University of ghent
Faculty of Economics and Business administration
Academic year 2009-2010
OPTIMIZATION OF FLEET DESIGN FOR
CYCLIC INVENTORY ROUTING PROBLEMS
Master thesis presented to obtain the degree of Master in Applied Economics: Business engineering
Wouter Bonte under the leadership of
University of ghent
Faculty of Economics and Business administration
Academic year 2009-2010
OPTIMIZATION OF FLEET DESIGN FOR
CYCLIC INVENTORY ROUTING PROBLEMS
Master thesis presented to obtain the degree of Master in Applied Economics: Business engineering
Wouter Bonte under the leadership of
parts of it for personal use. Every other use is subject to the copyright laws, more specifically the source must be extensively specified when using from this thesis.
De auteur en promotor geven de toelating deze scriptie voor consultatie beschikbaar te
stellen en delen ervan te kopi¨eren voor persoonlijk gebruik. Elk ander gebruik valt onder de beperkingen van het auteursrecht, in het bijzonder met betrekking tot de verplichting uitdrukkelijk de bron te vermelden bij het aanhalen van resultaten uit deze scriptie.
Ghent, May 2010
The promoter The author
Before I start with this dissertation, I want to emphasize that I could not have done this on my own. Rome was not built in one day and neither was this work. That is why I want to pay some attention to everyone who had any contribution.
First of all I want to thank my promoter, dr. B. Raa. Without him I could not have started this work in the first place. He also pushed me in the right direction when necessary.
Second, I do not want to forget my parents, my girlfriend, colleagues at Ghent University and other friends who helped me in one way or another.
Thank you,
Wouter Bonte
List of Figures vi
List of Tables viii
1 The cyclic inventory routing problem 1
1.1 Introduction . . . 1
1.2 The cyclic inventory routing problem . . . 3
1.3 Solution approach . . . 6
1.3.1 Route design . . . 6
1.3.2 Fleet design . . . 7
2 Literature review and contribution 10 2.1 Introduction . . . 10
2.2 Literature review . . . 10
2.2.1 Strategic inventory routing problem . . . 10
2.2.2 Vehicle minimization for periodic deliveries (VMPD) . . . 12
2.2.3 Models and algorithms for the cyclic inventory routing problem . . . . 13
2.2.4 Fleet and route design for cyclic distribution problems . . . 14
2.2.5 Aligning frequencies in cyclic delivery scheduling . . . 15
2.2.6 Cyclic scheduling of multiple tours with multiple frequencies for a single vehicle . . . 16
2.3 Contribution . . . 17
3 Modelling approach 18 3.1 Introduction . . . 18
3.2 Modelling approach . . . 18 3.3 Cost structure . . . 20 3.4 Mathematical Model . . . 21 3.5 Illustrative example . . . 23 4 Solution approach 28 4.1 Introduction . . . 28 4.2 General procedures . . . 28 4.2.1 Infinite horizon . . . 28
4.2.2 Lower bounds to the required number of vehicles . . . 29
4.2.3 Lower bounds to the total cost . . . 32
4.3 Constructive heuristics based on fixed cycle times . . . 34
4.3.1 Insertion heuristic - Avoid adding a new vehicle . . . 34
4.3.2 Insertion heuristic - Levelling . . . 37
4.3.3 Insertion heuristic - Best fit with startday . . . 38
4.4 Constructive heuristics taking into account cycle time alignment . . . 39
4.4.1 Insertion heuristic - Avoid adding a new vehicle . . . 40
4.4.2 Insertion heuristic - Levelling . . . 42
4.5 Improvement heuristics . . . 42
4.5.1 Remove and insert . . . 43
4.5.2 Ejection chain . . . 47 4.6 Metaheuristic extensions . . . 50 4.6.1 Tabu search . . . 50 4.6.2 Genetic algorithm . . . 54 5 Computational results 64 5.1 Introduction . . . 64 5.2 Design of experiments . . . 64
5.3 Priority rule analysis . . . 67
5.4 Evaluation . . . 69
5.4.1 Constructive heuristics based on fixed cycle times . . . 70
5.4.3 Remove and insert improvement heuristics . . . 72
5.4.4 Ejection chain improvement heuristics . . . 73
5.4.5 Tabu search heuristics . . . 74
5.4.6 Genetic algorithms . . . 75
5.4.7 Comparative analysis of the 6 groups . . . 77
5.4.8 Regression model and significant interaction effects . . . 82
5.5 Conclusions . . . 96
6 Conclusion 97 6.1 Concluding summary . . . 97
6.2 Recommendations for further research . . . 99
A Pseudo Codes 102 A.1 Introduction . . . 102
A.2 Constructive heuristics based on fixed cycle times . . . 102
A.3 Constructive heuristics taking into account cycle time alignment . . . 105
B Performance of priority rules 109 B.1 Introduction . . . 109
B.2 Performance overview . . . 112
1.1 Replenishing a set of customers from one single depot . . . 3
1.2 Inventory path with deterministic demand and continual replenishment possi-bility . . . 4
1.3 Distribution plan with fixed cycle times . . . 8
1.4 Distribution plan with fixed cycle times . . . 8
2.1 Greedy Scheduling (Campbell and Hardin, 2005) . . . 13
3.1 Distribution schedule - Illustrative example (fixed cycle times) . . . 25
3.2 Optimal distribution schedule - Illustrative example . . . 26
4.1 Lower bounds for the illustrative example . . . 33
4.2 Ejection chain principle . . . 49
4.3 Tabu search to escape from local optima . . . 52
4.4 Genetic Algorithm Model . . . 55
4.5 First subsolution . . . 55
4.6 Second Subsolution . . . 56
4.7 Representation . . . 56
4.8 Illustration of calculating the degree of diversity of a generated solution . . . 58
4.9 Two-point crossover example . . . 60
4.10 Cycle Crossover example . . . 61
4.11 Mutation example . . . 62
5.1 Overview of the different cost trade-offs per category . . . 78
5.2 Coefficients of the linear regression model (SPSS 17.0) . . . 83
5.3 Global effect of the number of tours . . . 85
5.4 Interaction between the factors NT and CR . . . 86
5.5 Interaction between the factors NT and DT . . . 87
5.6 Interaction between the factors NT and VC . . . 88
5.7 The global effect of the vehicle cost . . . 90
5.8 Interaction between the factors VC and DT . . . 91
5.9 Global effect of the duration of the tours . . . 93
5.10 Global effect of cost ratio . . . 95
5.11 Interaction between the factors CR and DT . . . 96
6.1 Choosing a variant of the CIRP out of many IRP versions and dividing it into two parts: route and fleet design . . . 98
6.2 Overview of the developed heuristic procedures . . . 100
B.1 Friedman test for priority rules . . . 112
3.1 Mathematical model for distribution scheduling (fleet design) . . . 22
3.2 Tour characteristics for the illustrative 6-tour example . . . 24
3.3 Cost rates for the illustrative 6-tour example . . . 24
3.4 Optimal costs for the illustrative 6-tour example (fixed cycle times) . . . 25
3.5 Average idle time for the illustrative 6-tour example (fixed cycle times) . . . 26
3.6 Old and new cycle times for the illustrative 6-tour example . . . 26
3.7 Optimal costs for the illustrative 6-tour example . . . 27
3.8 Average idle time for the illustrative 6-tour example . . . 27
4.1 Infinite horizon . . . 29
4.2 Pseudo code lower bound based on relatively primes . . . 31
5.1 Factors of the 10×24 Factorial Design . . . 65
5.2 Four different priority rules . . . 68
5.3 Mean costs and p-value (Friedman test) for the 4 priority rules . . . 68
5.4 Mean costs and p-value (Wilcoxon signed ranks test) for CF1 and CF2 . . . . 70
5.5 Characteristics CF1 and CF2 . . . 70
5.6 Mean costs and p-value (Wilcoxon signed ranks test) for CV1 and CV2 . . . 71
5.7 Characteristics CV1 and CV2 . . . 71
5.8 Mean costs and p-value (Friedman test) for RI1 till RI4 . . . 72
5.9 Characteristics RI . . . 73
5.10 Mean costs and p-value (Friedman test) for E1 till E4 . . . 73
5.11 Characteristics E . . . 74
5.12 Mean costs and p-value (Friedman test) for T1 till T4 . . . 74
5.13 Characteristics T . . . 75
5.14 Mean costs and p-value (Friedman test) for G1 till G6 . . . 75
5.15 Characteristics G . . . 76
5.16 Reliability of the genetic algorithms, expressed by CV in % . . . 77
5.17 Heuristic selection within each group for comparative analysis . . . 78
5.18 Mean costs and p-value (Friedman test) for the six groups . . . 78
5.19 Characteristics of the six groups . . . 80
5.20 Overview of number of times that a category gives the best result . . . 80
5.21 Overview of number of times that the heuristic gives the best result . . . 81
5.22 Overview of the best lower bound for the 8 groups (NT = low) . . . 82
5.23 Global effect of the number of tours . . . 84
5.24 Effect of the number of tours when CR is low . . . 85
