Solving a bi-objective multi-product vehicle routing problem with heterogeneous fleets under an uncertainty condition

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1 Solving a bi-objective multi-roduct icle routing roblem with heterogeneous fleets under an uncertainty condition Reza Tavakkoli-Moghaddam 1*, Zohreh Raziei 1, and Siavash Tabrizian 2 Received: Acceted: Abstract: This aer resents a novel bi-objective multi-roduct caacitated icle routing roblem with uncertainty in demand of retailers and volume of roducts (UCVRP) and heterogeneous icle fleets. The first of two conflict fuzzy objective functions is to minimize the cost of the used icles, fuel consumtion for full loaded icles and shortage of roducts. The second objective is to minimize the shortage of roducts for all retailers. In order to get closer to a real-world situation, the uncertainty in the demand of retailers is alied using fuzzy numbers. Additionally, the volume of roducts is alied using robust arameters, because the ossible value of this arameter is not distinct and belongs to a bounded uncertainty set. The fuzzy-robust counterart model may be larger than the deterministic form or the uncertain model with one aroach and it has with further comlexity; however, it rovides a better efficient solution for this roblem. The roosed fuzzy aroach is used to solve the bi-objective mixed-integer linear roblem to find the most referred solution. Moreover, it is imossible to imrove one of the objective functions without considering deterioration in the other objective functions. In order to show the conflict between two objective functions in an excellent fashion, a Pareto-otimal solution with the ε-constraint method is obtained. Some numerical test roblems are used to demonstrate the efficiency and validity of the resented model. Keywords: Caacitated icle routing roblem, Bi-objective model, Robust otimization, Fuzzy otimization, Multile roducts. * Corresonding author. tavakoli@ut.ac.ir 1. Professor, School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran 2. M.Sc. Student, School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran 2. M.Sc. Grad., Deartment of Industrial Engineering, Sharif University of Technology, Tehran, Iran 207 Vol.3/ No.3/ Summer 2015

2 Solving a Bi-Objective Multi-Product Vehicle Routing Problem with Heterogeneous Fleets under 1. Introduction A icle routing roblem (VRP) was one of the most crucial and essential issues in recent years. The objective of this roblem is obtained through otimal route and distribution of icles in different nodes in a network. Vehicles start their travel from the deot and after serving all vertices (nodes), return to the deot (excet in an oen VRP). There are different tyes of a VRP according to different tyes of real-world alications. There are several assumtions in formulating VRP. Some considered assumtions in this study are include: a caacitated VRP considering the load on each route does not exceed the caacity of a icle. A multi-commodity VRP that is considering different tyes of roducts transorted on each route. A heterogeneous or homogeneous fleet considers different tyes or caacity for icles. Also, a multi or single-deot VRP is one of the critical assumtion. One of the most regular objective functions is to minimize the cost. For instance, the cost of fuel consumtion with resect to the distance between each node is considered in a few aers. It is one of the most imortant issues in the comany to save the fuel cost in transortation. Some aers discussed about the effect of distance travel on the fuel consumtion, [Kara, Kara and Yetis, 2007] roosed a caacitated VRP with a new cost function based on the distance and the load of a icle. The emirical analysis of [Sahin et al, 2009] discussed for the truck with the caacity of 20 tons, when the fuel cost for 1000 km traveled included 60% of the total cost. Therefore, it is imortant to reduce the fuel consumtion at the oerational level. Hence, the travel distance is one of the critical factors effecting on the fuel consumtion when it is considered the full loaded icle caacity. Also, one of the roblems dealing with the VRP is inventory decision. The inventory-routing roblem (IRP) is regarded as a medium-term roblem; however, the VRP is a short-term roblem [Moin and Salhi, 2007]. The first study on the IRP was introduced by [Golden, Assad and Dahl, 1984] to identify a routing roblem take into account the inventory assumtions. One of the critical issue is to integrate inventory management with a routing decision. As recent studies on this roblem [Yu, Chen and Chu, 2008] roosed a model with slit delivery and homogeneous fleet. They used a Lagrangian relaxation method that is 208 decomosed in two sub-roblems; inventory and routing roblems in order to solve the large-scale roblem. [Raa and Aghezzaf, 2009] roosed an integrated model of inventory management and icle route with a fleet size in a ractical case in the context of vendor-managed inventory (VMI) with a cyclic aroach. The aim is to minimize the average distribution and inventory cost. [Coelho, Cordeau and Laorte 2012] roosed and IRP model with a more ractical case in the context of VMI. They assumed that roducts could be transshied from a sulier to a customer and a customer to another customer. [Li et al, 2014] investigated a novel model for the IRP for a gasoline distribution industry, which minimizes travel time. They rovided tabu search and an adated US algorithm as a art of a solution rocedure to obtain a better solution. In the last two decades, the alications of this roblem under uncertainty have increased in both ractitioners and researchers area. This uncertainty existed in the board sectrum of the theories that include stochastic, fuzzy and robust uncertainty. Fuzzy uncertainty is associated with the ambiguity of linguistic statements. These ambiguous situations may occur in objectives or in the arameters. It is often haening in arameters, such as demand and time windows. [Cheng, Gen and Tozawa 1995] roosed a VRP model with time windows and considered due-time instead of fixed time windows. Their fuzzy membershi function shows the degree of customer satisfaction of the service time. [Wang and Wen, 2002] investigated a VRP model for a Chinese ostman roblem and considered fuzzy time windows. [Tang et al, 2009] roosed bi-objective VRP model with fuzzy soft time windows and considered a linear and concave fuzzy membershi. Their objective functions are to minimize the routing cost and maximize the overall customer satisfaction level. [Ghannadour et al, 2014] roosed a multi-objective dynamic VRP with fuzzy time windows. The objective functions are to minimize the total travel distance and waiting time on icles and maximize customer references for service. They roosed a genetic algorithm with three basic modules to solve this roblem. Other studies have considered demand and travel time as fuzzy arameters. [Guta, Singh and Pandey, 2010] solved a multi-objective VRP with fuzzy time windows. The objective functions are to minimize the fleet size, maximize the average

