FORMING DELIVERY ROUTES WHILE PROCESSING THE STOCHASTIC FLOW OF REQUESTS FOR FORWARDING SERVICES

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1 TRANSPORT PROBLEMS 2017 Volume 12 ssue 4 PROBLEMY TRANSPORTU DO: /tp Keywords: delivery routes; stohasti demand; request for transportation servies; servie level Vitalii NAUMOV Craow University of Tehlogy Warszawska 24, Krakow, Poland Corresponding author. vnaumov@pk.edu.pl FORMNG DELVERY ROUTES WHLE PROCESSNG THE STOCHASTC FLOW OF REQUESTS FOR FORWARDNG SERVCES Summary. Formation of rational delivery routes is the main way to inrease the effetiveness of lient servies for freight forwarding ompanies. The main part of the requests for transport servies omes from oasional (n-onstant) lients. The set of those requests forms a stohasti flow. For stohasti requests flow, the delivery routes are formed in the proess of the requests reeipt. Therefore, the standard approahes for merging of requests from multiple lients into routes, based on linear programming tehniques, ant be used in suh onditions. An algorithm of formation under the stohasti demand onditions of suh delivery routes, whih allow serviing of two or more shippers, is proposed in the paper. The author has developed a speialized software to support deisions made by dispathers of forwarding ompanies. 1. NTRODUCTON While making deisions onerning ustomer servie, the transport ompanies must take into aount many fators that determine the main parameters of the tehlogial proess and features of its organization. The enterprises of different transport modes have to solve the problem of merging the onfliting interests of shippers, onsignees and arriers. The main problem that arises while serviing the transport ompanies lients is making unreasonable deisions by serviing organizations, whih signifiantly redue the effetiveness of the serviing proess. The basi diretive of solving this problem is to develop and implement optimization tools to support deisionmaking proesses. One of priorities for ontemporary logistis management siene is the task of the effetive delivery routes development, whih is alled vehile routing problem (VRP) [1]. By the proess of forming the routes for the delivery of goods, we understand the hoie of suh sequenes of the shippers detour for the vehile that provide the optimal value of the effiieny riterion under kwn parameters of transport demand. The generally aepted approahes to reating the optimal routes are based on the methods of linear and dynami programming, ombinatorial methods and heuristi analysis [1-3]. The algorithms of the delivery routes sheduling are implemented for situations in whih the transportation demand has deterministi nature [2] for the apaitated VRP with equal demands [4, 5] and VRP with time window onstraints [6]. However, the most hallenging problem in logistis management is dealing with unertainty in logistis systems. As is mentioned in [1], ustomer demand an never be foreasted exatly, travel times will never be ertain and mahine and vehiles will break down. That is why development of new methods of logistis management, whih take into aount as muh unertainty as possible, is needed. For the ase of stohasti demand for transport servies, whih is typial for the ontemporary transportation markets, there are proposed models that take into aount ertain limitations and parameters: metaheuristi model taking into aount existing restritions on loading [7]; the routes

