Journal o Economc and Socal Research 5 (2), -2 A Bnary Partcle Swarm Optmzaton Algorthm or Lot Szng Problem M. Fath Taşgetren & Yun-Cha Lang Abstract. Ths paper presents a bnary partcle swarm optmzaton algorthm or the lot szng problem. The problem s to nd order quanttes whch wll mnmze the total orderng and holdng costs o orderng decsons. Test problems are constructed randomly, and solved optmally by Wagner and Whtn Algorthm. Then a bnary partcle swarm optmzaton algorthm and a tradtonal genetc algorthm are coded and used to solve the test problems n order to compare them wth those o optmal solutons by the Wagner and Whtn algorthm. Expermental results show that the bnary partcle swarm optmzaton algorthm s capable o ndng optmal results n almost all cases. Key Words: Partcle swarm optmzaton; Lot szng; Genetc algorthm; Evolutonary Algorthms. Jel Codes: C6, C6.. Introducton The lot-szng problem attracted attenton because o ts mpact on the nventory levels and, hence the nventory holdng cost and the setup/orderng cost. It s bascally concerned wth ndng order quanttes mnmzng the total cost o lot szng decsons. Lot quantty mght be ether an amount o purchase or producton dependng on the problem doman on hand n order to Management Department, Fath Unversty, 345 Buyucemece, Istanbul, Turey. Emal: tasgetren@ath.edu.tr Department o Industral Engneerng and Management, Yuan Ze Unversty No 35 Yuan-Tung Road, Chung-L, Taoyuan County, Tawan 32, R.O.C.
2 M. Fath Taşgetren & Yun-Cha Lang meet the net requrements o the customer demand. In lot szng problems, tme horzon s dened as gven tme bucets n whch quantty decsons are generally gven at the begnnng o each tme bucet. Lot szng decsons are made n such a way that all customer requrements are met at the end o the tme horzon. In general, lot quanttes are determned as the total requrement or a number o perods n whch the total cost s mnmzed. It balances the tradeo between the orderng and the holdng costs. In other words, t depends on the requrement n the current perod plus the requrements or the uture perods. So the order quantty s determned by groupng the net requrements or a number o perods ahead. There exst derent technques to determne the lot quanttes. Most popular one s the lot-or-lot where whatever needed s ordered. One other strategy s to order a xed order quantty, whch s common n ndustry, at each perod regardless o any varaton n the demand requrements. Another technque s to cover the net requrements or a number o uture perods, called xed perods. In addton, t s also possble to combne the derent strateges together. Several actors should be consdered when lot-szng decsons are gven. These actors are orderng cost, holdng cost, shortage cost, capacty constrants, mnmum order quantty, maxmum order quantty, quantty dscounts and so on. Combnaton o these actors results n derent models to analyze and derent soluton procedures are used dependng on the model employed. The model and ts soluton procedure can be made complcated by consderng these actors n the models, whch can be classed as capactated or uncapactated, sngle-level or mult-level, sngle-tem or mult-tem models. In ths paper, we consder the uncapactated, sngle-tem, no shortages-allowed and sngle-level lot szng model and solve t by a bnary partcle swarm optmzaton (PSO) algorthm. The rest o the paper s organzed as ollows: The next secton ormulates the problem, ollowed by a lterature revew. Then the bnary PSO algorthm appled to lot szng problem s gven along wth a tradtonal genetc algorthm, ollowed by some expermental results. Fnally, concluson and uture wor s presented.
