A Bi-objective VRPTW Model for Non-adjacent Products

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1 American Journal of Science, Engineering and Technology 2017; 2(1): 1-5 htt://wwwscienceublishinggroucom/j/ajset doi: /jajset A Bi-objective VRPTW Model for Non-adjacent Products Mohammad Hossein Sarbaghi Yazdi, Farhad Esmaeili Deartment of Industrial Engineering, Mazandaran University of Science and Technology, Babol, Iran address: hsarbaghi@yahoocom (M H Sarbaghi Y), esmaeilifarhad@gmailcom (F Esmaeili) To cite this article: Mohammad Hossein Sarbaghi Yazdi, Farhad Esmaeili A Bi-objective VRPTW Model for Non-adjacent Products American Journal of Science, Engineering and Technology Vol 2, No 1, 2017, 1-5 doi: /jajset Received: October 30, 2016; Acceted: December 26, 2016; Published: January 12, 2017 Abstract: Vehicle Routing Problem (VRP) with time windows is a generalization of the classic VRP Secifically, every customer must be met in a certain time window Sometimes in the real life, it is not ossible to carry different roducts simultaneously In other words, these roducts are non-adjacent This aer resents a comrehensive model for the vehicle routing roblem with time windows and the ossibility of delivery slit of non-adjacent roducts The roosed model is an extension of VRP considering the rofit in a bi-objective otimization model Keywords: VRP, Time Window, Bi-objective, Non-adjacent Products 1 Introduction Today, transortation lanning is one of the imortant fields in oerations research, industrial engineering, and civil engineering The main objective of this field is to minimize the cost of transortation of goods and materials from the roducers to the consumers 11 Vehicle Routing Problem VRP is one of the imortant roblems has been raised in the field of logistics management and transortation systems VRP is a generalization of traveling salesman roblem (TSP), which is considered as a vehicle VRP is aimed at determining the otimum route of the vehicle so that to meet the demands of all customers where all constraints are satisfied and transortation cost is minimized In classic VRP, the start and the end of all routes is a secific warehouse or a central deot The VRP is generally defined on a comlete grah G=(V, A, D) Where V={V 0, V 1, V 2,, V n } is the set of nodes (vertices) and A={(V i, V j ): i j} is set of arcs that connect node i to j V 0 indicates central deot and V n reresents the customer n d reresents distance or cost between node i and node j q i is demand in node i Tyically, the objective is to minimize the total travel cost Dantzig and Ramser [1] roosed the VRP for the first time After that, many variants of VRP have been considered: VRP with time windows (VRPTW), icku and delivery (VRPPD), multi deot VRP (MDVRP), and so on 12 Vehicle Routing Problem with Time Window If each customer must be visited during a secific time window, we face with VRPTW In fact, time window is a time interval with earliest time (b i ) and the latest time (e i ) that a customer ermits to start receiving service Some of alications of the VRPTW include ostal deliveries, refuse collection, bank deliveries, school bus routing, etc Over the ast 25 years, many researchers have worked on VRP with time windows VRPTW belongs to NP-Hard roblem [2], the same as the classic VRP, so many metaheuristics have been used to solve it, eg Simulated Annealing (SA) [3, 4], Genetic Algorithms (GA) [5, 6], Tabu Search (TS) [7, 8] and Ant colony Otimization (ACO) [9] Some authors like Yu et al [10] resented a hybrid aroach to solve VRPTW, a mixture of ACO and TS Küçükoglu and Öztürk [11] resented a hybrid algorithm which includes tabu search and simulated annealing In the last three decades, researchers have mixed VRPTW with other modes that are close to real-word roblems, for instance VRPTW with Slit Delivery (VRPTWSD) [12, 13] Dror and Trudeau [14] resented slit delivery roblem for the first time In VRPSD, unlike the classic VRP, more than one vehicle can suly demand of each node [15] El- Sherbeny [16] has reviewed literatures in VRPTW and its combination to other VRP modes He studied various methods of solving VRPTW