5.25 Effect of the number of tours when CR is high . . . 86
5.26 Effect of the number of tours when DT is low . . . 86
5.27 Effect of the number of tours when DT is high . . . 87
5.28 Effect of the number of tours when VC is low . . . 88
5.29 Effect of the number of tours when VC is high . . . 88
5.30 Global effect of the vehicle cost . . . 89
5.31 Effect of the vehicle cost when tour durations are low . . . 90
5.32 Effect of the vehicle cost when tour durations are high . . . 90
5.33 Effect of the vehicle cost when the number of tours is low . . . 91
5.34 Effect of the vehicle cost when the number of tours is high . . . 92
5.35 Global effect of the duration of the tours . . . 92
5.36 Global effect of the cost ratio . . . 94
5.37 Effect of the cost ratio when tour durations are low . . . 95
5.38 Effect of the cost ratio when tour durations are high . . . 95
A.1 Pseudo code for the insertion heuristic “Avoid adding a new vehicle” . . . 102
A.2 Pseudo code for the insertion heuristic “Levelling” . . . 104
A.3 Pseudo code for the insertion heuristic “Avoid adding a new vehicle” . . . 105
B.1 Priority Rule: Degree of cycle time fit . . . 110
B.2 Priority Rule: Cycle time . . . 110
B.3 Priority Rule: Duration versus cycle time . . . 111
The cyclic inventory routing
problem
1.1
Introduction
Managing different processes in a company as independent entities is a main barrier to im-prove a company’s performance. This lack of process integration within as well as between companies has been an important subject for research over the last decades. Consequently companies started to recognize the need for integration and collaboration. Many initiatives that originate from this new approach fit within the scope of Supply Chain Management (SCM). SCM has been evolving rapidly and becomes difficult to describe or define in a nut-shell. The prominent idea is the collaboration between different stages in the supply chain. One can approach SCM from a transportation and logistics perspective (Tan, 2001). Ac-cording to this perspective, SCM opts to build and manage integrated logistics systems in order to fulfill customer requirements (Mentzer et al., 2001). This integration enables channel members to compete as a unified entity.
SCM prescribes to incorporate business logistics of the entire value chain into the strategic decisions. Besides the strategic ones, supply chain managers face some challenging operational issues. One needs to manage the total flow of materials, products and information from the start till the end of the value chain. Among other things, this means that for each channel member, decisions have to be made about when, how much and how frequent to replenish
the subsequent stage of the supply chain. In other words, an effective distribution plan has to be constructed.
In the past, most of the time the customer organized the replenishment independently by placing orders based on current inventory levels, expected future demand and expected lead times of their supplier(s). The irregularities of the orders and effects like the bullwhip effect (Lee et al., 1997) make this supply strategy harmful for every channel member. As the supplier does not know exactly when his customers will place their orders, he ought to design and probably redesign a distribution schedule on a daily basis. Additionally, the irregularities lead to uncertainty on the lead time. Therefore customers will have to keep up a considerable level of safety stock ending in less efficiency. Moreover, if worse comes to worst, safety stock might not suffice, so that out-of-stock situations occur, possibly ending in unsatisfied and churning end customers.
As mentioned earlier, people have recognized this flaw and eliminate it by managing the supply chain as a whole (SCM) in order to remove much uncertainty. Customers work together with their suppliers in a way that benefits both. Stock levels are made available for the supplier, e.g. by implementing Electronic Data Interchange (EDI) technology (Mukhopadhyay et al., 1995). This way suppliers have a clearer view of the complete situation, which gives them increased flexibility in deciding whom to visit when. Moreover, the fact that suppliers are able to determine an efficient distribution plan reduces the level of safety stock kept up at the customers. However, a distribution plan consists of more than just decisions on when replenishments will occur. The supplier also needs to design well-reasoned vehicle routes. Thanks to effective information flows, the supplier is able to integrate decisions about whom to replenish when as well as decisions about vehicle routes. Vehicle routes can be determined more optimally and clients’ stock levels can be managed more efficiently, which results in a better overall performance finally ending in more satisfied and more loyal end-customers. This type of collaboration between supplier and client, whether within or between companies, is called ‘Vendor Managed Inventory’ (VMI) or ‘Supplier Managed Inventory’ (SMI)(Waller et al., 1999). VMI is a contemporary phenomenon that integrates distribution and inventory management seamlessly.
systems. As explained above, giving the supplier the ability to decide the timing and quantity of deliveries for customers clearly gives the supplier greater flexibility and room for increased efficiency. How to take full advantage of this increased flexibility, though, is still under intense investigation because of the size and the complexity of the problems that come along with VMI systems. It is in this context that the newly proposed solution methodologies of this thesis have to be understood.
1.2
The cyclic inventory routing problem
One of the emerging problems in managing a VMI system is called the ‘Inventory Routing Problem’(IRP). The objective of the IRP is to determine a distribution plan that minimizes the total cost (distribution and stock holding costs) without causing any stock-out situations at the customers. Thus, IRPs arise when inventory as well as routing considerations are included in a model to effectively capture the features of a distribution system (Larson and Webb, 1995).
Figure 1.1: Replenishing a set of customers from one single depot
Consider a distribution system in which one supplier replenishes a set of customers, denoted byS and indexed byj from one single depot, denoted ∆. Each customer consumes a certain
amount of products per unit of time, which is called the demand rate, denoted by d and indexed byj,j ∈S. Also the storage capacity is relevant in inventory management. The stock capacity of each customer is denoted byκ and indexed by j,j ∈S and is expressed in units. All customers should be sustainably replenished from the depot without ever stocking out.
In literature the IRP has several variants according to the problem characteristics. A key assumption in this work is that the demand ratesdjare constant over time. Thus, a customer
will consume the same number of products per unit of time in the future as he does at the moment, e.g. 100 units per day.
dj is constant, ∀ j∈S (1.1)
This assumption has a substantial influence on how to approach the IRP. As all relevant customer characteristics, storage capacityκj and demand rate dj, remain the same, the
envi-ronment for the supplier is static. This way an effective solution for the IRP will also be valid in the future. More specifically, presume one developed an efficient vehicle route to replenish a subset of customers, denoted bySi, Si ⊆ S. The vehicle delivers quantities, denoted byqj,
j ∈ S that cover the various customer demands for a certain period of time, denoted by T.
qj = T·dj, ∀j∈Si (1.2)
Figure 1.2: Inventory path with deterministic demand and continual replenishment possibility
A period of time T later, a next replenishment consisting of the same quantities will be needed since the demand rates are constant. Thus, the vehicle route and the delivered quantities will still be valid and can be used in the future. When a solution can be maintained and repeats after some fixed period, it is called a cyclic solution. It is obvious that cyclic solution approaches result in cost efficient solutions that are stable and predictable over time. This
predictability reduces operational complexity and variability in a high degree, for the supplier as well as for the customer (Raa, 2006).
