A). Step 3. Check the reann works that have not been processed yet and roup the wth the slar new orders. Step 4. Splt the roup based on due dates. If a roup has several dfferent due dates, re-roup the nto due dates. Step 5. Set the value of predeterned paraeters that are needed n calculaton. Step 6. Set the nuber of batch equal wth the nuber of te. Step 7. Solve the proble usn the proposed odel. Step 8. Splt the batch usn Sub Alorth for Batchn and Sequencn. Step 9. Use the result as revsed schedule. Step 10. Check the status of all new orders. If entre works are rejected, then back to the ntal schedule. If at least one of the new orders s accepted, then use the revsed schedule. Step 11. Fnsh. b. Sub Alorth for Batchn and Sequencn Step 1. Set N = + 1. Step 2. Solve the proble of batchn and sequencn usn the forulaton. Step 3. Note the result of total producton cost. Step 4. Check feasblty of schedule based on due date of each order. Stop searchn f t s not feasble, then o to Step 7. Otherwse, o to Step 5.
Total Cost A ult due date batch scheduln odel 321 Step 5. Check the nuber of batch. If t has reached the nuber of te, then o to Step 7. Otherwse, o to Step 6. Step 6. Set N = N + 1. Back to Step 2. Step 7. Use the schedule wth nu total producton cost as the result. Step 8. Fnsh. 5 Nuercal Experence There are two knds of products that wll be processed on three staes flow shop, product A and product B. The values of the paraeters as shown n Table 1 and Table 2. Ths stuaton presented deternstc crcustance wthout dsturbances that ay nterrupt the process. The result of ths proble s stated as the ornal schedule, can be seen as shown n Table 3. Table 1 Paraeters for achnes t A B s 1 4 6 2 2 2 10 12 8 3 3 8 10 4 2 C1, Table 2 Paraeters for products Product n d C2, C3, A 5 800 5 1 B 10 1000 8 4 Table 3 The resultn batch for statc envronent N TC N TC 2 56.502 9 48.350 3 52.058 10 48.258 4 50.672 11 48.198 5 49.548 12 48.170 6 48.960 13 48.158 7 48.564 14 48.260 8 48.456 15 48.390 The soluton shows that the relaton between total producton cost and batch sze s convex. Splttn batch nto sall sze wll reduce the total producton cost untl the nu pont s reached and after that the nuber of batch wll ncrease the total producton cost, as shown n F.2. The ornal schedule of ths proble s perfored as shown n Table 4. 58 000 56 000 54 000 52 000 50 000 48 000 46 000 0 5 10 15 20 The Nuber of Batches F. 2 Total producton cost for dfferent batch sze
322 Tenku Nuranun et al. Table 4 The ornal schedule Startn N Q Ite Pont F M B 1 1 A 1 2 6 2 8 18 3 18 26 2 2 A 1 8 16 2 26 46 3 46 62 3 2 A 1 18 26 2 54 74 3 74 90 4 1 B 1 28 34 2 82 94 3 94 104 5 1 B 1 36 42 2 102 114 3 114 124 6 1 B 1 44 50 2 122 134 3 134 144 contnued 7 1 B 1 52 58 2 142 154 3 154 164 8 1 B 1 60 66 2 162 174 3 174 184 9 1 B 1 68 74 2 182 194 3 194 204 10 1 B 1 76 82 2 202 214 3 214 224 11 1 B 1 84 90 2 222 234 3 234 244 12 1 B 1 92 98 2 242 254 3 254 264 13 1 B 1 100 106 2 262 274 3 274 284 For dynac stuaton, suppose that there s an order of eht unts te B arrves at 154 of te perod and needed to delver on 450 of the te perod. The proposed alorth wll be used to solve ths stuaton. Step 1. The current work that are processn on the shop floor at nute 154 s Q7 on the achne-3. Step 2. The avalabltes of achne are A1 = 58, A2 = 154, and A3 = 164. Step 3. The reann work that have not been fnshed to process s te B, slar type wth the new order. Step 4. Due date of the new order s thter than the current work. Then, to dffer t, the new order wll be labelled as te C. Step 5. Set the value of predeterned paraeters that are needed n calculaton. Step 6. Set N = 2 Step 7. The resultn value of total producton cost for N = 2 are showed n Table 5. Step 8. The resultn batch s presented on Table 5. Step 9. The resultn schedule s presented on Table 6. Step 10. Check the status of all new orders. If entre works are rejected, then back to the ntal schedule. If at least one of the new orders s accepted, then use the revsed schedule. Step 11. Fnsh.
