a tabu search approach for the livestock collection problem

Computers & Operations Research 35 (2008) 3213–3229www.elsevier.com/locate/cor

A tabu search approach for the livestock collection problemJohan Oppen∗, Arne LZkketangen

Molde University College, 6402 Molde, Norway

Available online 1 March 2007

Abstract

We present a problem dealing with transportation of live animals to slaughterhouses. The problem is taken from the Norwegianmeat industry, and may be viewed as a vehicle routing problem extended with inventory constraints to ensure a smooth productionflow at the slaughterhouse. In addition, several constraints to ensure animal welfare have to be met. These include duration limitsfor how long animals can stay on the vehicle as well as rules for mixing different animal categories. In this paper, we show that thisreal-world problem can be solved heuristically, even if it is large both in size and complexity. A tabu search based solution methodfor the problem is presented, and results from computational testing are given, including comparisons with manual solutions fromtoday’s planning system.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Vehicle routing problem (VRP); Rich VRP; VRP with side constraints; VRP with inventory constraints; Livestock transportation; Tabusearch

1. Introduction

The project “Transportation of living animals—reduced transportation costs, good animal welfare and first-classmeat quality” is a co-operation between the Norwegian Meat Research Center, Gilde Norsk KjZtt, Fatland and MoldeUniversity College. The Norwegian Meat Research Center is a center for competence and emergency control in theNorwegian meat industry. Gilde Norsk KjZtt and Fatland are Norwegian meat companies. They both run slaughter-houses, and they also cut and process the carcasses into finished meat products. The project, which was launched in2003 and stipulated to last for five years, focuses on the collection of animals for slaughter. The main goal of the projectis to develop a decision support system (DSS) to reduce transportation costs and at the same time secure good animalwelfare and meat quality, as these are three main factors for the profitability of both farmers and the meat industry.Tomasgard and HZeg [1] describe an optimization model for production processes that are later in the value chain formeat products. It would be beneficial to integrate larger parts of the value chain in the same planning system to avoidsuboptimization, but this paper concentrates on a part of the value chain where optimization-based planning systemsare almost non-existing.

The livestock collection problem (LCP) may informally be described as the problem of constructing the best possibleset of routes for transportation of living animals from farmers to one slaughterhouse. Each route must be feasibleaccording to constraints on duration, vehicle capacity, mix of animal types and visiting order of the customers. Thewhole solution as a set of routes must adapt to both the production plan, see Section 2.2.5, and the lairage capacity at the

∗ Corresponding author. Tel.: +47 71214240; fax: +47 71214100.E-mail addresses: [email protected] (J. Oppen), [email protected] (A. LZkketangen).

0305-0548/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.cor.2007.02.021

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3214 J. Oppen, A. L�kketangen / Computers & Operations Research 35 (2008) 3213–3229

slaughterhouse.Ability to get solutions for the livestock collection problem will be an important part of the DSS referredto above, and the contribution of this paper is to describe the problem and give a solution method. The rest of the paper isdivided as follows. In Section 2 follows a closer description of the livestock collection problem. An overview of solutionmethods is given in Section 3, followed by a description of our tabu search (TS) approach in Section 4. In Section5 we describe test instances and our computational results. Conclusions and suggestions for future work are given inSection 6.

2. Problem description

The livestock collection problem may be viewed as a vehicle routing problem (VRP) extended with constraintsregarding production and inventory. In this section we first present the basic VRP. We then give a more detailedpresentation of the livestock collection problem, and discuss what known extensions to the VRP should be includedin this setting. In the end we add constraints to model the connection between the routing part and the inventory andproduction part of the problem.

2.1. The vehicle routing problem

The VRP deals with the allocation of transportation tasks to a fleet of vehicles, and the simultaneous routing for eachvehicle. The VRP was first described by Dantzig and Ramser [2], and is a computationally hard optimization problemwith high industrial relevance. The classical VRP is defined on a graph G = (N, A) where N = {0 . . . n} is a vertexset and A = {(i, j) : i, j ∈ N} is an arc set. Vertex 0 is the depot; the other vertices are the customers. The travel costbetween customer i and j is defined by cij �0 and di is the demand for customer i. The vehicles are usually identical,each with a capacity q. The goal is then to design a least cost set of routes, all starting and ending at the depot, whereeach customer is visited exactly once. The total demand of all customers on a route must be within the capacity q.This classical formulation is often referred to as the capacitated VRP or CVRP. If cij �= cji for at least one pair ofcustomers, e.g. due to a one-way road, we have an asymmetric VRP.

2.2. The livestock collection problem

Our version of the livestock collection problem deals with transportation of live animals from farmers to slaughter-houses, and includes both a routing part and an inventory problem at the slaughterhouse. An exhaustive descriptionof inventory problems can be found in [3]. Gribkovskaia et al. [4] have treated a simplified version of the livestockcollection problem, and we are not aware of other research dealing with this kind of combination of a VRP and aninventory problem. There exists a large body of research where the VRP is combined with other logistic problems,such as location routing [5] and inventory routing [6]. While location routing combines vehicle routing with facilitylocation, the location of the slaughterhouse is already decided in the LCP. Instead, the inventory part of the LCP addsa timing aspect to the problem, so that we have to consider both when the routes arrive at the slaughterhouse as wellas what they bring into the inventory. In the inventory routing problem, on the other hand, one has to decide both whatcustomers to serve, how much to bring to each of them and how to do the routing. In the LCP, we consider the ordersas data, meaning that we know both where to go and what to collect.

Several extra constraints must be added to the basic VRP model to have a problem formulation that corresponds wellto the real-world problem. Toth and Vigo [7] describe a rich variety of extensions to the classical VRP, many of theseare used in our modeling of the LCP. In the following we give a brief presentation of the necessary constraints, moredetails can be found in [8], which is a pure problem description with no focus on solutions.