3 Reza Tavakkoli-Moghaddam, Zohre Raziei, Siavash Tabrizian grade of satisfaction, minimize the total travel distance, and minimize the total waiting time for icles. [Cao and Lai, 2010] roosed an oen icle routing roblem (OVRP) model. They considered fuzzy demands and used a fuzzy chance-constrained rograming model based on a fuzzy credibility theory to solve this model. Some aers have considered travel time as fuzzy arameters. [Kuo, Chiu and Lin, 2004] roosed a VRP model with time windows and fuzzy travel time. They used ant colony otimization (ACO) to solve this roblem. [He and Xu, 2005] considered a single deot VRP with fuzzy demand and robability travel times between customers that minimizes travel time by considering caacity and arrival time constraints on the greatest degree. Because of the comlexity of an uncertain VRP, they used a genetic algorithm to solve it. [Zheng and Liu, 2006] roosed a single-deot VRP model with time windows and considered travel time as fuzzy variables. They designed the integrated fuzzy simulation and genetic algorithm to solve this roblem. [Xu, Yan and Li, 2011] roosed a bi-objective VRP model with soft time windows and fuzzy random environment. The objective functions are to minimize the total travel time and maximize the average satisfaction level of all customers. In order to obtain the equivalent cris, the concet of the fuzzy random exected value is used. Some recent studies have considered a dynamic condition. [Ghannadour, Noori and Tavakkoli-Moghaddam, 2013] considered the customer satisfaction level by using the concet of fuzzy time windows with the revious assumtions. Also, the first objective function is to minimize the total traveling distance and waiting time on icles and the second one is to maximize the customer reference for service. Some recent studies have used the concet of robust otimization in a VRP model. This concet is used to comensate for the limitation of a stochastic aroach for this roblem, because this aroach does not increase the comlexity of the roblem. [Montemanni et al, 2007] roosed a new extension for solving travel salesman roblem by robust otimization. They alied a robust deviation criterion to imrove otimization over an interval data roblem. [Sungur, Ordónez and Dessouky, 2008] roosed a VRP model with demand uncertainty to introduce robust otimization to solve the roblem. The aim is to minimize transortation costs with satisfying all demands. [Sungur et al, 2010] roosed an 209 uncertain VRP model. They considered scenariobased stochastic rogramming for uncertainty in customers and robust otimization for uncertainty in service times. [Li, Xu and Zhou, 2012] rovided chance-constrained rogramming for a fuzzy demand in a VRP model. Also, they found the lowest total mileage in the model by using the imroved tabu search algorithm. [Gounaris, Wiesemann and Floudas 2013] roosed caacitated VRP model with uncertain demand. They used robust otimization to solve the roblem by develoed robust rounded-caacity inequalities for two broad classes of the demand caacity. [Agra et al. 2013] roosed a VRP model with time windows and roosed a new formulation of a robust roblem. The first formulation extends the well-known resource inequality formulation by considering adjustable robust otimization and the second aroach formulation generalizes a ath inequalities formulation for the uncertain context. [Solano- Charris, Prins and Santos, 2014] considered a robust VRP with uncertain travel time with discrete scenarios. The objective function is to minimize the worst total cost in each scenario. They used a genetic algorithm to solve this roblem for small and medium-sized roblems. [Sun and Wang, 2015] roosed a VRP model with uncertainty in the demand and transortation cost. They used robust otimization considered situations ossibly related to bidding and caital budgets to solve the roblem. To summarize the contribution of this aer, a bi-objective caacitated VRP with uncertainty (UCVRP) is resented. Also, an assumtion of multi-roduct and heterogeneous icles (with different caacity) is considered. The fuzzy objective functions are considered that minimize the traditional cost by considering the fuel consumtion cost due to the traveled distance for fully loaded icles as the first objective. According to the constraints of inventory management in the model, the second objective is to minimize the shortage of roducts (i.e., negative inventory). Two tyes of uncertain arameters (e.g., demand of retailers and volume of roducts) are considered in the model. The uncertainty in the demand of retailers is due to the linguistic ambiguity of eole, so it is exressed as fuzzy arameter. Also, the uncertainty in the volume of roducts is exressed as robust arameter, because the level of uncertainty of the volume of roducts belongs to the small bound without knowing