2 74 V. Naumov development methods based on the model of ant olonies [8]; robust algorithms, addressing unertainty [9]; multi-agent approah for solving the dynami VRP [10]; et. During the past years, there have appeared new approahes that aim to solve the routing problems in real-time mode, among whih the following works should be highlighted: the routing implementation with the onsideration of restritions plaed by the drivers operation modes [11], forming the delivery routes using the adaptive algorithm for searhing the nearest arrival point [12], using the iterative searh algorithms for harmonization of vehiles and load points funtioning proesses [13], solving the apaitated vehile routing problem with stohasti demands applying the simulated annealing algorithm [14], and implementing an integer L-shaped algorithm for the apaitated VRP with stohasti demands [15]. Pratial use any of the mentioned routing models offered by ontemporary authors envisages development of a speialized software that implements the relevant algorithms. Existing approahes to forming the different types of argo delivery routes ould be used in the ase of deterministi demand parameters. However, the argo delivery proesses in the ontemporary transportation market are haraterized by the delivery requests flow with dynamially hanging parameters. The respetive routing tehniques should be applied in this ase while proessing the requests flow [16-20]. t requires making the deisions about delivery routes onfiguration during short time intervals for the atual demand parameters with the onsideration of given rolling stok harateristis. 2. DESCRPTON OF THE MODEL FOR THE DELVERY ROUTES FORMATON Solving the problem of the delivery routes forming for the dynamially hanging set of requests requires the definition of the effiieny riteria for the hoie of optimal or near-optimal route onfigurations, mathematial formulation of the problem for the delivery routes formation, and the appropriate software implementation of the algorithms for the routes forming. The results of the preliminary studies allow us to suggest that as an effiieny riterion of the serviing proess for the ontemporary transport and forwarding ompanies, it is appropriate to use the servie level oeffiient R a ratio of the servied requests number to the total number of reeived requests ( R 1). As a riterion for evaluation of formed routes, the dynami oeffiient of the vehile apaity utilization γ d ( γ 1 d ) ould be used. The proess of making the deision on forming delivery routes is being made after the next request arrives on the basis of available information about the requests reeived earlier. Thus, an algorithm for forming the delivery routes should be iterative, where the identifier of the next iteration beginning is the time moment t i when the next request appears in the database. For the newly reeived request, on the basis of information on requests reeived earlier, the paired request is seleted. t is obvious that the best variant for the pair of request, formed in a route, would t be suh a ase when γd 1,0. f for the requests, ontained in a database at the moment t i, suh an optimal pair was t found, then the searhing proedure should be ontinued, iteratively dereasing a value of γ t d to its low bound γ d. n general form, the desribed approah for solving the problem of routing under onditions of the dynamially hanging database is presented in Fig. 1. n the presented algorithm, as an input value (the proedure argument), we use the request r i on transport servies, whih was reeived at the moment t i ; it is also assumed that the information about all the requests reeived earlier and t servied is kwn. The proedure of searhing the paired request from the set of t servied earlier requests begins from the highest value of the apaity utilization dynami oeffiient: γd 1. Searh of the pair r i is implemented in a speial funtion, the parameters of whih are the analyzed request r i and the given level γ d. n ase the pair was found, the algorithm returns to the formed route ρ { r, r }, ontaining i i i

3 Forming delivery routes while proessing the stohasti 75 the urrent request and the paired one. f the pair was t found, the allowable value of the apaity utilization dynami oeffiient is dereased on the speified value δ γ : γd γd δ γ, (1) then the proedure of the pair searh is implemented for the urrent request and the new, smaller value of the oeffiient γ d. Begin C A B Request r i γ γ d d γd 1 Searh of the paired r i for γ d γ γ δ d d γ The paired r i is found The route ρ { r, r } i i i End C A B Fig. 1. Algorithm of the method for formation of the delivery routes under the onditions of dynamially hanging requests database This step is repeated until suh a route is formed, whih is haraterized by the γ d value of the apaity utilization dynami oeffiient, or the ritial value γ d of the oeffiient is ahieved, meaning that formation of the route for the request serviing is eomially inexpedient. n suh a manner, the algorithm onsiders the onstraints on the delivery routes effetiveness. 3. MPLEMENTATON OF THE ALGORTHM SOLVNG THE PROBLEM The algorithm of the proposed heuristi approah for forming the delivery routes under onditions of dynamially hanging database was implemented using the PHP programming language (the basi ode is available at n the proposed program implementation, the searh of the paired request in a set of t servied $ ir _ pair $ idx,$ gamma. previously requests is performed by iterative funtions pair idx and The pair $ idx funtion for the request in a flow with an index $idx performs the searh of the paired request in the set of reeived earlier requests in the ase of pendulum route with the loaded ir _ pair $ idx,$ gamma funtion for the reverse run for the deliveries of the first lass argo. The request in a flow with the $idx index allows to determine the paired request from the set of requests that were reeived earlier and were t servied, under the ondition that the $gamma parameter of the funtion is t greater than the dynami oeffiient of the vehile s apaity utilization for the formed $ ir _ pair $ idx,$ gamma return an index of the paired request an a flow route. The pair idx and