The resulting problem that needs to be solved is a “Cyclic Inventory Routing Problem” (CIRP). A cyclic distribution plan has to be developed in order to minimize the total cost. As it concerns a cyclic solution, one rather speaks in terms ofcost rates, i.e. costs per period of time. The cost structure used in this work consists of five components.
1. A fixed cost per vehicle expressed in euro per period, denoted by ψ.
2. The variable transportation cost is a second cost component.
3. A fixed dispatching cost of ϕ∆euro is allocated.
4. A fixed cost of ϕj euro is allocated for each delivery made to customerj ∈S.
5. For every customer j ∈S, a holding cost ofηj euro per unit per period is allocated.
As one could already have derived from the cost structure, another crucial assumption con-cerns the endogeneity of the number of vehicles needed to execute the distribution system. Until recently, designing the fleet of the vehicles has never been part of the problem. Here we assume a homogeneous fleet of vehicles with a capacity of κ units. The incorporation of the fleet size increases the value of the final solution considerably. Value should be comprehended as the capability to support management and/or operational decisions more effectively. On the other hand, the complexity of the problem increases significantly.
Before proceeding, it is important to grasp the involved limitations of the assumptions men-tioned above. First of all, this CIRP version considers a single depot. Though it covers one-to-one, one-to-many and many-to-one VMI systems, many-to-many VMI systems need an extension to multiple depots. Second, costs for loading and unloading a vehicle are fixed. This can be justified since these costs are usually small in comparison with other costs. Still, in certain situations these costs can account for a relatively big part of total costs, e.g. fragile goods that have to be shored individually. In such cases, the assumption of these costs being fixed should be abandoned as the total cost of a vehicle route will depend significantly on the number of products needed to be loaded. Third, the CIRP model described above deals
with only one product. Problems consisting of multiple products are applicable as long as demand rates can be expressed in a common unit and the different products can be trans-ported together, i.e. in the same vehicle at the same time. If this is not the case, then some extra transportation constraints should be added. Fourth, demand rates are supposed to be constant over time, which implicates a cyclic solution approach. A cyclic solution approach is not valid in all situations, but it is considerably robust, as Raa (2006) proved in a practical application. Yet, if demand rates are not constant at all, another approach should be followed. Fifth, the number of vehicles are included as a part of the problem. The number of vehicles is usually constant at short notice, but becomes variable in the long run. A long-term vision fits better within the framework of strategic SCM and leads to more valuable solutions, as mentioned earlier. Finally, transportation costs usually consist of a fixed part and a variable part, both depending on the capacity of the vehicle. In this work we use a transportation cost that is completely variable in function of the duration. Nevertheless, this does not have a negative impact on the general suitability of our approach.
1.3
Solution approach
As will become clear out of the literature review, covered in the next chapter, there are several approaches appropriate to tackle the (C)IRP. In this section, the solution approach that will be followed is briefly explained. The problem is divided into two main sections. First, customers are allocated to a certain periodic route (route design). Then, each route is periodically allocated to a vehicle (fleet design). One notices that a route might not always be conducted by the same vehicle.
1.3.1 Route design
Route design has been subject for research for a long time and good, acceptable solutions are available and will not be investigated further in this work. However, the methodology of the route design can not be simply ignored. For a given set of customers, the performance evaluation of the fleet design definitely depends on how routes are designed. Therefore a short description of the methodology followed in this work concerning route design is reported.
dividing customers into subsets, taking into account several constraints. First, each route covering a customer subset should take no longer than a full working day. Second, the cumulative demand of the customer subset over one day should not exceed the vehicle capacity κ. As it concerns a cyclic solution approach, a cycle time1for each route has to be determined and can vary from one day to the maximal cycle time, denoted by Tmax, resulting from
the limited vehicle capacity and customer storage capacities. Each route has an optimal cycle time, denoted by Teoq, that gives the best possible trade-off between the distribution
costs (loading, transportation and unloading) and the customer’s stock holding costs. An evolutionary algorithm consisting of a savings and insertion heuristic, is currently applied to solve this partitioning problem of customers into subsets. We refer to Raa and Dullaert (2007) for a more detailed explanation of both procedures.
Obviously, the required fleet size to execute these routes is not yet known. Consequently, the optimal cycle times cannot take the fixed vehicle cost into account. Nevertheless, Raa and Dullaert (2007) incurred a little modification to reflect this extra cost. Routes with a higher cycle time are more adequate, since they utilize less time on the vehicles. Therefore not the optimal cost rates are used. To incorporate the preference for higher cycle times, the optimal cost rates are divided by a factor (1 + 0.1·Teoq). Including this factor in both heuristics gives
a trade-off between the objectives of designing inexpensive routes and designing routes with higher cycle times.
1.3.2 Fleet design
Fleet design forms the key subject of this work. Each route has a certain individual cycle time and has to be added to a distribution plan. As a result the complete distribution plan will be cyclic and its cycle time will be given by the least common multiple of all individual cycle times. Each iteration of a route at a particular day of the cycle has to be assigned to a vehicle in a way that minimizes total costs. The constraint that no vehicle is allowed to work more than 8 hours on any day in the cycle has to be taken into account.
Until recently, route and fleet design have been regarded as two independent subparts, that collectively lead to a final solution. Once customers were assigned to routes (route design), all
1
routes were completely fixed. Though, better solutions can be obtained when the individual cycle times of the different routes can still be modified when starting to design the fleet. A short example illustrates this issue. Imagine route design results in two routes, both lasting 1 day to complete, with aTeoq of 2, respectively 3 days. When these cycle times are fixed, the
minimum number of vehicles needed to complete all routes is 2 vehicles. However, as shown in figure 1.3 and 1.4, when modifying the cycle time from 3 days to 2 days (route B), one vehicle can be spared.
Figure 1.3: Distribution plan with fixed cycle times
Figure 1.4: Distribution plan with fixed cycle times
Although reducing the number of required vehicles will most of the time result in a better, thus less expensive solution, this might not always be the case. One need to realize that deviating from the optimal cycle time increases the individual cost of that route and is only justified when that increase does not exceed the amount of vehicle cost savings as a consequence of that cycle time modification. The example given above can illustrate this issue. Suppose a vehicle implies a fixed cost of e200/period, then the modification of the cycle time of route B to 2 days is only justified if the extra route cost does not exceede200/period.
When the cycle times of the various routes are fixed when designing the fleet, minimizing total costs when designing the fleet comes down to minimizing the number of vehicles. However, this way many alternative and perhaps better solutions are not verified. Thus, allowing the opportunity to align cycle times is advised as this increases the number of possible solutions, enlarging the chances of obtaining a better solution.
cost taking into account the aligning of the various cycle times. The different methodologies are then evaluated, both on optimisation performance and calculation time performance.
The remainder of this work is classified as follows. Chapter 2 reflects on what has been studied about this subject in the past and how this thesis will contribute. A brief review of several relevant articles and/or books is given. Chapter 3 describes the modelling approach that has been followed throughout this thesis. In chapter 4 all aspects of the various solution methodologies are covered. In chapter 5 the computational results of the different solutions, based on a set of benchmark instances, are explained. Finally, chapter 6 provides some conclusions and recommendations for additional research.