4 Solving a Bi-Objective Multi-Product Vehicle Routing Problem with Heterogeneous Fleets under ossible value. As a result, a new model is roosed by considering two uncertainty aroaches and fuzzy objective functions in this study. The rest of this aer is organized as follows. Classifying and reviewing the literature in Section 1. In Section 2 introduces the deterministic formulation of the model. Section 3 resents a fuzzy aroach used in the objective functions and demand arameter and then rewrites fuzzy formulation of the model. Section 4 resents a robust otimization aroach for an uncertain volume of roducts and rewrites the final form of the model. Numerical results and sensitive analysis are rovided in Section 5, and we conclude this study in Section Problem Statement In this aer, we consider the bi-objective multiroduct VRP with heterogeneous fleet and retailer s inventory assumtion. We assume that the icles are driving the same seed with full load on each tour. Additionally, we consider an uncertain demand for retailers, volume of roducts and objective functions. Additionally, the aim of this model is to minimize the total cost and shortage of roducts (i.e., negative inventory) with fuzzy objectives. The total cost consists of used icle, fuel consumtion and shortage cost as lack of inventory in each demand oint. It is notable that the cost of fuel consumtion is calculated with resect to the travel distance, because the seed of full loaded icles are considered the same on each route. 2.1 Assumtions The main assumtions of the resent model are listed below. Considering the caacitated icle routing roblem (CVRP). Considering different tyes of roducts. Considering a icle with the different caacity (i.e., heterogeneous fleet). Considering an inventory olicy. Considering fuel consumtion as a art of cost function in the first objective. Minimize the shortage of roducts as the second objective. Demands of retailers are assumed to be in a fuzzy nature. Volumes of roducts are assumed to be in a robust nature. Considering the fuzzy objective functions. Figure 1 rovides an examle of a mathematical model in order to illustrate this roblem with the above assumtions. This samle roblem is described for nine retailers, three full loaded icles with different caacity and same seed and three tyes of roducts. According to the described model, the tours constructed from a central deot to visit retailers are shown. Also, the icles which choose to carry the amount of three tyes of selected roducts are shown. Figure 1. Illustrative examle of the given roblem 210

5 2.2 Model Formulation Alied sets, indices, arameters and modeling variables are as follows: Sets 1,..., P Set of roducts v 1,..., V Set of icles c 1,..., C Set of retailers Parameters dem c Reza Tavakkoli-Moghaddam, Zohre Raziei, Siavash Tabrizian The amount of demand of retailer c for roduct ca v d cc ˆ c v ca vol f v k fuel c c zl 1, zu 1 zl 2, zu 2 zll, zuu v, M Caacity of icle v Distance between retailer c and c Cost of icle v Caacity of roduct in central deot Volume of roduct Amount of fuel consumtion of icle v er unit distance and unit icle weight Density er each tye of roduct Price of fuel consumtion er unit distance er unit icle weight Cost of shortage (i.e., negative inventory) for roduct Fuzzy multi objective arameters Uer bound and lower bound of right hand side of fuzzy constraints Robust arameters Decision variables x vc 0 u vc 0 inv c inv c invn c vcc ˆ z v Amount of roducts delivered to retailer c by icle v Amount of roducts when entering in retailer c by icle v Amount of inventory levels of roduct for retailer c Amount of ositive and negative inventories of roduct for retailer c If icle v is assumed to travel from retailer c to c Otherwise If icle v is used Otherwise 211

