Fakultät Wirtschaftswissenschaften

Bundesdeutscher Arbeitskreis für für Betriebswirtschaft e. V. Umweltbewusstes Management e. V.

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A MIXED-INTEGER PROGRAMMING MODEL FOR A VEHICLE ROUTING PROBLEM IN THE FOOD RETAILING INDUSTRY Von Felix Tamke

Dresden, Juli 2018

Herausgeber: Die Professoren der Fachgruppe Betriebswirtschaftslehre

A mixed-integer programming model for a vehicle

routing problem in the food retailing industry

Felix Tamke

July 18, 2018

Abstract

We consider a rich vehicle routing problem occurring at a large Ger-man food retailer. The company has to distribute goods from a centralwarehouse to a number of hypermarkets on a daily basis. To accom-plish this task, it engages several carriers, which supply the necessaryamount of trucks and drivers. Payment of carriers is regulated by distincttransport tariffs for the supply of each market and each commodity class.The latter result from four different transport temperatures. Moreover,compartmentalized vehicles are used. A last-in-first-out loading policyis applied, since rearranging of cargo on the load bed is prohibited plusunloading is only possible from the rear end. Furthermore, splitting deliv-eries is allowed and backhauls are required to return used loading devices.We describe extensively the mentioned problem as well as the tariff-basedcost function. Furthermore, a mixed-integer program is presented whichcontains all relevant features.

Keyword: Routing, Retailing, Transport tariffs, Multiple Compartments

1 Introduction

1.1 Problem description

The German food retailing market is dominated by discounters. In combinationwith a small willingness to pay of the customers, this leads to a high pricecompetition among the companies and consequently, small margins (Rehder andStange, 2015). Thus, cost savings are vital. An important part of the logisticscosts are caused by the supply of the outlets (Kuhn and Sternbeck, 2013). Hence,efficient transportation planning to reduce logistics costs is essential for foodretailers and therefore, will be further investigated in this paper.

To contribute to this field of research, we describe, model mathematically,and solve exactly a real-world vehicle routing problem (VRP) occurring at alarge German food retailer. The company has to distribute its goods from acentral warehouse to circa 120 hypermarkets (in the following only referred to asmarkets) on a daily basis. As a full range trader, the set of goods offered by the

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retailer is very heterogeneous, ranging from frozen food to clothes. In terms ofdistribution, especially the transport of food is subject to strict legal regulations.As shown in Table 1, this leads to four different commodity classes (productcategory) regarding the transport temperature. Furthermore, the goods aretransported on loading devices which accumulate at the markets. Since thelocal storage space is limited, those devices need to be carried back to thecentral warehouse in accordance to a fixed schedule. However, a backhaul issolely possible from the last market on a route to reduce loading and unloadingoperations.

Table 1: Commodity classes and transport temperatures

Commodity class Transport temperature Example products

1 ≤ −18 Frozen food2 −1.5 to +2 Cheese, cold cuts, meat3 +13 to +17 Fruits, vegetables, plants4 No requirements Non-food items, textiles

The retailer performs the delivery process in cooperation with several freightforwarders. Each of them has a dedicated area with a number of markets to sup-ply, whereby the areas do not overlap. The collaboration is organized as follows.All markets order products for the next day from the central warehouse. After-wards, within a time frame of two hours, the staff of the warehouse schedules theroutes for the next day and passes these on to the different freight forwarders.Subsequently, those have to provide the necessary capacities, i.e. vehicles anddrivers, on the next day and execute the tour. Thus, the planners do not haveto consider a maximum number of trucks available or legal requirements forthe drivers’ working hours. Moreover, as long as no backhaul is performed,the vehicles are not obliged to return to the warehouse as they belong to theforwarders. However, the planners have to take heed of other particularities.The freight forwarders operate vehicles whose trailers can be divided into upto three compartments with flexible sizes as shown in Figure 1. Thus, they areable to carry different commodity classes at once. Yet, cooling units are justpositioned at the front and rear end of the trailer. As a consequence, chilledgoods must not be loaded into the second compartment. Furthermore, dividerscan only be shifted in discrete steps of size three. Additionally, unloading thetrailer is solely possible from the rear end and rearranging of goods on the loadbed is undesired. Therefore, a last-in-first-out (LIFO) unloading is required.

