Multi-objective Optimization For The Dynamic Multi-Pickup and Delivery Problem with Time Windows

Multi-objective Optimizati on For The Dynamic Multi-Pickup and Deli very Problem with Time Windows I. Harbaoui Dridi (1) (2) , R. Kammarti (2) (1) LAGIS : Ecole Centrale de Lille, Villeneuve d’Ascq, France (2) LACS : Eco le Nationa le des Ingénieurs d e Tunis, Tunis - Belv édère. TUNISI E imenharbaoui@ gmail.co m , kammarti.ryan@pla net.tn P. Borne (1) , M. Ksouri (2) (1) LAGIS : Ecole Centrale de Lille, Villeneuve d’Ascq, France (2) LACS : Eco le Nationa le des Ingénieurs d e Tunis, Tunis - Belv édère. TUNISI E p.bor ne@ec-lille.fr, Mekki.Ksouri@i nsat.rnu.tn Abstract — The PDPTW is an optimization vehicles routing problem which must meet requests for transport between suppliers and customer s sat isfying precedence, capacity and time constraints. We present, in this paper, a genetic algorithm for multi-objective op timization of a dynamic multi pickup and d elivery problem with time windows (Dynamic m-PDPT W). We p ropose a brief literature review of th e PDPTW, present our approach based on Pareto domin ance m ethod and l ower bounds, to give a satisfying solution to the Dynam ic m -PDPTW minimizing the compromise between total travel cost and total tardiness time. Keywords— vehicle ro uting, pickup a nd delivery, time windows, genetic algorithm. I. INTRODUCT ION The vehicle routing proble ms are studied because o f the growing passenger and freight tod ay. Some studies are pri marily oriented to wards solvin g the vehicle rou ting prob lem (VRP: Ve hicle Rout ing Problem). Another maj or area of resear ch focused on an important variant of VRP is the PDPT W, where in addition to the existence o f ti me constraints a nd capacit y constraints on the vehicle, th is pr oble m involves a clients set and a suppliers set geographicall y located. Each ro uting will also satisfy precede nce constrain ts to ensure that a customer should not be visited before his supplier. The P DPTW is the Pickup a nd Deliver y Proble m with T ime W indows, is d ivide d into two categor ies: 1-PDPT W (a vehicle) and m-PDPTW (multiple vehicles). T hese c an also be treated in two versio ns: static or dynamic In this paper we present a liter ature review of the P DPTW followed our multi-ob jective app roach, minimizing the compro mise bet ween total travel cost and total tardiness time, b ased on genetic algorithms a nd lo wer bounds, which treats the dynamic m-pdptw by incorpor ating un expected while routing have bee n planned and that t heir operatio n has started. II. LITERATURE REVI EW A. Veh icle routing pro blem The Vehicle Rout ing P roble m (VRP) rep resents a multi-goal co mbinatorial optimization prob lem which has b een the subjec t of many work a nd variations in the liter ature. [1 ][2] The V RP p rinciple is: given a depo t D and a set of custo mers ord ers C = (c1, ... , Cn), to b uild a package routing, for a finite n umber of vehicles, beginning and e nding at a dep ot. I n these r outi ng, a customer must be ser ved onl y o nce b y a single vehicle and ve hicle capacity transport for a routing should not be exceeded. [ 3] The Meta he uristics were a lso ap plied to solve the vehicle routi ng prob lem. Among these methods, we can include ant colony algorithms and genetic algorit hm whic h were u sed for the resolution of DVRP . [4] [5] Savelsbergh and al have s hown tha t the VRP is a NP-hard problem [6]. Sinc e the m-PDPTW is a generalization of t he VRP it’s a NP-hard p roblem. B. The PDP TW: P ickup and Delivery Problem with Time Windo ws The P DPTW is a variant o f VRPTW w here in addition to the ex iste nce o f tim e constrai nts, t his problem implies a s et of customers and a s et of suppliers geographica lly located. Every routing must also satis fy the pre cedence constr aints to ensure that a c ustomer should not be visited before his supplier. [ 7] A d ynamic approac h f or resolving the 1-PDP without and with time windo ws was developed by Psaraftis, H. N co nsidering o bj ective function as a minimization weighting o f th e total travel time and the non-custo mer satisfaction. [8] Jih, W