A Bnary Partcle Swarm Optmzaton Algorthm or Lot Szng Problem 3 2. Problem Formulaton The mathematcal ormulaton o the lot szng model consdered n ths paper s gven as ollows: ( Ax ) n mn = ci + subject to : I = (2) I + x Q I = R I (4) Q (5) x {, } (6) () (3) Where n =number o perods A =orderng/setup cost per perod c =holdng cost per unt per perod R =net requrement or perod Q =Order quantty or perod I =projected nventory balance or perod x = an order s placed n perod, = otherwse. x In the objectve uncton (), a penalty A s charged or each order placed along wth a penalty c or each unt carred n nventory over the next perod. Equaton (2) guarantes that no ntal nventory s avalable. Equaton (3) s the nventory balance equaton n whch the order quantty covers all the requrements untl the next order. Equaton (4) satses the condton that no shortages are allowed. And nally, equaton (5) shows the decson varable to be ether (place an order) or (do not place an order). It x should be noted that ntal nventory s zero, I =, such that x = by equaton (3) R >. Because o the mnmzaton nature o the problem, Q
4 M. Fath Taşgetren & Yun-Cha Lang the endng nventory at each perod s mnmzed to avod the penalty charge c, partcularly I n =. 3. Lterature Revew Derent soluton procedures are avalable to determne the lot quanttes. The most popular one s the economc order quantty, EOQ (Mennell, 96). Theoretcally, EOQ mnmzes the orderng and holdng cost, but assumes that the requrements are constant or statonary rom perod to perod. For the case o dynamc requrements n whch the requrements are sgncantly varable over the perods, Slver and Meal (973) presented a heurstc, so called Slver-Meal heurstc. The heurstc tres to mnmze the orderng and holdng costs per unt o tme. Other heurstcs are common n most producton text boos such as Least Unt Cost, LUC and Part Perod Balancng, P. See Spper and Buln (997) or detals. Wagner and Whtn, WW (958) provded a dynamc programmng to solve the problem optmally. 4. Bnary PSO Algorthm For Lot Szng Problem Partcle Swarm Optmzaton (PSO) s one o the evolutonary optmzaton methods nspred by nature whch nclude evolutonary strategy (ES), evolutonary programmng (EP), genetc algorthm (GA), and genetc programmng (GP). PSO s dstnctly derent rom other evolutonary-type methods n that t does not use the lterng operaton (such as crossover and/or mutaton) and the members o the entre populaton are mantaned through the search procedure. In PSO algorthm, each member s called partcle, and each partcle les around n the mult-dmensonal search space wth a velocty, whch s constantly updated by the partcle s own experence and the experence o the partcle s neghbors. Snce PSO was rst ntroduced by Kennedy and Eberhart (995, 2), t has been successully appled to optmze varous contnuous nonlnear unctons. Although the applcatons o PSO on combnatoral optmzaton problems are stll lmted, PSO has ts mert n the smple concept and economc computatonal cost. The man dea behnd the development o PSO s the socal sharng o normaton among ndvduals o a populaton. In PSO algorthms, search s
A Bnary Partcle Swarm Optmzaton Algorthm or Lot Szng Problem 5 conducted by usng a populaton o partcles, correspondng to ndvduals as n the case o evolutonary algorthms. Unle GA, there s no operator o natural evoluton whch s used to generate new solutons or uture generaton. Instead, PSO s based on the exchange o normaton between ndvduals, so called partcles, o the populaton, so called swarm. Each partcle adjusts ts own poston towards ts prevous experence and towards the best prevous poston obtaned n the swarm. Memorzng ts best own poston establshes the partcle s experence mplyng a local search along wth global search emergng rom the neghborng experence or the experence o the whole swarm. Two varants o the PSO algorthm were developed, one wth a global neghborhood, and other one wth a local neghborhood. Accordng to the global neghborhood, each partcle moves towards ts best prevous poston and towards the best partcle n the whole swarm, called gbest model. On the other hand, accordng to the local varant, called lbest model, each partcle moves towards ts best prevous poston and towards the best partcle n ts restrcted neghborhood (Kennedy, 2). Kennedy and Eberhart (997) also developed the dscrete bnary verson o the PSO. PSO has been successully appled to a wde range o applcatons such as power and voltage control (Abdo, 22), mass-sprng system (Brandstatter & Baumgartner, 22), and tas assgnment (Salman et al., 23). The comprehensve survey o the PSO algorthms and applcatons can be ound n Kennedy et al. (2). In ths paper, we use the global varant wth bnary verson appled to the smple lot szng problem. A. PSO algorthm Pseudo code o the general PSO s gven n Fgure. In a PSO algorthm, populaton s ntated randomly wth partcles and evaluated to compute tnesses together wth ndng the partcle best (best value o each ndvdual so ar) and global best (best partcle n the whole swarm). Intally, each ndvdual wth ts dmensons and tness value s assgned to ts partcle best. The best ndvdual among partcle best populaton, wth ts dmenson and tness value s, on the other hand, assgned to the global best. Then a loop starts to converge to an optmum soluton. In the loop, partcle and global bests are determned to update the velocty rst. Then the current poston o each partcle s updated wth the current velocty. Evaluaton s agan perormed to compute the tness o the partcles n the swarm. Ths loop s termnated wth a stoppng crteron predetermned n advance.