Another peculiarity arises from the contracts negotiated among the retailerand the freight forwarders. The prices which have to be paid by the retailerare not proportionate to the distance traveled, but depend on the commodityclasses transported as well as the markets visited on a tour. In particular, thecontracts contain only transport tariffs for direct tours from the warehouse tothe markets. There, a distinction is made between tours without (open) or withbackhauls (closed). Hence, eight freight rates exist for each market.

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Truck 3 2 1

unloading(LIFO)

dividerpallet space

cooling unit

Figure 1: Structure of a vehicle with three compartments

The remainder is organized as follows. First, we discuss related literaturein subsection 1.2 and demonstrate that the problem has not been consideredas a whole. In section 2, the tariff-based real-world cost function is modeledmathematically and a mixed-integer programming model for mentioned problemis presented. Finally, an outlook on further research directions is given in thelast section.

1.2 Related literature

Resulting from the variety of different features, the described problem can beclassified as a rich vehicle routing problem (RVRP). RVRP extend classical VRPvariants (see Toth and Vigo (2014) for a collection of classical variants) by com-bining multiple characteristics to solve real-wold applications. This problemclass has attracted a lot of interest in recent years due to various reasons. Onthe one hand, classical VRP variants are nearly explored. On the other hand, arising practical need as well as improving technical capabilities are pushing aca-demics towards more complex problems (Hartl et al., 2006; Hasle and Kloster,2007). The increasing interest in this topic can also be observed in attemptsto define the term RVRP based on literature surveys (Lahyani et al., 2015b;Caceres-Cruz et al., 2015). In the following, present VRP variants will be spec-ified shortly.

One of the most important characteristics is the employment of vehicles withmultiple compartments. Their application as well as the cohesive use of a LIFOpolicy are part of the broader class VRP with loading constraints (Iori andMartello, 2010; Pollaris et al., 2015). Former arises in various industries, e.g.in the distribution of petroleum to gas stations (Cornillier et al., 2008; Vidovicet al., 2014) or cattle food to farms (El Fallahi et al., 2008) and the collection ofwaste (Muyldermans and Pang, 2010; Henke et al., 2015), milk (Caramia andGuerriero, 2010) or olive oil (Lahyani et al., 2015a). In the context of transporta-tion planning for food retailers, multi-compartment vehicles have been used, forexample, by Chajakis and Guignard (2003) and Ambrosino and Sciomachen(2007). Recently, Hubner and Ostermeier (2018) have published an approachusing the model formulation for the multi-compartment VRP introduced byDerigs et al. (2011) to support the transportation planning of another largeGerman food retailer. However, this one operates its own fleet of vehicles withfour compartments. All of them are accessible from the rear end since dividers

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are positioned lengthways. Thus, a LIFO policy is not necessary as in all otherapproaches. LIFO loading and unloading are often part of pickup and deliveryproblems (Battarra et al., 2010; Benavent et al., 2015; Cherkesly et al., 2016).However, rearrangement of cargo is rarely prohibited but usually associated withhandling costs (Veenstra et al., 2017). Moreover, Henke et al. (2015) are the firstto consider trailers with flexible and discretely variable compartment sizes andconsequently, introduce the multi-compartment vehicle routing problem withflexible compartment sizes.

Furthermore, since the requested quantities are high and it is economicallybeneficial, the demand of each commodity class can be split between differentvehicles. This defines the VRP with split deliveries, which has been studiedextensively (Dror et al., 1994; Archetti and Speranza, 2012). In addition, thereturn of empty loading devices after delivering the goods creates the VRPwith backhauls (Goetschalckx and Jacobs-Blecha, 1989; Toth and Vigo, 1997).However, as backhauls are solely possible from the last market in a tour, thepresent variant is a simplified modification of the VRP with backhauls.