We illustrate our version of the LCP by giving a small example, this will then be used to explain important constraintsand features of the problem as well as its solution. Fig. 1 shows where the slaughterhouse, three vehicle drivers and19 customers (producers) are located. The slaughterhouse is shown as a solid square, the drivers’ homes as opensquares and the customers are shown as circles. In this example, we assume a euclidean topology where it is possibleto travel between any pair of locations by following a straight line. In the real world, we have to use road data tocompute distances between locations. We use different colors to keep track of different animal types to be collected,see Section 2.2.1, as well as the health status of each farm, see Section 2.2.3. In addition to the data for the examplegiven in Table 1, we also need estimates for vehicle speed, time needed to load and unload vehicles, etc.

J. Oppen, A. L�kketangen / Computers & Operations Research 35 (2008) 3213–3229 3215

Fig. 1. Locations.

A solution for an instance of the LCP consists of a set of planned routes describing how the collection of animalsshould be carried out. Time information from the route plan, together with the slaughter plan for the instance, will giveinformation about the expected inventory levels for the different animal types during the time horizon. As an exampleof what a solution may look like, we show a plan for Tuesday for our example in Fig. 2. Similar plans exist for Sundayand Monday, and together these constitute a solution for our small example. In addition to the graphical view shownhere, one also needs a more detailed written description of the solution. This typically includes tour information: whenand from where should the tours start, when and in what order should the different orders be served and when shouldthe vehicles return to the slaughterhouse.

2.2.1. Mixing rulesTransportation of live animals requires that several rules to secure animal welfare are followed. Animal welfare

affects meat quality and thus has both economical and ethical aspects. Animals are divided into types and categories,where the types by and large correspond to the different species. Bovine refers to cattle, while ovine is used for sheepand goats. Each type is divided into categories by size, gender, whether bovines have horns or not and whether sheepare sheared or not. Mixing of animals of different types in the same compartment in a vehicle is not allowed. Neitheris it allowed to mix animals with and without horns or animals of substantially different size. These rules are supposedto protect animals from being stressed or injured during transport.

In our example, we have only two types of animals, bovine and pigs. Customers with bovine for collection are shownas light blue circles, those who have pigs are shown in red. We consider all bovines to be young bulls without hornsand all pigs to be slaughter hogs. We are then left with only one mixing rule in this particular case: we cannot haveboth bovine and pigs in the same compartment.

Problems where different types of loads have to be kept in separate compartments during transport are often referredto as loading problems, see [9,10]. Most of the research in this area deals with problems where all compartments ofthe vehicle are accessible all the time, such as transport of petroleum products to gas stations. In the LCP, this is notthe case, see Section 2.2.4.

2.2.2. DurationAnother constraint that treats animal welfare is that no animal can be transported continuously for more than 8 h.

This means that no tours can be planned to last longer than 8 h, measured from the time of the first loading to the timeof unloading at the slaughterhouse. In our example, we may use customer 19 to illustrate this rule. The time requiredto load 60 pigs and go back to the slaughterhouse is estimated to be 7 h 25 min, which means that only one shortstop on the way back to the slaughterhouse is possible within the time limit. A vehicle may thus have to return to theslaughterhouse with less than a full load, because there is no time to make additional stops to fill up the vehicle.


Table 1Data for LCP example

Slaughter plan

Day Bovine Pigs

Sunday 0 0Monday 0 220Tuesday 45 110

Inventory capacities

Bovine Pigs

30 200

Vehicles

Vehicle number Configuration Total capacity

Front section Mid-section Rear section Bovine Pigs

1 1 or 2 comp. 1 or 2 comp. 1 comp. 14 792 2 comp. 2 comp. 2 comp. 15 1023 2 comp. 2 comp. 2 comp. 15 102

Orders

Customer Bovine Pigs Health status

Cust. 1 0 10 NormalCust. 2 3 0 InfectedCust. 3 9 0 NormalCust. 4 4 25 NormalCust. 5 0 50 NormalCust. 6 2 0 NormalCust. 7 0 5 NormalCust. 8 14 0 NormalCust. 9 0 20 NormalCust. 10 0 5 NormalCust. 11 7 0 NormalCust. 12 4 0 NormalCust. 13 0 25 BreedingCust. 14 0 20 NormalCust. 15 0 80 NormalCust. 16 0 10 NormalCust. 17 2 0 NormalCust. 18 0 20 NormalCust. 19 0 60 Normal

2.2.3. Visiting order and health statusWhen vehicles travel between farms to collect animals, one has to take precautions to avoid the spread of diseases.

Pig herds with boars (males) used for breeding purposes need extra protection, so these may only be visited by emptyand clean vehicles. This means they need to be visited as the first stop of the tour. Customer 13 in our example hasa breeding herd of pigs, and must therefore be first on the tour. This is indicated by printing the customers name inlight blue. Herds that are infected with some disease should be the last stop on the tour, the vehicle is then cleanedat the slaughterhouse before it is used again. Customer 2’s herd is infected with a disease, the name of the customeris printed in red, and this must be the last stop on the tour. A vehicle cannot visit more than one breeding herd ormore than one infected herd per tour, see Section 4.2. These constraints may easily lead to longer tours and therebyextra costs.


3

Driver 1

Driver 2

9

11

10

12

13

14

16

Driver 3

Inventorylevels

0 0

tuesday

Fig. 2. Routes for Tuesday.

Sigurd et al. [11] describe and propose a solution method for a pickup-and-delivery problem with time windows andprecedence constraints. The application is transportation of live pigs between farms in Denmark according to a numberof veterinary restrictions. To avoid the spread of diseases, the vehicles must visit the farms in a non-decreasing orderof health levels. A low health level corresponds to a healthy livestock.