6 M. Fath Taşgetren & Yun-Cha Lang Intalze parameters Intalze populaton Evaluate Do{ Fnd partclebest Fnd globalbest Update velocty Update poston Evaluate }Whle (Termnaton) Fgure. Smple PSO algorthm The basc elements o PSO algorthm s summarzed as ollows: Partcle: s a canddate soluton n swarm at teraton. The partcle o the swarm s represented by a d-dmensonal vector and can be dened as = x, x,..., x, where x s are the optmzed parameters and x d 2 d s the poston o the th partcle wth respect to d th dmenson. In other words, t s the value d th optmzed parameter n the th canddate soluton. Populaton: pop s the set o n partcles n the swarm at teraton,.e. pop =,,..., 2 n Partcle velocty: V s the velocty o partcle at teraton. It can be descrbed as = V v, where s the velocty wth respect to v, v,..., v 2 d d d th dmenson. Partcle best: s the best value o the partcle obtaned untl teraton. The best poston assocated wth the best tness value o the partcle obtaned so ar s called partcle best and dened as = pb pb pb wth the tness uncton,,..., 2 d Global best: GB s the best poston among all partcles n the swarm, whch s acheved so ar and can be expressed as GB = gb, gb,..., gb wth the tness uncton GB 2 th d
A Bnary Partcle Swarm Optmzaton Algorthm or Lot Szng Problem 7 Termnaton crteron: t s a condton that the search process wll be termnated. In ths study, search s termnated when the number o teraton reaches a predetermned value, called maxmum number o teraton. B. Soluton representaton Soluton representaton o partcle,, or the bnary PSO s gven n Fgure 2. Ths representaton s due to Hernandez and Suer (999). Each partcle has d dmensons reerrng to as the number o perods n the lot szng problem. The dmenson ndcates an order s placed or partcle n perod d at x d teraton. In other words, s a bnary value such that lot szng x x = d d decson s gven, x = otherwse. R denotes the net requrements or the d d perod d. v s the velocty o the partcle n perod d at teraton. Q s d the lot sze o the partcle n perod d at teraton and s the nventory balance o partcle n perod d at teraton. Then the calculaton o the cost o the orderng plan s trval as shown n Fgure 2 assumng the c=$ per unt per perod and A=$ per order. d 2 3 4 5 6 R d 6 4 5 8 7 x d v d 3.8 2.9 3. -.7 -.2 3. d d d d Q 6 9 5 I 6 5 7 ci 6 5 7 Ax I d ( ) C 6 5 7 48 Fgure 2. Representaton o the Soluton d
8 M. Fath Taşgetren & Yun-Cha Lang C. Intal Populaton A populaton o partcles s constructed randomly or the bnary PSO algorthm or lot szng problem. The values o dmensons are establshed randomly. For each dmenson o a partcle, a bnary value o or s assgned wth a probablty o.5. In partcular, else U (,) x d >.5, = then x d = Velocty values are restrcted to some mnmum and maxmum values, namely = V, V = 4, where V = V. The velocty o partcle n the d [ ] [ 4 V mn max ] mn v d max th dmenson s establshed by V + ( V V )* rand () =. Ths lmt enhances the local search exploraton o the problem space. Populaton sze s twce the number o dmensons. As the ormulaton o the lot szng problem suggests, the objectve s to mnmze the total orderng and holdng costs, the tness uncton value or the partcle s gven as ollows: d = Ax j = j + ci j. mn max mn D. Fndng new solutons Snce the bnary verson o the PSO algorthm s employed n ths study, we need to use two useul unctons or generatng new solutons, namely a sgmod uncton to orce the real values between and, and a pece-wse lnear uncton to orce velocty values to be nsde the maxmum and mnmum allowable values. So whenever a velocty value s computed, the V,, s ollowng pece-wse uncton, whose range s closed nterval [ ] used to restrct them to mnmum and maxmum value. V max, h v = v, d d V, mn v > V d max v d Vmax v < V d mn mn V max Ater applyng the pece-wse lnear uncton, the ollowng sgmod uncton s used to scale the veloctes between and, whch s then used or convertng them to the bnary values. That s,
