Heuristic Algorithm for Location-Allocation Problem Based on Wavelet Analysis in Integrated Logistics Distribution

Transcription

1 Heuistic Algoithm fo Location-Allocation Poblem Based on Wavelet Analysis in Integated Logistics Distibution Qian Zhang Business School, HuaQiao Univesity Quanzhou, FuJian, 36202, P.R. China o iangpei Hu Institute of Systems Engineeing, Dalian Univesity of Technology Dalian, Liaoning, 6023, P.R. China Abstact A eliable model fo Location-Allocation (LA) is poposed based on 0- mixed intege pogamming model pesented by Baumol-Wolfe. A heuistic algoithm (HA) based on wavelet analysis is used to solve LA poblem, which incopoates diffeent customes demands into a minimum wap-net. It indicates that customes ae allocated to one facility o one cluste if they satisfy that the distance is neaest to the facility. The model in the method involves the espective distance and demand of the diffeent customes, which can solve eal plana LA. A case study using compute simulation shows that the HA system achieves significant impovement compaed to a ecent LA solution. Keywods: Location-Allocation (LA), wavelet analysis, heuistic seach, minimum wap-net method, logistics systematic optimization. Intoduction All companies that aim to be competitive in maket should pay attention to how thei oganizations deal with the entie supply chain. In paticula, companies should focus on inceasing the efficiency of thei logistics opeations. It is impotant fo entepises to develop optimal systems of logistics distibution unde e-commece though integating thei logistics activities. Location Allocation poblem (LA) is an impotant banch of outing optimization in integated logistics systems, which can be involved in all logistics distibution copoations. Duing the last two decades o so, many scholas have developed moe efficient poblem solving techniques using the concept of integated logistics systems, including optimizing models and applying associated algoithms fo solving these poblems. Vehicle Routing Poblem (VRP) and location-allocation (LA) wee developed and analyzed in logistics systematic optimization. Alan T. Muay and Ross A. Gead [] intoduced the capacitated egionally constained p- medium poblem, which incopoates egional equiements in a LA famewok, in addition to ensuing that maximum capacity limitations ae maintained. Diin Gong et al. [2] discussed an extension of the LA model, which has capacity constaints, and poposed an Evolutionay Stategy (ES) combined with efficient taditional optimization techniques. Some scholas used the Beneges method that ensues the convegence fo the model with the constaint equations [3-7]. This helps because the model and solution fo LA ae constucted in a way that povides fo futhe LRP in integated logistics. Location Routing Poblem (LRP) is one of the poblems in integated logistics optimizations. The LRP can be defined as follows. A feasible set of potential facility sites and locations and expected demands of each custome ae given. Each custome is to be assigned to a facility, which will supply its demand. Vehicles dispatched fom the facilities and opeated on outes that include multiple customes cay out the shipments of custome demand. Thee is a fixed cost associated with opening a facility at each potential site. A distibution cost associated with any outing of vehicles includes the cost of acquiing the vehicles used in the oute, and the cost of delivey opeations. The cost of delivey opeations is linea with the distance taveled by vehicles. The LRP is to detemine the location of the facilities and the vehicle outes fom the facilities to customes to minimize the sum costs of the location and distibution in this way that the vehicle capacities ae not exceeded. Many methodologies have been utilized to solve the LA poblem. The most obvious methodologies ae the tanspotation/assignment, and the linea pogamming fomulations. Many eseaches have applied intege and mixed intege fomulations. Othes incopoated stochastic functions to account fo such distibutions as demand and /o supplies ae not deteministic. Baumol and Wolfe [8] solved the location poblem fo minimum total delivey cost with nonlinea pogamming. They gave a model of 0- mixed intege pogamming. The

