The Choice of Depots in Oil Truck Scheduling

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The Choice of Depots in Oil Truck Scheduling

Willem Broekema

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Preface

As final project for my study of Econometrics and Operations Research, I was given the opportunity to analyse and implement improvements to the trip scheduling program shortrec (developed by ortec), focusing on its use in the oil business. As a result, more efficient schedules are calculated, resulting in significant cost savings.

Let me hereby thank Niels Oudenaarde, Sander Ploos van Amstel and Ruud Gruson from ortec for providing the opportunity to do the final project un-der their guidance. I am looking forward to continue working on interesting problems at ortec.

I am also grateful to the Department of Econometrics and Operations Research at the University of Groningen – in particular Wim Klein Haneveld, the supervisor of this final project, and Marloes Cremers, the second reader. The proofreading by Hanke ter Hoeven has also led to various improvements.

Finally, I would like to thank my parents for always supporting me.

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Introduction

At ortec I was assigned the task to look into an inefficiency issue in the trip scheduling software shortrec, as reported by customers in the oil business. Below the problem will be sketched, and the general structure of this thesis is outlined.

In the Netherlands oil is transported from storage depots to gasoline stations by means of trucks. First, each truck is loaded at a depot. After that, deliveres at a number of gasoline stations take place. These two steps are repeated a number of times each day.

The depots at which a truck loads during a day are not predefined. To the contrary: each time a truck must be loaded, a different depot can be used. For each truck, the specific depots it will load at are determined based on the driving time and costs associated with each possible depot. The “choice of depots” is the problem to this thesis.

The truck schedules are determined by the planning departments of oil companies, often one day ahead. shortrec is used by some oil companies as a tool that supports in the creation of schedules.

In general, the trip schedules proposed by shortrec are considered to be of high quality. However, in certain cases the schedule calculated by shortrec was found to be somewhat inefficient, in that it was possible to improve it (with respect to cost) by making a few manual changes.

After analyzing the problem, we found that when it comes to chosing depots for a truck to load at, shortrec was not very clever. By analyz-ing the current approach and possible improvements, we were able to have shortrec make better choices. This made the observed inefficiencies dis-appear, leading to a decrease in the cost of trips, which ultimately lead to measurably more efficient schedules.

The rest of this thesis is structured as follows:

In Chapter1, the logistics of the oil business are described, focusing on the transport of oil from depots to gasoline stations by means of truck. The explanation will discuss both the process (the logistics chain) and the aspects relevant from an operational point of view (costs, delivery schemes).

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first place the most important features of shortrec, and how they are typically used, are described.

After this explanation, we turn to the technical aspects of Shortrec. The algorithmic steps taken by shortrec when it is asked to calculate an efficient trip schedule will be described.

Finally, the research questions of the thesis are posed.

In Chapter3, the problem of chosing the depots at which a truck loads, when most other aspects of the truck’s trips are already established, is explained in detail. Particular attention is paid to various issues encountered in practice that complicate the formulation a mathemati-cal model of the choice problem. Viewed mathematimathemati-cally, it turns out that the problem of chosing the best depots can be approximated with a variant of the Knapsack Problem.

In Chapter 4, the solution approach formerly taken by shortrec is described, including an evaluation of its performance based on real-world data. Subsequently, improvements will be suggested. These new variants were implemented in shortrec, and their performance was also evaluated. Based on the results, a recommendation is made regarding the best approach to be taken by shortrec.

In Chapter 5, some background is given on Knapsack Problems. For readability, the chapters before this one will refer to section in this chapter instead of including the information in-line.

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Chapter 1

Truck Scheduling in the Oil

Business

Between the drilling of crude oil and the selling of it as an end-user product, the oil has traveled a long distance and undergone several transformations. This chapter describes that interesting journey, focusing on the chain that ends with the selling of oil at gasoline stations in the Netherlands.

First we will describe the logistics involved, starting with a brief expla-nation of the exploration and refinery of oil. The focus is on the distribution of oil from a refinery via storage depots to gasoline stations, using trucks. Then we will introduce the problem of scheduling vehicles and how the soft-ware product shortrec, developed by ortec, helps planners in solving this problem.