A Bnary Partcle Swarm Optmzaton Algorthm or Lot Szng Problem 9 sgmod v = d v + e d. So, new solutons are ound by updatng the velocty and dmenson respectvely. Frst, we compute the change n the velocty v d such that v = c r + pb x c r gb x d d d 2 2 d d Then we update the velocty v d such that v = h v + v. d d d by usng the pece-wse lnear uncton Fnally we update the dmenson d o the partcle such that x d, =, (,) U < sgmod v otherwse d. The complete computatonal low o the bnary PSO algorthm s gven below: Step : Intalzaton Set =, n=twce the number o dmensons Generate n partcles randomly as explaned beore, {,,2,..., n} =,,..., x x x 2 d. =, where Generate the ntal veloctes o all partcles randomly, { V,,2,..., n} =, where V = v, v,..., v. v s generated randomly wth v d = V + mn 2 d ( V V )* rand () max mn d Evaluate each partcle n the swarm usng the objectve uncton,. For each partcle n the swarm, set =, where = =, =,..., = pb x pb x pb x 2 2 d d = n,,2,..., along wth ts best tness value, gbest Set the global best to, GB = mn, =,2,..., n wth [ gb, gb gb ] GB =,..., 2 d
M. Fath Taşgetren & Yun-Cha Lang Step 2: Update teraton counter = + Step 3: Update velocty by usng the pece-wse lnear uncton v = c r pb x + c r gb x d d d 2 2 d v d = h v + v d d c and c 2 are socal and cogntve parameters and r and r 2 are unorm random numbers between (,) Step 4: Update dmenson (poston) by usng the sgmod uncton x d, =, (,) U < sgmod v d otherwse Step 5: Update partcle best Each partcle s evaluated agan wth respect to ts updated poston to see partcle best wll change. That s, I then else <, =,2,..., n =, =,2,..., n =, =,2,..., n Step 6: Update global best, =,2,..., n, =,2,..., n, =,2,..., n gbest gbest else GB = mn lbest =,,2,..., n < gbest GB GB, then gbest GB = GB = gbest GB GB gbest gbest Step 7: Stoppng crteron
A Bnary Partcle Swarm Optmzaton Algorthm or Lot Szng Problem I the number o teraton exceeds the maxmum number teraton, then stop, otherwse go to step 2. 5. Numercal Example In order to llustrate how the bnary PSO algorthm solves the lot szng problem, the ollowng problem wth d=5 perods, c=$, A=$ s constructed as well as the net requrements gven n Fgure 2. The complete computatonal low or llustraton purpose s gven below: Step : Intalzaton =, and n=3 The ntal partcles n the swarm generated randomly are,, and,, and correspondng veloctes V, V, and, V 2 3, whch are gven as ollows: d 2 3 4 5 2 3 V 2 V 2 3 V 3 x d v 2.9.6 -.8 3.5 -.6 d x 2 d v 3.8 2.9 3. -.7 -.2 2 d x 3 d v -3..5-2.7 2..4 3 d Detals o the computaton o the tness uncton or the partcles are gven n Fgure 2. Now consderng the rst 5 perods o Fgure 2, the tness values o the partcles n the swarm are: = 58, = 47, 2 2 = 44 3 3 For each partcle n the swarm, set the partcle best to: = pb = p, pb = x,..., pb = x 2 2 d d wth = = 58
2 M. Fath Taşgetren & Yun-Cha Lang = pb = x, pb = x,..., pb = x 2 2 2 22 22 2d 2d 2 wth = = 47 2 2 = pb = x, pb = x,..., pb = x, 3 3 3 32 32 3d 3d wth 44 3 = 3 3 =, so d 2 d 3 d Set the global best to: d 2 3 4 5 ( ) pb 58 pb 2 47 pb 3 44 gbest GB = mn wth GB [ gb, gb,..., gb ] = = 44 3 3 =. So 2 d d 2 3 4 5 gbest (GB ) GB gb d 44 Step 2: Update the teraton number = + = Step 3: Update velocty by usng the pece-wse lnear uncton. Assume c = c = r = r =.5. As an example or partcle, second dmenson s updated as ollows: v 2 v 2 2 v 2 =.5 *.5 pb 2 =.5 *.5 2 ( ) +.5 *.5( ) ( ) +.5 *.5( ) ( + ) = h(.6 + (.25 )) =. 35 = h v 2 v 2 x 2 gb 2 =.25 x 2