Literature review and contribution
2.1
Introduction
The inventory routing problem has been approached in several ways. A first distinction can be made between papers handling about the cyclic inventory routing problem and those that do not. Since a key assumption in this work is that customer demands are deterministic and constant, only cyclic solution approaches over an infinite planning horizon are relevant. In this chapter some articles and books are reviewed in order to get in touch with former research and to express their impact on the ideas in this work.
Previous approaches, such as Viswanathan and Mathur (1997), do not consider fixed vehicle costs in their cyclic solution approaches for integrated distribution and inventory management. Most of the time one vehicle is used for only one tour and one assumes that enough vehicles are available. As mentioned earlier, in the long term the vehicle fleet becomes variable and fleet sizing has to be incorporated.
2.2
Literature review
2.2.1 Strategic inventory routing problem
Larson (1988) introduced the so-called Strategic Inventory Routing Problem (SIRP), in which one is attempting to optimally configure the system over the long run. Tactical IRPs, on the other hand, concern the efficient scheduling and routing of vehicles over the near term, given
fixed resources (Bell et al., 1983). Larson motivates his choice for a strategic approach by the fact that there is usually a significant amount of time between the purchase or lease agreement of the vehicles on the one hand and their availability for logistic operations on the other hand. In Larson’s strategic IRP the objective is to minimize the required fleet size, taking into account constraints related to limited storage capacity, limited vehicle capacity and transportation characteristics (e.g. waterways, draft limitations). To solve this he uses a heuristic “savings” algorithm, denoted by “SIRSA” (Strategic Inventory/Routing Savings Algorithm), which assigns customers to clusters. For replenishment, all customers of a cluster are visited in a single tour. Larson realized this is not always optimal and suggested to investigate this in additional research.
Being a strategic model, Larson ignores vehicle scheduling constraints and justifies it by saying that the final solution does not necessarily need to be globally optimal. Moreover worst-case scenarios are used and the minimum required fleet size has often to be rounded up to the closest integer. For the actual operational routing of the selected fleet, he refers to adaptive tactical scheduling and routing procedures, not cyclic deterministic procedures, because these allow for more flexibility in the short term. Still, Larson realized that considerable work must be devoted to the integration of the results of such a strategic model with scheduling and related concerns of tactical models.
Larson and Webb (1995) generalize Larson’s savings heuristic for SIRP in order to eliminate some inefficiencies. In Larson’s approach all customers of a cluster are visited in a single tour for replenishment, which often is suboptimal. Some customers are visited more frequently than necessary. These frequent visits are caused by the inclusion in the cluster of other customers who need to be visited more frequently. Larson and Webb introduce the concept of routesets and use two additional decision variables, called period and phase of individual customer replenishment. For any cluster of customers, a routeset consists of a number of component routes arranged in a specific order. Hence, not all customers of a cluster are included in every route of the routeset. Any routeset is completely defined by the number of routes it contains and the period and phase of each customer. Theperiod of a customer is the number of routes between successive replenishments and thephaseof a customer is the number of routes in the routeset before the first route that replenishes this customer. Introducing
these variables enlarges the set of feasible replenishment schemes, so the resulting heuristic is a generalization of SIRSA. Computational results show that the period/phase approach generally yields significant reductions in average vehicle requirement.
Both articles bring some useful insights in approaching CIRPs. They identify the required fleet size, but they lack to determine the actual vehicle routing as such. Note that Larson and Webb (1995) do explicitly consider the assignment of routes to vehicles but only in order to identify the required fleet size.
2.2.2 Vehicle minimization for periodic deliveries (VMPD)
Campbell and Hardin (2005) consider the problem of minimizing the number of vehicles required to make strictly periodic, single destination deliveries to a set of customers, under the initial assumption that each delivery requires the use of a vehicle for a full day. Gallego and Simchi-Levi (1990) evaluate the long-run effectiveness of direct shipping and conclude that direct shipping is at least 94% effective over all inventory routing strategies whenever minimal customer capacity is 71% of vehicle capacity. The IRP is known to be NP-hard1, but Campbell and Hardin investigate whether the problem is still NP-hard when all routes are restricted to be single stop. They refer to this problem by “vehicle minimization for periodic deliveries” (VMPD). They analyze the complexity of the problem and prove that the problem and many of its variants are NP-hard. Therefore they developed a (pseudo)polynomial, greedy algorithm that can be used for any instance of the problem and gives optimal solutions for some special cases (figure 2.1).
Next, they generalize the problem by including the possibility of doing multiple trips per day for a vehicle. In other words, it is not required anymore that each route occupies a vehicle for a complete day. The complexity of this problem increases drastically due to the bin packing aspects that come along. The modified algorithm makes use of a first fit rule, similar to the structure of the well known bin packing heuristic.
Campbell and Hardin are among the first that take into account the actual routing of the vehicles. That is why some of the ideas in this work reflect their findings. However, they start 1When a decision version of a combinatorial optimization problem is proved to belong to the class of NP-complete problems, then the optimization version is NP-hard (Atallah and Blanton, 2009).
Figure 2.1: Greedy Scheduling (Campbell and Hardin, 2005)
from fixed customer cycle times, which limits the range of possible solutions. Although people can argue that fixed cycle times are more realistic, this is usually not the case when working in a VMI system, based on the cooperation of multiple supply chain members. Customers, whether internal or external, will be eager to consider alternative delivery periods when both parties can benefit.
2.2.3 Models and algorithms for the cyclic inventory routing problem
Raa (2006) presents a novel modelling and solution approach for the cyclic inventory routing problem with constant consumer demand rates. He is among the first that actually takes into account the whole picture. A three-way trade-off is made between transportation, holding and fixed vehicle costs. He introduces a new routing concept, called “distribution patterns”, which allows a single vehicle to make multiple tours. The issue of assigning tours to vehicles is thus also explicitly solved. He uses a well-reasoned cost structure, which is largely adopted in this work. He also attaches importance to not only maximum, but especially minimum cycle times, something which was usually neglected in the past.
First, customers are partitioned over one or multiple vehicles. Second, all customers of a certain vehicle are further partitioned over one or more tours. Third, the actual tours are constructed by determining the order in which the customers are visited. Fourth, an iteration frequency is calculated for each tour of a certain vehicle. Fifth, a feasible delivery schedule is designed for each vehicle and the cost rate is determined.
To implement this solution approach he presents several solution methods that try to obtain global cost minimizing solutions. First, he presents a sequential insertion heuristic, in which customers are inserted one by one at minimal cost. Second, a parallel savings heuristic is presented. Besides these constructive heuristics, an improvement heuristic is also presented that is based on a remove and reinsert principle. Next, these heuristics are embedded in two metaheuristic frameworks.
Raa (2006) has made a significant contribution to former existing literature by being the first tackling the cyclic inventory routing problem in all his glory, not ignoring particular subparts of the problem. That is why some remains of Raa’s approach will be find in this work. Meanwhile, the routing concept of distribution patterns has shown not to be flawless. Distribution patterns imply that each iteration of a tour is done by the same vehicle. This constraint will be released in this work, in order to increase the flexibility, aiming at decreasing the required fleet size.
2.2.4 Fleet and route design for cyclic distribution problems
Raa and Dullaert (2007) present in this paper a new solution approach for the problem of constructing a cyclic distribution plan for replenishing customers with constant demand rates from a single depot. The approach consists of two phases, route design and fleet design. Unlike Raa (2006) they make it possible to allow the different iterations of the same route to be made by different vehicles. This is accomplished by designing routes before they are assigned to vehicles. They present provisional results which show that the new approach is very fast, but requires a more powerful, meta-heuristic approach to fully exploit the benefits of the generalized modelling framework.
