Optimization in Allocating Goods to Shop Shelves Utilizing Genetic Algorithm under the Introduction of Sales Probabilities

Transcription

1 Journal of Communcaton and Computer 2 ( do: / / D DAVID PUBLISHING Optmzaton n Allocatng Goods to Shop Shelves Utlzng Genetc Algorthm under the Introducton of Sales Probabltes Asam Shtara u Hguch 2 Dasue Taeyasu 3 and Kazuhro Taeyasu 4. Ta Corporaton Arnet Shzuoa Cty Japan 2. Faculty of Busness Admnstraton Setsunan Unversty 7-8 Ieda-naamach Neyagawa Osaa Japan 3. Graduate School of Culture and Scence The Open Unversty of Japan 2- Waaba hama-dstrct Chba Cty Japan 4. College of Busness Admnstraton Tooha Unversty 325 Oobuch Fuj Cty Shzuoa Japan Abstract: How to allocate goods n shop shelves maes great nfluence to sales amount. Searchng best ft allocaton of goods to shelves s a nd of combnatoral problem. Ths becomes a problem of nteger programmng and utlzng genetc algorthm may be an effectve method. Revewng past researches there are few researches made on ths. Formerly we have presented a paper concernng optmzaton n allocatng goods to shop shelves utlzng genetc algorthm. In those papers the problem that goods were not allowed to allocate n multple shelves and the problem that goods were allowed to allocate n multple shelves were pursued. In ths paper we eamne the problem that allows goods to be allocated n multple shelves and ntroduce the concept of sales profts and sales probabltes. Optmzaton n allocatng goods to shop shelves s nvestgated. Epanson of shelf s eecuted. Utlzng genetc algorthm optmum soluton s pursued and verfed by a numercal eample. Varous patterns of problems must be eamned hereafter. Key words: Dsplay genetc algorthm optmzaton shelf.. Introducton Dsplayng method n the shop maes nfluence to sales amount therefore varous deas are devsed. What nd of tems should be placed where n the shop how to gude customers to what asle n the shop are the bg ssues to be dscussed. Searchng best ft allocaton of goods to shelves s also an mportant ssue to be solved. In ths paper we see how to optmze n allocatng goods to shop shelves. As for allocatng good to shop shelves followng tems are well nown [-3]. Shelf heght s classfed as follows. Shelf of 35 cm heght: Customers can see the Correspondng author: Kazuhro Taeyasu Ph.D. professor research felds: maretng and optmzaton. E-mal: taeyasu@fj.tooha-u.ac.jp. whole space of the shop. Specalty stores often use ths type. Shelf of 50 cm heght: Female customers may feel pressure to the shelf heght. Ths heght may be the upper lmt to loo over the shop. Shelf of 80 cm heght: It becomes hard to loo over the shop. Therefore t should not be used for sland dsplay (dsplay at the center or nsde the shop. Net we show the followng three functons of shelf for dsplay: ( Ehbton of goods functon (2 Stoc functon (3 Dsplay functon Effectve range for ehbton s generally sad to be 45 cm-50 cm. The range of 75 cm-35 cm s called golden zone especally. For the lower part under 45