Due to the cooperation with freight forwarders, the vehicles do not returnto the depot if no empties are returned. This property leads to the open VRP(Li et al., 2007) and is common in cooperation with external service providers(Zachariadis and Kiranoudis, 2010). Additionally, the negotiated contracts in-duce the mentioned transport tariffs. In general, transport tariffs can dependone different attributes like weight, volume or type of product and often resultin complex cost functions (Irnich et al., 2014). Although they are implementedin some tools (Braysy and Hasle, 2014) and quite common in Germany (Drexl,2012), to the best of our knowledge, there is no scientific approach which con-siders real-world transport tariffs in the context of vehicle routing.

In conclusion, it can be stated that there are different approaches to solveVRPs with some of the described characteristics. However, there is no existingscientific approach combining all of the features mentioned above, especially thetariff-based cost function.

2 Mathematical description

2.1 General notation

The introduced problem is modeled on a complete directed graph G(V,A) withthe set of nodes V and the set of arcs A. The warehouse is denoted by vertex0 and the set N = V \ {0} contains all nodes corresponding to markets. Thedistance between two nodes i and j, i.e. the length of an arc (i, j), is denotedby δij . All vehicles of the set K have a set of compartments C and a capacityof Q pallets. Each vehicle is able to perform one route maximal. Hence, vehiclek is identical to route k. Dividers of trailers can be shifted in steps of sizesbase. Every market i has for each commodity class p a nonnegative demanddip, a price for an open route poip and a price for a closed route pcip. The setof all commodity classes is P and all product categories which need cooling are

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denoted by subset CP ⊂ P . Furthermore, all markets i have a nonnegativenumber of necessary backhauls Ri.

2.2 Real-world cost function based on transport tariffs

Due to the transport tariffs, the practical relevant cost function considered herediffers a lot from that one commonly used in theoretical work. For a betterunderstanding, it will be explained in greater detail hereinafter. The costs ofa route k consist of four components. The first part is the base price P base

k .This corresponds to the most expensive market/product category combinationin terms of the open tour price carried on the route. With the binary variablesyikp which attain one, if commodity class p is carried on vehicle k to market i,it can be modeled as

P basek = max

i∈N

(maxp∈P

(yikp · poip)). (1)

Secondly, the retailer has to pay the carrier an extra charge P backk for a route

on which empties are returned to the depot. Since the price of an open route isalready included in the base price, this extra charge is composed of the differencebetween the price of a closed route and an open route. Similar to the baseprice, the most expensive market/product category combination transported isdecisive for P back

k as well. In addition, it may only be taken into account, if abackhaul is performed from market i on route k. Hence, having binary variablesuik expressing this condition, it can be formulated as

P backk = max

i∈N

(maxp∈P

(yikp · uik · (pcip − poip))). (2)

Since the product of two binary variables is nonlinear but easy to linearize, itcan be modeled by the use of auxiliary binary variables bikp as

P backk = max

i∈N

(maxp∈P

(bikp · (pcip − poip))). (3)

Additionally, for each customer visited after the first one, a stop price of β hasto be paid. On that account, binary variables wik have to be introduced, whichattain one, if the vertex i is visited by vehicle k. Thus, this part can be statedas (∑

i∈Nwik − w0k

)· β. (4)

However, the actual length of a route is not considered yet. As a consequence,two markets which are wide apart could be combined into one route and there-fore, the real costs of a route, which are defrayed by the carrier, could exceedthe paid price easily. To avoid this situation, the fourth part of the cost functiontakes a so-called detour dtk of a route k into account. The detour is defined asthe length of the tour from the depot to the last market minus the maximum

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distance between the depot and any market visited on this route. To determinethe length of a route, binary routing variables xijk are added. They becomeone, if vehicle k travels directly from node i to j and zero otherwise. Hence, thedetour can be defined as

dtk =∑i∈V

∑j∈N

xijk · δij −maxj∈N

(wjk · δ0j). (5)

However, a detour is already included in the prices up to a permitted detourdtperm. Yet, if dtk is greater than dtperm, the base price is multiplied with theratio of exceeding detour to the maximum distance between the depot and anymarket visited on this route. This leads to the bipartite detour price P detour

k :

P detourk =

{0 , if dtk ≤ dtperm

P basek · dtk−dtperm

maxi(wik·δ0i) , if dtk > dtperm.(6)

