Impact of Postponement and Sectorisation Strategies in Inventory Routing Problems Dissertation

(1)

Dissertation

Impact of Postponement and Sectorisation Strategies

in Inventory Routing Problems

Supanan Phantratanamongkol

Student ID: 130488747/ S2658275

Newcastle University Business School

MSc Operations and Supply Chain Management

First Supervisor: Dr. G. Pang

University of Groningen Faculty of Economics and Business

MSc Technology and Operations Management

(2)

Abstract

(3)

Acknowledgements

First of all, I would like to express my deepest gratitude to my supervisor, Dr. Gu Pang, who shared with me valuable knowledge on the topic and support through the steps of the dissertation. Her insights on vehicle routing problems allowed me to explore new research topics. I would also like to thank Dr. Jan Riezebos for his feedback on the dissertation and for the directions of the literature research, which helps improve this work.

I also would like to acknowledge the contribution of both Newcastle University and University of Groningen for providing access to the abundance of resources and an excellent working space with a brilliant atmosphere.

I am also indebted to my family members and friends who always provide me with precious emotional support and bottomless encouragement. Without their continued support, I would have struggled more throughout the time of being away from home.

(4)

1. Introduction

1.1 Background and Motivation

The Inventory Routing Problem (IRP) aims to minimise the average distribution cost during the planning horizon while satisfying the requirements of all of the customers based on their usage (Campbell et al., 1998). It can be considered as an extension to the Vehicle Routing Problem (VRP), which concerns the optimisation of routing sequence for the allocated customers to fulfil their orders (Abdelhalim and Eltawil, 2013), since IRP adds the element of inventory holding cost into the decision.

IRP has gained in popularity over time since it was first studied by Bell et al. (1983). There have been many application of IRP; with applications in maritime logistics being the most well grounded (see Ronen, 1993 and Christiansen et al., 2004; 2006 for reviews). Other applications include the distribution of gas (Bell et al., 1983; Golden et al., 1984; Campbell and Savelsbergh, 2004), groceries (Custódio and Oliveira, 2006), and automobile components (Blumenfield et al., 1987).

A more recent popular application of IRP lies in Vendor-Managed Inventory (VMI) situations. This is as a result of increased in competition level in the supply chain (Bertazzi and Speranza, 2012; Coelho et al., 2014). This is because IRP allows the supplier to minimise the number of stockouts while realising the potential savings in distribution costs (Campbell et al., 1998). Many studies agree that VMI is a win-win situation where the supplier can obtain a reduction in overall distribution cost while the customer can achieve savings on ordering costs (Coelho et al., 2012a; Coelho et al., 2012b; Goetschalckx, 2011). According to Coelho et al. (2014), there are simultaneous decisions to be made by the suppliers with regard to the customer’s demand and time spent on distribution, and the integration of these onto specific routes. In terms of the solution methods, exact methods, heuristics, metaheuristics, and continuous approximation approaches are proposed to solve IRPs. These methods will be explained in Section 2.2 in more detail.

(7)

combinations of the seven criteria: time horizon, structure, routing, inventory policy, inventory decisions, fleet composition, and fleet size (Coelho et al., 2014). The remaining 23 papers propose extensions to the basic IRP. Interestingly, these extensions aim to broaden the scope of the IRP, such as adding production decision, or developing a model to handle multiple products at once, rather than optimising the existing components: routing and inventory. Since the customers in most logistics contexts require repeated visits over time, it is evident that the frequency of distribution plays an important role. Suppliers can benefit from the flexibility in their distribution service frequency when they apply a postponement strategy.

Postponement horizon (or accumulation time) refers to the time between customer visits. This strategy allows flexibility in the delivery time. It is proven by Pang and Muyldermans (2013) that postponed delivery services can save travelling cost (i.e. distance) based on certain routing settings. The benefit of a postponement strategy is further supported by studies within the VRP context (referring to the Periodic Vehicle Routing Problem (PVRP) (see for example Wen

et al., 2010; Kurz and Zäpfel, 2013), waste management (Pang and Muyldermans, 2013), and

reverse logistics (Beullens et al., 2004). Nevertheless, the focus on inventory decision process is still lacking.

(8)

In an attempt to contribute to the existing literature of IRPs, the postponement and the sectorisation strategies are investigated in this dissertation. The aim is to extend the study of Pang and Muyldermans (2013), in which the authors investigated the impact of postponement on routing decisions by assuming the same accumulation times for all clients located in a single service region. More specifically, this dissertation examines parameter-dependent optimal levels of postponement (or accumulation times) in the context of IRPs when both routing and inventory holding costs are minimised.

The motivation of this study can be twofold. First, it can assist large retailers operating centrally controlled distribution centres in making decisions on distributing and replenishing inventories to its local stores in a more cost-effective manner. Second, the insights of this study can guide distribution service providers to plan their logistics strategically by allocating appropriate distribution strategies to clients located in different sub-areas (sectors or zones).

1.2 Research Aims and Objectives

The aim of this study is to find the optimal accumulation time that results in the near-optimal trade-off between routing and inventory holding costs.

Adapting from the study of Pang and Muyldermans (2013), the impact of postponement on the reduction in total costs is dependent on:

(1) Sub-service regions (client locations)

This parameter represents the sub-regions in which the clients are located. When applying the sectorisation strategy, the entire service region is partitioned into several smaller service regions. Hence, clients located in different sub-service regions (different client locations) require different replenishment intervals.

(2) Client demand rates

(9)

smaller demand vs. bulkier demand rates) in order to satisfy the inventory requirement.

(3) Client density

Different numbers of customers within the same (original) service area result in a difference in client density within each sub-service region. Different replenish intervals need to be considered for sub-service regions with different client densities.

(4) Service requirements

The ratio of routing cost over inventory holding cost has an impact on the levels of postponement. For example, clients with higher inventory holding costs (i.e. when routing cost becomes less dominant) may require more frequent replenishment of inventories.

Therefore, in this dissertation the above parameters are to be examined in order to explore their impact on the optimal level of postponement, as well as the impact on the inventory routing costs.

To sum up, this study aims to find the optimal level of postponement (i.e. the maximum accumulation time between two service intervals required by a client) for a ring-radial geometry when both routing and inventory holding costs are minimised. More specifically, the impact of these aforementioned parameters on the optimal level of postponement, as well as the impact on the inventory routing costs will be studied.

1.3 Outline of the Dissertation

(10)

(11)

2. Literature Review

Section 2.1 provides an overview of IRPs. The available approaches to solve the problem will be analysed along with examples in Section 2.2. Section 2.3 and Section 2.4 discuss the relevance of using a postponement strategy together with the application of sectorisation strategy.

2.1 Overview of IRPs

The IRP focuses on the distribution of a product from a facility to a set of customers over a specific period of time. The customer has a demand and a limited capacity to hold inventories delivered by a fleet of vehicles having a capacity constraint. Taking into account both the routing and inventory decisions, the main objective of the IRP is to minimise the average distribution cost during the planning horizon while satisfying the requirements of all customers (Campbell et

al., 1998). IRP can be considered as an extension to the Vehicle Routing Problem (VRP) since it

integrates components of the logistics decisions namely, inventory management, vehicle routing, and delivery scheduling while the VRP only focuses on the optimisation of routing cost. IRP was first introduced in the seminal paper of Bell et al. (1983) where the authors considered the minimisation of transportation cost in response to stochastic demand. The study of Bell et al. (1983) aimed to fulfil inventory levels of the customer without taking into account the inventory costs.