While describing the shortrec heuristic, the problem of choosing de-pots for vehicles to load at enters the stage. This is a subproblem that has to be solved many times in order to create a trip schedule. Solving this subproblem as well as possible is necessary in order to end up with an ef-ficient trip schedule. The problem of choosing depots, from here on called the Depot Assignment Problem, is the raison d’ˆetre of this thesis.

The chapter ends with the research questions regarding the Depot As-signment Problem, as formulated by ortec for this final project. These questions will guide the rest of this thesis.

1.1 Oil Distribution Chain

Crude oil, found in the outer layer of the Earth, is formed from decayed remains of prehistoric animals and plants.1 _{Under influence of pressure and}

heat these remains become liquid and accumulate in porous rocks. Even-tually an oil field is formed, located thousands of meters below the Earth’s

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crust and possibly hundreds of kilometers across. There are more than 40,000 known oil fields, the largest of which are located in the Middle East. North America, Russia and Antarctica also have large reserves. So-called oil rigs are the installations used to drill for and pump up crude oil from oil fields. They can be either land-based or offshore.

In order to be usable as fuel, the crude oil needs to be processed in an oil refinery. The largest refinery of Europe is located near Rotterdam; it is owned by Shell. Some other oil companies operating in the Netherlands are BP, Texaco, Q8 and Total. By means of boats, trains or pipelines, the refined oil is transported from the refinery to inland storage depots in the Netherlands and neighbouring countries; this is called the primary transport. The transport from a depot to gasoline stations is done using trucks on normal roads; this is called the secundary transport. The depots are acting as regional distribution centers. In the Netherlands, each oil company uses about ten of these depots. In§1.2the role of depots in secundary transport is explained in more detail. Finally, There are a few thousand gasoline stations in the Netherlands. The focus of this thesis is on the secundary transport.

Later on we will discuss trip schedules for a complete truck fleet. For now, we just mention that the quality of schedules is based partly on their total cost. The total cost is made up of the costs of each individual truck. For that reason, we will now already mention various types of costs involved in the secundary transport.

1.2 Depots

In the Netherlands, storage depots are used as hubs located in the distrib-ution chain in between the refinery and gasoline stations. Their locations are spread over the country in order to allow efficient distribution to all customers.

Most depots are owned by a single oil company, though some are partly owned by several companies, while still others are owned by neutral com-panies that handle the storage of oil but not its drilling, refining or further distribution.

In principle, an oil company uses its own depots as the points from which secundary transport takes place. However, it is possible to use other depots, too, when an exchange contract has been established. Such a contract gov-erns the quantities of the different types of oil available at a depot for use by the other party, and the price that will be charged for its use. Often it is valid for a period of one year. The contract could take the form of company

A getting the right to load a certain quantity at a depot owned by B, and

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1.3. GASOLINE STATIONS 3 where the other party is underrepresented. Alternatively, the use of depots could be traded against a sum of money. The price to pay would consist of the costs of the oil itself, plus a premium.

For oil companies, there are costs associated with the transport of oil to, and storage of oil at depots, and also the transport of oil from depots to gasoline station. Some of these costs are the costs of extracting the oil, processing it in a refinery, and the primary transport to the depot; the costs incured by storage and handling of the oil at the depot; and finally the costs of secundary transport.

For different depots, the associated costs may differ. For that reason, the locations of depots (which is a strategic, i.e. long-term decision) and the decision on which depots are used by which trucks on any day (a tactical, i.e. short-term decision) are important aspects for all oil companies.

The depot near the Shell refinery in Rotterdam has very low costs for just the loading, but the costs of the secundary transport must be added to it to allow a fair comparison of the costs of different depots. Then it turns out, that for many location in the Netherlands the depot with lowest total cost is not the one in Rotterdam.

A depot can have associated time windows, indicating during which hours vehicles can load at it. However, most depots are open all day.

1.3 Gasoline Stations

Gasoline stations located along roadways form the “points of sale” of fuel. Most gasoline stations are owned by one of the large oil companies. Some act independently, although still relying on an oil company for supply.