2.2.4. Vehicle fleetThe vehicle fleet is heterogeneous, as the vehicles differ both in size and how they can be divided into compartments.

Almost all vehicles used for animal transport in Norway are divided horizontally into three sections by permanentlypositioned walls or gates. Most vehicles are outfitted to stack pigs and ovine in two tiers, while bovines in many casesneed the full height of the vehicle. An “average” vehicle has a capacity of about 15 cows or bulls, 90 slaughter hogs or150 lambs. This means that a compartment typically can take five cows or bulls, 16–18 pigs or 25 lambs. When onlyone animal type is transported at the time, which is normally desired for animal welfare reasons, it is relatively easyto calculate the legal load of the vehicle. Sometimes two or three animal types are transported by the same vehicle,and then the loading problem becomes more cumbersome, see Section 2.2.1. This situation typically occurs when it isdifficult to compose full loads of only one animal type. Fig. 3 shows how a vehicle may be divided into sections andcompartments.

Vehicle 2 and 3 in our example have enough height to carry bovine in the lower compartment and pigs or ovine inthe upper compartment at the same time, while vehicle 1 is lower and may only stack animals in two tiers if there arepigs or ovine in both compartments of a section. Vehicle 1 also has only one compartment (no lift) in the rear section.The upper compartment in a section must always be loaded before the lower, as the floor in the upper compartmentis raised and lowered by a lift. When the upper compartment is used, the floor is first lowered. The animals are thenbrought into the compartment, and the floor is raised. The lower compartment can then be filled with other animals.

Vehicle 2 and 3 in our example both have a larger capacity than vehicle 1, as can be seen in Table 1. It should alsobe noticed that these two vehicles may carry 51 slaughter hogs in addition to 15 bulls, as the upper compartments arealways available. In order to fully utilize these vehicles capacity, one has to consider the sequence in which the animalsare loaded. If bovines are loaded into a section, the upper compartment will be blocked and cannot be accessed untilthe bovines are unloaded. In other words, the sequence in which the orders of a tour are served may affect the actualcapacity of the vehicle.


Fig. 3. A vehicle is divided into sections and compartments.

The smaller height of vehicle 1 makes it impossible to have both bovine and smaller animals in the same section of thevehicle. This makes the loading order less important for this vehicle, as bovine will not block any upper compartmentin the same section. One should note, however, that putting one animal type into the lower compartment of a sectionblocks the section in front of the one that is being loaded. This also leads to situations where the loading order affectsthe vehicle capacity.

The vehicles are used for up to four tours each day, the first tour each day usually starts from the driver’s home.

2.2.5. Today’s practice on registration and planningAnimals for slaughter are reported to the slaughterhouse by the farmer; normally this has to be done by the week

before wanted collection. The day of collection may or may not be fixed right away, this depends on several factors.Plans are usually made for one week at the time, then adjustments are made day by day if needed. The plans for animalcollection and production must of course fit together, and several factors govern the construction of these plans for agiven period of time. These factors include the number of animals registered for collection during the time horizon,production capacity at the slaughterhouse, vehicle capacities and demand for meat products. It is not necessarily thesame factors that restrict different companies, as some may have capacity to collect and process all registered animals,while others utilize their full production capacity most of the time. If the amount of registered animals is large, acompany may decide to postpone some of the collection to a later week.

The geographical area from which the slaughterhouse collects animals is divided into smaller districts. In somedistricts one vehicle has a monopoly, while in others two or three vehicles operate. Some vehicles operate in two orthree districts. This system thus often predetermines what vehicle will pick up what orders, it normally also excludesthe possibility to construct routes where vehicles collect animals from districts different from their own.

As our project focuses on the transportation part rather than on the production part of the slaughterhouses’ area ofoperation, we view the slaughter plan as an input. We also treat the set of animals to be collected during the timehorizon as given, thus we do not include the selection of what to collect by the current plan and what to postpone, inthe problem. For our small example, the plan is to slaughter all the bovines on Tuesday, while we are going to slaughter220 pigs on Monday and 110 on Tuesday. All orders must be serviced during Sunday, Monday and Tuesday, but weare free to decide what day to collect from each farm.

It is also important to realize that in the real world, unforeseen changes happen every day. In our setting, this meansthat new orders occur that have to be served on short notice, i.e. animals that for some reason need to be collected forslaughter right away. Because of urgent orders and the fact that there is some deviation between what is reported by thefarmers and what is actually delivered, there is usually a difference between how the collection of animals is plannedand what actually happens.

2.2.6. Inventory constraintsWhen vehicles arrive at the slaughterhouse, the animals are unloaded into a lairage, where they are kept until they

are slaughtered. The animals cannot have more than one overnight, and no slaughtering is done during the weekend.This means that no animals can be kept in lairage over the weekend. If an animal type is to be slaughtered the next


day, some animals are usually kept overnight to be able to start the production early next morning. This means someanimal collection is done on Sundays, even if there is no slaughtering that day. No animals can be lairaged over theweekend, see Section 2.2.6, so less collection is needed on Fridays. The lairage has a certain capacity which cannotbe exceeded, neither should the lairage run empty, as halting production is very costly. It is important to note that theinventory constraints are global, and not associated with single routes.

In our example, we have a capacity of holding 30 bovines and 200 pigs in the inventory. Bovines will not beslaughtered on Monday, so we cannot collect any of them on Sunday. We will have to collect at least 15 bovines onTuesday in order not to overfill the inventory from Monday to Tuesday. We can collect up to 200 pigs on Sunday, therest may be brought in on Monday without exceeding the inventory capacity.