2.1.1 Characteristics of IRPs

(12)

Table 2.1: Characteristics of the IRP (adapted from Coelho et al. (2014))

Characteristics Possible Options Time Horizon Finite Infinite

Structure One-to-one One-to-many Many-to-many

Routing Direct Multiple Continuous

Inventory Policy Maximum level Order-up-to level

Inventory Decisions Lost sales Back-order Non-negative

Fleet Composition Homogeneous Heterogeneous

Fleet Size Single Multiple Unconstrained

The time horizon is the planning period of the IRP. The structure variants stem from the fact that the number of suppliers and customers can vary. The most common case is one supplier with several customers. Direct routing means that there is only one customer within the route while multiple routing allows more customers to be served. It can also be continuous where there is no central depot. The most common inventory policies are the maximum level (ML) and the order-up to level (OU). As defined by Coelho and Laporte (2013), the deliveries under the ML policy are more flexible since the supplier can deliver any order as long as it respects the customer inventory capacity. The latter policy imposes a constraint on the supplier, as the delivery must bring the inventory level up to its maximum capacity. The inventory decision affects inventory management. If it is possible for the inventory level to fall below zero, back ordering is enabled. The corresponding demand will then be served during the next cycle. Conversely, if back ordering is impossible, the unmet demand is taken as lost sales. Though, there could be penalty costs for both cases. The non-negative inventory decision is used when there is a deterministic demand. Moreover, the fleet can comprise of either homogeneous or heterogeneous vehicles. The number of vehicles within the fleet can be limited to one or any specified number or even unconstrained.

2.2 Approaches in Solving IRPs

(13)

et al., 2007; Raa and Aghezzaf, 2008). In addition, there are other solution approaches such as

metaheuristic (see Aziz and Moin, 1970; Hemmelmayr et al., 2010; Pisinger and Ropke, 2010) and continuous approximation (see Erlebacher and Meller, 2000; Beullens et al., 2004; Shen and Qi, 2007), all of which will be discussed in the following Sections 2.2.1 – 2.2.4.

2.2.1 Exact Algorithms

These algorithms are more suitable with smaller problems with constraints such as limited number of customers or shorter time periods (Edirisinghe and James, 2014). Among other approaches, Archetti et al. (2007) proposed the first branch-and-cut algorithm, which become a basis for later developments. Recently, Coelho and Laporte (2013) proposed a solution to multi-vehicle VRP using the branch-and-cut algorithm and were able to solve the problems with up to 50 customers over 6 time periods. However, Coelho and Laporte (2013) stated that the time taken to solve was intractably long and even exceeded the 24 hours time constraint for a more complex instance.

2.2.2 Simple Heuristics

Simple heuristic follows a set of rules and provides a near-optimal solution to a specific problem. It uses simple neighbourhood structures such as interchanges to explore the solution space of the IRP and usually decomposes the problem into hierarchical sub-problems (Coelho et

al., 2014). The benefit of this method is that less computational effort is required. However, as

Voss (2001) states, the optimality from heuristics cannot be guaranteed. This critically limits the scale of the problem that can be solved, which is not entirely practical for IRP context (Cordeau

et al., 1997). The current solutions available for the IRP cover one-warehouse, multiple-retailers

instances under a deterministic demand rate (Burns et al., 1985; Anily and Federgruen, 1993). Some models are extended to allow the handling of multiple products (Viswanathan and Mathur, 1997; Jung and Mathur, 2007).

2.2.3 Complex Heuristics: Metaheuristics

(14)

produces high-quality solutions (Soriano and Gendreau, 1996; Toth and Vigo, 2003). Some better-known examples of metaheuristics are simulated annealing, tabu search, and generic algorithms (see Sherbeny, 2010 for a detailed explanation). Nonetheless, according to El-Sherbeny (2010), the superiority of the method is apparent in examples of large dimensions with the sacrifice of high computational time. Also, there are many efforts to improve the procedure using techniques such as parallelisation and hybridisation between two metaheuristics (see for example, Gehring and Homberger, 2002; Bianchi et al., 2006). More recent research lies in the metaheuristic domain covering many sub-approaches such as the use of local search (see Qin et

al., 2014), adaptive large neighbourhood search (see Coelho et al., 2012b), and greedy

randomised adaptive search (see Campbell and Savelsbergh, 2004). Most of these studies focus on a finite time horizon and a one-to-many structure in order to determine a policy that minimises the expected costs (Coelho et al., 2014).

2.2.4 Continuous Approximation (CA)

Continuous approximation aims to solve problems based on concise summaries of data and analytic models (Langevin et al., 1996). According to Newell (1973), the objective of CA is to analyse problems, which are finite-dimensional, by converting them (approximately) into problems involving continuous functions. The advantage of this method is that near optimal solution can be obtained without exhaustive computation (Daganzo, 2005). Also, as noted by Hall (1986), the method is useful to develop models that are easily interpreted and comprehended. In case the function is discrete, the continuous model can closely approximate it if the number of discrete objects is large enough.

(15)

2.3 Postponement in IRPs

Postponement is considered as a business strategy where cost is minimised and benefits are maximised by delaying activities within the supply chain until the latest moment possible (Pagh and Cooper, 1998). This can be applied to either the manufacturing or distribution context. According to Pagh and Cooper (1998), postponement within the distribution context can be achieved using either place, or time. Place postponement concerns with the location of the warehouse or the main distribution centre while time postponement involves the decision on the delivery schedule in order to fulfil the customers demand. It is possible to withhold or delay the delivery until the last minute. The length of delay is called the postponement horizon and can also be referred to as the accumulation time, which is the time period between two visits of a customer. It is usually expressed as a time unit such as hours, days, and weeks. Considering the use of a postponement strategy within the manufacturing context, manufacturers can hold unfinished goods in inventory for appropriate final customisation depending on demand signals (Rietze, 2006). Within a broader supply chain context, the postponement strategy leverages the conventional demand forecasting. This is because more accurate forecasts are achievable over a shorter time period (Lee and Whang, 1998). For a more detailed literature review on different aspects of postponement, readers can refer to van Hoek (2001).

Focussing on the context that is closely related to the IRP, the postponement strategy is commonly used in different distribution applications such as periodic vehicle routing problems (PVRP) (see for example Wen et al., 2010 and Kurz and Zäpfel, 2013), waste collection management (Pang and Muyldermans, 2013), and reverse logistics (Beullens et al., 2004).

(16)

postponement under different integration policies such as backhauling and mixing in a reverse logistics setting. Their work concludes that the optimal postponement level is affected by different parameters, for instance, the location of the depot and customer density. Based on these studies, and the practical study by McMillen and Skumatz (2001) with a confirmed savings of 20% in the routing cost achieved when the collection frequency is reduced, the value of postponement is evident.