Gasoline stations can be seen as the customers for the products offered by the depots. Each gasoline station stores its supply in large underground tanks. As an example, the sizes of the tanks – called tank layout – of one of the busiest and largest stations of the Benelux is shown in Table1.1. The

max stock and safety stock denote “control quantities” used for planning

deliveries, as described below. As this gasoline station is located near a major highway, many of the customers are trucks. Therefore, six of the eight tanks available contain diesel as product.

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tank product capacity max stock safety stock avg demand 1 plus 30,000 28,500 1,451 3,629 2 euro 30,000 28,500 3,052 7,631 3 diesel 30,000 28,500 3,146 4 diesel 30,000 28,500 3,146 5 diesel 40,000 38,000 17,259 135,672 6 diesel 40,000 38,000 17,259 7 diesel 50,000 47,500 18,722 8 diesel 50,000 47,500 18,722

Table 1.1: Sample tank layout at a large gasoline station. Volumes are in liters, demand in liters per day.

overfilled and no product will be left onboard the truck.

The safety stock denotes the minimum volume that should be in the tank at all times. It is used as regular stock when demand is higher than expected or when the delivery of an order has a small delay. This should prevent situations in which customers can not be serviced. In this example, the safety stock is about one half of the daily average demand.

Because the pump connected to the tank is located slightly above the bottom, some amount of liquid at the bottom is not physically reachable. This is called the bottom stock.

The six diesel tanks are coupled into three pairs; this arrangement follows from the way in which the tanks are located in the ground. The rate of demand within a pair is equal, as the tanks are physically connected, but it differs between pairs. This pairing is called siphoning.

The total demand per day is on average 147,000 liters As the delivery trucks have a capacity of around 40,000 liters (see §1.4), this station will receive supplies nearly 4 times each day.

tank product capacity max stock safety stock avg demand

1 euro 10,000 9,500 176 176

2 plus 30,000 28,500 917 917

3 diesel 20,000 19,500 728 728

Table 1.2: Sample tank layout at a small gasoline station.

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1.4. TRUCKS 5

1.4 Trucks

The trucks used to transport oil from depots to gasoline stations have differ-ent capacities, but most common in the Benelux are trucks with a capacity of around 40,000 liters, give or take a few thousand.

To allow one truck to transport different product types at the same time, the capacity is divided among several compartments of different sizes. The number of compartments and their sizes differ per truck, but for any truck the lay-out is fixed. In a typical arrangement, the total capacity of 40,000 liters is spread over 5 compartments, with sizes of 5, 7, 7, 9, and 12 thousand liters.

There are restrictions on the total weight of the load and on the way the weight can be divided among the axes of the trailer. Also, there can be restrictions on the use of a compartment. For example, certain compart-ments must be filled either below 25 % or above 75 % during transport. The reason is, that a half-filled compartment poses danger as the load could start shaking.

It is usual in the Netherlands, though less common in Belgium and Lux-embourg, that on any day a truck is shared by two drivers. In that case, after the first driver has carried out his shift, the truck is driven to a depot that forms the end location of his trip. That depot forms the starting point of the shift of the second driver. If a truck is shared this way, usually there are two shifts on a day, each having a duration of about 8 hours.

Associated with the deployment of trucks and drivers are fixed and vari-able costs; the latter vary with distance or time. The exact numbers will vary per country, driver age, truck type, working hours, et cetera, but to give an impression: the labor costs for one driver can be about _{¿ 45,000} per year, and the costs of a truck about ¿ 50,000. The total costs for a driver–truck combination are then around¿ 95,000 per year.

1.5 Orders

An order, consisting of gasoline types and their requested amounts, precedes the actual delivery. A sample order for a gasoline station is shown in Ta-ble1.3.

product volume

euro 10,000

plus 4,000

diesel 19,000

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The orders for a gasoline station are composed either manually by the management of the station, or generated automatically. In the latter case, the supply tanks located at the station are equipped with gauges measuring the stock level. The measurements are sent to the oil company, which in turn determines the moment at which an order is placed and the corresponding order volumes. This is called Vendor Managed Inventory; it is more efficient as it automates manual work.