We have not taken into account the possibility of running empty during the day, even if the longest tours arrive atthe slaughterhouse in the afternoon. In practice, most tours collect animals quite close to the slaughterhouse, whichprovides a smooth supply of animals for the production process.

3. Solution methods

The VRP has become one of the most widely studied problems in combinatorial optimization, and much effort hasbeen put into developing both exact and heuristic methods to solve the VRP. The classical VRP is known to be stronglyNP-hard.

3.1. Solution methods for the VRP

3.1.1. Exact methodsAn exact solution method is able to find the optimal solution of a problem and to prove the optimality of the obtained

solution. Exact solution methods for the VRP include branch and bound and branch and cut [7]. These techniques areknown to consistently solve benchmark instances with up to 50 customers in reasonable time.

3.1.2. Heuristic methodsHeuristic solution methods cannot guarantee that an optimal, or even feasible, solution is found; neither can optimality

easily be proved when an optimal solution is actually found. The reason for using heuristics is the assumption that asuitable heuristic should be able to find reasonably good solutions relatively quickly. This can be very useful in caseswhere exact methods perform poorly due to the size or structure of the problem. Bräysy et al [12,13] give an overviewof heuristics for the VRP. This includes heuristics for rich models where many of the possible extensions are included,which is important in our setting.

3.2. Solution methods for the LCP

Not much has been done to solve VRPs extended with inventory constraints for the depot. Gribkovskaia et al. [4]solved a simplified version of the livestock collection problem with eight orders using CPLEX, and found that addingthe ninth order made the problem too hard for CPLEX to find a feasible solution in reasonable time. They formulated theproblem as a mixed integer problem and used a simplification by dealing only with one animal category, a homogeneousvehicle fleet and no precedence constraints.

4. A TS-based heuristic for the LCP

We propose a TS-based heuristic for the LCP, and our algorithm is based on a TS algorithm for VRPs with timewindows described by Cordeau et al [14]. Their method has been extended and adjusted to deal with the constraintsdescribed in Section 2.2.

TS is a local search based metaheuristic introduced by Fred Glover. The main ideas are to avoid recently visited partsof the solution space and to guide the search towards new and promising areas. Non-improving moves are allowed toescape from local optima, and attributes of recently performed moves are declared tabu or forbidden for a number ofiterations to avoid cycling. TS does not rely very much on randomness, but tries to be “intelligent” in its way to perform


the search. TS has been successfully adapted and applied to numerous discrete optimization problems, VRP included.See Glover and Laguna [15] for more information about TS.

4.1. Time and inventory

The time horizon for a problem instance is one or more days, usually six (one week, starting on Sunday).For each tour, we keep track of the time the vehicle leaves the driver’s home or the slaughterhouse, when the vehicle

arrives and leaves each of the farms, when the vehicle arrives at the slaughterhouse and when the vehicle is ready forthe next tour after unloading and cleaning.

Time is measured in minutes, and for each tour the time from the first animals are loaded until they are unloaded atthe slaughterhouse is computed. If this is more than 8 h, the duration constraint for the tour is violated. If the vehiclereturns to the slaughterhouse at a later hour in the evening than some specified point in time representing the end of theworking day, the end-of-day constraint for the tour is violated.

The slaughter plan tells how many animals of each type are to be slaughtered each day. The inventory level from theprevious day plus the animals received during the day must be sufficient to meet the slaughter plan for the day, and theinventory level must always be between a minimum safety level and the lairage capacity.

4.2. Moves

A tour has a legal starting and ending location if it satisfies the following: the first tour of each day starts at thedriver’s home, the following tours start at the slaughterhouse. All tours end at the slaughterhouse.

Let S be the set of all solutions that satisfy the following constraints: every tour has a legal starting and endinglocation, every order belongs to exactly one tour and two tours that use the same vehicle are separated in time. Inaddition, no tour visits a breeding herd after having visited a non-breeding herd, and no tour visits a non-infected herdafter an infected herd.

This means that we allow for intermediate solutions that may violate some of the constraints explained inSection 2.2:

• One or more tours in a solution may collect more animals than the vehicle’s capacity allows for.• One or more tours may violate precedence constraints by visiting more than one breeding or infected herd.• One or more tours may violate the 8-h rule because the time from the first loading until the arrival at the slaughterhouse

exceeds 8 h.• One or more tours may arrive at the slaughterhouse after the working day has ended.

These four types of constraint violations are controlled by penalties, see Section 4.3. Allowing for intermediateinfeasible solutions during the search has been used successfully by Cordeau et al, see [14], and both [16,17] show thatdynamically adjusted penalties work well to get a mix of feasible and infeasible solutions during the search.

Let A(s)={(i, j, k, l) : order i is visited by vehicle j on day k and tour number l} be an attribute set associated witheach solution s ∈ S. The neighborhood N(s) of a solution s is defined by applying an operator that removes an attribute(i, j, k, l) from A(s) and replaces it with a different attribute (i, j ′, k′, l′), where j �= j ′ ∨ k �= k′ ∨ l �= l′. This meansa move consists of moving an order from its current tour to a different tour. The neighborhood size is |N | = n(m − 1),where n is equal to the number of orders and m is equal to the total number of tours for all vehicles. When an orderis removed from a tour, the tour is reconnected by linking the predecessor and successor of the removed order. Theinsertion of an order into a tour is done so as to minimize the increase in the length of the tour, but without changingthe order of the orders already in the tour. If the order to be inserted represents a breeding herd or an infected herd, itis inserted as the first or last order, respectively.

We use three different variants of the move type explained above:Global moves are moves where the full neighborhood is explored. When such a move is evaluated, we look for the

best overall move without considering the fact that moving an order from one day to another may lead to violation ofinventory constraints. To be able to maintain a satisfactory balance between the amounts of animals that are collectedeach day, and thus control the inventory levels, we also need more restricted moves.