Pang and Muyldermans (2013) have conducted a study on the Capacitated Vehicle Routing Problems (CVRPs) where the distributor or the collector is able to decide on the appropriate time to service the customers to fulfil the demand accumulated over time. They prove that the use of postponement strategy mainly reduces the local routing distance. In other words, the benefit of postponement strategy is greater when the local routing distance dominates radial routing distance. Using the continuous approximation model, Pang and Muyldermans (2013) find that there are several parameters affecting the value of postponement. They conclude that the benefits from postponement are larger when the client density is lower and when the depot is more centrally located in the service region. As the vehicle capacity (or maximum accumulation time) becomes bigger, the benefits of postponement are greater. However, these benefits reduce when client demands become bulkier.

2.4 Sectorisation Technique in IRPs

(17)

one were to cover irregular shaped region with rectangular shaped zone and thus, some pattern of concentric rings of zones around the source may prove to be more suitable. This statement is also agreed upon by Haimovich and Rinnoy Kan (1985). The work of Daganzo (1984b) has been given much attention and some of more notable studies are that of Newell and Daganzo (1986) and Langevin and Soumis (1989). Newell and Daganzo (1986) have studied the problem instance where the vehicles have limited capacity. They have used ring-radial geometry as a base region and developed a method to divide zones. Langevin and Soumis (1989) used the same geometry and successfully devised a procedure to determine the zones and the average number of points to visit regardless of the capacity of the vehicles. For further examples of the use of sectorisation strategy and the benefit of reduction in distribution costs, readers can refer to Anily (1994), Aghezzaf et al. (2006), and Jung and Mathur (2007).

2.5 Summary

(18)

3. Methodology

The objective of this chapter is to provide an overview on the solution approach and the solution procedures applied in this study. For the detailed explanation and discussion of both, readers can refer to Chapter 4.

This study aims to find the parameter-dependent optimal levels of postponement (or accumulation times) in the context of IRPs when both routing and inventory holding costs are minimised. Most importantly, it examines the impact of these key aforementioned parameters (see Section 1.2 for an overview), namely sub-service regions (client locations), client demand rates, client density, service requirements, on the optimal level of postponement, as well as their impact on the inventory routing costs. Therefore, the hypothesis of this study states as follows. The application of the postponement and sectorisation strategies can reduce total inventory routing costs, yet the benefits depend on the settings of these key parameters. The detailed explanations about the parameters and experimental scenarios are provided in Sections 4.2 and 4.3, respectively.

(19)

The overview of the solution procedures is described as follows.

Step 1: Establishing Experimental Scenarios

In order to fully understand the impact of the postponement strategy, different scenarios are to be generated and solved in order to derive the impact of certain parameters in the IRPs. These parameters include the sub-service regions, client locations, client demand rates, client service requirements, client densities, and number of clients. In addition, the optimal postponement levels that minimise total costs are to be determined. The detailed configurations of each scenario can be found in Section 4.3.

Step 2: Solving Scenarios using the Continuous Approximation Model

A continuous approximation model of Daganzo (1984a; 1984b) is chosen as a solving mechanism. This is because the model is the most comprehensible since it is displayed as a cost function allowing the user to gain an in-depth insight into the trade-offs between different logistics costs, and to obtain reasonable solutions without exhaustive computation. Specific scenarios are solved by inputting different parameters obtained in Step 1. By inputting the assigned values for specific parameters corresponding to each scenario into the cost function equations in Excel, the results are obtained.

Step 3: Presentation of Results

In Step 3, the results obtained in Step 2 are to be visualised. The following diagrams are plotted to fulfil this purpose:

-‐ Routing cost curves,

-‐ Inventory holding cost surface diagrams, -‐ Total cost surface diagrams, and

-‐ Optimal postponement horizon curves.

(20)

determine the optimum level of postponement. The explanations of individual graphs can be found in Section 4.6.

Step 4: Analysis of Results

(21)

4. Computational Experiments

Chapter 4 explains the computational experiments in greater detail. It begins with the specification of assumptions (see Section 4.1) followed by a set of parameters, including the characteristics of the entire service region, the customer demand rates, the numbers of clients (the client densities), vehicle capacities and time periods (see Section 4.2). Furthermore, Section 4.3 reports the experimental scenarios and Section 4.4, discusses the solution procedures in more depth.

4.1 Assumptions

In order to solve the problem, certain assumptions are made regarding the different parameters in each scenario. The assumptions made in this study are as follows:

(1) The problem involves a single product.

(2) The central depot has an infinite amount of product available to be distributed. (3) Each customer is visited exactly once and no demand split is allowed.

(4) Each vehicle route starts and ends at the depot, which is centrally located. (5) The load in each vehicle satisfies its capacity constraint.

(6) The maximum capacity of each vehicle is equal to or bigger than any individual client’s demand rate in a certain period of time.

(7) The distribution plans are made such that the service requirements of the customers are fulfilled; therefore, there is no occurrence of stockout cost.

(8) The inventory holding cost is considered and it is incurred by the clients. (9) Travel times and speeds are known.

4.2 Parameters 4.2.1 Service Region (A)

(22)

Figure 4.1: Service Region Configuration

In order to utilise the sectorisation strategy, the service region presented in Figure 4.1 is divided into concentric rings of different radius based on the method of Langevin and Soumis (1989). Readers can refer to Section 4.4 for more details on the procedure.

4.2.2 Depot Location

In line with the procedure developed by Langevin and Soumis (1989), the depot is centrally located in the service region (see Figure 4.1).

4.2.3 Customers Demand Rate (r)

There are two types of normalised demands. These are:

(1) Constant demand (CD); and (2) General demand (GD)

(23)

4.2.4 Number of Customers (n) in a Service Region

Three population sizes are used as inputs for the mathematical model for both types of demand (CD and GD). These are 1,000; 3,000; and 5,000 customers. Subsequently, the customer density within each sub-service region can be computed by:

2π∂ r rdr

r_k rk-1

= n

Where ∂ r represents the density as a function of radius, which is a linear function decreasing from the centre to the periphery with the density at the centre twice as much as the density at the periphery. rk is the outer radius of the sub-region (ring), and rk-1 is the internal

radius of the sub-region.

4.2.5 Time Period (T)

The time period (T) is the period of interest, which can be the operating hours of the truck drivers or the planning horizon of the vendor. It comprises of the local travel time (T1), handling

time (T2), and line-haul time (T3). T1 is the travel time for the vehicle to visit all the customers in

a considered zone, which depends on the speed (v) of the vehicle. In this study, the average speed of 30 km/h is assigned to this parameter. As for the handling time (T2), it is the time that

takes for the operator to successfully handle the product for the customer. It is calculated based on the fixed handling time per client base (τ). In this study, it is assumed that τ = 5 minutes. Finally, T3 is the time to travel between the depot and the inside radius of the sub-region. In the

experiment, there are two different time periods considered. These are T = 2, and T = 4 hours in order to investigate the impact of the time period on the total inventory routing cost.

4.2.6 Vehicle Capacity (Q)

(24)

GD1 (GD2), the maximum allowable demand is 5 (7) times larger than the case with constant demand, so the capacities are obtained by multiplying the original capacities by 5 (by 7). Subsequently, the following parameters can be obtained.