As explained in 1.3 we use this breakdown (route and fleet design) in this work. However, Raa and Dullaert (2007) partition fleet design into two subsections. First, they determine
the number of vehicles required for making all routes with their given cycle time. Second, they align the cycle times of the routes to reduce the required fleet size, based on modifying the “critical” route and all routes with higher cycle times. The “critical” route is the route that caused the last vehicle to be added to the fleet. Moreover, in the second part, they do not take explicitly into account the actual cost increases by deviating from the optimal cycle times. They only include a priority rule which gives priority to routes with higher cycle times, because the relative deviation from the optimal cycle time is smaller.
In this work, we leave the approach of partitioning fleet design into two subsections and try to include the cycle time alignment simultaneously, taking explicitly into account the actual cost increase caused by deviating from the optimal cycle times. We reject the concept of the “critical” route as specified above. Focusing on the tour that caused the last vehicle to be added is maybe the most obvious choice, but is rather blunt and oversimplified in our opinion. The need for adding the last vehicle originates from all previously inserted tours and it is very unlikely that one tour is responsible. Therefore we have chosen to follow a more general approach in deciding about which tours to modify.
2.2.5 Aligning frequencies in cyclic delivery scheduling
Raa and Aghezzaf (2009) investigate a new heuristic that aligns delivery frequencies of mul-tiple tours that have to be made by a single vehicle, in order to construct a feasible schedule. Thus, as in Raa (2006) each iteration of a tour is made by the same vehicle. The heuristic ap-proach consists of two iterative phases. First, tour cycle times are selected, starting from the optimal cycle times. Second, the feasibility of a given combination of cycle times is checked by constructing a schedule. As soon as a feasible schedule is found, the procedure stops and the cycle time combination is returned.
Tours are sorted in order of increasing cycle times and decreasing durations in case of a tie. In the first phase, the cycle times are adjusted as follows. All routes in the order starting from the “critical” tour are modified so that their cycle times become a divisor or an integer multiple of the least common multiple of the routes before the critical tour. The second phase consists of two steps. First, a “critical” tour is identified by comparing the cycle times two by two. If two tours have relatively prime cycle times (which means that they have to be
scheduled on the same day at certain moments) and cannot be made together on the same day without violating the driving time restrictions, no feasible schedule exists and the tour with the higher cycle time is marked as the critical tour. The procedure returns to the first phase to align the routes. When no two-by-two conflicts occur in the first step, the second step is started. This step includes designing an actual schedule using the best-fit insertion heuristic, adapted from Raa et al. (2009). As soon as a tour cannot be feasibly inserted, this tour becomes the critical one and the procedure returns to the first phase.
Since each iteration of a tour is done by the same vehicle, this heuristic cannot be completely applied to our variant of the problem. However, some insights about cycle time alignment are useful and will be adopted in some of our alignment approaches.
2.2.6 Cyclic scheduling of multiple tours with multiple frequencies for a single vehicle
Raa et al. (2009) tackle the cyclic multi-frequency multi-tour scheduling problem with driving time restrictions. A best-fit insertion heuristic is proposed. This heuristic inserts the tours one by one, meanwhile extending the time horizon when necessary. All tours are sorted in order of decreasing frequencies, adding the rule that no two tours with the same frequency are added to the schedule consecutively. The best-fit rule says that from all possible days for the first iteration of a tour, one has to choose that day for which the cumulative remaining time is minimal. Thus, for each startday, the tour is inserted in the schedule and the remaining time on each day the tour is scheduled, is totalled. The startday for which this sum is minimal, is chosen. This way bigger blocks of idle time are left for the remaining tours.
Although the cyclic multi-frequency multi-tour scheduling problem is NP-hard, a branch-and-bound algorithm is implemented to optimally solve a set of test instances. The proposed heuristic approach is then evaluated by comparing the results with the optimal solution.
Even though the problem described in this paper is significantly different, it has some dual aspects compared to our problem. In their version of the problem, the relative frequencies of the different tours are known and the minimal number of days has to be determined in order to make it possible that a single vehicle can make all the tours. In our work on the other hand, we have to determine the number of required vehicles to execute a set of tours within
a certain number of days (the length of the schedule).
2.3
Contribution
The existing literature on cyclic inventory routing problems ignored the incorporation of fixed vehicle costs for a long time. However, as reported above, recently some papers do consider fixed vehicle costs and the assigning of tours to vehicles. But usually one assumes that each iteration of a tour is made by the same vehicle, which diminishes the scheduling flexibility considerably. One exception to this is the paper of Raa and Dullaert (2007).
This work aims at utilizing the extra flexibility, created by leaving open the possibility of having multiple vehicles doing iterations of the same tour. This might lead to better solutions and enlarges the practical relevance of cyclic inventory routing.
Modelling approach
3.1
Introduction
The literature review of the previous chapter shows that there are multiple ways of approach-ing the cyclic inventory routapproach-ing problem. This chapter presents the modellapproach-ing approach that is followed. In section 3.2 the general model is explained. In section 3.3 the cost structure is clarified more thoroughly. Section 3.4 translates this in a mathematical model and section 3.5 illustrates the approach by means of an example.
3.2
Modelling approach
We consider a set of tours that have to be assigned to a certain vehicle, iteration by iteration. Each touri visits a disjunct subset of customersSi from a single depot. This way all subsets
form a partition of the setS. Each tour can be assigned to each vehicle as all tours start from and arrive at the common depot. The cyclic inventory routing problem obviously implies that each tour has to be repeated periodically. The time between two consecutive iterations is called the cycle time of a tour, denoted by Ti. Yet, not all cycle times are feasible for a
tour as lower and upper bounds have to be taken into account. There are two kinds of lower bounds. First, a vehicle needs some time to complete the tour since the vehicle has to be loaded, has to visit all customers in the subset and has to be unloaded at each customer. This way a tour cannot be repeated before it is finished. Second, some customers do not want to be replenished too often and place a restriction on their delivery frequency. Due to capacity
restrictions, the cycle time is also bounded from above. Again, there are two kinds of capacity restrictions. The first restriction is related to the limited vehicle capacityκ, identified by the maximum quantity a vehicle can distribute in one single tour. Therefore, a vehicle has to replenish the customers before they have consumed this full vehicle load. The limited storage capacity of the customers needs also to be taken into account. If the maximum delivery quantity resulting from the vehicle capacityκ exceeds the storage capacity κj of a customer
j ∈ S, the maximum cycle time decreases.
Note that the fair-share rule is applied to divide the vehicle load over the different customers in a tour i. This way each customer j ∈ Si receives a quantity qj that covers the demand
for the same period of time, i.e. the tour cycle time Ti. For example, assume we have two
customers in a tour that have a demand rate of 15, respectively 45 units per day. Then they will always receive 25%, respectively 75% of the vehicle load.
A tour i lasts a certain amount of time to complete, i.e. loading the vehicle, visiting all customers and unloading the vehicle at each customer. We denote this duration by δi. The
time for driving is obviously constant as the subset of customers remains the same over time. However, time for loading and unloading the vehicle depends on the cycle time as the quantity
qj delivered to a customerj is related to the cycle time of the tour, as shown in equation 1.2.
In most cases these deviations are very small and therefore are neglected in this model.
All vehicles can make multiple tours on a day, as long as the driving time restrictions are not violated. In some cases like raw material replenishment, driving time restrictions might not be present as vehicles can make a tour at any time of the day. However, deliveries can only be made during the day. When driving time restrictions apply, we assume that any tour cannot last longer than the daily amount of time available. Moreover, the cycle time of a tour should be an integer number of days, such that a tour cannot occur more than once a day. Existing models often consider each iteration of a tour to be made by the same vehicle. In this model we reject this assumption and allow different vehicles to execute iterations of the same tour.