4 76 V. Naumov of onseutive requests, if suh a request was found, or the value 1, if the paired request was t found. pair $ idx funtion is presented in Fig. 2. An algorithm of the At the stage of initialization of auxiliary variables, the pair $ idx funtion algorithm assigns the value 1 to the pair variable, whih is used to store an index of the paired request, and if suh a request is found, and the value 0 to the variable-ounter i, whih is used to refer to unproessed requests reeived earlier. At the next stage, the algorithm determines an iming request r (the request with the idx index) and the first request r p from the set of unproessed requests in a flow (the request with the index i 0 ). The proposed program implementation of the algorithm uses the input _ reqs array to store the requests from the iming flow. Begin А B C D ndex idx r p.o R = r.s R pair = 1; i = 0 r = input_reqs [idx] pair = i r p = input_reqs [ i ] i = i + 1 idx = i i < n Σ r p.s R = r.o R pair А B C D End Fig. 2. Algorithm of the pair $ idx funtion The pair $ idx funtion algorithm defines the r p request as the paired one, if the following onditions are met: 1) the index of the iming request r is t equal to the index of the request r p, i.e. in the ase of the same request, proessing is eliminated: pair i ; 2) the geographial segment of the onsignee o R of the request r p oinides with the geographial segment of the onsigr s R for the request r : rp. or r. sr ; 3) the geographial segment of the onsigr s R of the request r p oinides with the geographial segment of the onsignee o R for the request r : rp. sr r. or.

5 Forming delivery routes while proessing the stohasti 77 f all of the desribed onditions are evaluated to be true, then the i value is assigned to the variable pair and is returned by the proedure. f any of the onditions is t satisfied then the value of the ounter inreases by 1. f t all of the requests from the set of reeived earlier requests are heked, then the algorithm proeeds to hek parity of the next request, otherwise the algorithm terminates eturns to 1 (this indiates that the paired request was t found). ir _ pair $ idx,$ gamma funtion is presented in Fig. 3. An algorithm of the This algorithm orresponds to the struture of the pair $ idx funtion algorithm. The key differene is ontained in the blok heking the requests parity: the algorithm determines the request r as the paired one, if the following onditions are evaluated to be true simultaneously: p 1) the index of the iming request r is t equal to the r p request index (in the ase of the same request, omparison is eliminated): similar to the appropriate ondition of the pair $ idx funtion; 2) the value of the dynami oeffiient of the vehile apaity utilization γ p for the potential route is t less than the speified value gamma. Begin А B C idx, gamma γ p gamma pair = 1; i = 0 r = input_reqs [idx] pair = i r p = input_reqs [ i ] i = i + 1 pair = i i < n Σ Estimation of γ p pair А B C End Fig. 3. Algorithm of the ir _ pair $ idx,$ gamma funtion The proedure of the route forming is onsidered as a finished one if for the pair of request, the time restrition is ompleted. n the proposed implementation, this restrition is being onsidered serve _ req $ idx,$ pair funtion of the requests servie is alled for. The funtion when the parameter $idx refers to the index of the urrent request r and the $pair parameter to the index of $ ir _ pair $ idx,$ gamma funtions the paired request obtained as a result of the pair idx or the all. The _ $,$ serve req idx pair funtion algorithm onsiders onsequently the following ases for the pair of requests r p :

6 78 V. Naumov 1. The $pair parameter equals 1: in this ase for the request r the pair was t found, whih means that it ould be servied only using the pendulum route with an empty reverse run. n the proposed algorithm, solving the routing problem with suh a request is rejeted, beause the pendulum route with an empty reverse run is onsidered as an eomially inappropriate one. 2. The $pair parameter value is t equal to 1: for the request r the pair was found, whih makes possible forming of the effetive delivery route under ertain time onstraints: 2.1. The iming request r was reeived later than the paired request r p : t t, (2) ( ) ( p) r r where: ( с) ( ) t r, t p r the reeption time of the r p requests respetively, h The time interval between the requests r p exeeds the maximum time waited for the beginning of the paired request serviing: the requests r p are t formed into route, but remain in the list of requests, beause they ould be analyzed further for the lower allowable values of γ d : t t t, (3) ( ) ( ) ( ) r r w ( ) where: t p allowable waiting time for the start of the request r p serviing, h. w The time interval between the requests r p is less than allowable waiting time for the start of the paired request serviing: the requests r p are formed into the irular (or pendulum with loaded reverse run) route: ( ) ( ) ( ) t p p r tr t w. (4) 2.2. The iming request r was reeived earlier than the paired request r p : ( ) ( p) tr t r. (5) The time interval between the requests r p exeeds allowable time waited for the beginning of the iming request serviing: the requests are r formed into route but remain in the list of requests for further analysis: ( ) ( ) ( ) t p t t, (6) r r w where: t ( ) w allowable waiting time for the start of the request r serviing, h The time interval between the requests r p is less than allowable waiting time for the start of the request r serviing: the requests are formed into route: ( ) ( ) ( ) t p r tr t w. (7) The algorithms of the desribed proedures ould be onsidered as a series of alulations for the manual (t automated) methods of the argo delivery routes forming, but obviously more effiient way of their use is through the program implementation of the proposed algorithms as modules in the speialized information systems of deisions support for transport ompanies. 4. ESTMATNG THE BOUNDS OF THE PROPOSED ALGORTHM USE n order to estimate the area of the effetive use for the proposed algorithm, the study of dependene of the serviing level on the demand parameters has been onduted. Parameters of demand for freight forwarding servies ould be defined on the grounds of the logistis portal data, and the method of estimation of the demand parameters is desribed in the paper [21]. On the basis of the analysis of statistial data aumulated with the use of the logistis portal, the statistial parameters of demand should be defined the risk relation for time interval between the onseutive requests in a flow, median for the onsignments volume, and the expeted value of the waiting duration aepted by lients. The mentioned harateristis are the parameters of random variable desribing the respetive demand harateristis.