Table 4.1: Vehicle Capacities for Each Demand Rate

Vehicle Capacities for Constant Demand Vehicle Capacities for General Demand 1 Vehicle Capacities for General Demand 2 1 5 7 2 10 14 3 15 21 5 25 35 6 30 42 10 50 70 15 75 105 30 150 210 40 200 280 50 250 350 60 300 420 70 350 490 80 400 630 120 600 840 150 750 1,050 200 1,000 1,400

4.2.7 Summary of the Parameters

(25)

Table 4.2: Summary of the Parameters

Parameter Options

Service Region (A) Circular service region of radius 15 km

Depot Location Central to the service region

Customer Demand Rate (r)

Constant demand (r = 1) General demand 1 (r = 3) General demand 2 (r = 5) Number of Customers (n) 1,000 customers 3,000 customers 5,000 customers Speed (v) 30 km/h

Handling Time per Customer (τ) 5 minutes

Time Period (T) 2 hours

4 hours

4.3 Scenarios

The scenarios presented in this Section are generated based on different combinations of the parameters given in Table 4.1 and Table 4.2. The instances and the detailed explanations are as follows.

4.3.1 Scenario A

The first experimental instance is when customers are distributed over a circular service region of radius 15 km. The customers demand rate is unitary while the vehicle capacities for this scenario range from 1 to 200 units. The time period is 4 hours. There are three sub-scenarios based on the number of customers:

(26)

4.3.2 Scenario B

Scenario B is relatively similar to Scenario A but the time period is reduced by a half (e.g. from T = 4 hours in Scenario A to T = 2 hours in Scenario B). There are also 3 sub-scenarios according to the number of customers.

4.3.3 Scenario C

This scenario uses the same parameters as Scenario A but general demand rate, GD1.

4.3.4 Scenario D

As for this scenario, general demand rate, GD2 is used.

4.3.5 Scenario E

The settings of this scenario are the same as Scenario C with the reduction in time period from 4 hours to 2 hours.

4.3.6 Scenario F

General demand rate, GD2 is used in this scenario while other parameters remain the same as in Scenario E.

4.3.7 Summary of the Scenarios

Table 4.3: Summary of the Experiment Scenarios

(27)

C3 GD1 5,000 5 to 1,000 units 4 hours D1 GD2 1,000 7 to 1,400 units 4 hours D2 GD2 3,000 7 to 1,400 units 4 hours D3 GD2 5,000 7 to 1,400 units 4 hours E1 GD1 1,000 5 to 1,000 units 2 hours E2 GD1 3,000 5 to 1,000 units 2 hours E3 GD1 5,000 5 to 1,000 units 2 hours F1 GD2 1,000 7 to 1,400 units 2 hours F2 GD2 3,000 7 to 1,400 units 2 hours F3 GD2 5,000 7 to 1,400 units 2 hours 4.4 Sectorisation Procedure

In order to divide the original service region (see Figure 4.1) into concentric rings, the method proposed by Langevin and Soumis (1989) is applied. In this dissertation, each concentric ring is considered as a sub-service region. In other words, each ring (each sub-service region) becomes a part of a larger (original) service region. For each of these sub-service regions, the logistics provider has to decide the optimal service internal of each sub-service region. Langevin and Soumis (1989) consider a particular zone with its centre at the distance r* from the depot, then try to determine the width of that zone where the total distance per unit area would be minimised. This width is then used to determine the radii of the rings starting from the outer ring towards the centre since the contribution of the outer rings, with larger radii, are more important than that of the inner rings (Langevin and Soumis, 1989).

Starting from the outermost ring, the inner radius is calculated using a bisection method to solve the equation devised by Langevin and Soumis (1989):

3r_k – r_k-1+ r_k– r_k-1 τv 6∂ rk + rk-1

2 = Tv

(28)

clients in this study), v is the average speed of the vehicle, ∂ rk + rk-1

2 is the linear function of

the customer density at the centre of each ring, and T is the length of the time period.

The sectorised region will appear as follows:

Figure 4.2: Example of the Sectorised Sub-service Regions

Since the time period (T) and the number of customers (n) are the two main parameters that are adjusted, the resulting sub-service regions could have different numbers of rings. This means that the number of clients per ring as well as the customer density per ring also vary.

4.5 Costs

There are two main costs involved in the experiment. These are routing cost and inventory holding cost. By converting the travelling distance of the vehicles obtained from the continuous approximation model into cost functions and inputting different combinations of parameters, the total costs can be calculated.

4.5.1 Routing Cost

According to Daganzo (2005), the total distance travelled by the vehicle can be virtually decomposed into radial and traverse (local) distances and hence; can be calculated by:

Σ2r_i r

V δiRi + ti V_r

(29)

The term Σ2r_i r

V δiRi represents the total radial distance, where ri is the distance from

the depot to the outer radius of the ring i. Parameter r is the average customer demand, and V is the capacity of the vehicle. Parameter δi is the customer density of ring i, and Ri is the area of the

ith ring. As can be seen from the formula that the radial distance varies upon the radius, different distances are obtained for each ring. Additionally, since the total radial distance is the summation of the distance from each ring, the number of rings also plays an important role. The second term,

t_i V r

6 , represents the total lateral distance, which is the distance between customers travelled by

vehicles, where t_i is the average radius of the ring between its outer and inner radii.

In order to investigate the role of the postponement strategy in this study, the routing distance function of Daganzo (2005) described earlier is to be extended to incorporate the new parameter – the maximum accumulation time (denoted as tmax). This is the maximum postponed

period of time of a specific logistics activity. In this study, the maximum accumulation time is the maximum postponed delivery (or collection) time between two visits of a customer. This gives the logistics service provider the benefits of cost reduction while still satisfying the service requirement of the customer, that is, there is no occurrence of stockouts. Consequently, by multiplying the total distance by the routing cost coefficient, cr, and dividing by the maximum

accumulation time to normalise the distances traversed in the routing instances created; the routing cost can be obtained and is expressed as:

Σ2r_i _{V δ}r _iR_i + ti V

r 6

t_max *cr

4.5.2 Inventory Holding Cost

(30)

tmax*r*n

2 *ch

It can be seen that the cost increases with the total accumulation time (t_max), the demand rate of the customer (r), and the holding cost per unit per unit time, ch.

4.5.3 Total Cost

The total cost is obtained by summing up the above two cost elements. The resulting expression then becomes:

Σ2ri _{V δ}r iRi + ti V_r 6 t_max *cr + t_max*r*n 2 *ch

Since c_r and c_h are unknown constants, a sensitivity factor, α is introduced. It is defined as a ratio between the inventory holding cost and the routing cost (α = ch

c_r). Therefore, the

previous expression reduces to:

Σ2r_i _{V δ}r _iR_i + ti V r 6 t_max + t_max*r*n 2 *α

In the experiment, different values for the sensitivity factors are to be used in order to find the optimum levels of postponement in various scenarios described in Section 4.3.

4.6 Diagrams

(31)

inventory holding cost surface diagrams, total cost surface diagrams, and optimal postponement horizon curves.

4.6.1 Routing Cost Curves

These are plots of the routing cost as a function of the postponement horizon. For instances with constant customer demand (r = 1), then the maximum postponement horizon is equal to the vehicle capacity (tmax = Q). Consequently, the routing cost curves for constant

demands are plotted as a function of the vehicle capacity.