and al have d evelop ed an appr oach based on the hybrid genetic algorithms to solve t he 1- PDPTW, aim ing to minimize com binatio n of t he total cost and to tal waiting tim e. [9] Another genetic algorithm wa s d eveloped by Velasco, N and al to sol ve t he 1-P DP bi-obj ective in whic h t he total travel time must be minimized while satisfy i n prioritise the most urgent req uests. In this literature, the method prop osed to reso lve this proble m is based o n a No do minated Sorting Algorithm (NSG A-II). [10 ] Kammarti, R and al deal the 1-PDP TW, minimizing the co mpro mise between t he total tra vel distance, total waitin g time and total tardiness t ime, using an evolutionar y algo rithm with Special genetic opera tors, tabu search to pr ovide a set of viable solutions. [1 1][12 ] This work ha s been exte nded, in proposing a new appro ach based on the us e of lo wer bounds a nd Pareto dominance method, to minimize the compromise bet ween t he total travel distance and total tardiness time. [13][ 14] About the m-P DPTW , So l, M and al have proposed a b ranch and pr ice algorithm to sol ve t he m-PDPTW , minimizing the vehic les number required to satisfy all tr avel demands and the total travel distance. [15] Quan, L and al ha ve pr esented a construction heuristic based on the integration pr inciple with t he objective function, minimizing the total cost, including the vehicle s fixed costs and travel expenses tha t are pr oportional to the travel d istance. [16] A new metaheuristic based on a tabu algorith m, was develope d by Li, H and al to solve the m- PDPTW. [17] Li, H and al ha ve develop ed a “Squeaky wheel” method to solve the m-PDPT W with a local sear ch. [18] A genetic al gorithm was develop ed by Harbaoui Dridi. I and al treating t he m-PDPTW to minimize the tota l travel distance a nd the total transport cost [ 19]. T his w ork ha s been extended, in prop osing a ne w appro ach based on the use Pareto dominance method to give a set of satisfying solution s to the m-PDPT W minimizing total travel cost, total tar diness time and the vehicles number. [20] [21] One more appr oach based o n the genetic algorithm was al so developed by [ 22] to sol ving the DPDP TW . T his appro ach is evaluated usi ng a large number of prob lem instances with var ying degrees of d ynamism. Mitrovic-Minic S and al d escribe a double- horizon b ased heuristics for the d ynamic PDPT W. The Co mputational r esults o f this work s how the advantage of using a double -horizo n in conju nction with insertion and improveme nt heuristics.[23 ] Rusdiansyah. A a nd a l ha ve d eveloped a m odel and heuristic algorithm for Dynamic Pick Up and Delivery P roble m with T ime W indows for Cit y- Courier providers. Consider ing t he number of timeslots and Degree o f D ynamism ha s a d irect relationship to the time requir ed co mputational.[24] III. MAT HEMATICAL FOR MULATION Our proble m is characterized by the following parameters: • N : Set of customers, s upplier and dep ot vertices, • ' N : Set of custo mers and supp lier vertices, • N + : Set of supp lier vertices, • N − : Set of custo mers vertices, • K : Vehicle number , • ij d : Euclidian dista nce betwee n the vertex i and the vertex j . If ij d = ∞ then the road b etween i and j doesn’t exi st, • ijk t : T ime used by the veh icle k to travel from the verte x I to the verte x j , • [ , i i e l ] : Time windo w of the vertex i , • i s : Stopping time at the vertex i , • i q : G oods quantity of the verte x i request. If i q > 0, the vertex i is a supp lier; if i q < 0, the vertex i is a custo mer and if i q = 0 then the vertex was served. • k Q : Capacity o f vehicle k , • i = 0.. N : Predecessor verte x index, • j = 0.. N : Successor vertex ind ex, • k : 1..K: Vehicle inde x, • 0  =   1 If the vehic le trave l from the v erte x i to the v erte x j Xijk Else • i A : Arrival time o f t he vehicle to the vertex i, • i D : Departure ti me o f t he vehicle fro m t he vertex i, • ik y : The good s q uantity in t he vehicle k visiting the verte x i, • k C : Travel cost assoc iated with vehicle k, • A vertex is served o nly once, • There is