Local moves are moves that allow orders to be moved only between tours on the same day.


Dedicated moves are used to repair violated inventory constraints. These moves allow only orders containing a certainanimal type to be moved between tours on certain days.

The neighborhood size will be significantly smaller for the two latter move variants than for the global. The use ofthe different variants are explained in Section 4.6.

4.3. Move evaluation

For each solution s ∈ S, let c(s) be the total length of all tours in the solution, and let q(s) be the total violation ofcapacity constraints for the tours, measured as the total number of animals exceeding the vehicles’ capacities. Let r(s)

be the total number of precedence constraint violations. Let t (s) be the total amount of duration violations, and let u(s)

be the total amount of end-of-day violations. When moving from a solution s to another solution s′ ∈ N(s), possiblemoves are evaluated using

• �c(s′), change in total length,• �q(s′), change in capacity violations,• �r(s′), change in number of precedence violations,• �t (s′), change in duration violations,• �u(s′), change in end-of-day violations.

In order to have terms of approximately equal magnitude, the five terms listed above are normalized by dividingeach of them by a number that represents the largest change that normally occurs for that particular term in a move.These five numbers are represented by �c, �q , �r , �t and �u, respectively.

Solutions are then evaluated using a function f (s′) = �c(s′)/�c + ��q(s′)/�q + ��r(s′)/�r + ��t (s′)/�t +��u(s′)/�u, where �, �, �, � are positive parameters that are dynamically adjusted during the search. If f (s′)�0,moving to the new solution s′ is a non-improving move, if f (s′) < 0 the move is improving. Because the moveevaluation function includes changes in constraint violations, a move may be regarded as non-improving even if thetotal distance is reduced, and vice versa.

We also tried to include an additional penalty term for inventory constraint violation in the move evaluation function.This strategy was chosen to treat the inventory constraint violations in the same way the standard VRP constraintviolations are treated, see Section 4.6, which has shown to be a successful way of balancing the search between feasibleand infeasible solutions. It turned out, however, that this led to huge difficulties in finding feasible solutions at all. Itseemed very hard to get the different penalties to work together towards feasibility, instead they seemed to co-operateto increase infeasibility for all constraint types. We have not been able to establish if these problems are caused by themere number of penalty terms or by the global nature of the inventory constraints.

4.4. Diversification

In order to guide the search to new and possibly unvisited parts of the solution space, we use continuous diversification.If moving to a solution s′ ∈ N(s) is found to be deteriorating, that is, f (s′) > 0, this move is given a penalty p(s′) =��i/iter that is added to f (s′). Here, �i is the number of times order i has been moved during the search, iter is thenumber of iterations or moves performed so far and � is the diversification intensity. Setting � to zero corresponds tono diversification at all, while a larger value for � makes it more likely that an order that has been moved only a fewtimes is chosen to be moved in favor of an order that has been moved more frequently. Note that this penalty is addedto the move evaluation function only if f (s′) > 0, which means that it comes into play only when the search is tryingto escape from a local optimum.

4.5. Initial solution

Two different approaches have been tried to construct an initial solution for the livestock collection problem. Thefirst one failed due to problems with the inventory constraints. It seems to be important to find a starting solution wherethe amounts of collected animals for each day fit to the slaughter plan and the inventory capacities. We found it hard


to recover from a large amount of inventory constraint violations, and thus we pay more attention to this aspect in ourcurrent method for constructing an initial solution. This method consists of four steps:

1. clustering of orders,2. distribution of clusters throughout the days of the time horizon,3. assigning clusters to vehicles,4. route construction.

In the first step, orders are put together to make clusters that correspond approximately to truckloads. The ordersshould represent the same animal type to meet the desire for homogeneous loads, they should not be geographicallytoo far apart and they should have a total number of animals that is close to the average vehicle capacity. Both theheterogeneous vehicle fleet and the diversity in order size make it difficult to construct clusters that make up perfectloads.

In the second step, the clusters are assigned to days. Each day is assigned enough animals to cover the slaughterplan, and to provide animals for lairaging for the next day. The amount of animals in the lairage from one day to thenext varies, but at least half of the lairage capacity is usually used. The slaughterhouse needs to have animals for 1 or2 h of production in the lairage when the work starts in the morning, otherwise one may run out of animals to slaughterbefore the vehicles start to arrive with the first loads.

Clusters are assigned to days in a random way, but for Sunday we pick clusters that are relatively close to theslaughterhouse. This is done to meet the drivers’ wish for relatively short tours on Sundays.

When all clusters are assigned to days, they are assigned to the closest vehicle that has spare tours on that day. Thenumber of tours each vehicle may have each day in the initial solution depends on the distance from the driver’s hometo the slaughterhouse.

The last step in creating the initial solution is that each vehicle creates its own routes to service the clusters that havebeen assigned to it. There are normally from one to five orders in each cluster.

The initial solution may have tours that violate vehicle capacities, precedence constraints, duration constraints aswell as end-of-day constraints. All these constraint violations are relatively easy to recover from. The initial solutiondoes not violate the inventory constraints.

4.6. Tabu search

The TS starts from the initial solution and moves, in each iteration, to the first encountered non-tabu neighbor thatrepresents an improvement, that is, f (s′) < 0. If no such neighbor is found, the move that minimizes f (s′) is chosen.If f (s′) > 0, a non-improving move is performed, and the diversification mechanism explained in Section 4.4 comesinto play.

