Optimized ready mixed concrete truck scheduling for uncertain factors using bee algorithm

Transcription

1 Songklanakarn J. Sc. Technol. 37 (2), , Mar.-Apr Orgnal Artcle Optmzed ready mxed concrete truck schedulng for uncertan factors usng bee algorthm Nuntana Mayteekreangkra and Wuthcha Wongthatsanekorn* Department of Industral Engneerng, Faculty of Engneerng, Thammasat Unversty, Khlong Luang, Pathum Than, Thaland. Receved: 5 June 2014; Accepted: 26 February 2015 Abstract Ths research proposes a systematc model by usng bee algorthm to optmze ready mxed concrete truck schedulng problem from a sngle plant to multple szed recevers n a large search space usng uncertan factors of bee algorthm compared to genetc algorthm. The obectve s to mnmze the total watng duratons of RMC trucks. Four benchmark problems wth 3, 5, 9 and 12 constructon stes are evaluated. Furthermore, eght addtonal problems are created from the prevous four problems by varyng demands and travelng duratons, n order to prove the algorthm accuracy and effcency. Hence, a total of 12 problems would be solved usng both BA and GA. The smulaton results show that the BA approach can get lower total watng duratons and faster than GA for all problems. Ths research offers a more effcent alternatve for solvng RMC truck schedulng. Keywords: bee algorthm, genetc algorthm, ready mxed concrete truck schedulng, a large search space, an uncertan factors 1. Introducton In ths study, the focus s on how to delver Ready Mxed Concrete (RMC) to the customers constructon stes from the supplers or the batch plants effectvely. RMC s mxed accordng to customer s mxture recpe and t s ready to use once t s delvered at the constructon ste. Its usage has been expandng over the past several years because t s fast to soldfy and ts qualty s better than manually mxed concrete. There are many RMC manufacturers n the market and the materal cost s not much dfferent. Therefore, each manufacturer competes wth each other on the customer servce satsfacton. Customers are lookng for the vendor that can delver RMC accordng to ther requrements such as on-tme delvery. One maor constrant of RMC delvery problem s that RMC must be delvered to the constructon ste wthn certan tme wndow after producton. Ths s * Correspondng author. Emal address: [email protected] because RMC cannot be pre-manufactured and stored as an nventory at the plant due to ts quck soldfyng nature; RMC delvery problem s qute complex. Usually, the planner solves RMC delvery problem based on experence and ths can cause dssatsfacton from the customer f the delvery s late. Snce RMC delvery problem s qute complex, t draws nterests from many researchers. For example, Feng et al. (2000) generated problems and bult a systematc model to solve RMC schedulng problems usng genetc algorthms (GA) to mnmze the total wat duraton of RMC trucks. They then developed the RMC Dspatchng Schedule Optmzer program (2004). Lu and Lam (2005) also proposed optmzed concrete delvery schedulng usng GA. Graham et al. (2006) presented a neural network to solve RMC problems. Naso et al. (2007) used a hybrd GA to optmze schedules for ust-n-tme producton and delvery problem. Yan et al. (2008) presented a network flow model for an RMC carrer and employed a tme-space network technque. They then developed a soluton algorthm to mprove RMC system operatng (Yan et al., 2011; 2012). Srchandum and Rurayanyong (2010) developed Feng s research usng bee colony