o ne depot, • The cap acity constraint must b e respected , • The d epot is the start a nd the fi nish vertex for the vehicle, • The vehic le sto ps at ever y vertex for a period of ti me to allow the req uest processi ng, • If the ve hicle arr ives at a verte x i b efore it s time windo ws beginning date ei , it waits. The function to minimize is given a s follows: 1 1 0 2 2 c C d X k ijk ijk k K i N j N Minimize f c max( , D l ) X i i ijk k K i N j N λ λ   +   ∈ ∈ ∈   =   −     ∈ ∈ ∈   ∑ ∑ ∑ ∑ ∑ ∑ (1) Subject to: 1 2 1 1 N K x j N ijk i k = = = = ∑ ∑ , , ... (2) 1 2 1 1 N K x i N ijk j k = = = = ∑ ∑ , , ... (3) 1 0 , X k K i k i N = ∀ ∈ ∈ ∑ 1 0 , X k K jk j N = ∀ ∈ ∈ ∑ (5) 0 , , X X k K u N iuk ujk i N j N − = ∀ ∈ ∀ ∈ ∈ ∈ ∑ ∑ (6) 1 , , ; = ⇒ = + ∀ ∈ ∀ ∈ X ijk y y q i j N k K jk ik i (7 ) 0 0 , y k k K = ∀ ∈ (8) 0 , ; Q y i N k K k i k ≥ ≥ ∀ ∈ ∀ ∈ (9) , ; ; D D i N w N v N w v i i + − ≤ ∀ ∈ ∀ ∈ ∀ ∈ (10 ) 0 0 D = (11) 1 , , ; X e A l i j N k K ijk i i i = ⇒ ≤ ≤ ∀ ∈ ∀ ∈ (12) 1 0 , , ; ; = ⇒ ≤ + ≤ ∀ ∈ ∀ ∈ ≠ X e A s l i j N k K s ijk i i i i i (1 3) 1 ( ), , ; = ⇒ + ≤ − ∀ ∈ ∀ ∈ X D t l s i j N k K ijk i ijk j j (14) The constraint (2) and (3) ensure that eac h vertex i s visited o nly once b y a single vehicle. The constraint (4) and (5) en sure that the ve hicle ro ute beginning and fini shing i s t he depot. T he co nstraint (6) e nsures the routi ng contin uity b y a vehicle. (7), (8) and (9) ar e the cap acity co nstraints. T he preced ence constraints are gu aranteed by (1 0) and (11) . The constraints (12 ), ( 13) and ( 14) ensure compliance ti me wi ndows. Where i λ and c i are weights and scali ng coefficient s. IV. MULTI-OB JECT IVE APPROACH FO R THE DYNAMIC m-PDPT W A. Using ge netic alg orith m The principle of d ifferent genetic operatio ns such as c hromoso me co ding, the generatio n o f populatio ns as well a s p roced ures for corr ecting capacit y and p reced ence are detailed in our work [19] . In our case, w e will gener ate two types of populatio ns. A first pop ulation noted P node , which r epresents all nod es to visit with all vehicles, accord ing to the per mutation list co ding. The second populatio n noted P vehicle indicates nodes number vi sited b y each vehicl e. Whereas t hese t wo populatio ns and correc tion proce dures, w e o btai n the final p opulatio n P node / vehicle whose s hows an individual example in Fi g 1. 1 V 1 C 0 5 8 2 6 4 3 0 2 V 2 C 0 10 7 9 1 0 Figure 1: Example of Individual of the population P node / vehicle With: 10 N ' and K=2 = B. Mult i-criteri a ev alua tion A multi-crite ria pr oble m is defined as an optimization vector pr oblem, which seeks to optimize severa l compone nts o f a vector function cost. However, it is nec essary t o find solutio ns represe nting a p ossible co mpromise be tween the criteria. T he P areto optimality co ncept intro duced by the eco no mist V. P areto in the ninetee nth century is frequentl y used [ 25]. V. P areto formulated the follo wing co ncept: in a multi-criter ia pr oble m, there is a b alance so that we cannot i mprove one criterio n w ithou t dete riorating at le ast one o ther. T his bal ance has b een called Pareto optimal A solution is noted Par eto optimal i f it i s dominated by a ny o ther point in solutions s pace. These p oints are note d non-do minated solutions . This part has been de tailed in our work [20 ] [21] . (4) C. Calculation of lower boun ds Since we have tr ansfor med our multi-objective problem (MOP ) in a mono-objective p roblem (PMO λ ), as shown in Eq uation 1, which is to combine the differen t cost functions f i of the problem into a single ob jective function F , usually a linear [27 ]. The problem now, is to deter mine the various constants c i that are weights and scaling coefficients. The constant s c i are usually initialized to 1 ( *) f x i where ( * ) f x i is the optimal solution associated to the obj ective functio n f i . Because we ha ven’t information on t he op timal solutions as sociated with dif ferent co st function s f i for our problem we are faci ng th e calculatio n o f lower bounds to deter mine the scali ng co nstants c i . To do this, we