A Bnary Partcle Swarm Optmzaton Algorthm or Lot Szng Problem 3 Step 4: Update poston by usng the sgmod uncton Snce U x 2 = (,) =.99 < sgmod ( =.35 ) v 2 =.79 Ater completng all the calculatons or velocty and dmensons we have the ollowng partcles updated n the rst teraton: d V d 2 2d V 2 2d 3 3d V 3 3d d 2 3 4 5 x v 2.9.35 -.8 3.75 -.6 sg v d.95.79.4.98.7 u (,).2.99.9.6.34 x v 3. 2.7 2.25 -.27 -.9.96.9.9.43.29 sg v 2d u (,).89.99.2.2.63 x v -3..5-2.7 2..9 sg v 3d.5.82.6.88.7 u (,).2.98.9.34.49 Step 5: Update partcle best Each partcle updated s evaluated or rst ts new postons and then or ndng the partcle best as ollows: = 42, = 44, 2 2 = 44 3 3 Snce = 42 < = = 58, = = 42 wth =.
4 M. Fath Taşgetren & Yun-Cha Lang Snce = 44 < = = 47, = = 44 wth 2 = 2 2. 2 2 2 2 2 Snce = 44 = = = 44, = = 44 wth = 3 3 3. 3 3 3 3 3 Note that neutral moves are allowed. Ater completng the smlar comparsons or the other partcles, we have 2 d 3 Step 6: Update global best 2 3 d 2 3 4 5 ( ) pb d 42 pb 2 44 pb 3 d 44 gbest GB mn,, = 2 2 3 3 = = 42 3 3 gbest sn, gbest ce GB = 42 < GB = 44, then gbest gbest GB = GB = 42 wth GB = { gb, gb,..., gb } 2 d So, the global best s: Step7: Stoppng crteron else stop d 2 3 4 5 gbest (GB ) GB gb d 42 < max teraton, goto step2 2 3 2 3
A Bnary Partcle Swarm Optmzaton Algorthm or Lot Szng Problem 5 6. Expermental Results The bnary PSO algorthm presented or the uncapactated lot szng problem s coded n C and run on an Intel P4 2.6 GHz, 256 machne. The perormance o the bnary PSO s measured wth a tradtonal GA and the optmal Wagner and Whtn algorthm. For ths purpose, a tradtonal GA s coded n C and test problems are generated randomly or expermentaton. The GA s a tradtonal one wth a unorm crossover, smple nverson mutaton and a tournament selecton o sze 2. We used the ollowng parameters or the bnary PSO and the tradtonal GA. For the bnary PSO, the sze o the populaton n the swarm s taen as the twce the number o perods. Socal and cogntve parameters are taen as c = c2 = 2 consstent wth the lterature. For the GA, the populaton sze s the same as the bnary PSO. The crossover and the mutaton rates are.7 and. respectvely. Both GA and PSO algorthms are run or generatons/teratons. Frst test sut consstng o problem nstances wth net requrements or 5 perods s generated rom a unorm dstrbuton, UNIF(5,25), and the second test sut consstng o problem nstances wth net requrements or 5 perods s generated rom a unorm dstrbuton, UNIF(,25). The total 2 problem nstances are run or both GA and the bnary PSO wth holdng cost o c=$.5 and orderng cost o A=$ n order to compare the results wth those optmals by the Wagner-Whtn algorthm. For each problem nstance, replcatons are conducted. Mnmum, maxmum, average, and standard devaton are gven together wth the CPU tmes. As can be seen rom Table and 2, the bnary PSO produced comparable results wth GA, and t even produced better results. The GA was able to nd the 4 optmal solutons out o 2 whle the bnary PSO was able nd the 9 optmal solutons out o 2 even though the average standard devaton o the GA over 2 replcatons was slghtly better than the bnary PSO,.e., σ GA = 6.48, σ PSO = 6.94. In terms o the computatonal tme, the GA too approxmately seconds or each nstance whle PSO too 6 seconds, whch s computatonally expensve than GA. But PSO s good perormance on ndng optmal solutons more than GA compensates ts computatonal necency.