2 222 N. Mayteekreangkra & W. Wongthatsanekorn / Songklanakarn J. Sc. Technol. 37 (2), , 2015 optmzaton compared to GA and tabu search focusng on sngle plant delvery to 3, 5 and 9 stes for three concrete types. Schmd et al. (2010) used a hybrd soluton by ntegratng an nteger mult-commodty flow optmzaton and a varable neghborhood search. Sh and Wang (2012) proposed a schedulng model for RMC dspatchng usng GA. Zhang and Zeng (2013) presented a formulaton for RMC schedulng problems wth dependent travel tmes usng the local search algorthm. Su (2013) presented the fuzzy multobectve lnear program to analyze the cost-effectveness of vehcles. Hanf and Holvoet (2014) solved dynamc schedulng of RMC delvery problems usng delegate MAS that s a bo-nspred coordnaton machansm for optmzaton. Knable et al. (2014) proposed a mxed nteger programmng and a constrant programmng model to fnd effcent routes concernng concrete delvery problems. Recently, modern heurstc optmzaton has been pad much attenton by many researchers. Bee Algorthm (BA) s a typcal meta-heurstc optmzaton whch provdes a search process based upon ntellgent behavors of honey bees (Pham et al., 2005; 2006). It performs a type of neghborhood search combned wth random searches whch can effcently explore and explot nformaton from the mechansm tself. Ths research proposes an effectve optmzaton for solvng RMC schedulng problems from a sngle plant to multple szed recevers n a large search space wth uncertan factors usng BA compared to GA (Feng et al., 2004). Matheekrangkra and Wongthatsanekorn (2014) only attempted to solve the same problem for 3 and 5 constructon stes by BA. The obectve s to mnmze the total watng duratons of RMC trucks. Four benchmark problems wth 3, 5, 9 and 12 constructon stes are evaluated. Furthermore, eght addtonal problems are created from the prevous four problems by varyng demands and travelng duratons, n order to prove the algorthm accuracy and effcency. Hence, a total of 12 problems have solved by BA and by GA and the results are compared n terms of qualty solutons, algorthm effcences and accuracy. 2. RMC Truck Schedulng Problem Formulaton 2.1 Systematc model There are fve sub-processes for RMC supply process whch are materal praparaton, RMC mxng, qualty nspecton, delvery RMC and return to the batch plant. The same sub processes are terated agan for each RMC delvery to fulfll a customer s order. There are four parts to the systematc model, whch are nput parameters, decson varables, constrants and system output. Input parameters: These parameters are travelng tme, castng tme, mxng tme, and allowable buffer tme and requred number of RMC delveres. Decson varables: These decsons are dspatchng sequences of each RMC truck to dfferent constructon stes. Constrants: The watng tme for the arrvals of the RMC truck at the constructon stes must be less than the allowable buffer tme. In addton, the RMC truck capacty and number of trucks are lmted. System output: The solutons are total watng tmes of RMC trucks at constructon stes and RMC trucks dspatchng sequence. 2.2 Soluton structure The soluton structure s desgned so that all permutatons can be represented and evaluated. Frst, the length of the soluton s defned as total number of RMC trucks that wll be dspatched. For example, f there are three constructon stes that requre three, four and fve RMC trucks n the same nterval of tme perod, the length of the soluton would be twelve. Second, an array of random numbers s used to avod nfeasble solutons generated wthn the evoluton process. Fgure 1 shows the process of decodng a soluton wth random array. Ths soluton represents the dspatchng sequence nvolved wth constructon ste numbers 1, 2 and 3, whch requres three, four and fve RMC trucks respectvely. Here, Ste ID denotes each bt, correspondng to each constructon ste. The dspatchng sequence s then determned accordng to each bt s Ste ID and ts correspondng random number n ascendng order. For example, the smallest random number of the bt s 0.03 and the correspondng Ste ID s 2, whch ndcates the sequence startng wth assgnng the RMC truck to the constructon ste 2. Consequently, the dspatchng sequence of the strng s decoded to 2, 3, 2, 2, 1, 3, 3, 3, 1, 1, 2 and 3. The total soluton space of the dspatchng schedules can be determned by Eq. (1). For example, f there are fve constructon stes and each ste requres four delveres, the total soluton space s or ( )!/ (4!4!4!4!4!). Fgure 1. Example of the soluton structure

3 N. Mayteekreangkra & W. Wongthatsanekorn / Songklanakarn J. Sc. Technol. 37 (2), , TS m 1 m 1 k! ( k!) where: TS = The total soluton space k = The requred number of RMC delveres for constructon ste m = The number of constructon stes that request RMC delveres 2.3 Input nformaton RMC truck schedulng s evaluated for 3, 5, 9 and 12 stes n problem 1 4. In addtonal, we generate eght more problems by varyng demand and travellng tme of the orgnal four problems n Table 1. In case A, we ncrease delvery round and n case B we adust the travellng tme between (1) the plant and the constructon ste. Number of RMC truck and allowable buffer duraton are also modfed properly. 2.4 Ftness functon The ftness value of a dspatch schedule s determned by summng the total watng tmes (TWC) that each truck must wat to place concrete at a constructon ste. In addton, the process of castng concrete at a constructon ste could be nterrupted f an RMC truck s delayed longer than the allowable buffer tme. A penalty functon P s used to avod the nterupted schedule by calculatng one nterrupton as tme n mnutes of one day, as defned n Eq. (2). The ftness value F s the total watng tme at a constructon ste (mn) ncludng a penalty for the nterrupton number, as defned n Eq. (3) P (the number of nterruptons) (2) The ftness value F of a dspatched schedule s defned as Eq. (3): F P TWC (3) Table 1. Informaton of 12 problems. Problem Ste SCT CD TDG TDB ABD k Trucks Problem A Problem B Problem A & B TDG TDB TDG TDB ABD 1 8: : : : : : : : : : : : : : : : : k Trucks 1 8: : : : : : &25 7 8: : : : : :

4 224 N. Mayteekreangkra & W. Wongthatsanekorn / Songklanakarn J. Sc. Technol. 37 (2), , 2015 where, TWC s the total tme that RMC trucks wat to cast RMC at the constructon ste. A smple example to determne the ftness value of a dspatch schedule s descrbed n problem 1, as follows; Descrpton of problem 1 It s assumed that the plant owns fve RMC trucks, the concrete mxng tme (MD) s 3 mn per cubc meter, and the maxmum load a RMC truck can bear s 6 m 3. Informaton concernng dspatch operatons s lsted n Table 1. The RMC truck should arrve at constructon ste wthn an allowable buffer duraton (ABD ), and the next truck must arrve at the constructon ste on tme. At the plant, RMC needs some addtonal tme for mxng and loadng the RMC. Then, there s travellng tme to the constructon ste and a RMC truck may ncur watng or nterrupton tmes. Fnally, there s travellng tme back to the batch plant, and so RMC truck schedulng s assocated wth the departure tme from the plant(idt), the arrval tme at the constructon ste (TAC), the leavng tme from the constructon ste (LT), and fnally the arrval tme of the RMC truck upon ts return (TBB), whch are later descrbed n Eqs. (8) - (15). In practce, the dstance (km) from the plant to the constructon ste can be found by usng a Global Postonng System (GPS). In ths study, the average speeds of RMC trucks travelng to the constructon ste and returnng to the plant are assumed to be 20 km hr -1 and 30 km hr -1, respectvely. Therefore, the travelng tme from the plant to the constructon ste (TDG ) can be calculated usng Eq. (4), and the return tme from the constructon ste to the plant (TDB ) can be calculated usng Eqs. (5): TDG D 3 (4) TDB D 2 (5) where, D = The dstance from the plant to the constructon ste (km). Step 1: Determne the best departure tme of each truck from the batch plant. Ths should occur when the RMC truck leaves the plant as soon as the concrete s loaded. Therefore, the departure tme of each truck s determned by Eq. (6), and ths nvolves the departure tme of the frst dspatched RMC truck from the plant to each constructon ste by selectng the truck wth the mnmum leavng tme from the plant. For the frst delvery, each truck s needed to arrve at the constructon ste at start castng tme of the constructon ste (SCT ). Eq. (7) dentfes the deal departure tme of th dspatched RMC truck. The plant n problem 1 s assumed to have fve RMC trucks, so IDT for earlest delvery are determned only for = 1 to 5 and the rest wll be based upon the departure tmes of returned trucks n step 2: m 1 FDT mn SCT TDG (6) IDT FDT MD, 1 IDT MD, 2 ~ N 1 (7) Where, m k th m N k FDT SCT IDT MD 1 = Dspatched order of an RMC truck. = Number of constructon stes that request RMC delveres. = Requred RMC delveres to constructon ste = Dspatch sequence of RMC trucks to each constructon ste. = The total number of RMC delveres to all constructon stes. = Departure tme of the frst dspatched RMC truck. = The start castng tme of the constructon ste = Ideal departure tme of th dspatched RMC truck. = Concrete mxng tme for the th dspatched RMC truck. Step 2: Calculate the departure tmes for the remander of delveres for the returned RMC trucks. Ths s computed based upon departure tmes from the batch plant, arrval tmes at the constructon ste, leavng tmes from the constructon ste, watng tmes for each delvery and the returnng tmes to the batch plant, accordng to Eqs. (8)-(15). The frst departure tme of each RMC truck can be determned usng step 1. However, the number of trucks s lmted. It s possble that the delvery schedule s unfeasble. Therefore, only the departure tme of the frst fve dspatched RMC trucks can be determned by Eqs. (6) and (7). The rest of the delvery tmes can be computed when all trucks have returned from the constructon ste by Eqs. (8)-(15). Samples of RMC dspatch sequences are generated randomly as explaned n a soluton structure whch s llustrated as: [2, 3, 2, 1, 3, 1, 3, 3, 1, 2, 3, 2]. Ths represents a feasble soluton to a dspatch sequence of RMC trucks for 12 delvery tmes to constructon stes, as dsplayed n the smulated results of dspatch sequencng based upon the results n Table 2: SDT IDT, f c (8) l SDT mn[ TBB l MD ], f c N (9) TAC SDT TDG (10) PTF SCT or LT ( k 1 ) (11) WC PTF TAC (12) LT TAC WC CD, f WC 0 (13) LT TAC CD, f WC 0 (14) TBB LT TDB (15) Where,