2 model ensued minimum total costs. The weakness of thei model is that the fixed cost and the capacity s limit ae consideed, which makes it possible to choose much moe distibution centes. So the total cost will be inceased by this method. We modify the assumptions to develop a LA model that focuses on 0- mixed intege pogamming. In this pape a heuistic algoithm based on wavelet analysis is poposed fo LA, which combines the minimum wap-net taditional method fo choosing facilities. The intention of the poposed mixed method is to detemine the optimal position of Distibution Cente (DC) based on wavelet analysis theoy. An impotant and instuctive issue is how to eliminate noise and aise the accuacy of the solutions. Fist, the data is filteed by wavelet analysis, and then a heuistic algoithm is used to seach fo optimal (o nea optimal) solutions. The method incopoates both the distance and the demands among diffeent customes. The solution s pecision is impoved by wavelet analysis. As a esult, customes and thei chosen facilities ae detemined. The esult of compute simulation poves that this method is effective. The est of the pape is oganized as follows. Section 2 intoduces desciptions of LA and descibes the mathematical model of LA. Section 3 emphasizes on the solution of LA. Section 4 shows the computational esults and analysis. Finally, section 5 gives conclusions and futue wok. 2 Desciptions of LA Location-Allocation (LA) is one of the methods used fo poblems of integated logistics optimization. The LA can be defined as follows. A feasible set of potential facility sites and locations, and expected demands of each custome ae given. The amount and location of the facilities ae detemined in spatial positions accoding to the inteaction between candidate points fo customes and goods distibution [9]. In fact, the main popeties chaacteizing the facilities ae thei quantities, thei types and the costs associated with them. A geat numbe of obectives fo LA ae used to detemine the optimal spatial positions and the total distance. The obectives fo LA ae impotant fo planning in waehouses, plants, schools, hospitals, etail outlets and many othe public stuctues. John Cuent et al. [0] give a eview of LA. They put fowad the standad of the obective function fo classification, which consists of minimum cost, custome demand, maximum pofit, and othe condition. A typical desciption of LA poblem is shown in Fig.. The hypotheses fo mathematical model of LA [8][9] ae as follows. The candidate facility s position with the numbe (=,2,,R) is detemined, which is chosen with the maximum limit of R. The position and the demand epesented by D (=,2, N) of N customes ae also consideed, espectively. The factos ae focused on as follows. the tanspotation cost fom DC to node; the fixed and changeable costs fom DC to node; The limit of DC s capacity; the esticted amount of DC. Decision vaiables consideed ae as follows. x Z = = 0 if vehicle k goes fom custome i to custome i S, S,k V,i, 0 othewise if a facility is established at site, G othewise Fig. A typical location-allocation poblem Model paametes ae given as follows. G { =,..., m} = - the set of m feasible sites of candidate facility, H = i i = m +,...,m + N - the set of N customes to be seved, S = G H - the set of all feasible sites and customes (it is also efeed to as nodes), V = v k k =,..., K - the set of K vehicles available fo outing fom the facilities, C - aveage cost of pe distance fom node i to node,i S, S, { } { } { } { } i F - annual fixed cost of establishing and opeating a facility at site (=,,m),which is no elated to the DC s scale, q - aveage amount of goods demanded by custome, ( H ), k =Facilities =Customes Q - capacity of vehicle k ( k =,..., K), =Vehicle outes d i - distance fom node i to node, V - changeable costs coefficient with the candidate facilities (o called DC), W - ate of flow volume,

3 θ - the exponent fo flow volume, which is usually chosen /2, R - maximum quantity of the chosen facilities, D - total demand of node. Model of a special LA is defined as follows. The obective function of LA is f ( x) = min C q d + subect k V i H S G Z q H Z ipk G R S k V θ Z V ( W ) + to = 0 o = q S = 0 o Q D pk k i = 0 G F Z i H, k V, i H, k V, p S, G, i, S, k V, G. () In this model, the obective function minimizes the total costs of outing, establishing and opeating the facilities. Constaints (2) ensue that each custome is visited exactly once. Constaints (3) ensue that the vehicle capacity constaints ae not exceeded fo any of the vehicles used in outing, while constaints (4) guaantee that custome demands ae satisfied. Constaints (5) ensue that if vehicle k visits a point, it has to leave the point again (balance equation).. Constaints (6) give the limit of DC s amount. The last two of constaints ae the intege constaints []. 