6 M. Fath Taşgetren & Yun-Cha Lang 7. Conclusons and Future Wors We do realze that the problem presented here can be solved to optmalty by Wagner and Whtn (958) algorthm. Most research on the PSO concentrates on the contnuous optmzaton problems. The objectve o ths paper s to present the potental power o the bnary PSO to solve the bnary/dscrete optmzaton problems, whch has a varety o applcatons n the real world. Durng the past several years, the PSO has been successully appled to the contnuous optmzaton problems. To the best o our nowledge, ths s the rst reported applcaton o the bnary PSO appled to the lot szng problem. The results are encouragng to apply the bnary PSO to the combnatoral optmzaton problems ormulated as (, ) nteger programmng. In addton, we hope that ths wor leads to the applcaton o the PSO to the much more complex, NP-hard lot szng problems. The advantages o the PSO are very ew parameters to deal wth and the large number o processng elements, so called dmensons, whch enable to ly around the soluton space eectvely. On the other hand, t converges to a soluton very qucly whch should be careully dealth wth when usng t or combnatoral optmzaton problems. As a uture wor, the bnary PSO can be appled to the capactated, mult-tem, mult-level lot szng problems.
A Bnary Partcle Swarm Optmzaton Algorthm or Lot Szng Problem 7 Table. PSO Results WW PSO P Opt Best Max Avg Std P 42. 42. 4223.5 426. 6.45 P2 4354. 4354. 437.5 4357.55 5.39 P3 42. 42. 42. 42.. P4 478. 478. 478. 478.. P5 422.5 422.5 432.5 423.5 3.6 P6 4249. 4249. 4259. 425. 4.22 P7 469. 469.5 493. 47.85 7.43 P8 43. 43. 436. 43.5.58 P9 4433. 4433. 4435. 4433.2.63 P 424. 424. 448. 43.8 7. P 45. 45. 4527. 455.4 5.89 P2 4494. 4494. 453.5 452.65 3.33 P3 4424. 4424. 444. 4427.3 5.4 P4 448.5 448.5 454.5 453.8 23.3 P5 445. 445. 4492.5 4465.35 2.6 P6 4493. 4493. 45.5 45.75 8.32 P7 4353.5 4353.5 4354.5 4354..47 P8 442. 442. 4489. 4444.6 25.48 P9 445. 445. 4438. 442.8 9.6 P2 4397. 4397. 444.5 44. 6.6
8 M. Fath Taşgetren & Yun-Cha Lang Table 2. GA Results WW GA P Opt Best Max Avg Std P 42. 42. 427. 4228.8 7.7 P2 4354. 4354. 437.5 4359.8 6.78 P3 42. 42. 4.5 43.7 3.58 P4 478. 478. 42.5 489.7 2.28 P5 422.5 422.5 432.5 425.5 4.83 P6 4249. 4249. 4293. 4257.8 4.67 P7 469. 469.5 493. 47.85 7.43 P8 43. 43. 436. 432.5 2.42 P9 4433. 4433. 4438.5 4433.95.8 P 424. 424. 456. 437.75.82 P 45. 452. 454. 452.75.47 P2 4494. 4495.5 454.5 45.35 2.36 P3 4424. 4424. 4474. 4447.3 7.64 P4 448.5 4483. 45.5 4494.85 8.72 P5 445. 445. 4485. 446..26 P6 4493. 4498. 4559.5 452.25 23.52 P7 4353.5 4353.5 446.5 4373.95 9.82 P8 442. 4429. 4486. 445.85 22.5 P9 445. 445. 446.5 4428.75 8.33 P2 4397. 4397. 442.5 449.7.24
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