used the relaxation of variou s constraints. All computatio n do ne, we ob tain 1 f b and 2 f b successively represe nting a lower b ound of the total travel co st and a lower bound o f the total tardiness time. [26] 1 1 1 c f b (15) f C d b k min k . p . c k K     =   ∈  ∑ 1 0 2 2 2 2 c f f max( , D l ) X (16) s . c . f 0 b b i i ijk k K i N j N b  =    = −     ≠  ∈ ∈ ∈ ∑ ∑ ∑ D. Dynamic study The dynamic m-pdpt w is a n extensio n of the m- pdptw where b esides the tempo ral, precede nce and capacit y c onstraints, which impo ses thi s last o ne, the d ynamic m-p dpt w di stinguishes i tself by t he appeara nce of dynamic and urgent requests having planned the rou ting associated with each ve hicle. The planner will the n have t o satisfy the ne w requests besides t he former while opti mizing the chosen criteria a nd by resp ecting the imposed constraints. We will p resent o ur ap proach o f d ynamic adaptation, b y us ing two diff erent methods for the insertion o f new r equests. T he fir st met hod (Fig 2) consi sts in insert ing the dynamic no des from their appea rances on t he routi ng vehicle which minimizes the c hosen criteria , by satisfyin g this last one. B y keepi ng the sa me list o f re maining nod es to visit, after insertio n of d ynamic requests. The second method (Fig 3) is based on th e adaptation of the genetic operato rs and the correctio n p rocedures of pr ecedence and capa city initially c onceived for the static c ase to serve i n the dynamic case. While ensuring that ve hicles must not, unde r an y c ircumsta nces, visit a node already served. V1 Depot S1 S2 C1 S3 C 3 C2 Depot V2 Depot S4 S5 C4 C5 S6 C6 Depot V3 Depot S7 S8 S9 C8 C 7 C9 Depot V1 Depot S1 S2 C1 S3 C3 C2 Depot V2 Depot S4 S5 C4 C5 S6 C6 Depot V3 Depot S7 S8 S9 S10 C10 C 8 C7 C9 Depot Figure 2: F irst method of insertion from dy namic request Noting that the nodes which precede the appeara nce ti me o f the d ynamic r equest re main motionless. before reque st appearance after request a ppearance Time of req uest appearance (S 10, C10) Traffic of vehicles according to the already provided routing Appearance of the urgent requests at time t=t d Introduction of new requests and the vehicles positions at t = td Fix the nodes which precede the appearance time of dynamic requests t d Application of genetic operations and the correction procedures of precedence and capacity Update of the vehicles routing with th e new requests by verifying that a couple is visited by only one vehicle Figure 3: Se cond method of insertion f rom dynamic request Yes No We pre sent in Fig 4 our ap proach algorith m for the dynamic m-PDPTW. Figure 4: A pproach algorithm for the dy namic m-PDPTW E. Computatio nal results To test our approac h, w e use benc hmark problem instances generated by L i and Lim [17 ] from Solomon ’s ones [28 ] Corresponding to Solomon ’s c lassification of C1, C2, R1, R2, RC1 and RC2, their da ta sets were also ge nerated in six clas ses: LC1, LC2, LR1, LR2, LRC1 and LRC2. The L C proble ms are clustered whereas i n the LR pro blems, pr oviders a nd customers ar e rando mly gener ated. T herefore in the LRC pr oble ms the providers and the c ustomers are partially clustere d and p artially r andoml y distributed. W hile LC1, LR1 and LRC1 p roble ms have a short scheduli ng horizo n, L C2, LR2 and LRC2 have lon ger schedulin g one. [29] Each pr oble m includes severa l classe s. All t hese classes have ap proximatel y 100 suppliers and customers. T hey also contain the time windo ws, capacit y, the q uantities requir ed for eac h vertex and the coupling constr aints (supp lier, customer). In our w ork we will study the 8 classes that group the proble m LRC1 noted successively LRC101 until LR C108. Table I and II show the r esults of our simulation, by inserting it eve ry time t he couple of the d ynamic reque st. Of course, for ever y given solution, we note the co rrespo nding routi ng, crossed by eac h vehicle. TABLE I: Results for the LRC1 proble m with the First method of insertion fro m dynamic req uest LRC1 N sol N k 1 f 2 f ( ) F x LRC101 5 25 234467,71 63,95 53,66 216616,68 73 LRC101 47 9 