Each order has an associated time window that specifies during which hours the delivery should take place, or more exact, the moment it should

start. The length of this interval varies. Sometimes the moment of delivery is

unrestricted, which in practice means that it may happen anytime “between 6 am and 12 pm”. In other cases, especially at busy stations, the interval has a duration of just a few hours, for example “between 11 am and 1 pm”. Unloading an order takes time, consisting of a fixed and variable part. The fixed part consists of setting up equipment, while the variable part de-pends on the volume to unload (the quantity of the order) and the unloading speed. As an example, the delivery of a large order containing 36,000 liters (separated into different products) takes 15 minutes of fixed time. With an unload rate of 1,000 liters per minute, the total duration is 51 minutes. This duration the unloading takes depends on both the gasoline station and the truck used.

1.6 Frequency of Delivery

There is some variation in the rate at which a gasoline station gets deliveries. On average, a station is visited once every two or three days. The busiest stations, located near highways, get up to four deliveries each day, while the quietest stations are supplied about once every three weeks.

Gasoline stations always sell various products. It is most efficient to supply a gasoline station regularly with one fully-loaded truck (a

full-truck-load). This truck then carries of each product the quantity that is likely

to be sold in the time interval between this delivery and the next. This schedule minimizes the number of truck visits that have to be made in the long term.

ortec has developed an order management system, orion, that is used by oil companies to achieve an efficient delivery schedule consisting of mostly

full-truck-load orders. Customers have found that the delivery sizes and

dates calculated by this system lead to significantly lower average stock levels at the gasoline stations that they obtained by manual planning.

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1.7. VEHICLES AND TRIPS 7

1.7 Vehicles and Trips

The actual transport is carried out by trucks, or more correctly, vehicles. A vehicle is hereby defined to be a combination of a truck, a trailer and a driver. A vehicle carries out one or more trips in sequence. In each trip, the truck is loaded and orders are delivered.

In Figure 1.1, an example vehicle trip is shown. The horizontal position indicates the point in time; it is not a geographical location. The arrows indicate the movement of the vehicle.

start location start depot order 1 order 2 end depot end location

Figure 1.1: A vehicle trip.

This vehicle starts the trip at a start location. This is for example the home address of the truck driver, if the vehicle was stalled near his house during the night, or it could be a depot location. The vehicle is driven from the start location to the start depot, where it is loaded, and then two orders are delivered to customers.

After delivering the second order, the next stop is the end depot, where the vehicle could be preloaded for the next day, and finally the vehicle is driven to and stalled at the end location. It could be that the end location is the location of the end depot. The end location is the start location for the next trip of this vehicle. Sometimes the end location is the same as the original start location.

Note that a start depot is a depot that happens to be used as the first depot visited in a trip. It is not a type of depot: every depot can be a start

depot. The same is true for end depots and the reload depots introduced

below. The stops at which a start, reload or end depot is visited are called

depot stops.

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start location start depot order 1 reload depot order 2 end depot

Figure 1.2: A longer trip made up of two subtrips. The reload depot marks the end of the first and the start of the second subtrip.

1.8 Vehicle Scheduling

Scheduling the delivery of orders consists of determining which of the avail-able vehicles will be used, and the trips each vehicle makes.

The schedule is normally made by a planning department one day in advance. During the execution of a schedule, the planners may be notified of additional orders that have to be delivered the same day. These could be orders from small gasoline stations that place orders manually instead of being controlled by a Vendor Managed Inventory system; or it could be that demand at a gasoline station is much higher than expected so a shortage may arise. These extra orders have to be inserted in the running schedule, perhaps by moving some non-urgent orders scheduled for that day to the next day.

Often, it is not possible to include all orders in a schedule. This is caused by restrictions on drivers, trucks and deliveries. For example, drivers have a restriction on the total work time; trucks have limited capacity; and each order has a time window that specifies during which hours the delivery may take place.

When that happens, and some of the unplanned orders really need to be delivered, oil companies make use of external companies for delivering those orders. Those are neutral transport companies equiped with trucks without company logos – so-called white trucks. Each such company has several oil companies as customer. Invoking an external transport company is fairly expensive: the costs are 50 % to 100 % higher than when the trip would be executed by the company itself. Use of external companies is thus unattractive.