2 56 Optmzaton n Allocatng Goods to Shop Shelves Utlzng Genetc Algorthm cm goods are stoced as well as dsplayng. 2. Lterature Revew Revewng past papers there are many papers concernng lay out problem. As for the problem of the dstrbuton of equpment we can see Refs. [4-6] for the general research boo. There are many researches made on ths. Wu and amada et al. [7-9] handle ths problem consderng asle structure. Ito and Irohara [0] consders mult-floor faclty problem. Although there are many researches on correspondng theme as stated above we can hardly fnd researches on the problem of optmzaton n allocatng goods to shop shelves. Formerly we have presented a paper concernng optmzaton n allocatng goods to shop shelves utlzng genetc algorthm []. In those papers the problem that goods were not allowed to allocate n multple shelves and the problem that goods were allowed to allocate n multple shelves were pursued. In ths paper we eamne the problem that allows goods to be allocated n multple shelves and ntroduce the concept of sales profts and sales probabltes. Epanson of shelf s eecuted. Optmzaton n allocatng goods to shop shelves s nvestgated. Utlzng genetc algorthm optmum soluton s pursued and verfed by a numercal eample. The rest of the paper s organzed as follows: Problem descrpton s stated n Secton 3; Genetc Algorthm s developed n Secton 4; Numercal eample s ehbted n Secton 5 whch s followed by the remars of secton 6; Secton 7 s a summary. numercal eample we eamne usng these fve shelves. Frst of all we mae problem descrpton n the case there s only one shelf (case. Then we epand to the case there are multple shelves (case 2. ( Case : The case that there s only one shelf Although there are few cases that there s only one shelf t maes the foundaton for multple shelves case. Therefore we pc t up as a fundamental one. Suppose shelf poston s from to L (Fg. 2. Suppose there are N amount of = N. Set sales proft of goods as goods ( H. Table shows the sales probabltes when each goods s placed at each shelf poston. The values n ths table are wrtten for eample. Suppose goods are sold n the perod from t to t n. In addton a new goods s replenshed when goods s sold out. Set the accumulated sales probablty of goods n tme zone t shelf j and shelf poston n the table as HK t j. j Then the sales probablty K t / t of goods n n the perod wll be descrbed as follows. 50cm Fg. 75cm Shelf model. ~ 35cm 45cm Shelf poston 5 Shelf poston 4 Shelf poston 3 Shelf poston 2 Shelf poston 3. Problem Descrpton L Shelf model s constructed as Fg.. There are fve shelf postons. Shelf poston s manly to put bg and heavy goods ncludng stoc functon. Shelf poston 3 4 at the heght of the range 75 cm to 35 cm are the space of golden zone. Thus we can use shelves properly by assumng these shelves. In Fg. 2 Shelf poston. 3 2

3 Optmzaton n Allocatng Goods to Shop Shelves Utlzng Genetc Algorthm 57 Table Day of the Wee Sales probablty for each goods. Tme Zone (t Shelf j = Shelf j =2 Shelf j = m Shelf poston Shelf poston Shelf poston = = L = = L 2 = = L m (on. 0- (t = (t = (t = (Tue. 0- (t = (t = (t = (Sun. 0- (t = (t = (t = Fg. 3 = L = 3 = 2 Shelf poston under multple shelves. = L 2 = 3 = 2 = = 3 = 2 = = = j = j = 2 j = m L m n j Kt t = n / t= HK t j Ths can tae the value more than. For eample the value 2 means that 2 amount of goods were sold durng the perod. Set Beneft n the sales perod from t to t n as j P t / t ( = N ( j = m ( = L whe n n goods s placed at shelf j and shelf poston. Where beneft means: Beneft = SalesProbablty SalesProft Therefore ths equaton s represented as follows: j j Pt / t = K t t H n / ( n where j = because one shelf case s consdered here. Set as: = : Goods s placed at shelf poston. = 0 : Else Suppose only one goods can be placed at one shelf poston and also suppose that goods s allowed to allocate n multple shelf postons. Then constrants are descrbed as follows: ( = N ( L = 0 N = Under these constrants amze J = = (2 = ( = L (3 L N = = P t j / tn (4 (2 Case 2: The case that there are m shelves Suppose there are m shelves (Fg. 3. Set Beneft as:

4 58 j P t / tn Optmzaton n Allocatng Goods to Shop Shelves Utlzng Genetc Algorthm ( = N ( j m = ( = where goods s placed at shelf poston of shelf j. The sales perod s the same wth above stated (. Set j as: j = : Goods s placed at shelf poston of shelf j j = 0 : Else Suppose only one goods can be placed at one shelf poston and also suppose that goods s allowed to allocate n multple shelf postons. Then constrants are descrbed as follows. The sales perod s the same wth before. 0 ( N j = N j = = ( j m = ( j m Under these constrants 4. Algorthm N m L amze J = = j= = L j = ( L j = (5 = ( j P L j = (6 j t / tn j (7 We can mae problem descrpton as stated above although these are somewhat under restrcted cases. As far as only these are consdered as they are there s lttle dfference between these and the conventonal optmzaton problems. However as soon as the number of nvolved shelves becomes larger the number of varables dramatcally grows greater to whch the applcaton of Genetc Algorthm soluton and Neural Networ solutons may be approprate. There are varous means to solve ths problem. When that varable taes the value of 0 or the applcaton of genetc algorthm would be a good method. As s well nown the calculaton volume reaches numerous or even nfnte amounts n these problems when the number of varables ncreases. It s reported that GA s effectve for these problems [2-4]. A. The Varables Suppose the number of goods shelf poston and shelf are respectvely. In ths paper shelf poston s epanded from 2 to 3. Then the number of varables becomes three hundred. j = 0 ( = 20 ( j = 23 ( = 5 Therefore set chromosome as follows. X = ( L (8 B. Intalze populaton Intalzaton of populaton s eecuted. The number of ntal populaton s. Here set = 00. Set gene at random and choose ndvdual whch satsfes constrants. C. Selecton In ths paper we tae eltsm whle selectng. Choose P ndvduals n the order whch tae mamum score of objectve functon. Here set P = 20 D. Crossover Here we tae unform crossover. Set crossover rate as: P c = 0.7 (9 E. utaton Set mutaton rate as: P m = 0.0 (0 Algorthm of GA s ehbted at Table 2. Introducng the varable y s such that: y s = (