5 N. Mayteekreangkra & W. Wongthatsanekorn / Songklanakarn J. Sc. Technol. 37 (2), , SDT = Smulated departure tme of the th dspatched truck. TAC = Arrval tme of the th dspatched truck at constructon ste. = Index of the desgnated constructon ste, where 1 m. k = Order of the RMC trucks that arrve at the respectve constructon ste, wheren k 1 k for each constructon ste. l = The order of the trucks returnng to the batch plant. c = The number of RMC trucks stll at the batch plant. PTF = The startng tme of castng at constructon ste. WC 0 = Watng tme of the th dspatched truck at constructon ste. WC 0 = Watng tme for arrval of the th dspatched truck at constructon ste. LT = Leavng tme of th RMC truck at constructon ste. TBB = Returnng tme of th dspatched RMC truck back to the batch plant. Step 3: Determne the ftness value TWC from WC n Table 2, wheren the total watng tmes of RMC trucks wat for castng at constructon ste, summng the postve ntegers of WC, s 84 mn and the total watng tme of the constructon ste wat for truck arrvals, summng the negatve ntegers of WC, s 209 mn. The nterrupton of castng concrete s 4 tmes (marked as *), whch occurs when the watng tme for the arrval of a RMC truck s longer than the allowable buffer tme. Snce the nterrupton of castng concrete should be avoded, a penalty functon s appled accordng to Eq. (2). The nterm ftness value F of a dspatched schedule as defned n Eq. (3) s equal to ( ) mn. As a result, the algorthm re-generates a new feasble soluton and repeats the above steps untl t arrves at the optmal soluton. The trucks n problem 1 can be scheduled as shown n Fgure 2. In the mean tme, other problems are approached by usng the same process. The smulaton results wll be descrbed n the next secton. 3. BA for RMC Truck Schedulng Pham et al. (2005; 2006) frst ntroduced BA to solve optmzaton problem. BA s one of the optmzaton algorthms based on the behavor of honey bees. They use waggle dance to best locate food sources and locate new ones. In a colony of artfcal bees, bees are dvdng nto two groups. They are scout bees and employed bees. Scout bees are responsble for fndng new food sources and they move randomly around the hve. Once they return, those bees that found good food source go to the dance floor and perform the waggle dance. Durng the dance, they share the nformaton and communcate wth the employed bees whch on n the explotaton of the food source. The algorthm for BA can be descrbed as follows: NC = Number of teratons. n s = Number of scout bees whch could be defned as ntal feasble solutons. m B = Number of best selected stes out of n s vsted stes. e = Number of best stes out of m B best selected stes. nep = Number of bees recruted to fnd best e stes. nsp = Number of bees recruted for the other (m B -e) selected stes. ngh = Neghborhood search ratos, whch s the swap tme of solutons, t s requred to be an nteger. The rounded functon n Eq. (16) thus converts nto a real number to the nearest nteger: ngh ngh (16) NC max mn ngh round( ngh NC ) max Ths soluton represents RMC truck schedulng from a sngle plant to dfferent constructon stes. The ftness functon s obtaned by nterrupton tmes and total watng tmes. The procedure of BA, as shown n Fgure 3, can be summarzed as follows: max Table 2. Results of RMC parameter n problem 1. FDT = Mn [08:00-00:30, 08:00-00:25, 08:30-00:40] = 07: IDT 7:30 07:33 07:36 7:39 7:42 7:45 07:48 7:51 7:54 7:57 8:00 8: k SDT 7:30 7:33 7:36 7:39 7:42 8:53 8:57 9:23 9:28 9:53 10:11 10:35 TAC 7:55 8:13 8:01 8:09 8:22 9:23 9:37 10:03 9:58 10:18 10:51 11:00 PTF 8:00 8:30 8:30 8:00 8:55 8:20 9:20 10:02 9:43 9:00 10:28 10:48 WC * -17* * -23* -12 LT 8:30 8:55 9:00 8:29 9:20 9:43 10:02 10:28 10:18 10:48 11:16 11:30 TBB 8:50 9:25 9:20 8:54 9:50 10:08 10:32 10:58 10:43 11:08 11:46 11:50