The tours of the current solution are scanned sequentially to find the first order to move that leads to an improvement.In each iteration, the search starts from where the previously moved order was found, and all tours are scanned in acircular way. Because the first improving move is accepted, it may take several iterations to loop through all tours. If noimproving move is found, which corresponds to being in a local optimum, then all neighbors of the current solutionsare evaluated before a non-improving move is chosen.

The three different move types referred to in Section 4.2 are used to overcome the problems encountered in keepingfeasibility with respect to the inventory constraints, as described in Section 4.3. This is done in the following way. Thefirst 100 moves are restricted to local moves, that is, orders are moved only to others tours on the same day. In thisway, we do not affect the way the orders are distributed among the days. We then allow for 10 global moves, as wethink it is too restrictive only to move orders within the same day. By allowing only a small number of global moves,we avoid losing control of the inventory balance. When these global moves are performed, we check the inventorysituation throughout the time horizon. If there are any overflow or shortage situations, we perform dedicated movesuntil the inventory status is again feasible. A similar cycle is then repeated throughout the search: every 100 iterations,we allow for 10 global moves followed by the necessary number of dedicated moves to repair inventory constraintviolations caused by the global moves. Typically, we need between zero and five dedicated moves to restore inventoryfeasibility.

To prevent the search from going back to recently visited parts of the solution space, the attribute (i, j, k, l) that wasremoved from A(s) by the last move is declared tabu for tabu tenure iterations, where tabu tenure is the tabu length or


duration. During these iterations, it is not allowed to move order i back to vehicle j ’s tour l on day k. By the use of asimple aspiration criterion, a tabu move can still be chosen if this leads to a feasible solution that is the best found sofar in the search.

After each move, the values of the parameters �, �, � and �, see Section 4.3, are adjusted. If the current solution isfeasible with respect to vehicle capacities, the value of � is decreased to make it less costly to violate vehicle capacityconstraints. If the current solution does violate such constraints, � is increased to make it more costly. In the sameway, � is decreased if the solution violates precedence constraints and increased otherwise, � is adjusted accordingto whether duration constraints are violated or not and � is decreased or increased according to the violation of theend-of-day constraints. The long-term memory is updated by incrementing �i , where i is the order moved.

If the current solution is feasible and has a total length less than times the length of the best feasible solution foundso far during the search, the solution is considered good. Whenever a good solution is found, a 2-opt post-optimizationprocedure is applied to the tours of the solution. It would be a waste of time to apply the 2-opt procedure for all feasiblesolutions, but it is also important to apply it often enough to capture the solutions that become a new best solution aftera 2-opt procedure is applied. When the 2-opt procedure is applied, one must also check that the tour remains feasiblewith respect to vehicle capacity after a change is made, as changing the visiting and loading sequence may have aninfluence on the actual vehicle capacity, see Section 2.2.4. This means we may have to omit some distance-reducingmoves in the 2-opt and accept tours that are not 2-optimal.

The search continues until a preset time limit is reached, or until a preset number of moves are performed.

5. Computational testing

The TS algorithm described in Section 4 has been coded in C++ using Microsoft Visual Studio 2005, and the testsdescribed here have been run on a Dell Precision PWS380 computer with a 3.6 GHz Pentium(R) D CPU with 3.62 GBof RAM.

5.1. Real-world test instances

To test the performance of our algorithm, we have used data from week 7 in February and week 22 in May/June2006 from the slaughterhouse Fatland Jæren at HommersAAk, Norway. For both instances, we have information aboutwhat farmers were visited during the week and how many animals of each category were collected from each farmer.In addition, we have information about available vehicles (home location and capacity), approximate slaughter planand lairage capacities.

We have information both about what was planned and what actually happened at Fatland Jæren during theseparticular weeks. This makes it to some extent possible to compare our solutions with the plans from the manual systemthat is used today. We have also used these data to get estimates on vehicle speed and time consumption for the differentoperations, and to find deviations between the amount of animals the farmers report and the amount they actuallydeliver.

Some characteristics of the two instances are summarized in Table 2. In both cases, each order contains only animalsof the same type, and no order is larger than the capacity of the smallest vehicle type used. This is the reason why thereare more orders than farms, and this splitting is done in advance to avoid a situation where the algorithm has to splitsome of the orders. It is of course possible to pick up more than one order at the same location on the same tour if thatis desirable.

The data necessary to build the complete test instances were not available from Fatland’s information system, so afew days of manual registration work was needed. It is, however, of great importance to be able to test with real-worlddata, as a future system will have to solve real-world problems.

Table 2Real-world test instances

Instance Orders Farms Bovine Pigs Ovine Total Vehicles

Week 7 162 144 288 1438 176 1902 8Week 22 196 184 281 1328 288 1897 8


Table 3Random test instances

Instance Orders Vehicles Bovine Pigs Ovine Total

n150_v7_std 150 7 266 1388 175 1829n150_v7_ov 150 7 171 649 2219 3039n150_v5_std 150 5 266 1388 175 1829n150_v5_bov 150 5 403 825 84 1312n110_v5_std 110 5 230 1261 49 1540n110_v5_pig 110 5 96 1643 34 1773n110_v3_std 110 3 192 922 175 1289n110_v3_ov 110 3 129 557 1316 2002

5.2. Random test instances

In addition to the real-world instances described in Section 5.1, we have generated a set of eight random instancesfor the LCP. These instances are made in the following way: first, 12 orders and one vehicle are removed from theweek 7 real-world instance. This new instance is named n150_v7_std, its name indicating the number of orders, thenumber of vehicles and a mix of animal types close to the real-world instance from which it was made. Anotherinstance, n150_v7_ov, is then made by changing several of the orders containing bovine and pigs into orders containingovine. Two other instances are made by removing two more vehicles, then the orders from n150_v7_std are kept orpartly replaced with bovine orders to make up the instances n150_v5_std and n150_v5_bov, respectively. The lastfour instances all contain 110 orders, they have either five or three vehicles and the distribution of animal types varies.Whenever an order is changed, only the type and number of animals are changed, the location is not. Table 3 summarizessome of the characteristics of the random instances. In each case, the slaughter plan is changed to match the numberof animals.