7 Forming delivery routes while proessing the stohasti 79 Thus, the task of estimation of the demand parameters influening the effiieny of the routing method ould be formulated as the following dependene: R f m Q, λ T (8) where m Q sale parameter of the logrmally distributed random variable of the onsignment volume (median); λ sale parameter of the requests interval exponentially distributed variable (risk relation); μ T loation parameter of the rmally distributed variable of waiting time (the expeted value). Here we perform estimations of the dependene (8) for the following alternative approahes to routing proedures in the bounds of freight forwarding operations: the use of the proposed method of the pendulum and irular routes forming; forming the delivery routes with the use of the Clarke-Wright method in the real-time mode; the mixed variant of routing: at the first step, the delivery routes are being formed; then, for those request, for whih the effetive delivery routed ould t be defined, the serviing with the use of proposed routing proedure is being performed. For eah of the listed alternatives, the simulation experiment has been onduted, where the requests flow with the mentioned distributions of its parameters had been modelled and the respetive routing approah had been applied. As a result of regression analyses of the data obtained from the experiment, the following dependenies were established: RP 0,077 ln λ 0,097 lnμt, (9) R 0,203 m 0,087 ln λ 0,015 lnμ, (10) R D Q T 0,138 m 0,100 ln λ 0,091 lnμ, (11) M Q T where R P, R D and R M represent the serviing level for the proposed method, the Clarke-Wright method and the mixed method, respetively. The primary analysis of the dependenies (9) (11) allows us to assert that there exist suh values of the demand parameters for whih the hoie of the most effetive routing method from the onsidered alternatives would be ambiguous, e.g. there would exist areas for the effetive use for the onsidered methods. For the values of the requests flow μt 10 h and λ 0,2 h 1, there exist suh values of the onsignment volume median, for whih the proposed routing proedure would be the most effetive way of proessing; but also suh a range of values for the onsignment volume median ould be found, where the highest serviing level is being reahed by using the mixed proessing tehlogy; for the other range of the median values, the Clarke-Wright method implemented in the real time would be the most effetive solution (Fig. 4). n order to determine the bounds of the proessing alternative variants, the intersetion points of R f m Q, λ T should be estimated. the urves desribing the orresponding dependenies Values of the parameters in the intersetion points define the intervals of the demand harateristis, for whih the use of the tehlogy results with the highest level of the requests flow proessing. The bounds of these intervals ould be determined as roots of the following equations: RP mq, λ T RD mq, λ T, RP mq, λ T RM mq, λ T, (12) RM mq, λ T RD mq, λ T. R m, λ R m, λ with respet to eah of the parameters of Solving the equation P Q T D Q T demand for delivery, we obtain the following expressions for determining the boundaries of the areas of the most effiient use of the proessing tehlogies: PD mq 0,401 lnμt 0,050 lnλ, PD λ exp 7,952 lnμt 19,841 m Q, (13) PD μt exp 2,495 mq 0,126 ln λ,