3.3
Cost structure
As the ultimate objective is to design a distribution schedule that minimizes total costs, the cost structure that is adopted from Raa (2006) has to be clarified. The cost rate of a distribution schedule consists of five components.
1. A fixed cost per vehicle expressed in euro per day, denoted byψ. The number of vehicles that are required in a distribution schedule is denoted by Ω. An important remark is that the cost is allocated regardless of the activity of the vehicle. This reflects the opportunity cost and the time cost of a vehicle. A vehicle that stands still cannot be used for other destinations and is also depreciating. Moreover, there are some fixed costs that have to be incurred like wages, insurance, etc.
2. The variable transportation cost is a second cost component. If the transportation cost for completing a touri isθi per hour, the variable transportation cost per day is θi·δi
/Ti euro, denoted by CT,i.
CT,i=
1 Ti·
θi·δi (3.1)
3. Every time a vehicle leaves the depot to make a replenishment tour, a fixed dispatching cost ofϕ∆euro is allocated. It stands for the costs that are made for loading a vehicle.
4. The fourth cost is similar to the third and is the cost of delivery incurred at the customer. A fixed cost of ϕjeuro is allocated for each delivery made to customerj ∈ S. It reflects
the costs made for unloading the vehicle. The total dispatching costs per day, i.e. costs for loading and unloading the vehicle, for a tour i are denoted by CD,i.
CD,i = 1 Ti ϕ∆+ X j∈Si ϕj (3.2)
The sum of the variable transportation costs and the total dispatching costs is referred to as the transportation cost of a tour.
5. The last cost component is the stock holding cost at the customers. Since we are thinking in an integrated supply chain philosophy, this cost should not be neglected. For every customer j ∈ S, a holding cost of ηj euro per unit per day is allocated. The
equation 1.2. Thus, the average stock level at customerj amountsqj/2 units (cf. figure
1.2). The stock holding costs at customerj included in a touri areηj·dj·Ti/2 euro per
day. The sum of all stock holding costs at the customers of a touri is denoted byCH,i .
CH,i= Ti 2 X j∈Si (ηj·dj) (3.3)
The total cost of a distribution schedule in whichn tours are replenishing the set of customers
S is consequently given by the following formula. CTot= Ω·ψ | {z } A + n X i=1
CT,i+CD,i+CH,i
| {z }
B
(3.4)
Equations 3.1, 3.2 and 3.3 show that part B clearly depends on the specific cycle timeTi. One
might think that part A can be marked as a fixed cost, i.e. independent of the cycle times of the different tours, but this is wrong. The number of vehicles required in a distribution schedule depends definitely on the set of cycle times of all tours, as the illustrative example in section 1.3.2 shows.
Since the cost varies with the cycle times, an optimal cycle time Teoq,i exists for each tour
i. When constructing tours (route design) the required fleet size Ω is not yet known and fixed vehicle fleet costs cannot be taken into account. That is why the calculated optimal cycle time of a tour does not reflect the need for vehicles (part A in equation 3.4). The optimal cycle time gives the best trade-off between transportation costs and stock holding costs. Stock holding costs are usually small compared to transportation costs. Hence large delivery quantitiesqj and consequently long cycle times give better trade-offs. Since the size
of the delivery quantity is restricted by the vehicle capacity κ, very often the optimal cycle time is equal to the maximum cycle time.
3.4
Mathematical Model
A mathematical formulation for this model is added in table 3.1. Because of the periodicity, the infinite horizon can be reduced to the cycle time of the whole distribution schedule, denoted by Tsched. This cycle time is given by the least common multiple of all tour cycle
times. The binary variableXk
Table 3.1: Mathematical model for distribution scheduling (fleet design) MinimizeCTot subject to: n X i=1 Ximk ·δi ≤8 k= 1. . . Tsched m= 1. . .Ω (3.5) Ω X m=1 Tsched X k=1 Ximk = Tsched Ti i= 1. . . n (3.6) Ω X m=1 Ximk = Ω X m=1 Xk+Ti im k= 1. . . Tsched−Ti i= 1. . . n (3.7) Ω X m=1 Ximk ≤1 k= 1. . . Tsched i= 1. . . n (3.8) Tmin,i ≤Ti ≤Tmax,i i= 1. . . n (3.9) 1≤Ω≤n (3.10) X111 = 1 (3.11) Ximk ∈ {0,1} k= 1. . . Tsched i= 1. . . n m= 1. . .Ω (3.12) Ti∈Z+0 i= 1. . . n (3.13)
The objective is to minimize the total cost of the distribution schedule, while not violating any of the constraints. Constraint 3.5 states that a vehicle never has to drive more than 8 hours a day. Constraint 3.7 states that a tour has to be scheduled equidistantly, i.e. periodically, and constraint 3.6 adds the fact that each tour has to appear as frequent as its cycle time implies.
If for example a tour has a cycle time of 3 days and the total distribution schedule has a length of 60 days (Tsched), then this tour has to be iterated 20 times in the schedule. Constraint 3.8
restricts a tour to be iterated no more than once a day, while constraint 3.9 indicates that the cycle time of a tour needs to stay within the range of feasible values. Finally, constraint 3.10 limits the number of vehicles to the number of tours and constraint 3.11 fixes a starting point in the distribution schedule, without loss of generality.
The cycle times are variable which makes this model for building a distribution schedule non-linear and very complex to solve. Nevertheless, if the cycle times were fixed, it would still be very complex since it contains an assignment problem and a bin packing problem, known to be NP-hard. The illustrative example below, which contains only 6 tours, takes already about 20 minutes, to solve to optimality. Because of this computational complexity several heuristic approaches are proposed in chapter 4. We want to stress that solving the problem to optimality is not an issue that is investigated in this dissertation. The illustrative example below is solved to optimality by enumerating all possibilities that satisfy all constraints of the mathematical model in table 3.1. The calculation time is therefore not a meaningful indication but just a rudimentary sign of complexity.
3.5
Illustrative example
Here we present a small illustrative example and report the optimal solution. Consider the tours presented in table 3.2 as a result of the route design, which forms the input for building our distribution schedule. It consists of 6 tours replenishing a set of 20 customers. Table 3.3 shows the cost rates used in this illustrative example. For simplicity total dispatching costs, i.e. costs for loading and unloading, are ignored. Variable transportation costs are assumed to bee 50 per hour. All customers have the same stock holding cost of e0.15 per unit per day. Finally, the fixed vehicle costs amountse70 per day.
First, we discuss the case when the cycle time of each tour i is fixed to the optimal Teoq,i.
The infinite time horizon can be simulated by a distribution schedule of 84 days given by the least common multiple of all cycle times. Figure 3.1 shows 3 vehicles are required in the distribution schedule. Table 3.4 presents the resulting costs and table 3.5 shows the idle time of each vehicle.
Table 3.2: Tour characteristics for the illustrative 6-tour example
Tour id Teoq Tmin Tmax δ Pj∈Sidj (days) (days) (days) (hours) (units/day)
1 2 1 3 3 400 2 4 1 6 5 200 3 3 1 4 4 300 4 7 1 12 8 100 5 6 1 15 5 80 6 7 1 15 6 80
Table 3.3: Cost rates for the illustrative 6-tour example
Type Cost rate
ψ 70
θi 50 ∀i={1. . .6}
ϕ∆ 0
ϕj 0 ∀j={1. . .20}
ηj 0.15 ∀j={1. . .20}
As one can see from table 3.5 vehicles are left unused rather frequently. This might be an indication that by aligning the cycle times of the different tours, one or more vehicles can perhaps be saved. Moreover, from the distribution schedule one can derive that sometimes all vehicles are unused. Again this might be an indication that the schedule can be levelled in a better way. However, this can only be verified by actually looking at the solution with relaxed cycle times.