4.6.2 Inventory Holding Cost Surface Diagrams

The surface diagrams are three-dimensional graphs of the inventory holding cost incurred by the customers as a function of the maximum accumulation time and the sensitivity factor, α.

4.6.3 Total Cost Curves

As for this type of graph, the relationship between the total cost and the sensitivity factor α is shown. The curves are drawn from the lowest total cost obtained from the model corresponding to specific value of the sensitivity factor α.

4.6.4 Optimal Postponement Horizon Curves

(32)

5. Results and Discussions

5.1 Results

Section 5.1 presents the results obtained using the formulae explained in Chapter 4. The procedure starts by sectorising the service area into concentric rings based on the assigned parameters (see Section 4.3.7 for details). The total distance is calculated and substituted into the corresponding total cost formula to obtain the cost. The optimal level of postponement horizon for each scenario within this study is found by using an iterative process based on the range of the lowest total costs. This is because the cost function itself is a convex function with the lowest point at the optimal postponement horizon. One specific range of the lowest total costs together with one postponement interval is chosen per sensitivity factor in order to expand the range further to obtain the exact lowest total cost and its corresponding level of postponement.

Table 5.1 summarises the scenarios used to provide results presented in Figure 5.1 to Figure 5.10.

Table 5.1: Summary of the Originated Scenarios of each Graphs

(33)

Figure 5.1: Maximum Accumulation Time as a Function of α for each Ring (Constant Demand r = 1; T = 4 hours; n = 1,000 (left) and n = 5,000 (right))

Figure 5.1 shows the maximum accumulation time as a function of α for each ring. These are the results for the cases of constant demand (r = 1) with T = 4 and n = 1,000 (left), and n = 5,000 (right). It can be seen that the number of customers affects the number of rings (i.e. the number of sub-service regions). For the same time period (e.g. T = 4), increasing the number of customers results in more sub-service regions (or sectors). Also, the larger set of clients (n = 5,000) results in a higher client density within a ring (a sub-service region). This causes the difference in postponement horizon (tmax) levels between the same ring numbers. To be more

specific, considering the ring R1 of both scenarios n = 1,000 and n = 5,000, denoted as R1n=1000

and R1_n=5000, respectively. As shown in Chapter 4 that the density is represented by a linear function of the radius with the density at the centre of the service region twice as much as that of the periphery, the calculation of the client density (δ) for each ring is carried out by solving the integral (refer to Section 4.2.4), i.e. substituting the corresponding number of customers for each scenario. This will result in the linear function ∂ r specifically for each number of customers. In this case, the client density (δ) for R1n=1000 is 1.94 while it is 9.79 for R1n=5000. Consequently,

the maximum accumulation times for R1n=1000 (R2n=1000) and R1n=5000 (R2n=5000) are 16 (23) and

15 (20), correspondingly. 0 5 10 15 20 25 0.01 0.05 0.1 0.25 0.5 0.75 1 2 3 5 ₁₀ tmax Alpha

Maximum Accumulation Time as a

Function of α

R2 R1 0 5 10 15 20 25 0.01 0.05 0.1 0.25 0.5 0.75 1 2 3 5 ₁₀ tmax Alpha

Maximum Accumulation Time as a

Function of α

(34)

It is evident that the postponement levels for the area immediate next to the depot are noticeably lower. This could mean that a strategy such as JIT could be used for the sub-service regions within the vicinity of the depot while longer replenishment intervals are necessary for the more remote sub-service regions away from the depot. Furthermore, Figure 5.1 shows that, when holding cost dominates routing cost (i.e. larger α), the optimal level of postponement reduces. Thus, shorter accumulation times are required for clients with high inventory holding costs. Nevertheless, the postponement horizon becomes constant as α becomes larger.

Figure 5.2: Maximum Accumulation Time as a Function of α for each Ring (General Demand; T = 4 hours; GD1 (r = 3), n = 1,000 (top left) and n = 5,000 (top right); GD2 (r = 5), n = 1,000 (bottom left) and n = 5,000 (bottom right) 0 5 10 15 20 25 30 35 0.01 0.05 0.1 0.25 0.5 0.75 1 2 3 5 ₁₀ tmax Alpha

Maximum Accumulation Time as a

Function of α

R2 R1 0 5 10 15 20 25 30 35 0.01 0.05 0.1 0.25 0.5 0.75 1 2 3 5 ₁₀ tmax Alpha

Maximum Accumulation Time as a

Function of α

R3 R2 R1 0 10 20 30 40 50 0.01 0.05 0.1 0.25 0.5 0.75 1 2 3 5 ₁₀ tmax Alpha

Maximum Accumulation Time as a

Function of α

R2 R1 0 10 20 30 40 50 0.01 0.05 0.1 0.25 0.5 0.75 1 2 3 5 ₁₀ tmax Alpha

Maximum Accumulation Time as a

Function of α

(35)

As for the cases with non-constant demands, Figure 5.2 shows the maximum accumulation time as a function of α for each ring for the two different demands, GD1 (top) and GD2 (bottom), respectively. The results are obtained over the time period of 4 hours (T = 4) with n = 1,000 (left), and n = 3,000 (right). Figure 5.2 also demonstrates that postponement horizons are longer for customers with a higher demand rate (i.e. GD2). This could be attributable to the relatively larger size of the storage facilities of clients who require bigger demand and thus, could hold more inventories in correspondence with their larger orders.

Figure 5.2 confirms the observations made in the previous case where the demand is constant (see Figure 5.1). The optimal accumulation time reduces when the holding cost per unit per unit time plays a more important role (larger α or when holding cost dominates routing cost). Similar to the previous findings based on Figure 5.1, the interpretations allow the vendor to apply a JIT strategy for the customers close to the depot (sub-service region R1) and extend the replenishment period for the ones further away (sub-service region R2 and beyond). The remarks regarding the client density still hold for this scenario, as the maximum postponement horizons are shorter for the rings with higher client density (δ). As evident from the curves of the sub-region R2 for n = 1,000 and n = 5,000; the client density (δ) for R2n=1000 is 1.41 in comparison to

8.06 for R2_n=5000 and the maximum accumulation times are 33 and 29, respectively. This means that the service provider should provide more frequent deliveries to the sub-service regions with higher customer densities.

(36)

Figure 5.3: Routing Cost Curves for the Instance where r = 1; T = 4; n = 3,000

Figure 5.3 (left) delineates the routing cost curves for the case with constant demand, T = 4 and n = 3,000. The curves are plotted against the maximum accumulation time. It is noticeable that routing costs decrease with longer t_max. Considering the distance from the depot to each sub-service region (R1, R2 and R3), the further the sub-sub-service region is, the higher the routing cost becomes.