Unfortunately, this part of the cost function is neither linear nor convex andtherefore, not applicable for solving with a standard solver like Gurobi or CPLEX.However, with the exceeding detour

dtek ≥ dtk − dtperm (7)

of a route k as nonnegative decision variable, it is possible to approximate thedetour price by replacing P base

k /maxi(wik · δ0i) with a fixed penalty value γ.Hence, the costs of a route k can be determined by

P basek + P back

k +

(∑i∈N

wik − w0k

)· β + dtek · γ. (8)

2.3 Mixed-integer programming model

In addition to the already introduced decision variables, further ones are neededto describe the issue as a mixed integer linear program (MIP). For reasons ofclarity and comprehensibility, all variable definitions are listed in Table 2.

Using these and the general notation described above, the mentioned vehiclerouting problem can be formulated as MIP as follows:

min∑k∈K

[P basek + P back

k +

(∑i∈N

wik − w0k

)· β + dtek · γ

](9)

subject to

P basek ≥ yikp · poip ∀ i ∈ N, k ∈ K, p ∈ P (10)

P backk ≥ bikp · (pcip − poip) ∀ i ∈ N, k ∈ K, p ∈ P (11)

gk ≥ wik · δ0i ∀ i ∈ N, k ∈ K (12)

gk ≤ wik · δ0i + maxi∈N

(δ0i) · (1− fik) ∀ i ∈ N, k ∈ K (13)

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Table 2: Definition of decision variables

Variable Definition

xijk

{1, if vehicle k travels directly from vertex i to j

0, otherwise

wik

{1, if vertex i is visited by vehicle k

0, otherwise

yikp

{1, if product category p is transported to customer i by vehicle k

0, otherwise

zkpc

{1, if product category p is transported on vehicle k in compartment c

0, otherwise

vikpc

1, if product category p is transported on vehicle k in compartment cto customer i

0, otherwise

uik

{1, if empties are transported from market i to the depot on route k

0, otherwise

bikp

1, if product category p is decisive for backhaul price of market i onvehicle k

0, otherwise

fik

{1, if market i is the furthest market from depot on vehicle k

0, otherwise

hkpc number of base units used for product p in compartment c on vehicle kqikpc integer quantity of product category p delivered to market i by vehicle k in

compartment cPbasek base price of route k

Pbackk extra charge for a backhaul of empties on route kdtek exceeding detour of route kgk maximum direct distance of a market from the depot on route k

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∑i∈N

fik = w0k ∀ k ∈ K (14)

dtek ≥∑i∈V

∑j∈N

xijk · δij − gk − dtperm ∀ k ∈ K (15)

∑j∈V

xijk = wik ∀ i ∈ V, k ∈ K (16)

∑j∈V

xjik = wik ∀ i ∈ V, k ∈ K (17)

∑i∈S

∑j∈S

xijk ≤ |S| − 1 ∀ S ⊆ N, |S| ≥ 2, k ∈ K (18)

∑i∈N

qikpc ≤ hkpc · sbase ∀ k ∈ K, p ∈ P, c ∈ C (19)∑p∈P

∑c∈C

hkpc · sbase ≤ Q ∀ k ∈ K (20)

∑k∈K

∑c∈C

qikpc = dip ∀ i ∈ N, p ∈ P (21)

qikpc ≤ vikpc ·min {dip, Q} ∀ i ∈ N, k ∈ K, p ∈ P, c ∈ C (22)

yikp ≤ wik ∀i ∈ N, k ∈ K, p ∈ P (23)

wik ≤∑p∈P

yikp ∀ i ∈ N, k ∈ K (24)

yikp =∑c∈C

vikpc ∀ i ∈ N, k ∈ K, p ∈ P (25)

vikpc ≤ zkpc ∀ i ∈ N, k ∈ K, p ∈ P, c ∈ C (26)∑p∈P

zkpc ≤ 1 ∀ k ∈ K, c ∈ C (27)

∑p∈P

∑c∈C

zkpc ≤ |C| ∀ k ∈ K (28)

∑p∈CP

zkp2 = 0 ∀ k ∈ K (29)