6 yvc, zv, y, z, vcc ˆ vcc ˆ vcc ˆ Robust variables Fuzzy variable The Auxiliary variable for linearization 2.3 Mathematical model fuel z 1 Min cv zv c fv d c invn ˆ vcc ˆ cc ˆ c vv vv cccc P cc (1) z 2 Min invn c PcC s.t. 1 vvcc ˆ vcc ˆ c C, c>1 (3) cc ˆ vcc ˆ v V,c C (4) cc ˆ vccˆ xvc ca P (5) vvcc vol x ca z vc v v V (6) v PcC uvc ( u x ) vcˆ vcˆ vcc ˆ (2) Pv, Vcc,, Cc, 1 (7) u v1 x cc Pv vcˆ, V (8) z ˆ vcc ˆ v cc vvc, C (9) xvc M demc v V,cC (10) P cc ˆ invc xvc demc vv inv c inv c invnc Pc, C (11) Pc, C (12) 0,1 vcc ˆ cc ˆ, C (13) z v 0,1 v V (14) x vc 0 Pv, Vc, C (15) u vc 0 Pv, Vc, C (16) invc, invnc 0 Pv, V (17) inv c free Pc, C (18) Vol.3/ No.3/ Summer

7 Reza Tavakkoli-Moghaddam, Zohre Raziei, Siavash Tabrizian As mentioned before, this aer aims at minimizing the cost of urchasing the icle, fuel consumtion and shortage as the first objective function and minimizing the shortage of roducts (i.e., negative inventory) as the second objective function. Both of them are shown as the first and second equations. The first constraint as shown in Eq. (3) ensures that each retailer stands on just one route. Eq. (4) guarantees whether icle v enter in node c, it should leave this node. Eq. (5) ensures that the amount of delivering roduct tye does not exceed its caacity. Eq. (6) is the caacity of a icle constraint. Eqs. (7) and (8) are sub-tour elimination constraints based on the amount of roducts when entering in nodes and ensure that no route is aart from a central deot. Eq. (9) ensures that each icle is assigned to one route. Eq. (10) states that roduct tye is delivered to retailer c by icle v, if icle v is allocated the route of retailer c. Eqs. (11) and (12) are the inventory constraints stating that the amount of delivered roducts to a retailer (i.e., inventory level) is the difference between load delivering to retailers and the demand for them. On the other hand, it is the difference between the inventory level of retailer and the shortage of roducts in one eriod. 2.4 Linearization Ste Due to the non-linear term aeared in the aforementioned mathematical model, a linearization rocedure for Eq. (7) is alied as follows: uvc u x bigm(1 ) vcˆ vcˆ vcc ˆ (19) vcc,,, 1 3. Fuzzy Mathematical Programming The roosed model for the VRP resented in the receding section is a deterministic model. The fuzzy multi-objective mixed-integer linear rogramming (FMOMILP) is rovided in this section to overcome the rigidity of the deterministic model. The mathematical model in the fuzzy environment can be included vague goal and arameters. In the deterministic formulation, the total cost consists of the cost of the icle, cost of fuel consumtion and enalty cost for shortage of roducts (i.e., negative inventory). In the real world, most of these costs are not easily measured, because human ercetion is effected in this evaluation. When we face with this issue, the decision maker wants to reach some asiration levels of the objective functions and allow some violations in the arameters of constraints. In addition, the fuzzy objective functions, the demand is considered as fuzzy arameter reresented by intervals of tolerance. This aer uses a fuzzy aroach of [Zimmermann, 1978] and other [Chen and Chou, 1996] to deal with the imrecise arameters and fuzzy situation of objective functions. The linear membershi functions are emloyed for constraints and objective functions. This aroach is resented below. Max (20) s.t. min Minimum objective zr ( x) function (21) max Maximum zs ( x) function objective (22) ( x) gl Fuzzy constraint (23) g ( x) b Deterministic constraint (24) In the above model, min zr ( x), max ( x) and ( x) refer to a zs gl minimum objective, a maximum objective and fuzzy constraints, resectively. The membershi functions based on reference are linear as shown below: 1 z u r z r u z ( ) ( ) r z x r l ( ) u min x zr zr x z u l r z r z r z r 0 z z l r r (25) 1 z u s zs ( ) l max ( ) zs x z x s z l z ( x ) z u z u l s s s s zs zs 0 z z l s s (26) With two membershi functions for minimization and maximization objective functions, we can obtain an asiration level for the value of the objective functions. 1 g ( x) b gx ( ) b g ( x) b g( x) b d d 0 g( x) b d (27) 213