Measuring the quality of a schedule is not trivial. For example, take as potential criterium: the total number of trips the trucks make, according to the planning. Given the choice between a trip in which two customers are visited, and another trip in which five customers are visited, which one should be considered better? It is not possible to answer this question, because there are conflicting aspects: if both trips require an equal amount of worktime, then in the first trip two gasoline stations located relatively far away might get visited, and the truck might deliver relatively large orders. Yet in the second trip, several nearby stations could be visited, with each one receiving a relatively small delivery. Which of these two is most valuable?

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es-1.8. VEHICLE SCHEDULING 9 sential performance measures: (1) the total volume delivered according to the schedule; and (2) the costs per volume.

1. The total volume delivered

As many orders should be planned into the own truck fleet as possible (also in order minimize the use of the “wite trucks”). But as the orders have different sizes, the real measure is the total volume of the orders delivered: the higher the total volume delivered, the higher the quality of the schedule.

2. The costs per volume

The second quality criterium is the costs of executing the schedule. The costs consist of variable and fixed costs of the driver, truck and trailer. In order to compare costs, the quotient of costs and total delivered volume – the costs per volume – is often used as quality metric.

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Chapter 2

Planning with shortrec

The Dutch company ortec offers consultancy and software solutions for solving financial and logistical problems. One of the logistics products of-fered is shortrec, used for planning and optimisation of vehicle trip sched-ules. Since the start of its development around 1985, it has been used by customers in a variety of industries ranging from supermarket supply to prisoner transport.

Some other companies selling vehicle scheduling software in the Nether-lands are PTV-Ordis (offering TLNplanner) and Intris (offering Trip

Plan-ner).

2.1 Usage of shortrec

A screenshot of shortrec is shown in Figure 2.1. On the left a list of vehicles is shown, while the orders to be scheduled are shown at the right. The most important aspects of both vehicles and orders are shown:

For the vehicles, the columns show among others the truck number, the driver’s name, the number of orders that are scheduled in the truck currently, the total volume of the orders, the time needed to carry out the trip, and the costs per volume.

For orders, these are the order number, client name and city, ordered volume, the moment the delivery takes place (as calculated by short-rec).

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Figure 2.1: Screenshot of shortrec.

Typically, the process of scheduling starts with importing vehicle and order data into shortrec. Initially, none of the orders are assigned to vehi-cles. Users can then manually assign orders to vehicles, or the optimization engine in shortrec can calculate a schedule automatically. Usually, plan-ners first let shortrec create a schedule. The planplan-ners will then modify the order assignments they find undesirable.

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2.2. THREE-STEP HEURISTIC 13

2.2 Three-Step Heuristic

The shortrec approach to tackle the trip scheduling problem aims at find-ing a feasible schedule (i.e. a schedule in which all constraints are satisfied) which includes as many orders as possible with minimum total costs. It is a heuristic, which means that it does not calculate a schedule that is ‘op-timal’ in any sense. On the other hand, the resulting schedule is usually a reasonable solution, acceptible for the user.

The heuristic consists of three steps. The vehicle scheduling software offered by other companies probably follow a similar recipe. Below, we will first describe the three steps in the heuristic at a high level. These steps are not very shortrec-specific: other vehicle scheduling software probably uses comparable techniques. This is followed by an explanation of the technical details of the steps: the exact method taken by the software in order to carry out each of the steps.

Then, in _§2.2.5 we will describe a technical issue that is specific to shortrec. It is about the relation between the choice of depots in a trip on the one hand, and the feasibility check for a collection of orders assigned to a truck on the other hand. The reason for bringing this issue to attention, is that is sets the context within which shortrec solves the problem of chosing depots.

The consecutive steps are:

1. Find initial solution – Starting from an empty schedule, an initial feasible schedule is calculated. The goal of this step is to quickly plan a large number of orders in a fairly straightforward yet reasonably efficient way. This is done using an insertion heuristic that schedules vehicles one by one. The resulting schedule is called the initial solution. 2. Improve cost of solution – The initial solution is improved with respect to costs by applying a sequence of improvement methods to it. In the schedule resulting from this second step, the set of scheduled orders is the same as the set of orders scheduled in the initial solution but the orders are planned in a different way in order to end up with lower total costs.