(|C| − 1) · (xijk +∑p∈P

vikpc) +

c−1∑l=1

∑p∈P

vjkpl ≤ 2 · (|C| − 1)

∀ i, j ∈ V, k ∈ K, c = 2, . . . , |C|

(30)

∑k∈K

uik = Ri ∀ i ∈ N (31)

uik ≤ xi0k ∀ i ∈ N, k ∈ K (32)

bikp ≤ yikp ∀ i ∈ N, k ∈ K, p ∈ P (33)

bikp ≤ uik ∀ i ∈ N, k ∈ K, p ∈ P (34)

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yikp + uik − 1 ≤ bikp ∀ i ∈ N, k ∈ K, p ∈ P (35)

xijk ∈ {0, 1} ∀ i, j ∈ V i 6= j, k ∈ K (36)

wik ∈ {0, 1} ∀ i ∈ V, k ∈ K (37)

fik ∈ {0, 1} ∀ i ∈ N, k ∈ K (38)

yikp ∈ {0, 1} ∀ i ∈ N, k ∈ K, p ∈ P (39)

zkpc ∈ {0, 1} ∀ k ∈ K, p ∈ P, c ∈ C (40)

hkpc ∈ N ∀ k ∈ K, p ∈ P, c ∈ C (41)

vikpc ∈ {0, 1} ∀ i ∈ N, k ∈ K, p ∈ P, c ∈ C (42)

uik ∈ {0, 1} ∀ i ∈ N, k ∈ K (43)

bikp ∈ {0, 1} ∀ i ∈ N, k ∈ K, p ∈ P (44)

qikp ∈ N ∀ i ∈ N, k ∈ K, p ∈ P (45)

P basek , P back

k , gk, dtek ∈ R+

0 ∀ k ∈ K. (46)

The objective function (9) minimizes the sum of the costs of all routes. Con-straints (10) and (11) are the linearization of (1), resp. (3). Since P base

k andP backk have a positive sign in the minimization objective function, these inequal-

ities are sufficient. As opposed to this, constraints (12)–(14) are needed todetermine the maximum distance of a market, which is visited on route k, fromthe depot. Inequalities (12) are not enough to linearize term maxi∈N (wik · δ0j),because a higher gk would lead to a smaller objective function value. Thus, anupper bound on gk is set by constraints (13). The second term in (13) and con-straints (14) are needed to avoid conflicting lower and upper bound inequalities.The exceeding detour of a route k is considered in (15). Constraints (16) and(17) are the classical out- and indegree constraints. Subtours not containing thewarehouse are prohibited by subtour elimination constraints (18). The use ofdiscrete compartment sizes is established by inequalities (19). (20) guaranteethat the capacity of a vehicle is not exceeded. Complete demand satisfaction ofeach customer regarding all product categories is ensured by (21). Constraints(22) are linking variables qikpc and vikpc. Since the transported quantity can-not be greater than the demand or the vehicle capacity, the minimum of bothcan be used as big-M. The causal relationship that a commodity class can onlybe delivered to a market on a vehicle if the market is visited by this vehicle,is modeled by constraints (23). Constraints (24) are needed to suppress emptyroutes. Otherwise an empty truck could travel back and forth from the depot toa market. However, this would not change the objective value. The assignmentof a product category transported to a market to exactly one compartment isguaranteed by (25). Furthermore, constraints (26) are linking variables vikpcand zkpc. The single usage of one commodity class per compartment is assuredby constraints (27) and the maximum number of compartments occupied on avehicle by (28). The special structure of the trucks is taken into account in(29), since they prohibit cooled product in compartment two. Constraints (30)ensure that no goods for market j are loaded in a compartment closer to therear end of the vehicle (smaller index; see Figure 1) than products for market

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i, if market j succeeds market i on route k. Hence, the LIFO unloading policyis considered. The examples shown in Figure 2 with |C| = 3 shall clarify theseinequalities. As truck 1 (T) travels not directly from 2 to 1, x211 = 0. Hence,