3 A heuistic algoithm to slove LA based on wavelet noise eliminination (2) (3) (4) (5) (6) (7) (8) 3. Data filteing based on wavelet analysis Wavelet analysis is a signal analysis theoy of a new timefequency localization based on Fouie analysis. The wavelet analysis of signal is becoming inceasingly popula within the scientific community. With espect to taditional Fouie decomposition techniques, wavelet analysis allows signals having a lage bandwidth chaacteized by a small numbe of coefficients [2]. The mathematical expession of wavelet function 2 Definition :Suppose that Ψ ( w) L ( R) ( L 2 ( R ) is an actual amount of space with squae-integal, i. e., signal space is of finite enegy)and that its Fouie tansfom is Ψ ˆ ( W ). If Ψ ˆ ( W ) is satisfied with C Ψ Ψ ( W ) = w dw Ψ(w) is called the mothe wavelet. R < By zooming extension and tanslation, the mothe wavelet Ψ(w) can be obtained by wavelet sequences. Data filteing The signal gets decomposed into the supeposition of wave packets. They ae localized both in time and in fequency. These wavelet packets ae obtained by escaling a chosen Mothe Wavelet Ψ(t) with a pope scale to obtain a wavelet function. This allows a vey efficient decomposition of spiky signals and nonstationay signals in geneal. In pactice the popula Discete Wavelet Tansfom algoithm is usually used to pefom the wavelet analysis. One impotant application is that -D signal with noise used to be decomposed by wavelet analysis. A -D signal that contains noise can be expessed as follows: (9) s( i) = f ( i) + σ e( i), i =,..., n (0) whee f (i) is the eal signal, e(i) is noise, and s(i) is a signal with noise. In pactical engineeing, useful signals usually show some low-fequency signals o steady signals, and noise signals show high-fequency ones. So the pocess of eliminating noise is as follows. Fistly, signal is decomposed by wavelet, fo instance, theechannel decomposition. Fig. 2 shows decomposition pocess. Because noise is usually contained in CD, CD2, CD3 of high fequency, wavelet coefficients ae dealt with in theshold. Then signal is econstucted in ode to eliminate noise. The pupose is that signal s (i) eliminates estaining noise and the eal signal f (i) is ecoveed fom signal s (i). Wavelet s choosing and wavelet channel s definition The channels of wavelet decomposition geatly influence on the issue. If thee is a lowe channel, the high fequency in the signal leaves moe, howeve highe channel, and filte the useful low-fequency. It is detemined on the specific pupose, but expeiment on the histoical data to decide the channel. Meanwhile used to many expeiments, wavelet db3 is also used to decompose one channel to eliminate noise and econstuct the channel, fom which good esults can be obtained.

4 Fom Fig.3, the data of eliminated noise is smoothe than that of pimitive. CA3 CA2 CD3 CA Fig.2 Thee channels of wavelet decomposition S CD2 CD Meanwhile, both distances between customes and demands of them ae consideed to have a new distibuted elation. The elation indicates that special DC seves customes. The suitable facilities seving special customes ae chosen by this poposed algoithm. They satisfy the minimum total distance; i. e., the distance is accumulated between customes and thei facilities. Thus, the total cost is minimized. The definition of the heuistic functions Customes ae detemined to be seved by DC that is neaest to them. When the goods quantity is cetain, the total cost is minimized, that is to say, the total distance is the minimum one. The heuistic function is descibed as follows: N f ( x = HSZ = MINL i () ) i= Fig.3 the compaison between oiginal signal and filteed signal by wavelet analysis 3.2 The idea of the heuistic algoithm fo LA The fundamental idea of the heuistic algoithm fo LA contains seveal ules. The ule fo choosing a distibution cente is that customes ae classified into one cluste if thei distances ae neaest to DC. And then the customes in one cluste satisfy that the total demand of them in each DC is appoximately equal to seveal integal multiple of single vehicle s capacity. The specific ules fo choosing potential facilities ae as follows. The minimum wap-net ule is that one distibution cente is standad, which is linked to othe DCs though a staight line as