8 Solving a Bi-Objective Multi-Product Vehicle Routing Problem with Heterogeneous Fleets under where the left hand side of the fuzzy constraints is g( x ), the -th fuzzy value is b and the tolerance level that the decision maker can consider in the -th constraint of the fuzzy inequality is d. According to the fuzzy objective functions and constraints, the MILP formulation can be exressed as follows: max (28) ( z u z l ) z u z (29) ( z z ) z z (30) u l u ( dem u dem ) dem u c c c ( invc xvc ) vv, c (31) ( dem dem l c c) demc invc x, c vc vv (32) Constraints (3)-(6), (14), (8)-(10), (12) and (13). 0 1 (33) In order to obtain uer and lower bounds of two fuzzy objective functions, use the following rocedure. In minimization goals, z l can be i obtained by solving each of single objective linear rogramming. Also, in minimization goals, z u i is the non-ideal solution (i.e., maximum value). After calculating of uer bound of each objective, if the uer bound is infinite, eliminated the basic constraint (e.g., sub-tour elimination) of the roblem and calculated the uer bound of objective functions by minimizing them. 4. Robust Mathematical Programing Now, the robust counterart roblem for a fuzzy VRP is resented. The volume of each roduct tye is the other arameter of this model, which is faced with uncertainty. It is better to use a bound uncertainty set of data to solve this model efficiently in an uncertain condition. The exectation is that such a solution with a robust arameter will be efficient in this bound for every ossible outcome. According to the robust formulation [Bertsimas and Sim, 2004], the following constraint with uncertainty in arameter vol is considered, where J is the set of coefficients for this arameter and takes the value according to a symmetric distribution with the mean that is equal to the nominal value vol in the interval vol vol ˆ, vol vol ˆ. vol x ca z vc v v PcC Parameter is introduced that takes a value in interval 0, for each. The role of is adjusted the robustness of the roosed method against the level of rotection of the solution. The aim of this tye of robustness is rotected against that all cases u to change, and only one coefficient ( vol ˆ ) is changed by ˆ vol. The obtained robust solution is feasible deterministically, even if more than one change, the robust solution will be feasible with very high robability. The formulation is shown below: max cx s.t. vol x vc PcC max St SJ, S, tj\ S vol ˆ y vc ˆ vol y vc Pc C ca z v v yvc xvc yvc x0, y 0 The value of the second art of the left hand side of Constraint (36) can be comuted for each icle v and retailer c. Also, by considering 0, J, the robustness of the model against the level of rotection is more flexible than J. This model is equal to the objective function of the linear model as shown below: ( x *, ) max( vol ˆ x * i i vc zv ) PcC (39) z (40) v vv z v 0 1 (41) 214

9 With regards to the final form of the model introduced by [Bertsimas and Sim 2004], the new form of model is rewritten, which is established based on the dual of the model. First, the dual of the model is considered below: min vv z P (42) zv vol ˆ y cv, (43) vc cc (44) z 0 v v 0 (45) The objective value is obtained from the dual roblem is feasible and bounded for all 0, J. Now, the first model by considering the fuzzy objectives and fuzzy demands as well as a robust volume of roducts in constraints is rewritten. max (46) s.t. Normal constraints 1 vvcc ˆ vcc ˆ c C, c>1 (47) cc ˆ vcc ˆ cc ˆ vccˆ v V,c C (48) xvc ca vvcc P (49) uvc u x bigm(1 ) vcˆ vcˆ vcc ˆ vcc,,, 1 (50) u v1 x cc vcˆ Pv, V (51) z ˆ vcc ˆ v cc vvc, (52) xvc Mvcc ˆ v V,c C (53) P cˆ C inv c invc invnc Pc, C (54) Fuzzy constraints for objective function ( z u z l ) z u z (55) ( z z ) z z u l u Reza Tavakkoli-Moghaddam, Zohre Raziei, Siavash Tabrizian Fuzzy constraints for demand ( dem u dem ) dem u c c c ( invc xvc ) vv Pc, C (57) ( dem dem l c c) demc invc xvc vv Pc, C (58) Robust constraints for volume of roduct vol x z ca z vc v v v v PcC P v V (59) zv vol ˆ yvc cc Pv, V (60) xvc yvc Pv, Vc, C (61) (56) 215