The aim of this step: reduction of the total costs of the schedule. 3. Include additional orders – After applying the improvement

meth-ods (previous step), the distribution of the time orders over the vehicle has changed. If some orders that were initially assigned to some truck are now assigned to other trucks, truck capacity in the initial truck has become unused.

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that were up to now not included in the schedule. This is done using a so-called insertion heuristic.

Return to step 2 (so steps 2 and 3 will be repeated).

The aim of this step: increasing the number of orders included in the schedule, irrespective of the corresponding increase in cost.

After steps 1 and 2 have been executed, the pair of steps 3 and 2 can be repeated ad infinitum. However, after a certain point, executing these two steps will not improve the schedule any more, because there is neither room for additional orders (step 3) nor could there be found a way to decrease the costs of the schedule (step 2).

The sequence of steps usually taken is: 1, 2, 3, 2, 3, 2. In practice the heuristic is often stopped after executing step 2 for the third time, as it has been found that the quality of the schedule hardly improves by executing more iterations. The resulting schedule is called the final solution.

Sometimes the planning department makes minor changes to the final solution before it is accepted. These changes are often given in by aspects not included in the data based on which shortrec calculates the schedule. For example, drivers may have certain wishes regarding stop location.1 2.2.1 Creation of Trips for Vehicles

When shortrec calculates a schedule, basic operations are the assignment of an order to a vehicle, and the removal of an order from a vehicle. Both the Initial Solution and the Improvement Methods are implemented as a sequence of basic operations. After each basic operation, the feasibility of the resulting vehicle trip must be checked. Based on the outcome of this feasibility check, the order is either left in the vehicle (‘accepted’) or removed from it (‘rejected’).

To make matters confusing, the feasibility check in shortrec has much more functionality than just checking whether restrictions are violated. Con-cretely, the feasibility check:

determines the arrival and departure time at every stop in the trips of the vehicle;

chooses the depots that are used to load and reload at (out of the list of available depots).

1_{Also, some technical shortcomings in the shortrec heuristic can be overcome by}

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2.2. THREE-STEP HEURISTIC 15 The feasibility check has the “responsibility” to come up with outcomes for both problems (time calculations and depot choice) that don’t violate any restrictions (on driver work time, order time windows, or depot opening hours). Conceptually, time calculations and depots choice form a “result” of assigning orders to a vehicle.

We will come back to this issue in _§2.2.5. For now, it is important to realize that because of this concept, the heuristic can be defined just in terms of the basic operations (inserting orders in, and removing orders from, the trip of a vehicle).

2.2.2 Idea Behind the Initial Solution

The idea behind the trip construction method used to calculate the initial solution is the observation that the orders “farthest away”2_{from the vehicle’s}

start location are in general the most difficult to plan, because the required driving time takes up a relatively large part of the available time of a vehicle. Therefore, the first step in scheduling a new trip for a vehicle consists of selecting a far-away order. The trip is then extended by including orders in the neighbourhood of this first order as long as the total quantities of the orders fit the vehicle capacity, and as long as the vehicle does not exceed its maximum work time.

We will now describe technical details regarding the construction of the initial solution. This is intended as background information about the inner workings of shortrec.

2.2.3 Implementation of the Initial Solution

Both the vehicles and the orders are first grouped geographically. Orders will be assigned to vehicles that have a start and end location in the same geographical area. This grouping is an implementation concept as there is no such requirement in the problem formulation. It simply makes sense to let vehicles handle orders that are in their neighbourhood and it saves cal-culation time. Then, per geographical area, vehicles in the area are assigned to trips.

Starting with the unscheduled vehicle with the largest capacity, the un-scheduled order farthest away from the vehicle, but belonging to the same geographical area, is determined. This will be the first order in the first trip of the vehicle. Next, orders in the neighbourhood of the first order are inserted in this first trip, until no more orders can be added to it because the capacity of the truck would be exceeded, or because the maximum driving time would be exceeded.