2 2 Mohammad Hossein Sarbaghi Yazdi and Farhad Esmaeili: A Bi-objective VRPTW Model for Non-adjacent Products 13 Vehicle Routing Problem with Profit In classic VRP and TSP each node should meet once and it will fulfill the demand of all nodes, but in TSP with rofit (TSPP) and VRP with rofit (VRPP), there is not any obligation to serve all nodes and customers are served in accordance with the rofit of meeting In this roblem, each node gets a reward after meeting and a trade-off between cost and rofit in each node will indicate which nodes must been met [17] In TSPP and VRPP there are two objectives having obviously conflict: Maximizing rofit through meeting the maximum number of customers, which cause increase in distances and travel costs Minimizing travel distances that cause decrease in rofit of customer s meeting If two objectives were homogenous, for examle both be currency, we can solve them via converting to the singleobjective roblem As keller and Goodchild [18] mentioned, in many cases these two objectives are not commensurable and we should consider them as a bi-objective roblem Feillet et al [19] describe three classes of single-objective roblems that have roosed in TSPP: PTP (Profitable Tour Problem): their objective function is a combination of two objectives, ie minimizing travel cost minus rofit of customers meeting [17] OP (Orienteering Problem) or STSP (Selective TSP): their objective function is maximizing collected rofit such that travel costs do not exceed a redetermined value C max [20] QTSP (Quota TSP) or PCTSP (Prize Collecting TSP): their objective function is minimizing the total traveling distance such that rofit of customers meeting which should not be less than a fixed amount of P min [21, 22] It seems that VRPP has been less considered in the literature of VRP [23] Team OP (or TOP) is an extension of OP that has been studied by Chao et al [24] They resented a five-stes metaheuristic aroach based on simulated annealing to solve this roblem Lin and Yu [25] used a simulated annealing heuristic for TOP Boussier et al [26] used an exact method to solve this roblem Aráoz et al [27] solved a Prize Collecting Rural Postman in which the objective is minimizing the residuum of collected rofit and traveling costs 2 Proosed Model This aer resents a bi-objective model for VRP with time windows and slit deliveries The considered objectives are: To maximize the rofit of customers meeting: it considers secific rofit for each unit of roduct which has been met To minimize the costs, eg traveling cost, waiting time cost, or vehicle cost In fact, resented model is a kind of develoed VRPP As mentioned before, the considered two objectives are not homogenous In real life, various routes are not the same in view of transortation comlexity For instance, distance between customers 2 and 8 is similar to distance between costumers 4 and 6, but traveling time of these two routes are not the same due to their ath conditions, like seed constraint, traffic, etc Therefore, for each arc between two nodes we consider a difficulty factor and a rest time deending on the traveling time The time that each vehicle services a node deends on tye and amount of its demand 21 Assumtions and Characteristics of Model All vehicles start from central deot and traverse a subset of customers in a secified sequence Then they return to the central deot Each vehicle meets a customer for once Delivery slit is ossible so that demand of each customer (node) can be met by several vehicles Each customer has a time window to receive services If the vehicle arrives earlier than the lower bound of time window, it should wait until starting of service time A enalty is considered for vehicle tardiness in each node The number of available vehicles is given and vehicle costs are assumed to be zero if they are not in use The time that each vehicle serves a node (customer) deends on tye and amount of demand which is met by the vehicle All vehicles have a constant seed and fixed cost er distance (kilometer or mile) Difficulty factor of route is defined between 1 and 3 The roosed model is designed for non-adjacent roducts and each customer has determined demand Each vehicle transorts only redetermined roduct and it does not allow transorting any other roduct A defined award is considered for every unit of customer s demand which has been met It is not obligatory to meet all the demands Each vehicle should rest when it receives to a node and the rest time deends on distance assed from the revious node to the current one 22 Notations The notation we used in our model is resented below: Sets: Q= {1, 2,, i,, N} demanding nodes (oints) where oint 1 denotes central deot A = {( i, j) i, j Q, i j} arcs or connection between nodes V= {1, 2,, v,, U} vehicles K= {1, 2,, r,, O} roducts Parameters: P ir Profit of r-tye roduct in node i TC Traveling cost d Length of arc A where d 11 = 0, d = d i, j and ji