10 Solving a Bi-Objective Multi-Product Vehicle Routing Problem with Heterogeneous Fleets under xvc yvc Pv, Vc, C (62) 0 1 (63) x0, y 0 (64) zv 0, 0, xvc 0, uvc 0, invc 0, invnc 0 (65) {0,1}, z {0,1} vcc ˆ v 5. Extension of the Model with Decreasing the Load on Each Route Changing in the load of the icle has a direct imact on fuel consumtion. After delivery of roducts with large sizes (in the weight), the amount of fuel consumtion has decreased. However, in the medium and small sizes (in the weight) of the delivery rocess, the decreasing in the fuel consumtion is very low. So, the amount (66) of fuel consumtion can be considered fixed. In this section, the extension of the model is considered for assessing the effect of the load decreasing for delivered roducts in large sizes on the icle fuel consumtion. First, the deterministic form of the model is described as follows. Put u vcc ˆ vc then vcc ˆ fuel z 1 Min cvzv + c fvvol k d vv PvVc Ccˆ C vcc ˆ cc ˆ c invnc Pc C (67) s.t. u M (1 vcc ˆ ) vcc ˆ vc Pv, Vcc,, ˆ C (68) u M (1 vcc ˆ ) vcc ˆ vc Pv, Vcc,, ˆ C (69) ˆ ˆ vcc M vcc Pv, Vcc,, ˆ C (70) vcc ˆ 0 (71) Constraints (2)-(18) (72) In the Eq. (67) minimize the cost objective function includes the cost of urchasing the icle, fuel consumtion cost with considering the effect decreasing the load on fuel consumtion of a icle. Eqs. (68) - (71) are the linearization rocedure for cost of fuel consumtion term in the first objective function. Because the uncertainty in the volume of roducts is considered as robust arameter, it is needed to consider the robust formulation for this arameter in the objective function. Since the generating the robust model based on deterministic model is described before, in this section the relations are just reresented. The new model with fuzzy and robust arameters is shown bellow. Put A vcc ˆ fuel c fvvol k d cc ˆ then z 1 Min cv zv + c invn c (73) vv PcC A vcc ˆ vcc ˆ,,, ˆ PvVcc C (74) 216

11 Reza Tavakkoli-Moghaddam, Zohre Raziei, Siavash Tabrizian vcc ˆ (75) Constraints (68)-(72) (76) As described before, the robust model in Eq. (74) for mentioning robust factor is shown below. Max ˆ * z ˆ PvV cc cc vcc ˆ ˆ ˆ (77) s.t. (78) Pv VcC ˆ cc z vcc ˆ (79) 0 z 1 vcc ˆ (80) The dual of the above model is considered below: Min Pv V cc ˆ cc vcc ˆ z z aˆ * Pv, Vcc,, ˆ C vcc ˆ vcc ˆ vcc ˆ (82) 0 vcc ˆ Pv, Vcc,, ˆ C (83) z 0 (84) Now, the final form of the new model considering the fuzzy and robust constraints of original model is shown below. (85) a z Pv V cc ˆ cc vcc ˆ vcc ˆ PvV cc ˆ cc vcc ˆ (86) z a y vcc ˆ vcc ˆ vcc ˆ Pv, Vcc,, ˆ C (87) y y vcc ˆ vcc ˆ vcc ˆ,,, ˆ 217 (81) PvVcc C (88),, z 0 vcc ˆ vcc ˆ (89) Constraints (47)-(66) (90) 6. Comutational Results In the revious section, the given roblem was described in details. In this section, the numerical results of the deterministic, fuzzy, robust and fuzzy-robust model with Equations (1)-(66) are resented. The results is obtained by solving some instances via general algebraic modelling system (GAMS) /CPLEX. It is essential to create random intelligent various samle roblems in order to evaluate the mathematical model. For nominal exeriments, eight samle roblems with small and medium sizes are shown in Tables 2 and 3. The samle data are chosen from a uniform bound as shown in Table 1. It is notable that the cost of the three arameters in first objective function (e.g., c, c v and fuel c ) should be in the same scale. The cost scaled of fuel consumtion is different from the cost of the icle and shortage. It causes to ignore the effect of fuel consumtion cost. In order to overcome this issue, each arameter is normalized than the range of its data.