2_{An order is said to be “far away”, “in the neighbourhood”, etc, when the location at}

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After scheduling a trip this way, if time allows it, an additional trip is started for the same vehicle. When this happens, the end depot of the previous trip is also the start location of this new trip; it has become a

reload depot as described in _§1.7. This new trip is initialized by inserting the unplanned order farthest away from the depot. Again, additional orders are included in the trip until no more orders can be added.

After a trip have been assigned to a vehicle v1in this way, there can be an

unscheduled vehicle v2 that has a capacity smaller than that of the current

vehicle, but large enough to carry out the trip scheduled for v₁. (That is, the truck capacity of v2 is sufficiently large to store the amounts loaded at

the start and reload depots). If so, then the trip will be removed from v1and

assigned to v₂, so as to use the capacity more efficiently. Vehicle v₁, now removed from the schedule, will once again act as the unscheduled vehicle with the largest capacity.

After scheduling the first vehicle, the unscheduled vehicle with the largest capacity is determined. A trip is created in the same way, and so on until all vehicles are scheduled or all orders are included in a trip. Not all vehicles have to be planned: every unused vehicle implies cost savings. When there are remaining orders after scheduling trips for all vehicles, the geographical boundaries are removed and each of the remaining orders is assigned to any vehicle that can include the order in a trip. When there are no vehicles for an order, transport by a white truck must be arranged, as described in§1.8.

2.2.4 Improvement Methods

There are several improvement methods defined. Each method starts from a schedule; then by “shuffling the orders around” in a method-specific manner a new schedule is obtained in which the same orders are scheduled at lower costs.

The most important improvement methods are:3

 Exchange orders within trip

In a trip, the sequence of order deliveries is changed.  Exchange orders between trips

Orders are removed from one trip and inserted into another trip.  Exchange trips within vehicle

The sequence of trips of a vehicle is changed.  Choose cheapest vehicles

A vehicle is replaced by a cheaper vehicle.

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2.2. THREE-STEP HEURISTIC 17  Exchange trips between vehicles

A trip is removed4 from one vehicle and inserted into another vehicle, while a trip from the other vehicle is moved to the first.

 Exchange sequence of stops between trips

A sequence of orders (scheduled to be delivered immediately after each other) is removed from one trip and inserted into another trip.  Swap trips between vehicles

All trips are removed from a vehicle and inserted into another vehicle.  Reverse trip

The directon of a trip (and thus the sequence in which the orders in it are delivered) is reversed.

The effect of these improvement methods is typically a reduction of the total costs by 2 to 5 percent for oil truck schedules, which is financially significant.

2.2.5 Depot Choice and Trip Feasibility

To arrive at the Depot Assignment Problem, which is the central problem of this thesis, it is necessary to explain some more about the shortrec implementation. We already mentioned these issues briefly in§2.2.1.

 When creating trips, depots are not considered explicitly

The shortrec heuristic can be described in terms of two basic oper-ations: inserting an order into a vehicle, and removing an order out of a vehicle.

Concretely, in the description of the initial solution, it was stated that trips are created, initialized by inserting a “far-away” order, and that additional orders can be inserted into a trip. As for the improvement methods, we said that changes are made to orders and trips: orders are removed from trips and then inserted into other trips, trips are removed from vehicles and inserted into other vehicles.

Each trip must contain at least a start depot and end depot (and pos-sibly reload depots), yet choosing depots is not a basic operation.  Feasibility checks are executed all the time

Every time a trip is created or modified, the feasibility of the modified trip has to be determined. More correctly, the feasibility check is applied to the vehicle, not to the single trip, as all trips scheduled for the vehicle have to be taken into account. If the resulting trip is 4_{A trip is said to be “removed from (inserted into) the vehicle” when the trip is removed}

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infeasible, for example because it violates some restriction on capacity or work time, the change under consideration must be rejected. Whether or not a trip violates a restriction partly depends on which depots are chosen. For example, the test whether the total order quantities in a trip exceed the vehicle capacity, is independ of the depots used. But an example restriction where depot choice plays a role, is the restriction on maximum work time: a depot located far away from the start location of the vehicle could be infeasible as start

depot while a nearer depot would be feasible.