T

P 1 P 4 P 3

0 1 2

(a) Feasible loading

T

P 1 P 4 P 3

0 1 2

(b) Infeasible loading

Figure 2: Feasible and infeasible loading due to LIFO unloading policy

c = 2 : (3− 1) · (0 + 1) + 1 = 4 ≤ 2 · (3− 1) (47)

c = 3 : (3− 1) · (0 + 1) + 2 = 4 ≤ 2 · (3− 1) (48)

are the maximum cases of constraints (30) for figures 2a and 2b. Thus, noloadings are excluded. However, if we consider the opposite direction, then x121equals one and we get

c = 2 : (3− 1) · (1 + 1) + 0 = 4 ≤ 2 · (3− 1) (49)

c = 3 : (3− 1) · (1 + 0) + 1 = 3 ≤ 2 · (3− 1) (50)

for the feasible case in Figure 2a and

c = 2 : (3− 1) · (1 + 1) + 0 = 4 ≤ 2 · (3− 1) (51)

c = 3 : (3− 1) · (1 + 1) + 1 = 5 ≤ 2 · (3− 1) (52)

for the infeasible case in Figure 2b. Inequality (52) is not satisfied, since there aregoods of market 1 in compartment 3 as well as goods of market 2 in compartment2. Consequently, this loading violates constraints (30) and is prohibited. Furtherconstraints (31) – (35) incorporate the backhaul aspect, whereby all backhaulrequests have to be fulfilled (31) and a backhaul is only possible if the vehicletravels directly from this market to the depot (32). Inequalities (33) – (35) arenecessary to linearize equation bikp = uik · yikp. Finally, (36)–(46) define thedomains of the decision variables.

3 Conclusion and further Research

We have introduced and described real-world vehicle routing problem in thefood retailing industry. This problem incorporates different features such asvehicles with flexible compartment sizes, splitting of demands, and backhauls.Furthermore, a cost function based on product category dependent transporttariffs, which differs from common objective functions in vehicle routing, was

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presented. Subsequently, a mixed-integer program was introduced which con-tains all relevant features.

Future research should focus on extensive numerical studies. On the onehand, this allows for proving the proficiency of the presented model. On theother hand, managerial insight into benefits of various transport policies my begained by making different assumptions.

References

Ambrosino, D., Sciomachen, A., 2007. A food distribution network problem: acase study. IMA Journal of Management Mathematics 18 (1), 33–53.

Archetti, C., Speranza, M. G., 2012. Vehicle routing problems with split deliv-eries. International transactions in operational research 19 (1-2), 3–22.

Battarra, M., Erdoan, G., Laporte, G., Vigo, D., 2010. The traveling salesmanproblem with pickups, deliveries, and handling costs. Transportation Science44 (3), 383–399.

Benavent, E., Landete, M., Mota, E., Tirado, G., 2015. The multiple vehiclepickup and delivery problem with lifo constraints. European Journal of Op-erational Research 243 (3), 752–762.

Braysy, O., Hasle, G., 2014. Software tools and emerging technologies for vehiclerouting and intermodal transportation. Vehicle Routing: Problems, Methods,and Applications 18, 351.

Caceres-Cruz, J., Arias, P., Guimarans, D., Riera, D., Juan, A. A., 2015. Richvehicle routing problem: Survey. ACM Computing Surveys (CSUR) 47 (2),32.

Caramia, M., Guerriero, F., 2010. A milk collection problem with incompatibil-ity constraints. Interfaces 40 (2), 130–143.

Chajakis, E. D., Guignard, M., 2003. Scheduling deliveries in vehicles withmultiple compartments. Journal of Global Optimization 26 (1), 43–78.

Cherkesly, M., Desaulniers, G., Irnich, S., Laporte, G., 2016. Branch-price-and-cut algorithms for the pickup and delivery problem with time windows andmultiple stacks. European Journal of Operational Research 250 (3), 782–793.

Cornillier, F., Boctor, F., Laporte, G., Renaud, J., 2008. An exact algorithm forthe petrol station replenishment problem. Journal of the Operational ResearchSociety 59 (5), 607–615.

Derigs, U., Gottlieb, J., Kalkoff, J., Piesche, M., Rothlauf, F., Vogel, U., 2011.Vehicle routing with compartments: applications, modelling and heuristics.OR spectrum 33 (4), 885–914.