Total Cost A ult due date batch scheduln odel 323 The raph to show the convex objectve functon for dynac flow shop envronent as shown n F.3. Table 5 The resultn batch for dynac envronent N TC N TC 2 45.720 9 42.964 3 44.316 10 42.984 4 43.600 11 43.036 5 43.268 12 43.120 6 43.096 13 43.236 7 43.024 14 43.384 8 43.020 46 000 45 500 45 000 44 500 44 000 43 500 43 000 42 500 0 5 10 15 The Nuber of Batches F.3 Total producton cost for dynac envronent Table 6. The revsed schedule Startn N Q Ite Pont F B 1 2 C 1 60 72 2 162 186 3 186 206 2 2 B 1 74 86 2 194 218 3 218 238 3 2 C 1 88 100 2 226 250 3 250 270 4 2 C 1 102 114 2 258 282 3 282 302 5 2 B 1 116 128 2 290 314 3 314 334 6 1 B 1 130 136 2 322 334 3 338 348 7 1 C 1 138 144 2 342 354 3 354 364 8 1 B 1 146 152 2 362 374 3 374 384 9 1 C 1 154 160 2 382 394 3 394 404 6 Conclusons Ths paper addresses a ult due date batch scheduln odel on dynac crcustance for slar steps of the producton process to nze total producton cost. There are several cost eleents that s consdered n the proposed odel, they are setup cost, work-n-process (WIP) cost, nventory cost, and lost sale cost. Nuercal experence shows that the proposed odel and alorth are relable to solve the dynac scheduln proble for flow shop anufacturn. Future research as developent of ths odel and alorth s needed to solve the
324 Tenku Nuranun et al. ore coplex stuaton, e.. nvolvn penalty cost for the late fnshed jobs, consdern another dsturbance stuaton lke unavalablty of achnes, etc. References [1] D. Ouelhadj and S. Petrovc, A Survey of dynac scheduln n anufacturn systes, Journal of Scheduln, 12 (2009), no. 4, 417-431. http://dx.do.or/10.1007/s10951-008-0090-8 [2] M. hola M. Zandeh and A.A. Tabrz, Scheduln hybrd flow shop wth sequence-dependent setup tes and achnes wth rando breakdowns, The Internatonal Journal of Advanced Manufacturn Technoloy 42 (2009), no. 1-2, 189-201. http://dx.do.or/10.1007/s00170-008-1577-3 [3] L. Tan, W. Lu and J. Lu, A neural network odel and alorth for the hybrd flow shop scheduln proble n a dynac envronent, Journal of Intellent Manufacturn 16 (2005), no. 3, 361-370. http://dx.do.or/10.1007/s10845-005-7029-0 [4] S.Q. Lu, H.L. On and K.M. N, Metaheurstcs for nzn the akespan of the dynac shop scheduln proble, Advances n Enneern Software, 36 (2005), no. 3, 199-205. http://dx.do.or/10.1016/j.advensoft.2004.10.002 [5] R. Aoune, Mnzn the akespan for the flow shop scheduln proble wth avalablty constrants, European Journal of Operatonal Research, 153 (2004), no. 3, 534-543. http://dx.do.or/10.1016/s0377-2217(03)00261-3 [6]. uo, B. Wu and S. Yan, A job-nserton heurstc for nzn the ean flowte n dynac flowshops, Front. Mech. En., 6 (2011), no. 2, 197-202. http://dx.do.or/10.1007/s11465-011-0211-5 [7] E. Lodree, W. Jan and C.M. Klen, A new rule for nzn the nuber of tardy jobs n dynac flow shops, European Journal of Operatonal Research, 159 (2004), no. 1, 258-263. http://dx.do.or/10.1016/s0377-2217(03)00404-1 [8] K. Bulbul, P. Kansky and C. Yano, Flow shop scheduln wth earlness, tardness, and nteredate nventory holdn costs, Naval Research Lostcs (NRL), 51 (2004), 407-445. http://dx.do.or/10.1002/nav.20000 [9] A.H. Hal and H. Ohta, Batch-scheduln probles to nze nventory cost n the shop wth both recevn and delvery just-n-tes, Internatonal Journal of Producton Econocs, 33 (1994), 185-194. http://dx.do.or/10.1016/0925-5273(94)90131-7 Receved: February 12, 2016; Publshed: March 24, 2016