5 Optmzaton n Allocatng Goods to Shop Shelves Utlzng Genetc Algorthm 59 Table 2 Algorthm of mult-step tournament selecton method. Step Set mamum No. as g ma populaton sze as P crossover rate as p c mutaton rate as p m. Step 2 Set = t p p =. t for generaton No. and generate ntal soluton matr p ( ( j J ( p t for all soluton matr ( t p Step 3 Calculate Objectve functon ( Step 4 Set t = t + untl t > g ma. Step 5 Crossover Generate new ndvdual by crossover utlzng the method of above stated D. Step 6 utaton Reproduce by mutaton utlzng the method of above stated E. Step 7 Calculate objectve functon for reproducton of generaton t. Step 8 Selecton Net generaton s selected by eltsm. Go to Step 4. = ( p ( P = n generaton t. where s = + ( j 5 (2 when j = then Eq. (8 s epressed as: = y y L y (3 5. Numercal Eample ( 2 Now we eecute numercal eample usng POS sales data. Numercal eample s eecuted n Case 2 of 3 (2. Suppose the sales perod s 5 days for onday through Frday. Table 3 shows the unt sales proft H of each goods. Supposng a general daytme retal store we set openng tme to be 9 through 8 o cloc. Table 4 shows the sales probabltes of lot as an eample. Table 5 shows the sales probablty by shelf for each shelf poston. Table 6 shows the value n whch Table 4 and Table 5 are multpled. Table 7 shows the beneft Table n whch accumulated probablty of Table 6 and Sales Proft of Table 3 are multpled. In Table 5 shelf j =2 s located near the entrance therefore the table value reflects ths condton. Table 7 shows the beneft when each goods s placed at each shelf poston of each shelf. Epermental results are as follows. The epresson Eq. (8 s complcated. Therefore we use epresson by Eq. (3. A sample set of ntal populaton s 5 ehbted n Table 6. Convergence process s ehbted n Fg. 4. The problem s smple so combnaton of genotype for crossover saturates n the 70th generaton. Genotype n whch objectve functon becomes mamum s as follows: = ( Table 3 Unt sales prce and sales proft of each goods. Lot Sales prce H For Women For en For Kds

6 60 Optmzaton n Allocatng Goods to Shop Shelves Utlzng Genetc Algorthm Table 4 Sales probablty of Lot (Tme Zone. Day of the wee Tme Zone(t Sales probablty Day of the wee Tme Zone(t Sales probablty (on. (Tue. (Wed (Thu (Fr (Sat Table 5 Sales probablty of Lot (Shelf Poston. Shelf j = Shelf j =2 Shelf j =3 Shelf poston Shelf poston Shelf poston = =2 =3 =4 =5 = =2 =3 =4 =5 = =2 =3 =4 = Ths concdes wth the result of optmal soluton by the calculaton of all consderable cases therefore t concdes wth a theoretcal optmal soluton. We tae up smple problem and we can confrm the effectveness of GA approach. Further study for comple problems should be eamned hereafter. 6. Remars As there are few papers made on ths theme we constructed prototype verson before []. In ths paper we eamned the problem that allowed goods to be allocated n multple shelves and ntroduced the concept of sales profts and sales probabltes. An applcaton to the shop wth POS sales data was eecuted. We can see that genetc algorthm s effectve for ths problem. In practce followng themes occur. Sales probabltes should be arranged correctly. There are varous types of shelves correspondng to goods characterstcs (For eample cold storage goods. Furthermore genotype must be devsed n constructon when there are huge number of goods and shelves.