Since the routing cost curve of the sub-region R1 is considerably lower than the other two sub-regions (R2 and R3), another graph Figure 5.3 (right) is plotted to enlarge the routing cost for R1 confirming that routing cost decreases over longer accumulation time. It is noticeable that the routing cost of sub-regions (R1, R2, R3) becomes constant when the postponement horizon is longer. 0 10000 20000 30000 40000 50000 60000 1 3 6 15 40 60 80 150 C os ts Max. Postponement

Routing Cost Curves

(r = 1; T = 4; n = 3,000)

R3 R2 R1 0 20 40 60 80 100 1 3 6 15 40 60 80 150 C os ts Max. Postponement

Routing Cost Curves

(r = 1; T = 4; n = 3,000)

(37)

Figure 5.4: Inventory Holding Cost Surface Graphs for the Instance where r = 1; T = 4; n = 3,000

In order to investigate the behaviour of holding costs, Figure 5.4 reports the inventory holding cost surface graphs, which are three-dimensional representatives of the holding cost as a function of tmax and α. For the sub-service regions furthest away from the depot (e.g. the

outermost ring R3), as α increases, the inventory holding cost escalates and the maximum accumulation time significantly decreases. For instance, when α = 0.01, Inventory Holding Cost_R3 = 210.92, tmax = 23, and when α = 1, Inventory Holding Costs_R3 =

4,585.25, t_max = 5. The same trend can also be seen for the other two sub-service regions (R1 and R2). Nevertheless, the inventory holding costs of both sub-service regions R1 and R2 are

0.01 1 0 500000 1000000 1500000 2000000 1 3 ₆ 15 ₄₀ ₆₀ ₈₀ 150 A lp h a In ve n tor y H ol d in g C os t tmax

Inventory Holding Cost Surface

Graph (R3)

0.01 1 0 500000 1000000 1500000 1 3 ₆ 15 ₄₀ ₆₀ ₈₀ 150 A lp h a In ve n tor y H ol d in g C os t tmax

Inventory Holding Cost Surface

Graph (R2)

0.01 1 0 10000 20000 30000 40000 1 3 ₆ 15 ₄₀ ₆₀ ₈₀ 150 A lp h a In ve n tor y H ol d in g C os t tmax

(38)

considerably lower than that of the remote sub-service region R3. For example, α = 1, Inventory Holding Cost_R2 = 2,413.68, tmax = 4, and Inventory Holding Costs_R1 = 31.93, tmax = 2.

Figure 5.5: Total Cost Curves for the Instances where T = 4; n = 1,000 and 5,000 for r = 1 (left) and r = 5 (right)

Figure 5.5 displays the total cost curves for the cases where 1,000 and 5,000 clients are served with T = 4 for both constant demand (left) and general demand (GD2) (right). It is apparent that for both demand instances, the total costs are less when the number of customers is lower. Also, the total costs increase with higher demand rates.

0 20000 40000 60000 80000 100000 120000 140000 160000 180000 200000 0.01 0.1 0.5 1 3 10 tmax Alpha

Total Cost Curves (r = 1; T =4)

n = 1,000 n = 5,000 0 20000 40000 60000 80000 100000 120000 140000 160000 180000 200000 0.01 0.1 0.5 1 3 10 T otal C os t Alpha

Total Cost Curves (r = 5; T = 4)

(39)

Figure 5.6: Total Cost Line and Density of each Sub-Region where r = 1 (left), r = 3 (right); T = 4; n = 5,000; α = 0.5

Figure 5.6 reports the comparison between the density and the total cost for the instances where the service provider serves 3,000 customers with the time period T = 4 and the sensitivity factor α = 0.5 for both constant demand (left) and general demand (GD1) (right). It can be seen that for both demand cases, the total costs (red lines) reduce as customer density (δ) increases. The less remote sub-service region (R1) has higher client density. This also confirms the previous findings. That is, a JIT strategy should be applied to the sub-service regions having higher client densities.

0 1000 2000 3000 4000 5000 6000 7000 0 2 4 6 8 10 12 R3 R2 R1 T otal C os t D en si ty Sub-Service Region

Total Cost Line and Density of

each Sub-Region where

(r = 1; T = 4; n = 5,000; α = 0.5)

0 2000 4000 6000 8000 10000 0 2 4 6 8 10 12 R3 R2 R1 T otal C os t D en si ty Sub-Service Region

(40)

Figure 5.7: Total Cost Curves for the Instances where r = 1; T = 2; n = 3,000 (left), and r = 1; T = 4; n = 3,000 (right)

Figure 5.7 shows the comparisons of the total cost curves between the two instances with constant demand in order to serve 3,000 customers with the time periods T = 2 and T = 4, correspondingly. The shorter the time period, the more sub-service regions are needed (Figure 5.7, top, left). A greater number of sub-service regions help distribute the customers’ demand evenly over the entire service region. Thus, the total costs, both routing and inventory holding costs, are lower (Figure 5.7, bottom). Figure 5.7 also shows that the total cost for the sub-region R1 is substantially lower than other sub-regions (e.g. R2 onwards).

0 2000 4000 6000 8000 10000 12000 14000 0.01 0.05 0.1 0.25 0.5 0.75 1 2 3 5 ₁₀ T otal C os t Alpha

Total Cost Curves for Each Ring

(r = 1; T = 2; n = 3,000)

R6 R5 R4 R3 R2 R1 0 5000 10000 15000 20000 25000 30000 35000 0.01 0.05 0.1 0.25 0.5 0.75 1 2 3 5 ₁₀ T otal C os t Alpha

Total Cost Curves for Each Ring

(r = 1; T = 4; n = 3,000)

R3 R2 R1 0 10000 20000 30000 40000 50000 60000 0.01 0.05 0.1 0.25 0.5 0.75 1 2 3 5 ₁₀ T otal C os t Alpha

Overall Total Cost Curve

(r = 1; T = 2; n = 3,000 and

(r = 1; T = 4; n = 3,000)

(41)

Figure 5.8: Total Cost Curves for the Instances where 2x(r = 1; T = 2; n = 3,000) and r = 1; T = 4; n = 6,000

Figure 5.8 reports the total cost curves as a function of α for the case where the service provider performs twice the delivery (denoted as 2x) for constant demand with the time period of 2 hours, covering the total amount of 3,000 customers and the case where the delivery is made once over the period of 4 hours, covering the total of 6,000 customers. The total costs seem to be less in the case where the time period is shorter. The overall cost curves support the observation made earlier even when the vendor decides to make the delivery twice (T = 2 (x2)) to cover the same amount of customers. However, since it is evident that the costs involved are relatively similar, repeating delivery (i.e. delivering twice to cover the time period T = 4) might not be as beneficial in the situations where the service providers have a limited number of vehicles, and when the customers’ inventory holding cost dominates the routing cost.

0 20000 40000 60000 80000 100000 120000 0.01 0.05 0.1 0.25 0.5 0.75 1 2 3 5 ₁₀

Overall Total Cost Curve of

2x(r = 1; T = 2; n = 3,000)

and (r = 1; T = 4; n = 6,000)

(42)

Figure 5.9: Total Cost Curves for the Instances where r = 1; n = 3,000 and n = 5,000; T = 2 (left) and T = 4 (right)

Figure 5.9 shows the total cost curves for the cases of 3,000 and 5,000 customers having constant demand over the time period of T = 2 (left) and T = 4 (right). It is evident that, considering the two instances, T = 2 and T = 4, the same observation is found as the previous findings. That is, increasing the number of sub-service regions (e.g. when T = 2) results in reduced total costs. Nevertheless, the total cost in serving 3,000 customers is less than 5,000 customers regardless of the difference in the number of sub-service regions (rings).

0 10000 20000 30000 40000 50000 60000 70000 80000 90000 0.01 0.1 0.5 1 3 10 T otal C os t Alpha

Total Cost Curves (r = 1, T = 2)

n = 3,000 n = 5,000 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 0.01 0.1 0.5 1 3 10 T otal C os t Alpha

Total Cost Curves (r = 1; T = 4)

(43)

Figure 5.10: Total Cost Curves and Graphs of Percentage Improvement over T = 4 for the Instances with general demand where n = 3,000; T = 2 and T = 4; r = 3 (top) and r = 5 (bottom)

Figure 5.10 reports the total cost curves for the cases with general demands r = 3 (top) and r = 5 (bottom). The results are for both time periods of T = 2 and T = 4 in order to serve 3,000 customers. It can be seen that when the demand rate is high (i.e. GD2), the total costs of servicing the same number of clients are larger. For GD2, bin-packing problems become more difficult. With shorter time period (T=2), there is a greater number of sub-service regions. That is, having more sub-service regions reduces the overall costs. However, the percentage

0 20000 40000 60000 80000 100000 120000 140000 0.01 0.1 0.5 1 3 10 T otal C os t Alpha

Total Cost Curves

(r = 3; n = 3,000)

T = 2 T = 4 0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 0.01 0.05 0.1 0.25 0.5 0.75 1 2 3 5 10 P er ce n tage I mp rove me n t Alpha

Percentage Improvement

over T = 4

0 20000 40000 60000 80000 100000 120000 140000 0.01 0.1 0.5 1 3 10 T otal C os t Alpha

(44)

improvement over T = 4 seems to decrease more quickly with the higher demand rate (GD2). This is evident from the graphs on the right (Figure 5.10, top (GD1) and bottom (GD2)).

5.2 Discussion

Section 5.1 reports the results from the computational experiments. Based on these results, it can be seen that the optimal maximum accumulation times through which routing and inventory holding costs are minimised are affected by four main parameters. Section 5.2 presents more detailed insights into the impact of each of these parameters, including the sub-service regions, the time periods, the client demand rates, and the customer densities, on the optimal postponement horizons.

5.2.1 Impact of the Sub-Service Regions

Applying the sectorisation strategy proposed by Langevin and Soumis (1989), the entire circular service area is divided into concentric rings (in this study known as the sub-service regions, R1, R2 and so on). The results reveal that sub-service regions further away from the depot have higher maximum accumulation times. See for example, Figure 5.1 and Figure 5.2, the sub-region R2 (more remote from the depot) has a higher accumulation time than the sub-region R1 (nearer to the depot). This applies to both the constant demand (CD) and the general demand (GD1 and GD2) cases. This means that the service provider could apply a JIT strategy to the sub-service regions immediate next to the depot (e.g. R1) while prolonging the sub-service frequency for the more remote sub-service regions (e.g. R2, R3 and so on). As far as the total cost is concerned, having more sub-service regions helps reduce overall total cost. This observation holds true for both the constant demand (see Figure 5.7) and the general demand cases although the reduction in total cost is less substantial for the general demand case (see Figure 5.10).

5.2.2 Impact of the Time Period (T)

(45)

sensitivity factor α becomes bigger, as well as when the products become bulkier, the percentage improvement over a longer time period becomes less substantial (see Figure 5.10).

5.2.3 Impact of the Client Demand Rates (r)

Two types of demand are considered in this study. These are constant demand and general demand. Longer postponement horizons are required for customers having higher demand rates (i.e. bulkier products) (see Figure 5.1 and Figure 5.2). This can be attributable to the fact that clients having higher demand rates (or requesting bulkier products) also have larger capacities to hold such products, so they can order in large quantities (or bulky sizes) to account for the usage until the next shipment arrives.

5.2.4 Impact of the Client Density (δ)

(46)

6. Conclusion

This dissertation studied the Inventory Routing Problem, in which the aim was to investigate the benefits of two important strategies – the postponement strategy and the sectorisation strategy. The objective of this study was to find the optimal accumulation time, which minimised the total costs (i.e. the routing cost and inventory holding cost). The extensive computational experiment was carried out in order to investigate several scenarios where parameters, such as the size of the entire service region, depot location, customer demand rate, numbers of customers in a service region, customer densities in sub-service regions, time period and vehicle capacity vary. The results of this study confirmed that the application of the two strategies (postponement and sectorisation strategies) is beneficial in Inventory Routing Problems in terms of the reduction in total costs (i.e. inventory holding and routing costs). This research reports that the optimal maximum accumulation time is influenced by the settings of several important parameters, such as the sub-service regions, the client demand rate, and the customer density.

(47)

For sub-service regions having lower client densities, the service frequency can be postponed for a longer period of time.

(48)

References

• Abdelhalim, A.A. and Eltawil, A.B., 2013. ‘Recent Heuristics and Algorithms for Solving the Vehicle Routing Problems’, In Qi, E., ed. International Asia Conference on

Industrial Engineering and Management Innovation. Berlin: Springer-Verlag.

• Abrahamsson, M. (1993) ‘Time-Based Distribution’, The International Journal of

Logistics Management, 4(2), pp.75-84.

• Aghezzaf, E.H., Raa, B. and Landeghem, H.V. (2006) ‘Modeling Inventory Routing Problems in Supply Chain of High Consumption Products’, European Journal of

Operations Research, 169(3), pp.1048-63.

• Anily, S. (1994) ‘The general multi-retailer EOQ problem with vehicle routing costs’,

European Journal of Operational Research, 79, pp.451-73.

• Anily, S. and Federgruen, A. (1993) ‘Two-Echelon Distribution Systems with Vehicle Routing Costs and Central Inventories’, Operations Research, 41(1), pp.37-47.

• Archetti, C., Bertazzi, L., Hertz, A. and Speranza, M.G. (2012) ‘A Hybrid Heuristic for an Inventory Routing Problem’, INFORMS Journal on Computing, 24(1), pp.101-16. • Archetti, C., Bertazzi, L., Laporte, G. and Speranza, M.G. (2007) ‘A Branch-and-Cut

Algorithm for a Vendor-Managed Inventory Routing Problem’, Transportation Science, 41(3), pp.382-91.

• Aziz, N.A.B.B. and Moin, N.H. (1970) ‘The Applications of Metaheuristics in Inventory Routing Problems’, Malaysian Journal of Science, 26(0), n.p..

• Bell, W. J., Dalberto, L. M., Fisher, M. L., Greefield, A. J., Jaikumar, R., Kedia, P., Mack, R. G. and Prutzman, P. J. (1983) ‘Improving the Distribution of Industrial Gases with an On-Line Computerized Routing and Scheduling Optimizer’, Interfaces, 13(6), pp.4-23. • Bertazzi, L., Paletta, G. and Speranza, M.G. (2002) ‘Deterministic Order-Up-To Level

Policies in an Inventory Routing Problem’, Transportation Science, 36(1), pp.119-32. • Bertazzi, L. and Speranza, M.G. (2012) ‘Inventory Routing Problems: An Introduction’,

EURO Journal of Transport Logistics, 1(4), pp.307-26.

(49)

• Beullens, P., van Wassenhove, L.N. and van Oudheusden, D. (2004) Collection and Vehicle Routing Issues in Reverse Logistics. In Dekker, R., Fleischmann, M., Inderfurth, K. and Wassenhove, L.N., eds. Reverse Logistics: Quantitative Models for Closed-Loop

Supply Chains. Springer, pp.95-134.

• Bianchi, L. et al. (2006) ‘Hybrid Metaheuristics for the Vehicle Routing Problem with Stochastic Demands’, Journal of Mathematical Modelling and Algorithms, 5(1), pp.91-110.

• Blumenfield, D. E., Burns, L. D., Daganzo, C. F., Frick, M. C. and Hall, R. W. (1987) ‘Reducing Logistic Costs at General’, Interfaces, 1(17), pp.26-47.

• Burns, L.D., Hall, R.W., Blumenfield, D.E. and Daganzo, C.F. (1985) ‘Distribution Strategies that Minimise Transportation and Inventory Costs’, Operations Research, 33(3), pp.469-90.

• Campbell, A., Clarke, L., Kleywegt, A. and Savelsbergh, M. (1998) ‘The Inventory Routing Problem’, In Fleet Management and Logistics. Kluwer Academic Publishers, pp.95-113.

• Campbell, A.M., Clarke, L.W. and Savelsbergh, M.W.P. (2002) ‘Inventory Routing in Practice’, In Toth, P. and Vigo, D., eds. The Vehicle Routing Problem. SIAM, pp.309-30. • Campbell, A.M. and Savelsbergh, M.W.P. (2004) ‘A Decomposition Approach for the

Inventory-Routing Problem’, Transportation Science, 33(3), pp.469-90.

• Cattryssee, D. (2010) ‘Sectorization Strategies for Curb Side Waste Collection’, KU

Leuven [Online] Available at:

https://www.mech.kuleuven.be/en/cib/imresearch/pps/sectorization-strategies-for-curb-side-waste-collection (Accessed 28 May 2014).

• Christiansen, M., Fagerholt, K., Nygreen, B. and Ronen, D. (2006) ‘Maritime Transportation’, Transportation, 14, pp.189-284.

• Christiansen, M., Fagerholt, K. and Ronen, D. (2004) ‘Ship Routing and Scheduling: Status and Perspectives’, Transportation Science, 38(1), pp.1-18.

• Codd, E.F. (1972) ‘Further Normalization of the Data Base Relational Model’, In Data

Base Systems, Prentice-Hall, pp.33-64.

(50)

• Coelho, L.C., Cordeau, J. and Laporte, G. (2012b) ‘The Inventory-Routing Problem with Transshipment’, Computers and Operations Research, 39, pp.2537-48.

• Coelho, L.C., Cordeau, J. and Laporte, G. (2014) ‘Thirty Years of Inventory-Routing’,

CIRRELT, 48(1), pp.1-19.

• Coelho, L.C. and Laporte, G. (2013) An Optimised Target Level Inventory Replenishment

Policy for Vendor-Managed Inventory Systems. CIRRELT [Online] Available at:

https://www.cirrelt.ca/DocumentsTravail/CIRRELT-2013-05.pdf (Accessed: 30 May 2014).

• Coelho, L.C. and Laporte, G. (2013) ‘The exact solution of several classes of inventory-routing problems’, Computers and Operations Research, 40, pp.558-65.

• Cordeau, J.F., Gendreau, M. and Laporte, G. (1997) ‘A Tabu Search Heuristic for Periodic and Multi-Depot Vehicle Routing Problems’, Networks, 30(2), pp.105-19. • Custódio, A. and Oliveira, R. (2006) ‘Redesigning Distribution Operations: A Case Study

on Integrating Inventory Management and Vehicle Routes Design’, International Journal

of Logistics, 9(2), pp.169-87.

• Daganzo, C.F. (1984a) ‘The Length of Tours in Zones of Different Shapes’,

Transportation Research Part B, 18B(2), pp.135-45.

• Daganzo, C.F. (1984b) ‘The Distance Travelled to Visit N Points with a Maximum of C Stops per Vehicle’, Operations Research, 2, pp.393-410.

• Daganzo, C.F. (2005) Logistics Systems Analysis. 4th ed. Heidelberg: Springer-Verlag. • Dasci, A. and Verter, V. (2001) ‘A continuous model for production-distribution system

design’, European Journal of Operational Research, 129, pp.287-98.

• Edirisinghe, N.C.P. and James, R.J.W. (2014) ‘Fleet routing position-based model for inventory pickup under production shutdown’, European Journal of Operational

Research, 236(2), pp.736-47.

• El-Sherbeney, N.A. (2010) ‘Vehicle routing with time windows: An overview of exact, heuristic and metaheuristic methods’, Journal of King Saud University (Science), 22, pp.123-31.

(51)

• Gehring, H. and Homberger, J. (2002) ‘Parallelization of a Two-Phase Metaheuristic for Routing Problems with Time Windows’, Journal of Heuristics, 8, pp.251-76.

• Goetschalckx, M. (2011) ‘Vehicle Routing and Scheduling’, In Supply Chain

Engineering. Springer, pp.271-77.

• Golden, B.L., Assad, A.A. and Dahl, R. (1984) ‘Analysis of a Large-Scale Vehicle-Routing Problem with an Inventory Component’, Large Scale Systems in Information and

Decision Technologies, 7(2-3), pp.181-90.

• Haimovich, M. and Rinnoy Kan, A.H.G. (1985) ‘Bounds on Heuristics for Capacitated Routing Problems: I’, Mathematics of Operations Research, 10(4), pp.527-42.

• Hall, R.W. (1986) ‘Discrete models/continuous models’, Omega, 14(3), pp.213-20. • Hemmelmayr, V., Doerner, K.F., Harti, R.F. and Savelsbergh, M.W.P. (2010) ‘Vendor

managed inventory for environments with stochastic product usage’, European Journal

of Operational Research, 202(3), pp.686-95.

• Jahre, M. (1995) ‘Household Waste Collection as a Reverse Channel: A Theoretical Perspective’, International Journal of Physical Distribution and Logistics Management, 25(2), pp.39-55.

• Jung, J. and Mathur, K. (2007) ‘An Efficient Heuristics Algorithm for a Two-Echelon Joint Inventory and Routing Problem’, Transpotation Science, 41(1), pp.55-73. • Kurz, A. and Zäpfel, G. (2013) ‘Modeling cost-delivery trade-offs for distribution

logistics by a generalized PVRP model’, Journal of Business Economics, 83, pp.705-26. • Langevin, A., Mbaraga, P. and Campbell, J.F. (1996) ‘Continuous Approximation

Models in Freight Distribution: An Overview’, Transportation Research Part B, 30(3), pp.163-88.

• Langevin, A. and Soumis, F. (1989) ‘Design of Multiple-Vehicle Delivery Tours Satisfying Time Constraints’, Transportation Research Part B, 23B(2), pp.123-38. • Larson, R. (1988) ‘Transporting sludge to the 106-mile site: An inventory/routing model

for fleet sizing and logistics system design’, Transportation Science, 22(3), pp.186-98. • Lee, H. and Whang, S. (1998) ‘Value of Postponement’, In Ho, T. and Tang, C.S., eds.

Product Variety Management. Springer US, pp.65-84.

Impact of Postponement and Sectorisation Strategies in Inventory Routing Problems Dissertation

Dissertation