Now the assumption of fixed cycle times is released, increasing the number of alternative solutions. A first remarkable observation is that the cycle time of the entire schedule has significantly decreased. This shows that the actual cycle times are better aligned than the original ones. Only the first and second tour are not modified.
Obviously deviating from their optimal cycle times, increases the tour-specific cost rate by e 8.42. This increase still does not outweigh the additional fleet savings, which amount to e
Figure 3.1: Distribution schedule - Illustrative example (fixed cycle times)
Table 3.4: Optimal costs for the illustrative 6-tour example (fixed cycle times)
Type Cost per period (e) Transportation 345.83 Dispatching 0.0 Unloading 0.0 Holding 318.0 Tour-specific 663.83 Fleet 210.0 Total 873.83
140, since the new schedule makes use of only 1 vehicle instead of 3. The cycle time alignment is thus justified, resulting in a less costly schedule. Note that especially the transportation cost rate decreases, while the holding cost does increase. This can be easily explained by the fact that the cycle times of the modified tours have been adjusted upwards. This way
Table 3.5: Average idle time for the illustrative 6-tour example (fixed cycle times)
Vehicle Average Idle time (%)
1 38
2 77
3 97
Avg 71
Table 3.6: Old and new cycle times for the illustrative 6-tour example
Tour id Teoq New
1 2 2 2 4 4 * 3 3 4 * 4 7 8 * 5 6 8 * 6 7 8
these tours are executed less frequently resulting in lower transportation costs, but increased holding costs. Also note that if by some reason the fixed vehicle cost per period would drop toe4, the vehicles could not be saved profitably. Next, one can see that the vehicle stands less idle than in the previous solution. The schedule is thus better levelled.
Table 3.7: Optimal costs for the illustrative 6-tour example
Type Cost per period (e) Transportation 306.25 Dispatching 0.0 Unloading 0.0 Holding 366.0 Tour-specific 672.25 Fleet 70.0 Total 742.25
Table 3.8: Average idle time for the illustrative 6-tour example
Vehicle Average Idle time (%)
1 23
Solution approach
4.1
Introduction
Because of the complexity of the problem, solving to optimality is far too time-consuming. Heuristic solution methodologies that find qualitative solutions in a short while are needed. In this chapter several heuristic approaches are presented. Each approach is illustrated using a simple example that shows the procedure and possible flaws. The heuristic approaches can be categorized into three groups. The first group consists of constructive heuristics based on fixed cycle times (section 4.3). The second group contains constructive heuristics taking into account possible modifications of the cycle times (section 4.4). The pseudo codes of the constructive heuristics are included in appendix A. The third and last group consists of improvement heuristics which start from the solutions resulting from the constructive heuristics (section 4.5). Finally, some heuristics are extended with a metaheuristic approach (section 4.6). But first of all, in section 4.2 we mention some general procedures that are significantly relevant.
4.2
General procedures
4.2.1 Infinite horizon
The cyclic nature of our solution implies that the solution covers an infinite horizon. As mentioned in the mathematical model, the infinite horizon can be simulated by the cycle time of the whole schedule,Tsched. This cycle time is given by the least common multiple of
all tour cycle times.
Tsched=lcm(T1. . . Tn) (4.1)
In table 4.1 the validity of this assertion is easily established. Consider a variabledi,k to be 1 if touriis scheduled at dayk, otherwise consider it to be 0.
Table 4.1: Infinite horizon
∀i∈1. . . napplies : di,k =di,k+x·Ti k= 1. . . Ti x∈Z+ ∃ q∈Z+ :x·Ti=Tsched ⇓
di,k =di,k+Tsched
Rosen (1984) shows that the least common multiple of two integer numbers, aand b can be calculated as follows:
lcm(a, b) = a·b
gcd(a, b) (4.2)
The greatest common divisor of two integers, which are not equal to zero, can be determined by theeuclidean algorithm (Heath, 1908). This algorithm subtracts continuously the smallest number from the biggest number, until one number is reduced to zero. The other remaining number is then the greatest common divisor.
4.2.2 Lower bounds to the required number of vehicles
The final goal is to construct a feasible distribution schedule at minimal cost. As explained in 3.3, the cost rate of a distribution schedule depends on the cycle times of the different tours. Hence, the optimal cost rate of the eventual distribution schedule is fixed for a given set of cycle times. However, it is very time consuming to calculate this particular optimal cost rate. Therefore two lower bounds are presented which can rapidly give an accurate indication.
It is clear that for a given set of cycle times the tour-specific costs, i.e. total transportation and holding costs, can be easily calculated using equations 3.1, 3.2 and 3.3. The single cost component that causes calculation difficulties is the required number of vehicles. That is why
the lower bounds to the total cost can be reduced to lower bounds to the required number of vehicles.
Lower bound 1: the cumulative amount of work
On the one hand each vehicle can work 8 hours per day. The total amount of work, expressed in hours, that can be done by one vehicle in a distribution schedule is therefore 8·Tsched. On the other hand, each tour implies a certain amount of work, identified by its duration. The total workload of the tours in a distribution schedule, denoted by Λ, is given by equation 4.3 as each tour is iterated Tsched
Ti times over a time Tsched. Λ = n X i=1 δi·Tsched Ti (4.3)
A first lower bound to the number of vehicles needed to execute all tours can thus be calculated as in equation 4.4. LB1 = Λ 8·Tsched (4.4)
Suppose the cycle times to be fixed in the illustrative example, then Tsched equals 84, given
by the least common multiple.
Λ = 3·84 2 + 5· 84 4 + 4· 84 3 + 8· 84 7 + 5· 84 6 + 6· 84 7 = 581 LB1,Teoq = 581 8·84 = 1
A lower bound to the total cost, for the given combination of cycle times, is given by e 663.83 + 1· e 70 or e733.83. This ise140 less than the optimal cost rate with these cycle times.
Next, suppose the cycle times to be those from the optimal solution given by table 3.6. In this case the least common multiple drops to 8.
Λ = 3·8 2 + 5· 8 4+ 4· 8 4+ 8· 8 8+ 5· 8 8+ 6· 8 8 = 49 LB1,Topt= 49 8·8 = 1
A lower bound to the total cost is given bye672.25 + 1·e70 ore742.25, equal to the optimal cost rate.
Lower bound 2: the curse of relatively primes
The second lower bound to the number of required vehicles for executing the tours with a given set of cycle times, is based on the concept of relatively primes. According to Rosen (1984) two integers are relatively primes if their greatest common divisor is equal to 1. Tours with cycle times that are relatively primes are very inconvenient. As the example in 1.3.2 shows, two tours which take both a full day to be done and whose cycle times are relatively primes can impossibly be done by 1 vehicle. This way the number of required vehicles can rapidly surge. In order to anticipate these inconvenient cycle times, this lower bound takes them into account. Table 4.2 explains how this lower bound is calculated.