12 Solving a Bi-Objective Multi-Product Vehicle Routing Problem with Heterogeneous Fleets under Table 1. Source of random generations for samle roblems Parameters Corresonding random distribution Parameters Corresonding random distribution dem c Uniform (0,31) ca Uniform (200000,300000) ca v Uniform (200000,300000) vol Uniform (5,15) d cc ˆ Euclidean distance f v Uniform (5,15) c v Uniform (250000,300000) k Uniform (1,10) c Uniform (250000,300000) According to the discrete and continuous variables, the roosed model is a mixed-integer linear model. In order to validate the model and examined the result of the samle roblems for fuzzy, robust, fuzzy-robust and deterministic models, used GAMS software. In Table 2, dimensions and characteristics of small-sized instances with the result of the deterministic and fuzzy-robust model is shown. As resented in Table 2, the total cost of the samle roblem included, the cost of using icles, fuel consumtion and shortage of roducts for each retailer is between and in deterministic form. Also, shortage of roducts as the second objective is between 70 and 107. Table 2. Small-scale instances: Results for the deterministic and fuzzy-robust model Problem name Indices Deterministic Fuzzy-robust Obj1 Obj2 Obj1 Obj2 S01 ()4-(c)5-(v) S02 ()4-(c)6-(v) S03 ()4-(c)7-(v) S04 ()4-(c)8-(v) Min ()4-(c)5-(v) Mean ()4-(c)7-(v) Max ()4-(c)8-(v) (i): indices of roducts, (c): indices of retailers, (v): indices of icles In order to validate the exact solution of GAMS and erformance of model by increasing the comlexity of the roblem, four samle roblems with a medium size are created and otimized. The results of medium-sized roblems are shown in Table 3. Table 3. Medium-sized instances: results for the deterministic and fuzzy-robust model Problem name Indices Deterministic Robust-fuzzy Obj1 Obj2 Obj1 Obj2 M01 ()6-(c)8-(v) M02 ()6-(c)9-(v) M03 ()7-(c)9-(v) M04 ()7-(c)10-(v) Min ()6-(c)8-(v) Mean ()6-(c)9-(v) Max ()7-(c)10-(v) (i): indices of roducts, (c): indices of retailers, (v): indices of icles 218

13 Reza Tavakkoli-Moghaddam, Zohre Raziei, Siavash Tabrizian Figure 2. Otimum objective function for small-sized instances in comarison with fuzzy otimum results The results of the fuzzy and robust models for small instances are shown in Figures 2 and 3, searately. As characteristic of a fuzzy set theory, the solution of the fuzzy model is more stable than the deterministic form and shows the better insight into the model structure. So, the fuzzy results better than deterministic. Also, the result of the robust otimization model is better than the deterministic model, because this robust aroach [Bertsimas and Sim 2004] guaranteed that constraints are satisfied by changing data. Also, these results are the same for medium sizes. The results of comarison fuzzy-robust model with other medium samle roblems are shown in Figure 4. It is obvious that the result of both objective functions of the fuzzy-robust model are the best otimization value than the other ones. If the uncertainty of arameters is known by the most ossible value of them, it is efficient to use a fuzzy otimization aroach. If the uncertainty of arameters is unknown and only a small violation of them is known, it is efficient to use a robust otimization aroach. The most aroriate solution is achieved with these aggregation of aroaches for an uncertain model. In order to evaluate the flexibility of the resented model, we use a sensitive analysis rocedure in different conditions and measure the flexibility of the model. Therefore, various scenarios for robust model and fuzzy model are defined and imlementing sensitivity analysis rocedures. Figure 3. Otimum objective function for small-sized instances in comarison with the robust otimum results 219

14 Solving a Bi-Objective Multi-Product Vehicle Routing Problem with Heterogeneous Fleets under Figure 4. Otimum objective function for medium-sized instances in comarison with fuzzy otimum and robust otimum results In order to demonstrate the conflict between objective functions, we show the Pareto otimal solution. The augmented ε-constraint method is used to achieve this goal. Table 4 illustrates the ideal (Z I ) and nadir (Z N ) oints for objective functions. Also, the diagram of a Pareto front is shown in Figure 5. Table 4. Uer and lower bound Ideal and Nadir oints Objective 1 (Cost) Objective 2 (Shortage) Uer bound Z N = Z N =64 Lower bound Z I = Z I =56 According to the non-dominated solution shown in Figure 5, the first two oints of the Pareto front have the same value of the cost objective function, while the shortage of roducts objective function are different. Also, the two other oints are the same condition. Conflict among objective functions is not only clear in the Pareto frontier objective, but also is obvious in the model logically. According to the second objective function, if an unlimited amount of negative inventory for the second objective function is allowed, the number of delivered roducts by each icle (Constraints 11 and 12) is reduced. Also, the number of used icles and icle tris (Constraints 6 and 10) are reduced. So, the value of the cost objective function is reduced. Figure 5. Pareto front 220