8 80 V. Naumov PD where m Q, λ PD and μ PD T represent the effetive use boundaries between the proposed tehlogy of requests proessing and the variant of serviing with the Clarke-Wright delivery routes for the median of the onsignment volume, the risk ratio of the requests interval, and the mean waiting time, respetively, tons, h 1 and h. Fig. 4. Dependene of the serviing level on the onsignment volume median ( μ 10 T h, λ 0,2 h 1 ) On the grounds of the equation RP mq, λ T RM mq, λ T expressions for estimation of boundaries of the most effetive use of tehlogies:, we obtain the following PM mq 0,039 lnμt 0,163 lnλ, PM λ exp 0,239 lnμt 6,123 m Q, PM μt exp 25,643 mq 4,188 ln λ, (14) PM m Q where, λ PM and μ PM T represent the effetive use boundaries between the proposed tehlogy of requests proessing and the mixed variant of serviing for the median of the onsignment volume, the risk ratio of the requests interval, and the mean waiting time, respetively, tons, h 1 and h. R m, λ R m, λ are the following expressions for The roots of the equation M Q T D Q T estimation of the effetive use boundaries: DM mq 0,190 lnμt 1,170 lnλ, DM λ exp 5,276 mq 6,171 lnμ T, DM μt exp 0,855 mq 0,162 ln λ, (15) DM where m Q, λ DM and μ DM T represent the effetive use boundaries between the tehlogy of proessing based on the Clarke-Wright method and the mixed variant of serviing for the median of the onsignment volume, the risk ratio of the requests interval, and the mean waiting time, respetively, tons, h 1 and h.

9 Forming delivery routes while proessing the stohasti 81 The onduted studies on estimation of boundaries for the most effetive use of onsidered alternative proessing tehlogies allow us to onlude that a preliminary assessment of the appropriateness of the proposed serviing tehlogy use is needed for the given values of the demand parameters. Thus, the hoie of the most effetive variant of the requests proessing tehlogy should be done in the following way: vopt arg max Ri (16) i P, D, M where P, D, M a set of alternative variants of the requests proessing: P the variant of serviing on the grounds of the proposed method, D the variant of serviing with the use of the Clarke-Wright-formed delivery routes, M the mixed variant of serviing; and R i the servie level average value for the i-th variant of serviing. t should be mentioned that the set of alternatives for the used proessing tehlogy ould ontain other methods (or algorithms) for the delivery routes forming. 5. CONCLUSONS Due to the ompetitive harateristis of the transport servies market and its stohasti nature, there is a strong need for inreasing the effiieny of serviing proesses provided by forwarding ompanies to their lients. To ahieve the goal of inreasing the effiieny of forwarding enterprise, the most important omponent of the serviing tehlogy is proposed to be improved the routing proess. The proposed heuristis allow researhers and logistis managers to solve the problem of the irular and pendulum routes formation under onditions of dynamially hanging demand (requests database). Program implementation of the desribed algorithms ould be used in deision support information systems for transport and freight forwarding ompanies. The onduted studies on deteting the boundaries of the proposed algorithm usability have shown that there are suh values of demand parameters when the developed approah ould be onsidered as the most effetive tehniques of the requests flow proessing. The perspetive diretion of further studies is the development of proedures for the formation of irular routes with the number of freight owner pairs greater than 2. Suh ases usually appear during the small-lot argo deliveries in urban environments, so the proposed method of the delivery routes forming ould be applied in a framework of the ity logistis theory. Referenes 1. Simhi-Levi, D. & Chen, X. & Bramel, J. The logi of logistis: theory, algorithms, and appliations for logistis and supply hain management. New York: Springer Siene p. 2. Кожин, А.П. & Мезенцев, В.Н. Математические методы в планировании и управлении грузовыми автомобильными перевозками. Москва. Транспорт p. [n Russian: Kozhin, A.P. & Mezentsev, V.N. Mathematial methods in planning and management of road freight transport. Mosow. Transport]. 3. Bovy, P.H.L. On modelling route hoie sets in transportation networks: A synthesis. Transport Reviews Vol. 29(1). P Hasani-Goodarzi, A. & Tavakkoli-Moghaddam, R. Capaitated vehile routing problem for multiprodut rossdoking with split deliveries and pikups. Proedia Soial and Behavioral Sienes Vol. 62. P Wedyan, A.F. & Narayanan, A. Solving apaitated vehile routing problem using intelligent water drops algorithm. n: EEE nternational Conferene on Natural Computation P El-Sherbeny, N.A. Vehile routing with time windows: An overview of exat, heuristi and metaheuristi methods. Journal of King Saud University (Siene) Vol. 22. P

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