5.3. Finding search parameter values

Like most heuristic solution methods, our TS method for solving the LCP includes several parameters whose valueshave to be determined. Finding the best combination of parameter values is impossible in practice, as this requirestesting an enormous number of different combinations. In addition, some parameters are likely to be quite sensitive toinstance-specific data, and thus finding one single best value for such a parameter may be impossible. In our view, whatis needed is a set of robust parameter values. These values should give reasonably good performance for a wide rangeof problem instances, and they should be determined without too much effort. A few tests, as well as some experiencein using TS to solve VRPs, have resulted in the following.

Tabu tenure: The value for the tabu tenure is, in each iteration, chosen randomly from a set of integers. This strategyis found to give better results than using a fixed value for tabu tenure. We choose tabu tenure from the set of integersfrom 5 to 10, lower and upper bound included.

Infeasibility weights: Initially, the infeasibility weights �, �, � and � all have the value 10. After every iteration theirvalues are increased or decreased, see Section 4.6. We have found multiplicative adjustment to be the best strategy, andwe use the same adjustment policy for all four weights. A weight whose corresponding constraint is not violated inthe current solution is divided by 1.5, while the weight for a violated constraint is multiplied by 1.01. This means thatinfeasibility weights are decreased faster than they are increased.

Diversification intensity: Tests have shown that 0.001 serves as a good value for the diversification intensity parameter�. It should be noticed that this parameter is multiplied only by the relative move frequency of an order and that noother factors are used. Other researchers, see e.g. [18,19], also use scaling factors like problem size in their penalties.For the LCP, however, we have not found such scaling factors to have any significant positive effect.

When to apply 2-opt: We use 1.05 as the value for , which means that 2-opt is applied only on feasible solutionswith a total length not exceeding 1.05 times the length of the best found feasible solution.

Running time: Fig. 4 shows improvements found for the week 7 instance during 10 single 2-h runs of our algorithm.Each dot represents a new best solution found in each single run. As search time increases, fewer improvements are


6000

5500

5000O

bje

ctive

4500

4000

3500

0 20 40 60 80 100 120

Time in minutes

Fig. 4. Improvements found during the search.

Table 4Results of robustness testing

Setting Max. deviation

Standard 1.0312Tabu tenure (5–10) 3–7 1.0232

7–15 1.0489Divide (1.5) 1.3 1.0198

2.0 1.0219Multiply (1.01) 1.001 1.0766

1.1 1.0656Diversification (0.001) 0.01 1.0535

0.0001 1.0261

found. The steep descent phase is over in less than 10 min, but some improvements are found even during the secondhour in almost every run. For our subsequent testing purposes we have chosen to run the algorithm for 1 h, as most ofthe improvements should then be captured. We use a different random seed in each run, each test is run 10 times andwe report the best result found. The poorest result from the 10 runs is typically 3–5% above the best.

In practice, the running time will depend very much on the time available. If a solution is needed in just a fewminutes, one has to accept a solution that may be poorer than what may be achieved if the algorithm is allowed to rununtil the next day.

5.4. Robustness testing

The random instances described in Section 5.2 have been used to test the robustness of our algorithm.All instances arefirst solved using the parameter settings found in Section 5.3, different values are then tried for tabu tenure, adjustmentof infeasibility weights and diversification intensity. A summary of the results and the parameter values used are givenin Table 4.

The standard parameter values are given together with the name of the parameter in the leftmost column of thetable. The middle column gives the values tried for each parameter, and in the rightmost column the results are givenas the maximum deviation from the best result found for any of the eight instances. This means that e.g. changingthe diversification intensity parameter to 0.01, using the standard values for the other parameters, gives solutions thatdeviate at most 5.35% from the best solution found.


Table 5Test results

Method Week 7 Week 22

Total distance No. of tours Total distance No. of tours

Manual 4470 46 (5 tr.) 4513 40 (7 tr.)TS 3931 42 3573 38Improvement(%) 12 9 21 16

The standard parameter settings give a maximum deviation of 1.0312, thus we always find solutions with total lengthwithin 3.12% of the best found when we use the parameter values from Section 5.3. Changing some of the parametervalues may lead to small improvements, we found about 1% less deviation from the best results in three cases. Thealgorithm seems fairly robust with respect to both different instances and to changes in the parameter settings.

5.5. Test results

Our TS algorithm has been run on the data sets from Fatland, and Table 5 gives a summary of the results from theTS as well as numbers extracted from the manual plans.

The manually planned tours for the two weeks included the use of a trailer for five and seven of the tours, respectively.Our model does not include the use of trailers, but nevertheless suggests solutions with fewer tours and less total distance.

In our view, our TS method produces promising results, even if our model is still incomplete. It should be noted thatour method can guarantee plans that do not include any constraint violation, while today’s manual system in principleallows for the creation of route plans that violate rules. It is believed that in the future, it will be necessary to have asystem for planning with which one can prove that no plans are made that violate rules for animal welfare.

One important reason why we get quite good results is that we do not keep the geographical partitioning of today’ssystem, see Section 2.2.5, a partitioning we believe is an obstacle for the construction of better routes. In addition, theTS method chooses the pickup time without considering when the farmers want to be visited during the week. Timewindows are needed in the model, but it is also clear that today’s practice of assigning day for pickup when the animalsare registered reduces the set of feasible solutions quite much. It is probably not necessary to let the producers decidethe time for pickup to the same extent as reflected in today’s practice.