6 226 N. Mayteekreangkra & W. Wongthatsanekorn / Songklanakarn J. Sc. Technol. 37 (2), , 2015 Fgure 2. The Gantt chart of RMC truck schedules concernng problem 1. Step 1: Randomly generate ntal populatons of n scout bees. Then, set NC = 0 Step 2: Evaluate the ftness value of the ntal populatons, whch s defned by the number of nterruptons and total watng tmes based upon Eqs. (2) and (3). RMC truck schedule could be calculated as per Eqs. (4) - (15). Step 3: Select m B best solutons from step 2 for neghborhood searches n the next step based upon vstng stes. Step 4: Separate the m B best solutons nto two groups, whereby: Group 1 has e best solutons, and group 2 has m B -e best solutons. Step 5: Determne the scope of neghborhood searches for each best soluton (ngh) as shown n Eq.2 for both groups 1 and 2 best solutons. Step 6: Generate new solutons randomly around m B (group 1) and m B -e best soluton (group 2) wthn the scope of the neghborhood searches, as per step 5. Step 7: Evaluate the ftness value of new solutons and thus select the most approprate soluton for each patch. Step 8: Check the stoppng condtons. If satsfed, termnate the search, else NC = NC+1. Step 9: Assgn the n-m B populaton to generate new solutons. Go back to step 2: 4. Smulaton Results Fgure 3. BA process flow (Matheekrangkra and Wongthatsanekorn, 2014) The BA method has been appled to solve RMC truck schedulng havng twelve problems. The results were compared aganst GA, and all methods were performed wth 30 trals for each problem. The feasble soluton to each problem could be calculated as per Eq. (1), and the total solutons spaces n problems 1, 2, 3 and 4 are 27720, 18,918,900, and , respectvely. The software was mplemented usng MatLab languages on an Intel Core2 Duo 1.66 GHz Laptop wth 2 GB RAM under Wndows XP. The BA parameter settng usng tral and error methods was performed for each problem as shown n Table 3. The GA method was mplemented usng the same selecton, crossover and mutaton methods as Feng et al. (2004). The selecton s based on Roulette wheel selecton methods, the crossover s two ponts crossover, and the mutaton uses the selfmutaton technque. The best solutons to the 12 problems after 30 trals are shown n Table 4. Further analyse of an effectveness of BA s all shown n Table 5. Obvously, both methods can fnd optmum solutons wth hgh probabltes, however the percentage of achevng the optmum for BA s hgher than for GA. Hence, the standard devaton of the soluton for GA s hgher than