15 Reza Tavakkoli-Moghaddam, Zohre Raziei, Siavash Tabrizian 6.1 Analysis Based on the Uncertainty Level of the Demand The scenarios on reducing demand is considered and chosen samle roblem S02 as an examle. The result of reducing demand resented in Table 5. Two analysis is made for reducing the demand. First, the total cost is constant or decreases and the shortage of roducts decreases too. It is due to the fact that the icle number is constant. As a result, the shortage of roducts decreases means while the cost is constant or decreases. Second, costs decrease a lot and the shortage of roducts increases, which means that deciding to use fewer icles due to the demand reduction. So, the amount of the remained facilities are not caable of sulying more shortage of roducts. The results from the fuzzy model and robustfuzzy model are congruent with first exectation scenarios from the suggested model erformance. When the reduction of demands is increasing, the reduction rate of the objective function of the fuzzy-robust model is more than the fuzzy model. It reresents the best erformance of the fuzzyrobust model. These reduced rates for both models are shown in Figures 6 and Analysis Based on a Protection Level for Robust Parameters It is accetable to estimate the change in the objective functions with resect to the change in rotection level i of constraint i measure the validation of the robust otimization model. Hence, the rice of increasing or decreasing a change is evaluated at the rotection level for each constraint by the results of the objective functions in the robust model. Also, the results of the fuzzyrobust otimization model is obtained for each level of rotection. Moreover, the robability bounds of constraint violation is obtained [Bertsimas and Sim 2004]. In other words, if more than the uncertain arameters of the right i hand side of i-th constraint change, the robability of a violation of i-th constraint is at most these bounds. The result of change i on objective functions and robability bound of constraints for the robust and fuzzy-robust models are shown in Table 6. Table 5. Scenarios designed for the sensitivity analysis rocedure on demand Scenario Amount of demand Fuzzy Fuzzy-robust Obj1 (Cost) Obj2 (Shortage) Obj1 (Cost) Obj2 (Shortage) Demand Demand Demand Demand Demand Demand Figure 6. Reduced rate of the second objective function for both models in each scenario 221

16 Solving a Bi-Objective Multi-Product Vehicle Routing Problem with Heterogeneous Fleets under In the robust otimization model, the objective functions are increasing the rotection level; however, the trend of the objective functions in the fuzzy-robust model is not distinct. It is obvious that the otimal value of the objective functions in both models is affected when the rotection level is increased. For examle, in order to have a robability guarantee of a constraint violation at most 15.74%, it needs to reduce the first objective function of the robust model, 69%, while the reduction level for the fuzzy-robust model is 41%. The effect of the rotection level on an otimal value of the first objective function for the robust and fuzzy-robust models are shown in Figure 8. The results illustrate the better erformance of the fuzzy-robust model for each level of rotection. Also, Figure 9 illustrates the otimal value of the first objective function with resect to the robability bound of constraint violation. 6.3 Comarison Between Results of Original and Extension Models In order to show the effect of decreasing of icle load (for the model resented in Section 5), the result of the deterministic model described for small-sized instances are shown in Table 7. Also, the result of the fuzzy-robust model is shown in Table 7 to show the effect of the robust arameter in the objective function. Table 6. Results of the robust and fuzzy-robust solutions for Probability bound Robust Fuzzy-robust Obj1 Obj2 Obj1 Obj Figure 7. Otimal value of the robust and fuzzy-robust models as function of 222

17 Reza Tavakkoli-Moghaddam, Zohre Raziei, Siavash Tabrizian Figure 8. Otimal value of the robust and fuzzy-robust models as a function of the robability bound of the constraint Table 7. Comarision between the models for small-sized instances Problem Original model Extension model Deterministic Fuzzy-robust Deterministic Fuzzy-robust Obj1 Obj2 Obj1 Obj2 S S S S According to the results of Table 7, both objective functions considering load is worse than the original model for a deterministic form. In the fuzzy-robust model, adding the robust arameter to the objective function is changed without any secific behavior in both objective functions. 7. Conclusion In this study, we roosed a novel bi-objective caacitated icle routing roblem under uncertainty (UCVRP) and considered the multiroducts with heterogeneous fully loaded icles (i.e., different caacity) assumtions. The first objective function minimizes the cost of the used icles, fuel consumtion and shortage of roducts for each retailer. The second objective minimizes the shortage of roducts. The inventory constraints are considered in the model to obtain the shortage of roducts. We considered uncertainty in some arameters included the demand of retailers and volume of roducts. We utilized fuzzy otimization and robust otimization aroaches searately to coe with the uncertain arameters and objective functions. In order to overcome the ambiguous of linguistic for demand of retailers and most ossible amount of known demands in the real world, considered a fuzzy otimization model. Also, the uncertainty in the volume of roducts belongs to the bounded uncertainty set in the real world. Therefore, we considered a robust otimization model. Finally, an integrated model (i.e., fuzzy-robust model) resented to deal with both uncertainty arameters. Additionally, we coed with fuzzy uncertainty in the bi-objective mixed-integer linear roblem. The augmented ε-constraint method is used to demonstrate the conflict between objective functions. Also, an extension of model considering the effect of load of fuel consumtion is resented. Some samle roblems are generated and solved to demonstrate the efficiency of the fuzzy-robust model. Finally, we roosed a sensitivity analysis based on a reducing demand for the fuzzy and fuzzy-robust model and increasing rotection level for the robust and fuzzy-robust model. The results showed the efficiency of the integrated model. 223

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