170770, 53 1489 0,49 11 213386,84 211,4 LRC102 3 25 229200,37 64 ,2 53,47 233209,56 63,71 LRC102 37 8 183345, 04 914,9 0,49 12 224682,73 203,7 LRC103 2 25 220334,76 63,77 53,52 212968,59 64,77 LRC103 38 8 167587, 35 677 0,49 10 196233,34 121,5 LRC104 3 25 221505,56 64,21 53,47 246890,9 63,71 LRC104 32 11 176883,78 412,8 0,49 12 206176,92 66,75 LRC105 4 25 216597,48 70,07 53,66 248596,34 63,95 LRC105 39 9 177702,26 771,9 0,49 12 215801,04 83,11 LRC106 2 25 222740,68 64,14 53,39 231305,82 63,61 LRC106 38 9 176349, 14 644,8 0,49 12 205582,17 146,9 LRC107 5 25 234829,59 63,53 53,32 220063,68 64,31 LRC107 41 11 172050,35 286,6 0,49 12 211497,79 86,76 LRC108 4 25 236116,92 63,92 53,64 221455,42 64, 8 LRC108 39 11 170105,82 421,9 52,09 13 224198,4 63, 5 TABLE I: Results for the LRC1 proble m with the Second method of insertion fro m dynamic req uest Begin Step 1 : Create the initial population, (size n). Step 2 : Fill the interme diate population P node (size 2n) w ith individuals’ crossov er, mutation or copy . Step 3 : Corre ction procedure of Pre cedence and capacity . Step 4 : Create the 2 nd intermediate population P node/vehic le (size 2n * 2n) re presenting the routing of each vehicle. If Appearance of the ur gent requests at time t =t d Step 5: Fix the nodes w hich precede the appeara nce time of dy namic requests t d Where (the end criteri on is not satisfi ed) do Step 5.1 : I nsert the dynamic nodes in e ach individual from the population P node/vehic le Step 6.1 : Sort of po pulation by the minimum val ue of fitness (T otal tardiness time / To tal travel cost) Step 7.1 : Copy non-dominated solutions for the first method of insertion from dynam ic request End Where (the end cr iterion is not satisfied) d o Step 5.2 : I nsert the dynamic nodes in e ach individual from the population P node/vehic le Step 6.2 : Create the 3 th intermediate po pulation P node/vehicle dynamic (size 2n*2n * 2n) w ith individuals’ crossove r, mutation or copy, for t>t d Step 7.2 : Correctio n procedure of pre cedence and capacity and verify that a couple is visite d by only one vehicle. Step 8.2 : Sort of po pulation P n o d e / v e h ic l e d y n a m ic by the minimum value of fi tness (Total tardiness time / Total travel cost) Step 9.2 : Copy non-dominated solutions for the second method of insertion from dynam ic request End Else : Traffic of ve hicles according to the already provided rou ting End LRC1 N sol N k 1 f 2 f ( ) F x LRC101 4 25 210071 8,2 2,29 217638,09 0 LRC101 45 9 167285, 5 3989 0,49 12 212849,42 143 LRC102 4 25 224857,14 0 2,29 LRC102 43 8 183345, 04 849,8 0,49 14 221230,15 89,72 LRC103 5 25 209584,25 0 2,29 209112,54 12,87 LRC103 42 6 166937, 7 3892 0,49 10 202398,45 57,63 LRC104 10 25 216110,7 103,5 2,29 230606,14 1, 38 LRC104 39 11 213761,31 9, 08 0, 49 176834,1 84,95 LRC105 18 25 231586,07 1, 38 2, 29 211112,71 142,2 LRC105 36 9 177105,29 792, 6 0,49 12 212829,1 70,58 LRC106 3 24 220793,62 0 2,29 LRC106 35 9 174037, 5 1259 0,49 11 188774,51 59,49 LRC107 3 25 215355,6 0 2,29 LRC107 75 11 171035,53 222,6 0, 49 12 212836,6 6,01 LRC108 3 25 211749,64 0 2,29 LRC108 45 9 195396, 81 0 0, 49 11 167210,96 357,3 With: N sol : rep resents the number of non-do minated solutions. N k : repre sents the vehicles num ber used. Table s 1 and 2 sho w, for each st udy case, the solution which min imizes the total travel cost and the one who mini mizes the total tardiness ti me. Noting tha t the se t wo la st o nes are sorted a mong the not-do minated solutions. Adding as well a s for each case, w e use two different methods for co mputation the vehicles number used. We co nsider a vehicle n umber k ranging bet ween 1 and 25. We observe that o ur ap proach generate s a multiple number of sol utions minimizi ng the compromise bet ween t he total travel cost and the total tardiness t ime, that give flexibilit y of choice for the decision maker. We also observe that we obtain a total tardi ness equal to zero with a tolerable cost, by using the second method of insertion from dynamic re quest. These results j ustify the g enetic algor ithms utility. V. 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