7 N. Mayteekreangkra & W. Wongthatsanekorn / Songklanakarn J. Sc. Technol. 37 (2), , for BA. In terms of average CPU tme, BA runs faster n all problem szes except for the largest problems. BA could also take more tme to explot some problems, such as 4, 4A and 4B. Hence, t can be concluded that the BA approach outperforms GA n terms of effcency and accuracy, n ths research. In Fgure 4, GA and BA converge to optmal solutons n approxmately 7 and 30 teratons of problem 1; 5 and 987 teratons of problem 2; 100 and 1400 teratons of problem 3, and 12 and 213 of problem 4, respectvely. The results ndcate that BA can converge to optmum solutons faster than GA can n all 4 problems. Surprsngly, problem 3 requres a hger number of teratons than problem 4 to reach the optmal solutons for both GA and BA even though problem 4 has more stes to delver. Accordng to Karaboga and Akay (2009), BA produces new solutons by takng the dfference of randomly determned parts of the parent and choosng a soluton randomly from the populaton whle GA produces new solutons based on the current populaton usng crossover and mutaton operators. Another aspect s about keepng the best soluton n every teraton. For BA, the best soluton could be replaced wth new random soluton found by a scout bee whle the best soluton s always retaned n the populaton. These dfferences explan why BA could speed up the convergence search for local optmum faster than GA. Next, the best soluton of each tral were consdered and plotted, as shown n Fgure 5. Each dot represents the best soluton of each tral run for problems 1, 2, 3 and 4, respectvely. All ftness values could be obtaned usng BA wth dfferent optmal percentages dependent upon problem constrants, such as: avalable RMC trucks, allowable buffer tme, travellng dstance and RMC orders from the customer: Table 3. BA and GA parameter settng. Problem BA GA n s m B e nep nsp Popolaton sze Generaton Crossover Mutaton 1,1 A,1B ,2 A,2B ,3 A,3B ,4 A,4B Table 4. Optmal solutons of 12 problems. Problem Ste ID Total Interupton (Dspatchng Sequence) watng tme tme (mn) (mn) 1 [1, 2, 2, 1, 1, 3, 2, 3, 2, 3, 3, 3] A [2, 2, 1, 1, 3, 3, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 3, 1, 3] B [2, 1, 1, 1, 1, 2, 3, 1, 3, 2, 2, 3, 1, 3, 3, 2, 3, 2, 3] [4, 3, 4, 4, 4, 3, 2, 3, 2, 3, 5, 1, 1, 5] A [5, 1, 5, 2, 1, 4, 4, 2, 1, 4, 3, 2, 5, 3, 4, 5, 3, 4, 3] B [5, 1, 2, 4, 2, 4, 1, 1, 4, 5, 2, 3, 5, 4, 3, 3, 4, 5, 3] [7, 1, 5, 2, 8, 9, 6, 3, 9, 1, 8, 9, 6, 7, 2, 5, 8, 1, 3, 7, 8, 4, 2, 4, 8, 3, , 4, 4, 2, 7, 3, 4] 3A [5, 1, 9, 2, 7, 8, 4, 6, 3, 9, 1, 8, 5, 7, 2, 6, 9, 3, 1, 8, 6, 4, 7, 1, 6, 3, 9, 2, 5, 8, , 3, 1, 4, 9, 2, 6, 5] 3B [5, 1, 4, 8, 2, 9, 7, 4, 1,9, 6, 4, 5, 3, 2, 1,6, 8, 9, 7, 6, 3, 5, 9, 8, 6, 1, 2, 3, 7, 9, 0 0 1, 6, 5, 8, 3, 7, 2] 4 [5, 12, 1, 2, 7, 8, 9, 4, 6, 12, 11, 3, 4, 5, 2, 7, 1, 4, 8, 6, 3, 9, 12, 10, 8, 2, 11, 4 0 1, 3, 7, 12, 8, 9, 10, 11, 2, 3]. 4A [5, 12, 1, 2, 8, 7, 9, 4, 11, 3, 9, 4, 12, 7, 8, 10, 1, 9, 3, 4, 7, 5, 4, 11, 12, 8, 4 0 1, 9, 10, 6, 2, 3, 4, 7, 9, 5, 2, 3, 6, 8, 11, 10, 1, 6, 8, 3, 5, 6, 1, 8, 10, 2, 3, 6] 4B [9, 5, 12, 1, 11, 4, 7, 12, 9, 8, 3, 2, 9, 6, 7, 10, 12, 9, 1, 4, 5, 11, 8, 3, 6, 10, 7, 3, 9, 4, 2, 0 0 6, 1, 8, 3, 11, 5, 10, 2, 7, 4, 9, 3, 6, 1, 10, 8, 2, 4, 1, 8, 6, 5, 3]

8 228 N. Mayteekreangkra & W. Wongthatsanekorn / Songklanakarn J. Sc. Technol. 37 (2), , 2015 Table 5. Summary results of 12 problems. Problem Algorthm Max Avg Mn S.D. Avg CPU tme %Optmum (mn) (mn) (mn) 1 BA GA A BA GA B BA GA BA GA A BA GA B BA GA BA GA A BA GA B BA GA BA GA A BA GA B BA GA Fgure 4. A comparson of BA and GA convergence curves (Matheekrangkra and Wongthatsanekorn, 2014 for a. and b.)

9 N. Mayteekreangkra & W. Wongthatsanekorn / Songklanakarn J. Sc. Technol. 37 (2), , Fgure 4. Contnued Fgure 5. Best Soluton of each tral by GA and BA (Matheekrangkra and Wongthatsanekorn, 2014 for a. and b.) 5. Conclusons and Future Works Ths research proposes BA for solvng RMC truck schedulng problems for a large search space and uncertan factors whch are an NP-hard problem. The BA concept s to perform a neghborhood search combned wth random searches. Ths technque helps to explore and explot search spaces and acheves optmal effcency. In addton, GA s a search approach based upon natural selecton and genetc recombnaton. The algorthm works by choosng solutons from the current populaton and then applyng genetc operators such as mutatons and crossovers to mprove random solutons that can be changed to the worst solutons or trapped n local loops. The performance of BA was evaluated usng four benchmark and eght addtonal problems. The results show that the BA approach can fnd the optmal soluton better than GA n terms of effcency and accuracy for all 12 problems. Hence, ths research offers a more effcent alternatve for solvng RMC truck schedule. For future research, some companes have more than one plant located n dfferent areas n order to meet ncreasng customer demand, and there are many factors to consder

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