11

Drexl, M., 2012. Rich vehicle routing in theory and practice. Logistics Research5 (1-2), 47–63.

Dror, M., Laporte, G., Trudeau, P., 1994. Vehicle routing with split deliveries.Discrete Applied Mathematics 50 (3), 239–254.

El Fallahi, A., Prins, C., Calvo, R. W., 2008. A memetic algorithm and atabu search for the multi-compartment vehicle routing problem. Computers& Operations Research 35 (5), 1725–1741.

Goetschalckx, M., Jacobs-Blecha, C., 1989. The vehicle routing problem withbackhauls. European Journal of Operational Research 42 (1), 39–51.

Hartl, R. F., Hasle, G., Janssens, G. K., 2006. Special issue on rich vehiclerouting problems. Central European Journal of Operations Research 14 (2),103–104.

Hasle, G., Kloster, O., 2007. Industrial vehicle routing. In: Hasle, G., Lie,K.-A., Quak, E. (Eds.), Geometric Modelling, Numerical Simulation, andOptimization: Applied Mathematics at SINTEF. Springer Berlin Heidelberg,Berlin, Heidelberg, pp. 397–435.

Henke, T., Speranza, M. G., Wascher, G., 2015. The multi-compartment ve-hicle routing problem with flexible compartment sizes. European Journal ofOperational Research 246 (3), 730–743.

Hubner, A., Ostermeier, M., 2018. A multi-compartment vehicle routing prob-lem with loading and unloading costs. Transportation ScienceIn Press.

Iori, M., Martello, S., 2010. Routing problems with loading constraints. Top18 (1), 4–27.

Irnich, S., Toth, P., Vigo, D., 2014. The family of vehicle routing problems.In: Toth, P., Vigo, D. (Eds.), Vehicle Routing: Problems, Methods, andApplications. SIAM, Ch. 1, pp. 1–33.

Kuhn, H., Sternbeck, M. G., 2013. Integrative retail logistics: An exploratorystudy. Operations Management Research 6 (1), 2–18.

Lahyani, R., Coelho, L. C., Khemakhem, M., Laporte, G., Semet, F., 2015a. Amulti-compartment vehicle routing problem arising in the collection of oliveoil in tunisia. Omega 51, 1–10.

Lahyani, R., Khemakhem, M., Semet, F., 2015b. Rich vehicle routing problems:From a taxonomy to a definition. European Journal of Operational Research241 (1), 1–14.

Li, F., Golden, B., Wasil, E., 2007. The open vehicle routing problem: Algo-rithms, large-scale test problems, and computational results. Computers &Operations Research 34 (10), 2918–2930.

12

Muyldermans, L., Pang, G., 2010. On the benefits of co-collection: Experimentswith a multi-compartment vehicle routing algorithm. European Journal ofOperational Research 206 (1), 93–103.

Pollaris, H., Braekers, K., Caris, A., Janssens, G. K., Limbourg, S., 2015. Ve-hicle routing problems with loading constraints: state-of-the-art and futuredirections. OR Spectrum 37 (2), 297–330.

Rehder, L. E., Stange, K., 2015. Germany retail foods. Tech. Rep. GM15017,USDA Foreign Agricultural Service.

Toth, P., Vigo, D., 1997. An exact algorithm for the vehicle routing problemwith backhauls. Transportation Science 31 (4), 372–385.

Toth, P., Vigo, D., 2014. Vehicle Routing: Problems, Methods, and Applica-tions. Vol. 18. SIAM.

Veenstra, M., Roodbergen, K. J., Vis, I. F., Coelho, L. C., 2017. The pickup anddelivery traveling salesman problem with handling costs. European Journalof Operational Research 257 (1), 118–132.

Vidovic, M., Popovic, D., Ratkovic, B., 2014. Mixed integer and heuristics modelfor the inventory routing problem in fuel delivery. International Journal ofProduction Economics 147, 593–604.

Zachariadis, E. E., Kiranoudis, C. T., 2010. An open vehicle routing problemmetaheuristic for examining wide solution neighborhoods. Computers & Op-erations Research 37 (4), 712–723.

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