As long as manual planning is done, both geographical partitioning and assigning pickup day at once serve asnecessary means to keep the planning process manageable. For an automated planning system, on the other hand, theywill constrain the solution space and lead to poorer solutions than what would be possible without these restrictions.

5.6. The cost of additional constraints

In our view, one of the advantages of an automated planning system is the ability to generate plans for possiblescenarios to see the effect of situations that may occur. It may be valuable to know what the extra transportation costsin a given situation would be, in more extreme cases it may also be interesting simply to know if one can get feasibleplans. To illustrate this, we have run our test instances with an extra constraint that allows for homogeneous loads only,meaning that only one animal type is allowed on each tour.

The results from this test are shown in Table 6. Demanding only one animal type per tour leads to a rather largeincrease in total distance, while the number of tours is about the same. This corresponds well to the fact that it is easierto utilize the full capacity of the vehicles with only one animal type per tour, as the loading problem then more or lessdisappears. On the other hand, the distance increases because farmers who deliver more than one type of animals haveto be visited more than once, and because remote orders with different animal types that are close together cannot beserved on the same tour.

In a different test, we have manipulated the instance from week 7 to imitate a situation where some of the visitedfarms are infected with a disease. As explained in Section 2.2.3, only one infected herd can be visited per tour, and itmust be the last visit. If many infected herds have to be visited, this should lead to an increase in total distance traveled.In extreme cases many of the tours may visit only one farm. We have imitated two different types of situation, one


Table 6Cost for homogeneous loads

Method Week 7 Week 22

Total distance No. of tours Total distance No. of tours

Base case 3931 42 3573 38Home loads 4530 41 4502 40

Table 7Cost for “infected” scenarios

Base case Random Clustered

10% 20% 10% 20%

Dist. Tours Dist. Tours Dist. Tours Dist. Tours Dist. Tours

3931 42 3929 42 4287 43 4969 42 5179 45

Fig. 5. Normal route pattern.

where the infected herds are spread randomly geographically, and another where all herds in one part of the collectionarea are infected. In both cases we have two scenarios, one where 10% of all orders are from infected herds and onewhere 20% of the orders are of this kind.

Table 7 shows the results for this test. If the infected herds are spread randomly, increases in total distance aresmall. For the 10% random case, we actually get a marginally shorter solution than we did for the normal case. Ifthe infected herds are clustered, we get quite large increases in distance. For the 20% clustered case, we also seethat the number of tours needed increases significantly. These results show that the geographical distribution of theinfected herds has a large influence on the increase in transportation costs. If all herds in an area are infected, thisincrease may be significant even if the number of herds is quite small. This effect can be seen in Figs. 5 and 6,which show parts of the suggested route pattern for a “normal” day and for a day with many visits to infected herds.


Fig. 6. Routes visiting infected herds.

All the infected herds in Fig. 6 are in the north, which corresponds to the top of the figure. It should be noted that itis not the same set of customers that is served in the two cases, the purpose is to show typical changes in the routepattern. When several infected herds are visited, many routes have to add a visit to serve an order in the infected areaas the last stop. In a normal situation, many of these orders would have been served on the same tour, as can be seenin Fig. 5.

6. Conclusions and future work

We have described the livestock collection problem, which is a rich VRP extended with production and inventoryconstraints. Both model complexity and the size of real-life problem instances are likely to make exact methods fail tofind good solutions, and thus heuristic approaches are needed.

We have described a TS based algorithm for the LCP. Computational testing and comparisons with plans from today’smanual system show that our method constructs good solutions that do not violate the different constraints in the model.

Future work on this problem may include:Use of trailers: Some vehicles use a trailer on some tours, especially if the distances are long. It is not allowed to

leave a trailer containing animals behind and pick it up later, and some farms are not accessible if the vehicle uses atrailer. A similar problem is treated by Semet, see [20], but it is then allowed to leave the trailer behind and pick it uplater. The possibility to use trailers should be included in the model for the LCP.

Time windows: In the livestock collection problem, there are two kinds of time windows, and they occur for twodifferent reasons. The first type occurs because the farmer is not at home, ready to deliver animals, all the time.Therefore, he or she needs to be visited on certain days of the week or within certain time intervals during the day.The second type of time window is broader, typically two or three days, and occurs because the animals need to beslaughtered within a certain time interval in order to have the desired weight at the time of slaughter.

Further investigations are needed to find the real needs of time windows for the farmers, as today’s practice seemsto be too permissive in letting the farmers decide the time for collection.

Myopic focus: When a plan is made for several days at the time, one will often be more concerned about getting agood plan for the first day or two, while one only needs to know that a feasible plan exists for the last days of the timehorizon.


Replanning with minimal disturbance: Changes that occur during the planning horizon, see Section 2.2.5, are likelyto necessitate replanning. It will then be advantageous to be able to create new plans that deviate as little as possiblefrom the previous ones.

Diversification: The long-term goal of the project is to make a decision support system (DSS). It would then beadvantageous for the decision-maker to have several good, but different, solutions to choose from. The meaning ofdifferent then has to be defined more precisely, and a further investigation of how distance functions, see [21], may beused for this purpose would be interesting.

Acknowledgments

The work described here is part of the project “Transportation of living animals—reduced transportation costs, goodanimal welfare and first-class meat quality”, and is sponsored by the Norwegian Fund for Research Tax on AgricultureProducts. We thank Terje Wester and Jarle Dalane of Fatland Jæren,]ivind Bjurbeck, Halvor Hjelle and Erik Bergsengof Gilde, Stein W. Wallace of Molde University College and Inge Midtveit and Ole Alvseike of Norwegian MeatResearch Center for their help with the problem description and data collection.

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a tabu search approach for the livestock collection problem

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