7 Optmzaton n Allocatng Goods to Shop Shelves Utlzng Genetc Algorthm 6 Table 6 Sales probablty of Lot. Day of the wee (on. (Tue. (Sat. Tme Zone(t Shelf j= Shelf j=2 Shelf j=3 Shelf poston Shelf poston Shelf poston = =2 =3 =4 =5 =4 =5 =3 =4 =5 = =2 =3 =4 =5 Table 7 Beneft table. Shelf Shelf 2 Shelf 3 Shelf poston Shelf poston Shelf poston Lot

8 62 Optmzaton n Allocatng Goods to Shop Shelves Utlzng Genetc Algorthm (table 7 contnued Shelf Shelf 2 Shelf 3 Shelf poston Shelf poston Shelf poston Lot Table 6 A sample set of ntal populaton. = ( = ( = ( = ( = ( = ( Fg. 4 Convergence process of case 2. For these ssues epanded verson of the paper wll be bult hereafter consecutvely. As for constrants are relaed than those of ths paper. As for 2 epanson s easy to mae. As for 3 constructng genotype from the shelf sde would bear much more smple epresson.

9 Optmzaton n Allocatng Goods to Shop Shelves Utlzng Genetc Algorthm Concluson How to allocate goods n shop shelves maes great nfluence to sales amount. Searchng best ft allocaton of goods to shelves s a nd of combnatoral problem. Ths becomes a problem of nteger programmng and utlzng genetc algorthm may be an effectve method. Revewng past researches there were few researches made on ths. Formerly we had presented papers concernng optmzaton n allocatng goods to shop shelves utlzng genetc algorthm. In those papers the problem that goods were not allowed to allocate n multple shelves and the problem that goods were allowed to allocate n multple shelves were pursued. In ths paper we eamned the problem that allowed goods to be allocated n multple shelves and ntroduced the concept of sales profts and sales probabltes. Epanson of shelf was eecuted. Optmzaton n allocatng goods to shop shelves was nvestgated. Utlzng genetc algorthm optmum soluton was pursued and verfed by a numercal eample. Varous patterns of problems should be eamned hereafter. References [] Nagashma Shop Arrangement and Dsplayng (n Japanese. Nhon Jtsugyo Press. [2] Atsum S Shop Lay Out (n Japanese. Jtsumu Kyou Press. [3] Kohno H ang Charmng Shop (n Japanese. Paru Press. [4] Korte B. and Vygeu J Combnatoral Optmzaton. Sprnger. [5] Gen. and Cheng R Genetc Algorthms & Engneerng Desgn. JOHN WILE & SONS INC. [6] chalewcz Z Genetc Algorthms + Data Structures = Evoluton Programs. Sprnger. [7] Wu. and Appleton E The Optmzaton of Bloc Layout and Asle Structure by a Genetc Algorthm. Computers & Industral Engneerng 4: [8] amada T. Irohara T. and Fujawa H Desgn ethodology for the Faclty Layout Problem to Consder the Asle Structure and Intra-Department ateral Flow. JIA 55: -20. [9] amada T. and Irohara T Detaled Layout Desgn ethodology Usng ed Integer Programmng and Smulated Annealng. JIA 57: [0] Ito T. and Irohara T ult-floor Faclty Layout Technque Includng Determnaton of Detaled I / O Locaton. JIA 57: [] Taeyasu K. and Hguch Optmzaton n Allocatng Goods to Shop Shelves. The Journal of Economc Studes 54: [2] Aral. and asu Optmzaton of artme Contaner-Transportaton Networ through the Use of Genetc Algorthm. Japan Shp-arne Engneerng Socety 4: [3] Gen. Ida K. and L Solvng ult Objectve Sold Transportaton Problem by Genetc Algorthm. JIA 46: [4] Toyama H. Ida K. and Teramatsu C A Proposal of a Genetc Algorthm for Fed-Charge Transportaton Problem and Related Numercal Eperments. JIA 57: