Production Routing Problem with Multimodal Transportation

(1)

Multimodal Transportation

Master’s Thesis

Version 2 (repair)

Ruben Haring

S1999869

email: r.t.haring@student.rug.nl

Supervisors:

dr. E. Ursavas

dr. O.A. Kilic

University of Groningen

Faculty of Economics and Business Master Supply Chain Management

(2)

Productin Routing Problem with Multimodal

Transportaion

Ruben Haring

Abstract

This research introduces the production routing problem with multimodal trans-portation (PRPMT). This model minimizes the production, inventory and routing costs for proposed instances in one integrated planning. The multimodal trans-portation is added by introducing a second mode of transtrans-portation which costs are scaled by 2 factors. Extensive numerical experiments, with different values for these two factors, are conducted to create insights for policy makers. The results of the PRPMT shows that integrating a second mode of transportation leads to changes in all the sub processes, so including production and inventory planning. For policy makers this implies that plants needs to be more flexible for production in multiple periods. In addition, inventory levels are lower in general, which could lead to stock-out at customers. The results shows that implementing PRPMT is most effective for networks with higher in-between distances.

Acknowledgements

I would like to express my very great appreciation to dr. E. Ursavas and dr. O.A. Kilic for all the advice, meaningful guidance, extensive feedback and assis-tance during my research. I would like to offer a special thanks to all the people for supporting me during the last months, and especially to my colleagues at EventIn-sight for providing the extra time needed to finish my research.

(3)

1 Introduction 7

2 Literature review 9

2.1 Production Routing Problem . . . 9

2.2 Multimodal Transportation . . . 11

2.3 PRPMT . . . 12

3 Problem description and model formulation 14 3.1 Problem definition . . . 14

3.2 Two modes of transportation . . . 15

3.3 Decision making . . . 16 3.4 Model formulation . . . 16 3.5 Model . . . 18 4 Computational Experiments 20 4.1 Instances . . . 20 4.2 Routing costs . . . 22

4.3 Results one mode of transportation per instance . . . 22

4.4 Numerical experiments on PRMTP . . . 24

4.5 Results on production, inventory and routing costs . . . 25

4.5.1 Managerial impact . . . 28

4.6 Experiments of different values C and W . . . 28

4.7 Results for different in-between distance . . . 32

5 Conclusions 34 5.1 Summary of results . . . 34

5.1.1 Summary of managerial impact . . . 35

5.2 Limitations . . . 35

(4)

Production Routing Problem with Multimodal Transportation

Appendices 40

A Explanation of instance generation 41 B Full results on production, inventory and routing costs 43

(5)

3.1 Example of network representation with one plant and two customers 15 4.1 Graphical representation of routing during period 3 for ABS3 with

runtime of 600 seconds. . . 25 4.2 Overview of different values for C and W for set 1 compared to

pos-sible saving on total costs. . . 30 4.3 Overview of different values for C and W for set 2 compared to

(6)

List of Tables

4.1 Coordinates of instances of set 1, set 2, set 3 and set 4 . . . 21 4.2 Results one mode of transportation . . . 23 4.3 Average results on production, inventory and routing costs per set of

instances for C = 0.5 and W = 1.0 . . . 26 4.4 Average results on production, inventory and routing costs per set of

instances for C = 0.8 and W = 1.0 . . . 26 4.5 Average results on production, inventory and routing costs per set of

instances for C = 0.8 and W = 1.5 . . . 26 4.6 Results of different values for C and W for set 1 . . . 29 4.7 Results of different values for C and W for set 2 . . . 29 B.1 Results on production, inventory and routing costs for C = 0.5 and

W = 1.0 per instance . . . 44 B.2 Results on production, inventory and routing costs for C = 0.8 and

W = 1.0 per instance . . . 45 B.3 Results on production, inventory and routing costs for C = 0.8 and

W = 1.5 per instance . . . 46

(7)

Introduction

Nowadays, fast developments in technology, such as the internet, increase the glob-alization of the world economy and this leads to the fact that firms compete more internationally with each other. Due to this increase in international competition, the supply chain becomes a more intensive field of competition between firms. Often, the approach in supply-chain management (SCM) is to create a sub-planning for all of the different processes sequential of each other. The creation of every planning is only based on the output of the preceding planning. The most common performance measures of a supply chain are costs, often divided into distributing, manufacturing and inventorying costs (Beamon, 1999). The research of Lee and Billington (1995) at Hewlett Packard, Brown et al. (2001) at Kellog and the re-search of C¸ etinkaya et al. (2009) at Frito-Lay showed that the overall performance of the supply chain improved by implementing a SCM strategy with one integrated planning for all sub-processes. Optimizing an integrated planning of production, in-ventory, distribution and routing decisions is called the production routing problem (PRP) (Adulyasak et al., 2015).

This research focuses on the PRP and is going to extend into the domain of mul-timodal tranportation(MT). The past years MT had a big influence in the trans-portation sector. In 2016, freight transport inside the European Union consisted of the following modes of transportation: 49.3% by road, 32.3% by sea, 11.2% by rail, 4.0% by inland waterways, 3.1% by pipelines and 0.1% by air (Directorate - General for Mobility and Transport (European Commission), 2018). The use of these differ-ent modes of transportation makes choosing the best one an important elemdiffer-ent of distribution decision in every supply chain.

(8)

agreement for setting emission reduction targets, and, more recently, the Paris Con-vention. Policy and planning creators have started considering the sustainability aspect of supply chains more often. MT is one of the most important opportunities to improve the sustainability of supply chains (Iannone, 2012).

MT is adopted more widely and is an important factor into creating more sustain-able supply chains. Besides, the PRP shows potential to improve performance of a supply chain. These two are combined into the Production Routing Problem with Multimodal Transportation (PRPMT). In this research a basic mathematical model is developed to solve the PRPMT. Multimodality is modeled via a different cost structure for the distribution costs compared to one mode of transportation, more details about this are given in chapter 3.

Different numerical studies are ran to test different settings for the cost structure in the mathematical model. The numerical studies on the mathematical model are interesting to obtain further insights into which policies to use and on a larger scale, if implementing an PRPMT is useful at all. The aim of this research is to show the possibilities and results of the first basic PRPMT problem. To the best of my knowledge, the integration of MT into the PRP has not been researched before. Chapter 2 of this report provides an overview of the literature on the PRP and MT. Chapter 3 explains the problem definition and the model used in this research, to-gether with an explanation of how this model was obtained. Chapter 4 elaborates on the numerical and computational experiments together with the results of these ex-periments. Finally, chapter 5 describes the conclusion and discusses the limitations and the possibilities for further research of this study.

(9)

Literature review

This chapter starts with an explanation of the different aspects of the PRP and then a discussion is provided on how these aspects can influence the implementation in the PRPMT. This is followed by an overview of the concepts of MT and the consequences for the PRPMT. To conclude, this chapters brings these aspects together into the PRPMT.

2.1 Production Routing Problem

As stated in the introduction, the PRP is an application to optimize the production, inventory, distribution, and routing decisions in one integrated planning. For the PRP, it is important that there is one central decision maker that handles all the decisions, making it possible to align all of these decision into one planning. When inventory is managed from a central place, this is called Vendor Managed Inventory (VMI). The PRP is relevant for VMI and other modern practices of logistics, such as the just-in-time (JIT) principle (Qiu et al., 2017), and more specifically for business competition. Especially these modern practices are more common and a reason to use PRP in this research.

The PRP consist of two sub-problems, namely the lot-sizing problem (LSP) and the vehicle routing problem (VRP). Both of the problems are well known and researched in the field op supply chain management. The goal when solving LSPs is to minimize variable and fixed production costs and holding costs (Kazemi and Szmerekovsky, 2015). The results are decisions on when to produce what amount of a product. Most of the LSPs consider distribution cost to be fixed (Adulyasak et al., 2014), which is not real-life like. The goal of the VRP is to design optimal delivery routes from (multiple) depots to customers, where these depots and customers often have geographically scattered locations (Laporte, 1992).

(10)

to the PRP. The idea of integration started in 1993 when Chandra (1993) already proved the benefits of an integrated planning, and afterwards in 1994, Chandra and Fisher (1994) created a model that could lead to a cost reduction of 3% to 20% while using an integrated planning. The review article from Adulyasak et al. (2015) presents the developments in research of the PRP up to 2014. This article concludes that research has gained more attention in the past few years, but the exact solutions have been less extensively studied than the (meta) heuristics. An exact solution (al-gorithm) can solve small to medium instances but is not fit for large instances(Qiu et al., 2018). The small and medium instances mentioned in this article consists of 14 and 50 customers, respectively, and all with a single depot. Because this research will use an exact solutions as well, the focus is on small to medium instances. Heuristics are generally a trade-off between the quality and the computational time of the obtained solution. Some examples of heuristics are the greedy randomized adaptive search procedure (Boudia et al., 2007), tabu search (Armentano et al. (2011) and Bard and Nananukul (2009) and relax and fit (Miranda et al., 2018). All these heuristics gather better results than exact solution algorithms in limited time, often 10 of 20 minutes. The heuristic developed by (Zhang et al., 2017) based on dynamic route generation significantly outperforms other approaches on larger instances, with 2 plants, 240 customers, 20 vehicles and 56 and 168 time periods on the coarse and fine grid, respectively. For larger instances, heuristics offer the possibility to obtain an acceptable solution in the computational time given. Most of the heuristic developed focus on improving results in a given computational time of an already known problem. Because this research focuses on the development of a new problem and due to the limited time available for this research, heuristics are out of the scope of this research.

The review above shows that the PRP approach in SCM leads to a better perfor-mance of the supply chain as a whole, especially cost wise. Bard and Nananukul (2010) used price algorithms, Adulyasak et al. (2014) used branch-and-cut and Ruokokoski et al. (2010) created an exact solution with a branch-and-branch-and-cut algorithm using valid inequalities. Ruokokoski et al. (2010) solved up to 8 time periods with 80 customers or 15 time periods with 40 customers to optimality for one product. In real-life situations, supply chains often consist of multiple products, multiple production plants and at least 15 time periods. In addition, many have more than 80 customers or endpoints of deliveries.

Next to this, the traveling salesman problem (TSP), where the goals is to find the shortest route between multiple cities, is part of the PRP, and because the TSP is proved to be strongly non-deterministic polynomial-time (NP) hard Karp (2010), the PRP is NP-hard itself. This indicates that it takes a significant amount of

(11)

putational time to solve instances of a real-life size with an exact algorithm. For these reasons given for the classical PRP, the focus of this research is on creating a working model for the PRPMT and testing small instances for this model to show the general benefits of the PRPMT for supply chain performance.

2.2 Multimodal Transportation

MT is defined as transport of freight with the use of two or more modes of trans-portation (ITF et al., 2010). The study of Raoufi et al. (2013) concludes that the research in the area of multimodality is concentrated on hub-and-spoke networks. Hubs are the handling point of freight, and every spoke, called “customer” in this research, is allocated to a hub. These networks lead to a hub location problem, where the goal is to minimize the total cost, or in some cases, total distance, which relates directly to costs of the transportation. The hub location problem and the PRP have the same aim to minimize costs. The comparable structure of the net-works hints that implementing multimodality in PRP is possible.

Raoufi et al. (2013) did an extensive review on multimodal freight transportation in 2014. They distinguish two models in their research, network flow planning (NFP), which addresses the movement of commodities throughout the network, and service network design (SND), which includes all of the decisions made in the transportation service. The SND problems include the use of binary variables to define if a service is used (Raoufi et al., 2013). The use of binary variables is often used in PRP and these articles show the possibility of a connection between MT and PRP.

Jansen et al. (2004) incorporated a multimodal planning algorithm for the routing aspect of Deutsche Post World Net. Their research decomposed the network into six sub-problems and for the routing decisions, they used an intermodal transporta-tion structure. In intermodal transportatransporta-tion, two different modes of transportatransporta-tion are allowed in one delivery, while in MT, every delivery is made by one mode of transportation, but different modes of transportation are available. Jansen et al. (2004) showed that their intermodal transportation model benefits the supply chain performance. Their research is based on a real-life situation, by taking the railway connections and stations into account, whereas MT is more theoretical, because of-ten the connection between every customer and depot is possible to use. MT needs less input from real-life situations and therefore is used in this research and inter-modal transportation is ignored.

(12)

is a heuristic method and therefore not directly relevant; however, the results are promising and can be used for further research. The arc-based model is still adopted in this research, because it is directly related to the classical PRP, but the model can be expected to be (too) time consuming to run.

Kazemi and Szmerekovsky (2015) created a model in to minimize costs in a down-stream petroleum supply chain. In their model the decision on the mode of trans-portation is purely based on costs. The transtrans-portation of a certain amount is possible via 4 modes of transportation which all differ in costs, a binary variable indicates whether or not a certain mode of transportation is used. The model minimizes the overall costs, so the best option is picked. The binary variable structure on decision making can be usefull for this research because it is easy to model and comparable to the PRP.

2.3 PRPMT

Multiple papers proved the usefulness of PRP and MT seperately, but none of these papers used the combination of the two to improve the performance of a supply chain.

In the PRP a distinction is created between two elements, the LSP and the VRP, whereas LSP can be considered the production decision making part and VRP the routing decision making part. The inventory decision connects these two problem to one another and creates the PRP. Since 1993 the improvement in performance of supply chains with use an integrated planning has been researched and the benefit of such a planning model is proved multiple times.

MT has a large impact on sustainability and costs of a supply chain. Nowadays, the use of different modes of transportations is widespread. Multiple studies, mentioned before, showed the benefit of integrating multiple modes of transportation to the overall performance of a supply chain.

This study focuses on combining the routing decision from a MT model into a model solving the PRP. The decision on the production, the LSP part of the classical PRP, stays the same, but the VRP decisions are changed into a structure with possibilities of multiple modes of transportation. In the research, the link is created between the decisions making rules from a MT model, and this is connected to the PRP.

This research will focus on two modes of transportation, e.g. transportation by truck or train. No more than two modes of transportation are implemented to keep the model as simple as possible. The decision making will have the same structure as suggested by Kazemi and Szmerekovsky (2015) for the routing part. The novelty is creating two different cost structure for routing, one based on a truck and one based

(13)

on transportation by train. The model decides upon the best overall performance on a cost level. Other performance measures of the supply chain are ignored because it is not possible to implement them in a minimizing model. A basic numerical study will supply insights on whether the new model creates savings on costs.

(14)

Chapter 3 Problem description and model

formulation

In this chatper a general overview of the problem definition is supplied with special attention to the added value of this research. The situation is also presented graph-ically. Next, the model, and the original sources, together with the modifications, are described. Finally, the working principles of the model are stated, based on the constraints used.

3.1 Problem definition

The production routing with MT problem consists of the integrated planning of pro-duction, inventory management, distribution and routing. The goals of the decisions is to supply all the customers with the needed amount of the product(s) at a given time. The network includes nodes, which can be plants or customers. Every node has a x- and a y- coordinate. In figure 3.1 a very basic graphical representation is given of a network with three nodes, one plant (0) and two customers (1 and 2). Transportation is possible over arcs between nodes. In the basic PRP problem only transportation over one arc is possible between two nodes. This is represented in figure 3.1 by the filled lines (mode of transportation 1). Every node is connected to every other node for this mode of transportation.

All plants, in this research limited to 1 (assumption 1), can produce products in every time period. The amount of products is limited (assumption 2) and the pro-duction for every product is limited. All plants can hold inventory and this is limited by a maximum. All customers have a certain demand in every time period, which is constant over time, that needs to be satisfied. Every customer can hold inventory and this is limited by a maximum. All vehicles, in this research limited to one per mode of transportation (assumption 3), can hold a certain amount per product.

(15)

Figure 3.1: Example of network representation with one plant and two customers

Figure 3.1 provides an example of a small version of such a network with two cus-tomers and one plant, with all the routes possible for two modes of transportation.

3.2 Two modes of transportation

This research adds an extra mode of transportation. The graphical representation is given in figure 3.1. Every node is connected to another node, but now has two option for transportation, where the second mode of transportation is shown as a dashed line. The difference between the two mode of transportation are mainly in the structure of buildup of costs. For the first mode of transportation the costs are linear to the distances between the two nodes. In real-life this situation is compa-rable to a truck. The distance traveled is linear to the costs.

The second mode of transportation has fixed cost to use the mode of transportation and a factor linear to the distance. In a real-life situation this is comparable to a train. There are costs to load everything on a train or to use such a service, no matter what the distance traveled is, and there a costs for the distances covered. Of course, the costs linear to the distance for trains is lower than the costs linear to the distance for trucks. If it would be the same or higher, the train will never be taken, because it is always more expensive to take the train, considering the added fixed costs as well.

(16)

to go for multimodal transportation instead of intermodal transportation. Every vehicle, for every mode of transportation, will this way automatically start at the plant, visit one or multiple customers, and end at the plant.

3.3 Decision making

For the production decisions, for every plant, it is decided how much to produce during a certain time period. The costs for production are split into variable and fixed costs. Variable costs are for every unit produced and fixed costs which apply when there is production during a certain time period. Inventory decisions concern holding inventory at a certain node. Every node has certain holding costs, which are the costs to hold one unit for one period of time. The customers have a demand for every time period and product. This demand must be satisfied, either from in-ventory or from production.

For the routing decisions, deliveries are made by a vehicle from each mode of trans-portation. Every vehicle has a maximum capacity in unit loads of products, and every vehicle has a route that starts and ends at a depot. The costs for routing are divided into fixed costs to use a certain arc and costs per unit transported over a certain arc. These costs differ for each mode of transportation, which is important for this research, because the policy analysis focuses on these differences in cost. The following assumptions have been adopted for the modeling convenience. These assumptions were adjusted after the first initial test to reduce the computational time for the calculations:

Assumption 1 The number of plants is limited to one. Assumption 2 The number of products is limited to one.

Assumption 3 The number of vehicles per mode of transportation is limited to one.

Assumption 4 Only one vehicle can visit a certain customer during a period. This also holds for vehicles from different modes of transportation.

3.4 Model formulation

The PRPMT can be modeled as follows. The network created is a complete directed graph G = (N, A) where N represent the set of the plant and the customers indexed by i ∈ {0, ..., n} and A = {(i, j, r) : i, j ∈ N, i 6= j, r ∈ R}, where R = {1, .., r} is the set of possible modes of transport, represent the set of arcs. Node 0 represent

(17)

the plant and the set of customers is Nc = N \ {0}. A finite set of time periods

T = {1, ..., l} is defined. The total amount of vehicles M = PR

r mr where r is the

mode of transport. This leads to the following nomenclature: Parameters:

u unit production cost

f fixed production setup costs

hi unit inventory holding cost at node i

cijr cost to open arc from node i to j when using mode of transport r

aijr cost to transport one unit using arc from node i to j by mode of

transport r

dit demand at customer i at period t

C production capacity

V Cr vehicle capacity of vehicle for mode of transport r

V Qr amount of vehicles available for mode of transport r

Li maximum level inventory at node i

Ii0 initial inventory available at node i (before period 1)

Sets:

N Set of all nodes (including plant) {0, ..., n} Nc set of customers {1, ..., n}

T set of time periods {1, ..., l}

R set of all possible modes of transportation Decision Variables:

Pt production quantity in period t

Iit inventory at node i at the end of period t

wijrt load of a vehicle at the moment of using arc from node i to j in period

t using mode of transport r

yt equal to 1 if there is production at the plant in period t, 0 otherwise

zirt equal to 1 if customer i is visited in period t by mode of transport r,

0 otherwise, ∀i ∈ Nc

qirt quantity deliverd to node i in period t by mode of transport r

xijrt equal to 1 if the arc from node i to node j in period t using mode of

transportation r is used, 0 otherwise

Further, Mt = min{P C,Pl_j=tP_i∈N_cdij} and ˜Mirt = min {Li, V Cr,Pl_j=tdij} for

(18)

3.5 Model

The model used in this research is based on the models used in the research of Bard and Nananukul (2009) and Bard and Nananukul (2010). From Kazemi and Szmerekovsky (2015),some aspects of multiple modes of transportation have been adopted. For the objective function, the cost of transportation per unit of prod-uct was adopted from Kazemi and Szmerekovsky (2015). For cijr, aijr, zirt, qirt and

xijrt, the extra indices of the mode of transportation were added. The extra decision

variable wijrt was introduced in the model, adopted from Kazemi and Szmerekovsky

(2015). It is also important to notice that in this model, the delivery to customers can either come from the inventory at the end of the previous time period at the production facility or from the production in the current time period. This is differ-ent than in the models from Bard and Nananukul (2009) and Bard and Nananukul (2010) and is more similar to that of Kazemi and Szmerekovsky (2015).

minX t∈T  uPt+ f yt+ X i∈N hiIit+ X (i,j)∈A X r∈R

(cijrxijrt+ aijrwijrt)

  (3.1) subject to: I0,t−1+ Pt= X j∈Nc X r∈R qjrt+ I0t ∀t ∈ T (3.2) Ii,t−1+ X r∈R qirt = dit+ Iit ∀i ∈ Nc, ∀t ∈ T (3.3) Pt≤ Mtyt ∀t ∈ T (3.4) I0t≤ L0 ∀t ∈ T (3.5) Ii,t−1+ X r∈R qirt ≤ Li ∀t ∈ T, ∀i ∈ Nc (3.6) X i∈Nc qirt ≤ V Cr∗ z0rt ∀t ∈ T, ∀r ∈ R (3.7)

qirt≤ ˜Mirt∗ zirt ∀t ∈ T, ∀i ∈ Nc, ∀r ∈ R (3.8)

X

j∈N,j6=i

xijrt = zirt ∀t ∈ T, ∀i ∈ N, ∀r ∈ R (3.9)

X

j∈N,j6=i

(xijrt+ xjirt) = 2 ∗ zirt ∀t ∈ T, ∀i ∈ N, ∀r ∈ R (3.10)

z0rt ≤ V Qr ∀t ∈ T, ∀r ∈ R (3.11) X j∈N wjirt = qirt+ X j∈N wijrt ∀t ∈ T, ∀i ∈ Nc, ∀r ∈ R (3.12)

(19)

wijrt ≤ V Cr∗ xijrt ∀t ∈ T, ∀i, j ∈ N, ∀r ∈ R (3.13)

wiirt, wi0rt, xiirt, q0rt = 0 ∀t ∈ T, ∀i ∈ N, ∀r ∈ R (3.14)

Pt, Iit, wijrt, qirt ∈ Z+ ∀t ∈ T, ∀i, j ∈ N, ∀r ∈ R (3.15)

yt, xijrt ∈ {0, 1} ∀t ∈ T, ∀i, j ∈ N, ∀r ∈ R (3.16)

zirt∈ {0, 1} ∀t ∈ T, ∀i ∈ Nc, ∀r ∈ R (3.17)

z0rt∈ Z+ ∀t ∈ T, ∀r ∈ R (3.18)

(20)

Chapter 4 Computational Experiments

This chapter shows the experiments done using the model explained in the previ-ous chapter. First, the instances used in this research are described. Second, the results of the model being tested on a single mode of transportation are provided. Third, the results of the various computational experiments on two modes of trans-portation are given. All the computational experiments in this section were run on a computer with Intel Core i3-6100T @ 3.20 GHz with 8 GB of RAM, using the Windows 7 operating system and the optimization software of FICO, Xpress-IVE R

version 1.24.15 64 bit with Xpress Mosel version 4.2.0 and Xpress Optimizer version 30.01.04 (also called Xpress version 8.1).

The results are presented in three sections, concerning partial costs, costs for second mode of transportation and results for geographical orientation. Every section con-tains a brief discussion of the managerial impact of these results for policy makers.

4.1 Instances

Computational results are reported on instances of the benchmark proposed by Archetti et al. (2011). These instances provide all the necessary input for the model and consist of the following parameters: number of nodes (including the plant), horizon (number of time periods) and vehicle capacity. Next to this per node, for both the plant and the customers, the following parameters are supplied: x and y coordinate, start, maximum and minimum inventory level and holding cost per unit of inventory. Also the variable and fixed production cost are supplied for the plant and the demand per time period for every customer.

In the original instances sets Archetti et al. (2011) created four different classes. The first class is called the standard class and is the basis for every other class. These class are randomly generated and consists of 24 different parameter sets to randomly generate these instances, these details can be found in appendix A. For the other

(21)

Table 4.1: Coordinates of instances of set 1, set 2, set 3 and set 4 Node Set 1 Set 2 Set 3 Set 4

x y x y x y x y 0 143 99 287 199 1430 990 2870 1990 1 89 159 178 319 890 1590 1780 3190 2 76 314 153 629 760 3140 1530 6290 3 285 63 571 126 2850 630 5710 1260 4 401 235 802 651 4010 2350 8020 6510 5 16 310 33 621 160 3100 330 6210 6 267 401 534 803 2670 4010 5340 8030 7 249 123 498 247 2490 1230 4980 2470 8 477 238 955 476 4770 2380 9550 4760 9 374 194 748 389 3740 1940 7480 3890 10 277 101 554 203 2770 1010 5540 2030 11 445 14 890 28 4450 140 8900 280 12 312 450 624 901 3120 4500 6240 9010 13 421 213 842 426 4210 2130 8420 4260 14 79 71 159 142 790 710 1590 1420

classes the standard class is adopted and changed on one parameter. Using this methodology, one class with high production costs, one with high transportation costs and one with zero customer holding costs are created. This research focuses on innovation in the routing decisions and therefore the classes with high production costs and zero customer holding costs are considered less interesting than the class with high transportation costs.

These instances are available with 14, 50 and 100 customers. An initial run for the standard class of instances showed that for the instances with 14 customers 19 out of 24 instances are solved to optimality in 1200 seconds. None of the instances with 50 and 100 customers is able to solve to optimality in 1200 seconds. The lowest optimality gap obtained for 50 customers after 1200 seconds was 14.3%. Running instances with a longer computational time would take multiple days and therefore this research focuses on the instances with 14 customers.

(22)

are 10 times the coordinates of set 1 and the coordinates of set 4 are 20 times the coordinates of set 1. All the other parameters, e.g. holding costs or demand, are equal between corresponding numbers in the instances. This means that ABS1 is identical to ABS13, ABS49 and ABS61, except for the coordinates.

It is also noteworthy that the random generation of instances was only partially random. For example, ABS1 was identical to ABS2 except for the vehicle capacity and the coordinates of ABS1 up to ABS12 are identical. This makes the instances less comparable to real-life, due to the lack of variability. On the other hand, the instances are easier to compare to each other based on their parameters in common.

4.2 Routing costs

For the first mode of transportation, in real-life situations comparable to a truck, the cost structure is equal to the cost structure in Archetti et al. (2011). The costs of opening an arc between two customers or between a customer and the plant are equal to the Euclidian distances, the ordinary straight-line distance between two nodes, of the coordinates. No costs are inquired for the number of products trans-ported over a certain arc. Opening an arc always has the same, predetermined and fixed costs for every time period and number of products moved.

For the second mode of transportation, in real-life situations comparable to a train, the costs are dived into a fixed and a variable part. The fixed part is equal to a certain factor times the Euclidian distances. A factor, called C from here on, is introduced to differ the routing costs from the first mode of transportation for the fixed part, although the Euclidean distances are equal. Next to the fixed costs variable costs inquire, based on the amount of product transported. This amount is the same for every arc and therefore independent of the in-between distance of coordinates. The costs to transport one unit over an arc using the second mode of transportation is called W from here on.

4.3 Results one mode of transportation per

in-stance

For the test run using only one mode of transportation, R = 1 holds. The costs of transportation of a load over an arc was set to 0 for all arcs, so airt = 0 ∀t ∈

T, ∀i ∈ N, ∀r ∈ R. When only the first mode of transportation is used, the solutions should be identical to the solution obtained for the instances of Archetti et al. (2011) and other researchers, because this is actually the classical PRP. Qiu et al. (2018)

(23)

Table 4.2: Results one mode of transportation

Costs of Costs of Costs of Name Runtime (s) Total costs Gap (%) Production Inventory Routing

(24)

presented their results in such a way that they could be compared to the results obtained by this model and the results are (almost) identical. This validates the model used in this research . The optimization software was set to a limit of 1,200 seconds of maximum runtime. Table 4.2 shows the results, both on individual level of instances and as well as an average per set. Interesting to note is that sets with larger in-between distances of coordinate takes more runtime to complete. In set 1, 11 instances solve to optimality, in set 2 8 instances, in set 3 1 instance and in set 4 none of the instances solve to optimality. This indicates that sets with larger in-between distances takes longer to solve, due to the added complexity to minimize with higher costs for routing. The results obtained for one mode of transportation serve as the starting point of other observations and possible improvements.

4.4 Numerical experiments on PRMTP

For the experiments on different policies to implement the PRPMT the same in-stances ABS1-ABS24 and ABS49-ABS72 are used. The mode of transportation 1 is already described in the previous section. For the second mode of transportation the factor C and W are introduced, as described in section 4.2. First, some experiments were performed to decide upon logical values for C and W to run the model. Firs of all, for values of C it can be determined, using common sense, the value should be lower than 1. If we set C to a value higher than 1, the fixed costs of opening an arc are higher than the cost to transport over that arc using mode of transportation 1. Because the model minimizes total costs it than will always choose the cheaper option, in this case mode of transportation 1. Experiments showed that values higher than 0.8 for C leads to transport almost only with mode of transporta-tion 1 and values below 0.2 leads to transport only by mode of transportatransporta-tion 2. Therefore, in further experiments values between 0.2 and 0.8 are used for C.

For values of W it is less easy to determine a range that should be used by logi-cal reasoning. Multiple experiments are conducted to create a range for W. These experiments showed that values of 2 for W leads to the extreme favoring of mode of transportation 1 and values lower than 0.6 leads to extreme favoring of mode of transportation 2. Therefore, in further experiments values between 0.6 and 1.4 are used.

Figure 4.1 gives a graphical representation of the routing during period 3 for ABS3 with a runtime of 600 seconds, with C=0.5 and W=1.0. In this figure all transport carried out by a truck, mode of transportation 1, is shown by a filled line. The truck starts at the plant, 0 and dashed lines, and visits customer 1, 5, 2 and 14 to afterward return to the plant. The arc between customer 14 and the plant (0) has

(25)

Figure 4.1: Graphical representation of routing during period 3 for ABS3 with runtime of 600 seconds.

a load of 0, because the truck returns empty. This picture also shows the train is used for another route, illustrated with the dashed line. Interesting to notice is that the train is used for the longer distance haul and the truck is used for the short distance. This is logical, because with larger in-between distances factor C drops the fixed costs for opening an arc using mode of transportation 2 more than for short distance in absolute value.

Several numerical experiments are conducted using the model and different values of C and W and will be discussed in this chapter. The first experiment showed the differences in production, inventory and routing costs for the different modes of transportation, based on all 4 sets. The second experiment consisted of an analysis of different costs for the use of mode of transportation 2, based on set 1 and 2.

4.5 Results on production, inventory and routing

costs

(26)

Table 4.3: Average results on production, inventory and routing costs per set of instances for C = 0.5 and W = 1.0

Name Runtime(s) Total Cost Gap(%) [1] [2] [3] [4] [5] [6] [7] Avg. set 1 1200 57780 2.41 0.34 43700 -4.38 9687 19.22 4393 -6.73 Avg. set 2 1200 61023 4.10 1.64 43033 -5.09 10513 19.88 7476 6.16 Avg. set 3 1200 77821 5.63 16.48 41200 -1.23 13463 2.73 23158 40.06 Avg. set 4 1200 96776 5.88 26.47 40950 0.00 13814 -1.45 42012 45.47

Note: [1] saving of total costs(%),[2] Production costs, [3] saving on production costs (%), [4] Inventory costs, [5] saving on inventory costs(%) [6] routing costs, [7] savings on routing costs(%)

Name Runtime(s) Total Cost Gap(%) [1] [2] [3] [4] [5] [6] [7] Avg. Set 1 1138 57880 1.11 0.16 43450 -3.78 10017 16.47 4413 -7.23 Avg. Set 2 1200 62498 3.56 -0.74 42783 -4.48 10914 16.82 8800 -10.46 Avg. Set 3 1200 90817 8.73 2.53 40950 -0.61 13790 0.37 36077 6.62 Avg. Set 4 1200 122055 9.63 7.26 41200 -0.61 13724 -0.79 67131 12.87

Name Runtime(s) Total Cost Gap(%) [1] [2] [3] [4] [5] [6] [7] Avg. Set 1 1136 57908 0.87 0.11 42783 -2.19 10735 10.49 4390 -6.67 Avg. Set 2 1200 62180 2.50 -0.23 42783 -4.48 10844 17.35 8552 -7.35 Avg. Set 3 1200 93680 11.44 -0.54 42117 -3.48 12656 8.56 38908 -0.71 Avg. Set 4 1200 125969 12.62 4.29 42617 -4.07 12916 5.14 70436 8.58

(27)

consists partly of these parts, can be coupled directly to it. The goal of the objective function is to minimize the sum of these costs.

Table 4.3 shows the average results for the runs on instances of set 1–4 with C = 0.5 and W = 1.0. Table 4.4 and table 4.5 show the average results for the runs on instances of set 1–4 with C = 0.8 and W = 1.0 and C = 0.8 and W = 1.5, respec-tively. Table B.1, table B.2 and table B.3, shows the results per instance for these tests and can be found in appendix B.

The production costs, the inventory costs and the routing costs are compared to the corresponding costs for the model with one mode of transportation. Columns [1], [3], [5] and [7] show the savings on the partial costs compared to the one mode of transportation results. For example, [1] was calculated as follows:

(total costs, one mode of transportation)−(total costs, two modes of transportation)

(total costs, one mode of transportation) ∗ 100%

A positive value means that the costs for two modes of transportation are lower than the costs for one mode of transportation, and a negative value means that the costs for two modes of transportation are higher.

The results in table 4.3 illustrate that for all sets, the total costs are lower for the PRPMT than for one mode of transportation. The PRPMT only differs in rules from the original PRP problem for the routing decisions, the rules for deciding on production an inventory stayed the same. Table 4.3 shows that the results of the model with two mode of transportation and C = 0.5 and W=1.0 changes for the routing, production and inventory costs. The demand of all the customers is the same for the model with one mode of transportation as for the model with two modes of transportation. The variable production costs are equal to the total demand of all the customers times the costs of producing one unit. The variable production costs are the same for the model with one mode of transportation and the model with two modes of transportation. This means that all changes in production costs come from changes in the fixed production cost or, in other words, the number of periods during which production occurs. The production costs are higher for all the instances and different values for C and W, except for the average of set 4 for C = 0.5 and W = 1.0. This means that production occurs in a greater number of time periods for the sets with higher production costs.

(28)

C=0.5, W=1.0 for set 1.Second, when production occur more often, the inventory costs are lower. This can be seen when the savings on production costs are nega-tive, which is the case in all sets and values for C and W. The average inventory in units for customers and plants is lower than the original model, because the costs of inventory are lower and the holding costs are the same as in the original model.

4.5.1 Managerial impact

The impact of the introduction of a second mode of transportation does not only affect the routing process in the supply chain , but also the production and inven-tory processes. The results indicate that the production, on average, occurs more often, which requires the production plants to be more flexible by producing in more time periods. For all the sets of instances and the values of C and W, the costs of production are higher than in the original model. As discussed, higher production costs can only come from the higher fixed production costs, which means that pro-duction needs to happen more often at the plant. This is important knowledge for managers, because the factory needs to be more flexible and have the possibility to produce more often.

Inventory costs are in general lower for almost all sets and values of C and W, ex-cept for set 4. Set 4 has a very big optimality gap, and is therefore not realistic. Lower inventory costs means that there are less products in stock at the customer. Less inventory of products leads to earlier stock-outs for customers. In this model stock-out is not possible, the demand is known for all time periods in advance and the demand is considered constant. This is not realistic compared to real-life. Es-pecially changing demand could lead to even earlier stock-outs. Policymakers must consider these effects on all the processes in the supply chain when creating a new policy based on implementation of MT in the supply chain.

4.6 Experiments of different values C and W

Table 4.6 and table 4.7 show the results for the experiments for different values of C and W on set 1 and set 2, respectively. The percentage of routing is the percentage of routing compared to the total costs. The percentage of saving is the total costs of two modes of transportation compared to the total costs of one mode of transportation. Some experiments show negative values for the overall savings. This situation should be neglected, because it is still possible to not use mode of transportation 2 but only mode of transportation 1. In this case the results and costs would be the same, leading to a 0.00% of costs savings. The results that show negative savings all have a significant gap with the optimal solutions, which explains

(29)

Table 4.6: Results of different values for C and W for set 1

runtime total costs gap r = 1 r = 2 total routing % routing % saving 0.2C 1.0W 600 56871 3.54% 846 2950 3796 6.68% 1.90% 0.3C 1.0W 600 57293 3.30% 1021 3090 4111 7.18% 1.18% 0.4C 0.8W 600 57322 2.76% 830 3248 4078 7.11% 1.12% 0.4C 1.0W 600 57656 3.00% 1305 3051 4356 7.56% 0.55% 0.4C 1.2W 600 57777 2.99% 1796 2674 4470 7.74% 0.34% 0.5C 0.6W 600 57392 2.50% 813 3341 4154 7.24% 1.00% 0.5C 0.8W 600 57753 2.79% 1236 3200 4437 7.68% 0.38% 0.5C 1.0W 600 57796 2.62% 1989 2429 4418 7.64% 0.31% 0.5C 1.2W 600 57921 2.46% 2624 1843 4468 7.71% 0.09% 0.5C 1.4W 600 57981 2.37% 2828 1705 4533 7.82% -0.01% 0.6C 0.8W 600 57889 2.59% 2242 2238 4480 7.74% 0.15% 0.6C 1.0W 600 57859 2.05% 2576 1819 4394 7.59% 0.20% 0.6C 1.2W 600 57904 1.81% 3042 1398 4440 7.67% 0.12% 0.7C 1.0W 600 58064 2.06% 3011 1641 4653 8.01% -0.15% 0.8C 1.0W 597 57958 1.46% 3260 1236 4496 7.76% 0.03%

Table 4.7: Results of different values for C and W for set 2

(30)

the possibility of the existing negative savings. If the optimality gap were 0, the savings would be 0 or positive.

Figure 4.2: Overview of different values for C and W for set 1 compared to possible saving on total costs.

With the same solution, if C is higher the cost would be higher and if W is higher, the costs would be higher too. Because the model minimizes the overall costs, a lower value of C and/or W positively influences the savings possible.

Figure 4.2 shows the possible savings compared to the value of C for different values of W for set 1 and figure 4.3 shows the same, but for set 2. The result for set 1 shows that for C=0.6 having W=1.0 could lead to more possible saving than with W=0.8. It is easy to argue that more expensive routing could never lead to higher possible savings, although the difference is small. This makes this results neglectable. The results in table 4.7 and figure 4.3 show more promising results for set 2 compared to the results for set 1. The graphs show that changing the value of C has an almost linear effect for values for W of 0.8, 1.0 and 1.2. For C=0.5, changing the value for W from 0.6 to 0.8 shows a 0.96% of decrease in savings, changing W from 0.8 to 1.0 shows a 0.66% decrease in savings and changing W from 1.0 to 1.2 shows a 0.88% decrease. For C = 0.4, the change of W from 0.8 to 1.0 shows a 0.88% difference and from 1.0 to 1.2 shows a 0.87% difference. The results from C=0.6 can be neglected in this case due to negative values for the possible savings. To conclude, the change in the possible savings are comparable for increasing the value

(31)

Figure 4.3: Overview of different values for C and W for set 2 compared to possible saving on total costs.

of W by 0.2 for different values of C and W. Consequently, for both the value of C and the value of W, changing the input parameter for the buildup of costs for MT has a direct linear effect to the amount of savings possible.

4.6.1 Managerial impact

The results shown in this section could be considered as a guideline for implementing two modes of transportation. The different values of C and W guide decision-makers with advice regarding whether implementation of MT would lead to improvement in the total cost of the supply chain. Next to this, it shows to policy makers that the reduction of variable routing cost per load unit and fixed routing cost per arc opened are linear to the amount of savings possible.

(32)

4.7 Results for different in-between distance

As discussed in section 4.1, the instances in set 1 up to set 4 only differ in geograph-ical representation, as shown in table 4.1. The conclusion from section 4.1 was that set 2, 3 and 4 have coordinates which basically are identical to the coordinates of set 1 multiplied by a certain factor. For set 2 this factor is 2, for set 3 10 and for set 4 20. If both the x and y coordinates of two nodes are multiplied by factor 2, the Euclidian distance between these two nodes is also twice as large . The results for one mode of transportation show the difference in total costs between the sets. Large overall costs can be directly related to the higher in-between distances, be-cause this is the only difference between the 4 sets.

Table 4.3 demonstrates that, when two modes of transportation are implemented, possible savings of set with larger in-between distances are higher than the possible savings of sets with smaller in-between distances . The higher the in-between dis-tances, the higher potential savings by implementing two modes of transportation are. Reasoning by common sense will get you to the same conclusion Transporting products by a mode of transportation with lower costs per distance traveled, but with one-time setup costs will of course be more favorable for longer distances, be-cause the one time setup cost are lower compared to the total cost. Also, in real-life, often longer transport is more profitable to do by train.

The effects of extra costs of production, as discussed in section 4.4, are smaller for sets with larger in-between distances. This directly correlates to the significant amount of savings possible in the routing decision for larger in-between distances sets; the production must then be less flexible to still obtain a reduction of up to 25% in costs.

In addition, table 4.3 up to table 4.5 show that the possible savings in percentages of set 4 are more than 20 times greater than the savings for set 1, while the distances are 20 times larger. Because in-between distance is directly linked to routing costs, this indicates that implementing two modes of transportation has bigger effect on the total cost than only performing the same routes, but with lower costs. The higher routing costs for longer distances influence the other decisions in a positive way to lead to greater savings in general.

Furthermore, the difference between table 4.6 and table 4.7 clearly shows that for larger in-between distances, the effect of implementing two modes of transportation is greater. For all the test with different values for C and W, the possible savings of set 2 are always factor 3-5 higher than the possible savings of set 1. This does not hold for the last 4 combinations of C and W, but the results of these experiments can be neglected for at least one of the two sets, because it results in a negative

(33)

value for percentage of possible savings. Even for the values of C = 0.8 and W = 1.5, set 4 generates a solution with lower total costs when implementing two modes of transportation. This indicates that, even extreme high costs for the use of a train, it is possible to reduce the total costs, if the in-between distance is significantly large.

4.7.1 Managerial impact

Implementing two modes of transportation has more effect on the possible savings on total costs when the distances between the plant and customers and between customers themselves are larger. In all cases with the same value for C and W, the sets with larger in-between distances showed bigger potential of saving money by implementing the transport by train. The percentage of savings is more than factor 20 higher when the distance between nodes is factor 20 larger and transport by train is implemented.

(34)

Chapter 5 Conclusions

In this chapter a summary of the results obtained in chapter 4 is given, as well as the important limitations of this research and suggestions for further research.

5.1 Summary of results

The three analyses performed on the implementation of MT for the PRP show that it is possible reduce total costs. The implementation of a second mode of trans-portation, in this case transport by train, leads to changes in the planning of all different processes of the problem. First of all, the new model is tested for one mode of transportation and showed similar outcomes as other researches. This validates the model.

In general, as shown in table 4.3, 4.4 and 4.5, the cost of production raises in most cases, even up to 5.09% in one case. Extra costs in production always means that a plant needs to become more flexible, because the same amount is produced but the production is spread over more time periods.

In all cases the inventory costs drop using a second mode of transportation, up to 19.88%, compared to one mode of transportation. This indicates that the overall inventory is lower at the customers, which could lead to earlier stock out. The effect on saving highly depends on the in-between distance of the nodes of the model. Because the costs to open an arc between two nodes are equal to the Euclidian dis-tance, or a factor of the Euclidian distance. So between these two nodes, the costs of transportation are directly related to the distance. The results show that for sets with a higher in-between distance the possible savings are higher.

For the implementations of MT, policy makers have to take multiple things into account. First of all, implementing new transportation possibilities effect the pro-duction and inventory outcomes as well. New transportation possibilities demand more flexibility in the production planning and allowance for lower inventory levels

(35)

at the customers. This could lead to earlier stock out at the customers.

5.1.1 Summary of managerial impact

Every section of the results showed some insights for managerial impact. First of all, it is important to notice that implementing two modes of transportation, al-though it changes only the routing decisions, also influences the outcomes in other processes in the supply chain . In general, production needs to become more flexible and customer inventory shrinks, so attention should be paid to prevent stock-outs. The influence of the factors C and W, factors introduced to scale the costs of routing compared to the first mode of transporation, on the overall savings possible proved to be linear. This gives important suggestion that additional savings on the variable costs or the flexible costs of routing of a mode of transportation that needs to be implemented will benefit supply chain performance even further.

At last, this research proved that implementing two modes of transportation is es-pecially useful for long-haul transportation. Different sets with different in-between distances were tested which showed that implementing a second mode of transporta-tion on a network with factor 20 larger in-between distances, leads to savings higher than factor 20. This suggests, for policymakers, that implementing a PRPMT strat-egy is especially interesting for long-haul transport.

5.2 Limitations

Although the proposed model gives a good first indication of the effect of imple-menting multimodal transportation for the PRP, the results are far from optimal, especially for sets with larger in-between distances of nodes. The optimality gap on average is 8.60% for set 3 and 9.38% for set 4. This big optimality gap makes the results obtained less realiable and even lead to negative savings on total costs for some results. Negative savings proved theoretical impossible, so this makes the overall results especially untrustworthy.

(36)

5.3 Further research

Interesting topics for further research would be to study the implementation of more advanced exact solving algorithm and heuristics. The algorithm of Ruokokoski et al. (2010) solves problems with 80 customers and 8 periods to optimality in about 30 minutes. This is only for one mode of transportation, but still a far more complex model than presented in this research with well-performing algorithms.

Further research is needed to look at the developing of a heuristic for the PRPMT. A heuristic created by Zhang et al. (2017) showed solving capabilities up to 0.7% from the optimal value in 14% of the computational time. By developing a heuristic for the PRPMT it will be possible to solve larger instances, which are more realistic and comparable to real-world situations, in an acceptable computational time. A last topic for further research is intermodal transportation (i.e. using multiple modes of transportation to move a specific product from one location to another). Jansen et al. (2004) applied a model for intermodal transportation to a real-life situ-ation and following this research PRPMT can be adjusted to more real-life situsitu-ation when implementing intermodal transportation.

(37)

Adulyasak, Y., J.-F. Cordeau, and R. Jans

2014. Formulations and Branch-and-Cut Algorithms for Multivehicle Production and Inventory Routing Problems. INFORMS Journal on Computing, 26(1):103– 120.

Adulyasak, Y., J. F. Cordeau, and R. Jans

2015. The production routing problem: A review of formulations and solution algorithms. Computers and Operations Research, 55:141–152.

Archetti, C., L. Bertazzi, G. Laporte, and M. G. Speranza

2007. A Branch-and-Cut Algorithm for a Vendor-Managed Inventory-Routing Problem. Transportation Science, 41(3):382–391.

Archetti, C., L. Bertazzi, G. Paletta, and M. G. Speranza

2011. Analysis of the maximum level policy in a production-distribution system. Computers & Operations Research, 38(12):1731–1746.

Armentano, V., A. Shiguemoto, and A. Løkketangen

2011. Tabu search with path relinking for an integrated production–distribution problem. Computers & Operations Research, 38(8):1199–1209.

Bard, J. F. and N. Nananukul

2009. Heuristics for a multiperiod inventory routing problem with production decisions. Computers & Industrial Engineering, 57(3):713–723.

Bard, J. F. and N. Nananukul

2010. A branch-and-price algorithm for an integrated production and inventory routing problem. Computers and Operations Research, 37(12):2202–2217.

Beamon, B. M.

(38)

Bektas, T., M. Chouman, and T. G. Crainic

2010. Lagrangean-based decomposition algorithms for multicommodity network design problems with penalized constraints. In Networks, volume 55, Pp. 171–180. Boudia, M., M. Louly, and C. Prins

2007. A reactive GRASP and path relinking for a combined produc-tion–distribution problem. Computers & Operations Research, 34(11):3402–3419. Brown, G., J. Keegan, B. Vigus, and K. Wood

2001. The Kellogg Company Optimizes Production, Inventory, and Distribution. Interfaces, 31(6):1–15.

C¸ etinkaya, S., H. ¨Uster, G. Easwaran, and B. B. Keskin

2009. An integrated outbound logistics model for frito-lay: Coordinating aggregate-level production and distribution decisions. Interfaces, 39(5):460–475. Chandra, P.

1993. A dynamic distribution model with warehouse and customer replenishment requirements. Journal of the Operational Research Society, 44(7):681–692.

Chandra, P. and M. L. Fisher

1994. Coordination of production and distribution planning. European Journal of Operational Research, 72(3):503–517.

Directorate - General for Mobility and Transport (European Commission) 2018. EU transport in figures.

Iannone, F.

2012. The private and social cost efficiency of port hinterland container distribu-tion through a regional logistics system. Transportadistribu-tion Research Part A: Policy and Practice.

ITF, Eurostat, and UNECE

2010. Illustrated Glossary for Transport Statistics 4th Edition. United Nations Economic Commission for Europe.

Jansen, B., P. C. Swinkels, G. J. Teeuwen, B. Van Antwerpen de Fluiter, and H. A. Fleuren

2004. Operational planning of a large-scale multi-modal transportation system. European Journal of Operational Research, 156(1):41–53.

Karp, R. M.

2010. Reducibility among combinatorial problems. In 50 Years of Integer Pro-gramming 1958-2008: From the Early Years to the State-of-the-Art, Pp. 219–241. Boston, MA: Springer US.

(39)

Kazemi, Y. and J. Szmerekovsky

2015. Modeling downstream petroleum supply chain: The importance of multi-mode transportation to strategic planning. Transportation Research Part E: Lo-gistics and Transportation Review, 83:111–125.

Laporte, G.

1992. The traveling salesman problem: An overview of exact and approximate algorithms. European Journal of Operational Research, 59(2):231–247.

Lee, H. L. and C. Billington

1995. The Evolution of Supply-Chain-Management Models and Practice at Hewlett-Packard. Interfaces, 25(5):42–63.

Miller, C. E., A. W. Tucker, and R. A. Zemlin

1960. Integer Programming Formulation of Traveling Salesman Problems. Journal of the ACM, 7(4):326–329.

Miranda, P. L., R. Morabito, and D. Ferreira

2018. Optimization model for a production, inventory, distribution and routing problem in small furniture companies. Top, 26(1):30–67.

Qiu, Y., J. Qiao, and P. M. Pardalos

2017. A branch-and-price algorithm for production routing problems with carbon cap-and-trade. Omega (United Kingdom), 68:49–61.

Qiu, Y., L. Wang, X. Xu, X. Fang, and P. M. Pardalos

2018. A variable neighborhood search heuristic algorithm for production routing problems. Applied Soft Computing Journal, 66:311–318.

Raoufi, R., W. Nuijten, M. SteadieSeifi, T. Van Woensel, and N. Dellaert

2013. Multimodal freight transportation planning: A literature review. European Journal of Operational Research, 233(1):1–15.

Ruokokoski, M., O. Solyali, J. Cordeau, R. Jans, and H. Sural

2010. Efficient Formulations and a Branch-and-Cut Algorithm for a Production-Routing Problem. Transportation, G-2010-66:1–43.

Zhang, Q., A. Sundaramoorthy, I. E. Grossmann, and J. M. Pinto

(40)

Appendices

(41)

Explanation of instance generation

This appendix shows the full creation of the instances. This appendix is fully based on the article of Archetti et al. (2011). The instances are created in the following manner.

1. The instances have settings for amount of timeperiods (called horizon) and number of customers (14 in this research)

2. demand for every customer is constant over time and randomly generated on [5,25]

3. Maximum level of inventory is generated for everyh customer and is 2,3 or 6 times the demand

4. Starting level of inventory IL = L − d for every node

5. Inventory costs at customer are randomly generated on [1,5] or [6,10] 6. Inventory costs at plant are 3 or 8

7. Variable production costs are 10 times the inventory costs at the plant 8. Fixed production costs are 100 times the variable production costs (= 1000

times the inventory costs at plant)

(42)

Figure A.1: Overview of instances from Archetti

(43)

(44)

Table B.1: Results on production, inventory and routing costs for C = 0.5 and W = 1.0 per instance

Name Runtime(s) Total Cost Gap(%) [1] [2] [3] [4] [5] [6] [7]

ABS1 1200 40147 1.64 0.60 25200 0.00 10938 0.90 4009 3.47 ABS2 1200 40416 2.11 0.45 25200 0.00 10747 -2.48 4469 9.04 ABS3 1200 41192 3.67 0.21 25200 -13.51 10600 26.00 5392 -13.44 ABS4 1200 83218 0.75 0.58 67200 0.00 12274 0.58 3744 9.91 ABS5 1200 83485 0.97 0.67 67200 -13.51 12254 41.74 4031 -5.63 ABS6 1200 84167 1.51 0.33 67200 -13.51 12277 41.97 4690 -14.67 ABS7 1200 33200 3.77 -0.96 25200 -13.51 4095 40.99 3905 -4.33 ABS8 1200 33904 5.52 -2.75 22200 0.00 6956 0.37 4748 -24.42 ABS9 1200 33722 4.33 -0.97 22200 0.00 7106 0.03 4416 -7.97 ABS10 1200 72668 0.91 0.68 59200 0.00 9838 2.63 3630 6.01 ABS11 1200 73543 1.97 0.07 59200 0.00 9251 11.79 5092 -30.33 ABS12 1200 73695 1.80 1.99 59200 0.00 9907 16.73 4588 -12.18 Avg. set 1 1200 57780 2.41 0.34 43700 -4.38 9687 19.22 4393 -6.73 ABS13 1200 43674 3.98 1.96 25200 0.00 10851 1.69 7623 8.26 ABS14 1200 43561 4.02 3.83 25200 -13.51 10935 26.29 7426 10.12 ABS15 1200 44004 3.79 3.98 25200 -13.51 10791 26.55 8013 10.35 ABS16 1200 86614 2.33 1.22 67200 -13.51 12254 41.62 7160 4.41 ABS17 1200 87054 2.65 0.87 67200 -13.51 12282 41.81 7572 -0.81 ABS18 1200 87138 2.36 1.58 59200 0.00 21161 -0.01 6777 17.17 ABS19 1200 36706 7.60 -0.21 22200 0.00 7136 -2.82 7370 1.60 ABS20 1200 36981 8.45 -0.47 25200 -13.51 4461 37.15 7320 2.54 ABS21 1200 36215 4.83 3.40 22200 0.00 7042 0.93 6973 14.78 ABS22 1200 77249 4.16 -0.28 59200 0.00 9496 6.02 8553 -10.68 ABS23 1200 76529 2.76 1.26 59200 0.00 9746 7.07 7583 3.01 ABS24 1200 76545 2.24 3.45 59200 0.00 10004 15.92 7341 10.28 Avg. set 2 1200 61023 4.10 1.64 43033 -5.09 10513 19.88 7476 6.16 ABS49 1200 62241 8.99 18.18 25200 -13.51 12242 20.02 24799 35.70 ABS50 1200 60142 4.96 20.24 22200 0.00 16114 -2.84 21828 41.85 ABS51 1200 62802 6.49 20.73 22200 0.00 16288 1.42 24314 39.97 ABS52 1200 104261 6.06 11.35 59200 0.00 21226 -1.12 23835 36.31 ABS53 1200 102500 3.92 13.01 59200 0.00 21210 -0.21 22090 41.03 ABS54 1200 103782 2.95 14.36 59200 0.00 21287 0.29 23295 42.66 ABS55 1200 52481 9.44 22.09 25200 -13.51 4639 34.76 22642 40.49 ABS56 1200 52474 9.03 21.40 22200 0.00 7328 -3.07 22946 38.73 ABS57 1200 52834 5.86 24.51 22200 0.00 7290 -0.05 23344 42.36 ABS58 1200 92910 5.20 13.29 59200 0.00 10499 0.06 23211 38.02 ABS59 1200 93018 4.44 13.67 59200 0.00 11365 -3.17 22453 40.18 ABS60 1200 94404 0.17 15.54 59200 0.00 12066 -0.01 23138 42.87 Avg. set 3 1200 77821 5.63 16.48 41200 -1.23 13463 2.73 23158 40.06 ABS61 1200 78805 7.83 30.21 22200 11.90 15583 -26.36 41022 45.59 ABS62 1200 79184 6.05 29.95 22200 0.00 16355 -3.06 40629 45.80 ABS63 1200 81929 4.90 31.74 22200 0.00 16605 -0.55 43124 46.96 ABS64 1200 121941 6.12 21.43 59200 0.00 21093 0.10 41648 44.38 ABS65 1200 121452 4.22 21.81 59200 0.00 21194 -0.16 41058 45.23 ABS66 1200 124650 3.71 22.87 59200 0.00 21441 -0.46 44009 45.71 ABS67 1200 71195 9.96 31.58 25200 -13.51 5245 24.43 40750 45.60 ABS68 1200 70790 8.57 32.11 22200 0.00 7253 -1.94 41337 44.85 ABS69 1200 72918 4.57 34.04 22200 0.00 7301 -0.21 43417 46.44 ABS70 1200 111836 6.33 22.73 59200 0.00 10636 -1.25 42000 44.03 ABS71 1200 112395 5.33 22.65 59200 0.00 10949 1.79 42246 43.65 ABS72 1200 114216 2.93 25.02 59200 0.00 12107 -0.30 42909 47.07 Avg. set 4 1200 96776 5.88 26.47 40950 0.00 13814 -1.45 42012 45.47

Note: [1] saving of total costs(%), [2] Production costs, [3] saving on production costs (%), [4] Inventory costs, [5] saving on inventory costs(%) [6] routing costs, [7] savings on routing costs(%)

(45)

Table B.2: Results on production, inventory and routing costs for C = 0.8 and W = 1.0 per instance

Name Runtime(s) Total Cost Gap(%) [1] [2] [3] [4] [5] [6] [7]

ABS1 459 40266 0.01 0.31 25200 0.00 11037 0.00 4029 2.99 ABS2 1200 40440 0.61 0.39 25200 0.00 10756 -2.57 4484 8.73 ABS3 1200 41063 1.25 0.52 25200 -13.51 10592 26.06 5271 -10.90 ABS4 1200 83454 0.39 0.30 67200 0.00 12346 -0.01 3908 5.97 ABS5 1200 83831 0.67 0.26 67200 -13.51 12274 41.64 4357 -14.18 ABS6 1200 85037 1.59 -0.70 67200 -13.51 12256 42.07 5581 -36.45 ABS7 1200 32823 1.18 0.18 22200 0.00 6940 0.00 3683 1.60 ABS8 1200 33299 2.34 -0.91 22200 0.00 7039 -0.82 4060 -6.39 ABS9 1200 33729 2.63 -0.99 22200 0.00 7299 -2.69 4230 -3.42 ABS10 1200 72919 0.32 0.34 59200 0.00 9971 1.32 3748 2.95 ABS11 1200 73386 0.89 0.28 59200 0.00 9718 7.33 4468 -14.36 ABS12 1200 74313 1.42 1.16 59200 0.00 9971 16.20 5142 -25.72 Avg. Set 1 1138 57880 1.11 0.16 43450 -3.78 10017 16.47 4413 -7.23 ABS13 1200 44294 1.75 0.57 25200 0.00 11250 -1.93 7844 5.60 ABS14 1200 45260 3.48 0.08 25200 -13.51 11250 24.17 8810 -6.63 ABS15 1200 46998 5.88 -2.55 25200 -13.51 10871 26.01 10927 -22.25 ABS16 1200 87612 1.72 0.08 67200 -13.51 12294 41.43 8118 -8.38 ABS17 1200 89194 3.16 -1.57 67200 -13.51 12317 41.65 9677 -28.84 ABS18 1200 89787 3.01 -1.41 59200 0.00 21267 -0.52 9320 -13.91 ABS19 1200 37528 5.99 -2.45 22200 0.00 7157 -3.13 8171 -9.09 ABS20 1200 38019 6.66 -3.29 22200 0.00 6988 1.55 8831 -17.57 ABS21 1200 38065 4.89 -1.53 22200 0.00 7230 -1.72 8635 -5.54 ABS22 1200 76744 1.37 0.37 59200 0.00 10041 0.62 7503 2.91 ABS23 1200 77395 1.71 0.14 59200 0.00 10107 3.62 8088 -3.45 ABS24 1200 79080 3.06 0.25 59200 0.00 10200 14.27 9680 -18.31 Avg. Set 2 1200 62498 3.56 -0.74 42783 -4.48 10914 16.82 8800 -10.46 ABS49 1200 72808 10.47 4.29 25200 -13.51 12713 16.94 34895 9.53 ABS50 1200 73633 9.87 2.35 22200 0.00 16438 -4.91 34995 6.77 ABS51 1200 75774 8.49 4.35 22200 0.00 16622 -0.61 36952 8.76 ABS52 1200 118231 9.49 -0.52 59200 0.00 21265 -1.31 37766 -0.91 ABS53 1200 116743 7.82 0.92 59200 0.00 21237 -0.34 36306 3.08 ABS54 1200 117252 4.80 3.24 59200 0.00 21323 0.12 36729 9.60 ABS55 1200 65866 15.05 2.22 22200 0.00 7155 -0.62 36511 4.04 ABS56 1200 64495 10.88 3.39 22200 0.00 7228 -1.66 35067 6.37 ABS57 1200 64635 6.40 7.65 22200 0.00 7287 -0.01 35148 13.21 ABS58 1200 104950 7.76 2.06 59200 0.00 10561 -0.53 35189 6.03 ABS59 1200 105651 6.64 1.95 59200 0.00 11606 -5.36 34845 7.17 ABS60 1200 109763 7.14 1.79 59200 0.00 12044 0.17 38519 4.90 Avg. Set 3 1200 90817 8.73 2.53 40950 -0.61 13790 0.37 36077 6.62 ABS61 1200 104341 12.48 7.60 25200 0.00 12784 -3.67 66357 11.98 ABS62 1200 104994 11.32 7.11 25200 -13.51 14009 11.72 65785 12.24 ABS63 1200 109054 9.21 9.14 22200 0.00 16454 0.36 70400 13.41 ABS64 1200 145219 8.63 6.43 59200 0.00 21254 -0.66 64765 13.51 ABS65 1200 145099 6.67 6.58 59200 0.00 21400 -1.13 64499 13.96 ABS66 1200 150284 5.70 7.01 59200 0.00 21258 0.40 69826 13.86 ABS67 1200 93983 12.31 9.67 22200 0.00 7151 -3.03 64632 13.72 ABS68 1200 96937 14.74 7.03 22200 0.00 7325 -2.95 67412 10.06 ABS69 1200 100016 9.53 9.53 22200 0.00 7331 -0.62 70485 13.05 ABS70 1200 137817 10.98 4.78 59200 0.00 11515 -9.61 67102 10.57 ABS71 1200 135427 7.35 6.80 59200 0.00 12029 -7.89 64198 14.36 ABS72 1200 141490 6.61 7.12 59200 0.00 12174 -0.85 70116 13.50 Avg. Set 4 1200 122055 9.63 7.26 41200 -0.61 13724 -0.79 67131 12.87