thei pependicula bisecto. The minimum wap is delimited on the basis of these lines of inne points in evey distibution cente that is seved within these anges. This ule is shown in Fig. 4. The detailed division including seved customes with diffeent DC is given by a ule that the total demand of customes in one divided cluste is nealy equal to seveal integal multiple of single vehicle s capacity. The special customes belonging to some DCs that have been categoized into one cluste may be changeable in ode to aange DC with thei distibuted customes moe suitably. Fig.4 A simple desciption of minimum wap-net f 2 ( x) = HSZ 2 = q i p= = nq ( n =,2,3...,N; p =,2,...,i) L = N d i i= p k (2) (3) whee MINL i is the minimum one of the distances fom custome i to m DCs, N is the amount of custome points,d is the distance fom any custome i to any DC. 3.3 The analysis fo LA based on a heuistic algoithm by wavelet analysis The custome aggegate C is divided into clustes by the euclidean measue based on the minimum wap-net method. Optimizing facility location is pefomed in the following way, which has been extensively analyzed and used in location theoy to appoximate distances between two spatial coodinates. Fo the sake of simplicity, it is assumed that each facility has its own coodinates set of m candidate sites fo facility location, and no two facilities shae the same location, which ae denoted by PF (, Y ), PF 2 ( 2, Y 2 ),,PF i ( i,y i ),, PF m ( m,y m ). Evey two

5 coodinates of candidate sites fo facility locations ae denoted by PF i ( i,y i ) and PF (,Y ), and which satisfy the following expession: R i =/2[( i - ) 2 +(Y i -Y ) 2 ] /2 (4) whee R i is half of the distance between PF i ( i,y i ) and PF (,Y ). Rules fo classification ae given as follows. Rule ip <R i (5) C p can only be allocated to PF i if fomula (5) is tue. Rule 2 Customes C, C 2, C 3,,C l, C m ae classified in O i, if C, C 2, C 3,,C l, C m ae seved by DC i. The emaining customes C m ae moved into the DC that is neaest to DC i. This minimum wap-net method fo clusteing analysis incopoates the demand into elative distances among customes. It is useful fo solving pactical LA, in which the distance elated to demand with diffeent customes is consideed at the same time. Rule 3 The coodinates of customes ound a facility satisfy that the distance between the neaest two customes is less than that between the sub-neaest two points (Sub-neaest two points means that the two points ae not included in a cluste by measue of distance). Rule 4 If q > maxr i (6) whee q is the distance fom any custome C q to any facility PF, maxr i is the maximum one of half distances between any two facilities. Unde this ule, eithe these customes will not be seved, o else the candidate facility location will be abolished. Last two ules only elate to abnomal custome points. In the above ules,n is the numbe of customes,c=[c,c2,,cn], C(x,y ),C 2(x 2,y 2 ),,C p (x p,y p ),,C N (x N,y N ) expess custome points and thei coodinates, espectively. ip is the distance between any candidate facility PF i and any custome p. Tangent cycles O, O 2,, O i, O,, O m ae dawn with the adius R i, wheer,r 2,,R i,r,,r m ae thei adiuses, espectively. 3.4 Steps of the heuistic algoithm fo LA based on wavelet analysis Step The data is filteed by wavelet analysis. Wavelet db3 is used to decompose one channel to eliminate noise and econstuct it best. Step 2 Let m=r and points of customes ae chosen fom potential R facilities. Initial calculations fo R i and ip and heuistic function s value ae pefomed. And heuistic function s value is given to besthsz, which efeed to as the best heuistic function s value. Step 3 If ip <R i, the custome p falls into cluste O i. Remembe the minimum function value: opt_ f =besthsz. Othewise, etun to step. Step 4 Calculate the total demand satisfying that it is the integal multiple (seveal times) of single vehicle s capacity in any cluste as pecisely as possible. Step 5 If f<besthsz, and opt_f=f. Let m=r-, that is to say, one potential facility is eected. And epeat step to step 3, calculate R i and ip and heuistic function s value. Step 6 If f <besthsz, and opt_f=f. Goto step 4, and decide whethe the cycle is stopped o not. Note: The ule fo educing potential facilities is defined. The cost inceased in tanspotation fo custome is less than the cost of establishing and opeating one facility. If the ule holds tue, one facility is cut off. 4 Analysis of the