Table 4.2: Pseudo code lower bound based on relatively primes
lowerBound = 1 x = 2
setFound = true
While setFound and x ≤ numberOfTours setFound = false
For each set i of x tours
if all tour cycle times of set i are relatively primes
number = vehicles required to do all tours of set i on 1 day (bin packing problem) if number >lowerBound then lowerBound = number
setFound = true End for
increase x by 1 End while
Again the illustrative example with fixed cycle times is applied. First, we check for every pair of tours if their cycle times are relatively primes. In total there are 15 possible combinations of tours, denoted by (tour id1, tour id2). The pairs (1,2), (1,5), (2,5), (3,5) and (4,6) do not
pair number pair number (1,3) 1 (2,6) 2 (1,4) 2 (3,4) 2 (1,6) 2 (3,6) 2 (2,3) 2 (4,5) 2 (2,4) 2 (5,6) 2
The maximum number of required vehicles equals 2 and becomes the new lower bound. The variable x is increased to 3, so we take a look at all possible combinations of three tours. From all 20 combinations, only (1,3,4), (1,3,6), (2,3,4) and (2,3,6) result in relatively primes.
pair number pair number
(1,3,4) 2 (2,3,4) 3
(1,3,6) 2 (2,3,6) 3
The maximum number of required vehicles equals 3 and becomes the new lower bound. The variable x is increased to 4. However, no combination of 4 tours results in relatively primes. The process finally stops and gives a lower bound of 3 vehicles. The lower bound to the total cost, for the given combination of cycle times, is thus e663.83 + 3·e70 or e873.83, equal to the optimal cost rate for this set of cycle times (cf. table 3.4).
Next, suppose the cycle times to be those from the optimal solution given by table 3.6. No pair of tours results in relatively prime cycle times. The required number of vehicles remains 1. A lower bound to the total cost is thus given bye672.25 + 1·e70 ore742.25, equal to the optimal cost rate.
4.2.3 Lower bounds to the total cost
In the section above, two lower bounds are suggested to calculate the required number of vehicles, given the chosen cycle times of the tours. But if we want to have a lower bound to the total cost for a specific problem, we do not know the best cycle times in advance. That is why we need a lower bound that is cycle time independent.
times might be, for sure one will always need at least one vehicle. Moreover we do know the optimal cycle time of each tour. Calculating the cost of a schedule with one vehicle using the optimal cycle times of each tour, is a clear lower bound. The optimal cost of the schedule will never be lower. Based on table 3.3 and 3.4 this lower bound for the illustrative example of the previous chapter comes down to:
e663.83 + 1·e70 =e733.83
Next to this simple lower bound, one can opt to calculate the lower bounds of section 4.2.2 for each possible combination of cycle times. The cycle time combination that leads to the lowest lower bound (for each method separately), will be a lower bound to the total cost. Though this lower bound will probably be more aggressive, the calculation time will be significantly higher. Hence for larger problems it will not be possible to calculate these bounds in a limited amount of time.
For the illustrative example of the previous chapter there are 3·6·4·12·15·15 = 194.400 possible combinations of cycle times. The lowest lower bound that is found amountse733.83 for the method based on the total amount of work and e 740.67 for the method based on the curse of relatively primes. This means that for any combination of cycle times that is eventually chosen the total cost will definitely not be lower than the highest of these two numbers. Indeed the optimal cost rate, shown in table 3.7, is higher.
Figure 4.1: Lower bounds for the illustrative example
In chapter 5 we will evaluate the various solution methodologies. Comparing the results to the optimal solution will often be impossible as it simply takes too long to calculate the optimal solution. Therefore the lower bounds will play a crucial role in giving an indication of the gap between our solutions and the optimal one.
4.3
Constructive heuristics based on fixed cycle times
Although this work mainly focuses on the inclusion of cycle time modifications in the problem, three heuristic approaches are presented starting from fixed cycle times. There are two reasons why these more basic heuristics are developed. First, they can be used as a benchmark to compare further results with. Second, some ideas of these heuristics will come back in more complex heuristics later on.
The first two best-fit insertion heuristics insert the tours one by one, according to priority. Each time a certain tour has to be inserted, the ideal startday is calculated for that tour. The two heuristics differ in how this ideal startday is calculated. When the ideal startday is found, the tour is added to the schedule. Each day the tour has to be done, it is assigned to the first vehicle that has enough time available to execute the tour at that day. The third insertion heuristic has another approach which is explained later.
The priority rule that determines the sequence in which the tours are inserted into the sched-ule, has a great impact on the performance of the heuristic. In practice often the following priority rule is used: the lower the cycle time, the higher the priority. This makes sense as less frequent tours are usually easier to insert. In case of anex aequo, the tour with the largest duration gets the highest priority. Next to this rule, we elaborated three other priority rules. A statistical analysis of the performance of the different rules based on the benchmark problems is reported in section 5.3 and appendix B. Based on this analysis, we opt to put forward another priority rule in these insertion heuristics than the one that is commonly used in literature: the higher the ratio duration divided by cycle time, the higher the insertion priority. In case of a tie, the cycle time turns the scale.
4.3.1 Insertion heuristic - Avoid adding a new vehicle
This method is based on Campbell and Hardin (2005) and emphasizes the importance of keeping the number of required vehicles as low as possible, because usually the fixed vehicle cost is rather high in comparison to other cost components. Concerning a certain tour, the following analysis is done for each possible startday:
conflict is defined as a vehicle that is not able to execute the tour on that day, because not enough time is available.
The maximum amount of conflicts at one day, is reminded.
The startday which has the smallest maximum number of conflicts, is marked as the ideal startday. Note that this way a startday, which would result in the addition of a new vehicle, is postponed as long as possible.
Illustrative example. For each example throughout this chapter the same cost rates are used as in table 3.3. Dispatching and delivery costs are ignored to simplify the calculations.
Tour id Teoq Tmin Tmax δ
P
j∈Sidj
(days) (days) (days) (hours)
1 2 1 2 4 100
2 3 1 3 4 66.67
3 3 1 3 8 66.67
First of all the insertion priorities of the three tours are determined in the table below. For each tour the ratio duration divided by cycle time is calculated.
Tour id Teoq Duration Teoqδ Priority
1 2 4 2 2
2 3 4 1.33 3
3 3 8 2.66 1
The schedule is empty in the beginning so one tour (tour id = 3) can be inserted directly into the schedule. For the second tour (tour id = 1) the ideal startday has to be determined. Since its cycle time is 2 there are two possible startdays: day 1 and day 2. In case of day 1 the tour has to be scheduled on day 1, 3 and 5, while on day 2, 4 and 6 in case of choosing day 2. In the table below the conflicts for each day are shown.
Startday = day 1 Startday = day 2 day # conflicts day # conflicts
1 1 2 0
3 0 4 1
5 0 6 0
max 1 max 1
We notice that in case of starting on day 1, we need 4 hours while 0 hours are available on day 1. In case of starting on day 2, the same occurs at day 4. As both maximum number of conflicts are the same, the first startday is chosen.
The same analysis needs to be done for the next tour. Possible startdays are day 1, 2 and 3 as the (fixed) cycle time is 3 days. The ideal startdays are day 2 and 3. Hence, day 2 is chosen as the startday for the final tour.
Startday = day 1 Startday = day 2 Startday = day 3 day # conflicts day # conflicts day # conflicts
1 1 2 0 3 0
4 1 5 0 6 0
max 1 max 0 max 0
This approach results in a schedule with a total cost ofe485. From this total cost, the fleet cost accounts fore140. Note that a vehicle could be spared by modifying the cycle time of the first tour from 2 to 3 days. If the increase in tour-specific costs does not exceed the cost of one vehicle, this modification will result in a reduction of the total cost.
4.3.2 Insertion heuristic - Levelling
According to the reasoning that a levelled1schedule requires less vehicles, this method aims to achieve a more levelled distribution schedule. Therefore adding a new vehicle is not avoided if this results in a more levelled schedule. Again for each possible startday the schedule is analyzed:
Per day that the tour should be scheduled the number of conflicts is calculated. A conflict is defined as a vehicle that is not able to execute the tour on that day, because not enough time is available.
In contrast with the first method, we sum the total number of conflicts over the whole cycle.