3 American Journal of Science, Engineering and Technology 2017; 2(1): dii >> i > 1 b Difficulty factor for route i j, so that b =b ji t Traveling time of arc A where A ( i, j) e i Penalty for tardiness er each unit of time in node i a rv Binary variable which is equal to 1 if vehicle v is allowed to transort r-tye roduct, and 0 otherwise q ir Demand of node of i for roduct of tye r g r Necessary time to serve each roduct of tye r e Rest time that deends on assed distance lb i Earliest ossible time to start traveling from node i (lower bound of time window) ub i Latest ossible time to arrive in node i (uer bound of time window) R Fixed cost of using each vehicle V e Average seed of each vehicle M An enough huge number (M>> ) Decision variables: x v Binary variable which is equal to 1 if vehicle v goes from node i to j, 0 otherwise f irv Shortage of demand of r-tye roduct in node i that will be met by vehicle v w iv Waiting time of vehicle v in node i h iv The time that vehicle v sends to serve in node i re iv The time that vehicle v rests in node i s iv The time that vehicle v start to serve in node i 23 Mathematical Model The roosed mathematical model is a bi-objective otimization model reresented by equations (1)-(15) The first objective function maximizes the rofit of meeting of nodes (customers): Maximize ir firv (1) v V i Q r K The second objective function minimizes sum of three elements: 1) total assed distance, 2) fixed cost of using vehicles, and 3) total tardiness enalties in nodes: Mininmize v V i Q j Q + + v V j Q\{1} v V j Q\{1} TCd x Rx v 1jv ew i iv Equation (3) exresses each vehicle cannot visit a node more than once (2) xv 1 i Q (3) j Q Equation (4) guaranties each vehicle exit after arriving in a node xv = xv j Q (4) i Q j Q demand of tye r in node i, it definitely will meet node i xv firv i Q, r K (5) j N The following equation shows vehicle v can suly demand of tye r in node i f a i Q, r K (6) irv rv Each vehicle cannot serve more than its caacity: firvqir C v V (7) i N r K The following equation calculates the time which vehicle v serves node i hiv = grqir firv i Q (8) r K Equation (9) calculates the rest time of vehicle v in node j bd re jv = e xv j Q (9) ve i N The necessary time to traverse the arc of is calculates by equation (10) t b d = i Q, j Q (10) ve Equations (11)-(13) are related to time window Moreover, equations (11) and (12) revent forming any subset tours s + h + re + t + w + (1 x ) M S iv iv iv jv v jv i Q, j Q\{1} s + h + re + t + w (1 x ) M S iv iv iv jv v jv i Q, j Q\{1} (11) (12) lb s ub i Q (13) i iv i Equations (14) and (15) declare the variable tyes and their signs xv {0,1} i Q, j Q (14) f 0, w 0, h 0, re 0, s 0 irv iv iv iv iv i Q, r K 3 Solving the Proosed Model with Augmecon Aroach (15) A brief exlanation of AUGMECON aroach is resented here, an augmented version of the ε -constraint method First, consider the following multi-objective roblem: Equation (5) assures if vehicle v sulies a ortion of

4 4 Mohammad Hossein Sarbaghi Yazdi and Farhad Esmaeili: A Bi-objective VRPTW Model for Non-adjacent Products Max(,,, ) st x S 1 2 (16) Where S is the feasible region In the ε -constraint method, one of the objective functions gets otimized while other objective functions are considered as constraints: Max( ) st x S (17) We obtain different solution via changing the value of e i In AUGMECON, a develoed version of ε -constraint method, lexicograhic otimization is used to determine e i After calculation of e i, difference between uer and lower bound is divided into several arts The calculated values Table 1 Comutational results indicate allowable e i Pareto-otimal solutions increase as the number of iterations increases Then, the below function is solved using obtained values of e i Max( + es ( s + s + + s )) st s = e s = e x S ands > 0 i (18) Where S i are slack variables and es is a very small number (roughly between 10-3 and 10-6) For more information about AUGMECON aroach we refer the interested readers to [28] and [29] 4 Comutational Results We used GAMS software to solve the resented model for 8 different randomly-generated instances The results are shown in table 1 Instance Instance characteristics Number of nodes Number of roduct tyes Number of oints on the Pareto-otimal frontier Comutational time (seconds) Whereas the VRP belongs to NP-Hard roblems, if the roblem size (the number of arameters, variables and constrains) increases, the comutational time will increase exonentially So, it is not ossible to solve large-dimension roblems by GAMS software and heuristic or Meta-heuristic methods should be alied to solve these roblems 5 Conclusion In this aer, a bi-objective otimization model is resented for VRPTW with slit delivery to transort non-adjacent roducts Moreover, several real-world issues are considered, ie difficulty of route, rest time (deending on assed distance) and serve time (deending on tye and amount of demand) Finally, the resented model is solved using AUGMECON method in several instances and the results are discussed References [1] Dantzig, G and Ramser, J H The truck disatching roblem, Management Science, Vol 6, [2] Lenstra, J and Rinnooy, K Comlexity of vehicle routing and scheduling roblems, Networks, Vol [3] Baños, R, Ortega, J, Gil, C, Fernández, A and de Toro, F (2013) A Simulated Annealing-based arallel multi-objective aroach to vehicle routing roblems with time windows, Exert Systems with Alications, Vol 40, Issue 5, [4] Wang, C, Mu, D, Zhao, F and Sutherland, J W (2015) A arallel simulated annealing method for the vehicle routing roblem with simultaneous icku delivery and time windows Comuters & Industrial Engineering, Vol 83, [5] Ursani, Z, Essam, D, Cornforth, D and Stocker, R (2011) Localized genetic algorithm for vehicle routing roblem with time windows, Alied Soft Comuting, Vol 11, Issue 8, [6] Yang, B, Hu, Z-H, Wei, C, Li, S-Q, Zhao, L and Jia, S (2015) Routing with time-windows for multile environmental vehicle tyes, Comuters & Industrial Engineering, Vol 89, [7] Belhaiza, S, Hansen, P and Laorte, G (2014) A hybrid variable neighborhood tabu search heuristic for the vehicle routing roblem with multile time windows Comuters & Oerations Research, Vol 52,

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