computational esults In the following example six potential facilities and thity customs with diffeent demands ae analyzed. The same type of distibution vehicle that has a caying capacity of 30 tons seves each custome. The annual cost of establishing and opeating a facility F is 60 Yuan. The tanspotation cost C i is 2 Yuan pe ton, pe km. The data shown in Fig.3 is used to seach the optimal (o nea optimal) solutions by this algoithm. The pogamming is done by VC++. The esults of clusteing analysis based on wavelet analysis with minimum wap-net method and unning 3000 times ae given in Fig. 5 and table. PF C2 C22 C23 C24 C29 C30 PF2 C C5 C6 C2 C3 C4 C5 C6 C 7 C8 C27 C28 PF4 C2 C0 C C9 C20 PF6 C3 C4 C7 C8 C9 C25 C26 Fig.5 Results by a heuistic algoithm based on wavelet analysis with minimum wap-net

6 Poblem scale 6/30 Results by heuistic algoithm based on with minimum wap-net PF- C2,C22, C23,C24,C29,C30, 2PF2- C,C5, C6, C2, C3, C4, C5, C6, C7, C8,C27,C28, 3PF4- C2,C0,C,C9,C20, 4PF6- C3,C4,C7,C8,C25, C26. Function value Refeences [] Alan T. M., Ross A. G (997). Capacitated Sevice and Regional Constaints in Location Allocation Modeling. Location Science, 5(2): [2] Gong D.J., Gen M. S., Yamazaki G. J., u W. (997). Hybid Evolutionay Method fo Capacitated Location-allocation Poblem. Computes industial Engineeing, 33 (3-4): Table Simulation esults based on wavelet noise elimination Fom the simulation esults, the numbe of potential facilities that ae chosen depends on special conditions. When the location of customes ae widely scatteed and the demand of customes is vey lage, the fou potential facilities ae all chosen, and each chosen facility seves its special customes who ae neaest to it. The simulation esults ae shown in Fig. 5 and table. The value of obective function f (x) is Yuan. 5 Conclusions In this pape, the heuistic algoithm based on wavelet analysis fo mixtue choosing facilities is poposed. It combines custome s demand into minimum wap. This method can solve the LA poblem with many potential facilities and diffeent demand customes. A classification is given out, in which customes ae elated to thei divided DC. Using wavelet analysis technique, the solution s pecision is impoved. This method has contibuted to the pactical application of LA in integated logistics. The poposed method fo LA povides a way to solve LRP. The application to othe combinatoial optimization poblems should be investigated in integated logistics. 6 Acknowledgements This pape was suppoted in pat by Natual Science Foundation of FuJian Povince unde gant numbe A05009, in pat by Key Poect of FuJian Povince unde gant numbe 2005R032, in pat by Youth Foundation of Chinese Education by HuoYing-dong unde gant numbe 04009, in pat by Foundation of HuaQiao Univesity and in pat by Youth Foundation of Education Depatment in LiaoNing Povince unde gant numbe 2004F03, and in pat by National Natual Science Foundation of China unde gant numbe [3] Masood A. B (999). Combining the analytic hieachy pocess and goal pogamming fo global facility location-allocation poblem. Intenational Jounal of Poduction Ecnomics, 62: [4] Aleksanda M., Benad C.L., Alexande M (2000). Application of complex pogamming to escheduling of tansactions in deegulated powe maket. Poceeding of The 39th IEEE Confeence On Decision and Contol, Sydney, Austalia, [5] Boukas E. K., Benzaouia A (2002). Stability of discete-time linea system with Makovian umping paametes and constained contol. IEEE Tansactions on Automatic Contol, 47 (3): [6] B Liu, KK Lai (2004). Stochastic pogamming models fo vehicle outing poblems. Focus on computational neuobiology, 3-7. [7] G. Nagy and S. Salhi (2005). Heuistic algoithms fo single and multiple depot vehicle out-ing poblems with pickups and deliveies.euopean Jounal of Opeational Reseach,62. [8] Baumol W., Wolfe P(958). A waehouse location poblem. Opeation Reseach, 6(2): [9] John C, Min H. Schilling D A (990). Multi obective analysis of facility location decisions. Euopean Jounal of Opeational Reseach, 49: [0] Haison H (979). A planning system fo facilities and esouces in distibution netwoks. Inteface, 9(2): [] Hwang, Heung-Suk (2002). Design of supply-chain logistics system consideing sevice level. Computes and Industial Engineeing, 43(7): [2] E Dautbegovic, M Condon, C Bennan (2004). An Efficient Wavelet-based Nonlinea Cicuit Simulation Technique with Model Ode Reduction. High Fequency Postgaduate Student Colloquium,9-24