If a feasible depot assignment does not exist, a change made at a higher level (for example the insertion of an order, as part of the creation of the initial solution) must be reversed.

These two observations are related, as in shortrec feasibility of trips and selection of depots are linked. The depots of a trip are chosen while the feasibility check is carried out. Initially, a trip is characterized only by the orders it contains, not by the depots used. The depots enter the stage only when feasibility is being checked.

Concretely, this means that when the feasibility check has not deter-mined a restriction violation in the aspects independent of depot choice, it proceeds by choosing depots that will be used as start, end and reload depot. This task will from now on be called the Depot Assignment Problem. The following issues play a role:

1. the feasibility of the resulting vehicle schedule

2. the total costs of the vehicle (but only if the vehicle schedule is feasible) The maximum number of feasible trips is obtained when the depot nearest to the start location and the location of the first order is chosen as start depot. In other words, the depot that minimizes the travel time of the trip

hstart location → depot location → first order locationi should be chosen.5

For the reload and end depots, in a similar way the nearest depot can be chosen.

The costs of the use of depots were described in_§1.2. When the assign-ment resulting from choosing the nearest depots is feasible, it might not be the cheapest feasible assignment, and therefore be a suboptimal assignment with respect to costs.

2.2.6 Effect of Depot Choice Heuristic on the Schedule

The problem of assigning depots to trips has to be solved thousands of times during the calculation of a schedule. As part of the calculation of a solution 5_{In this thesis, when ‘distance’ is used, ‘travel time’ is really meant. When a location}

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2.3. RESEARCH QUESTIONS 19 the basic operations (insertion of an order into a vehicle, or removal from an order from a vehicle) are executed thousands of times. Each time, feasibility is checked, and as part of that, depots are chosen.

The heuristic used for choosing depots has an effect on the schedule as a whole, because if the choice of depots is not performed properly, trips could be considered ‘infeasible’ despite the existence of a feasible depot assignment. This will prevent the scheduling heuristic from finding certain (possibly very high-quality) schedule solutions.

When the heuristic for choosing depots sometimes leads to depot choices that are feasible, but have higher cost than an alternative choice that is also feasible, then the resulting overall schedule might have higher total costs than necessary.

It is important to realize that even if the depot assignment is not very efficient, the final solution might be quite reasonable due to the large number of order-vehicle combinations that are tried in the process of calculating a schedule. Nevertheless, it should be clear that improvements to the depot assignment heuristic will only have positive effects on the resulting schedule.

2.3 Research Questions

There have been complaints from users, who were able to manually include one or more additional orders in a schedule generated by shortrec (by shuffling the orders around somewhat), while shortrec itself was unable to include more orders in the schedule.

It was deemed necessary to analyze, and then hopefully improve, the way in which shortrec chooses depots as part of the feasibility check. Therefore the following research questions have been posed:

1. To get insight in the problem of chosing depots in trips, how can the

problem be formalized?

2. What approach is currently taken by shortrec to solve the problem,

and what is the quality of the solutions?

3. What alternative solution approaches are worth looking into?

To answer the first question, a model for the problem of chosing depots will be given in Chapter 3. We will assess the realism and completeness of this new model, and also look at the relation between this problem and standard problems described in the literature.

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a conclusion will be drawn regarding the best heuristic to use. This answers the second and third question.

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Chapter 3

The Depot Assignment

Problem

The heuristic used by shortrec to solve the vehicle scheduling problem was outlined in the previous chapter: the problem is tackled by first constructing an initial feasible schedule, and then improving this solution stepwisely until the final schedule is reached.

In the description, it was explained how the Depot Assignment Problem shows up many times during the calculation of a solution. This problem concerns the choice of depots at which a vehicle will load the fuel that is subsequently delivered to gasoline stations.

The problem will now be formulated and discussed in depth. First the core aspects of the problem will be discussed; these are the aspects that are included in the model that will be constructed subsequently. It will be shown that the Depot Assignment Problem shares similarity with the Knapsack Problem. We then evaluate the validity of the model by discussing various complicating issues that play a role in practice.

3.1 Core Problem Aspects

It is assumed that there is a vehicle1 _{for which a trip is scheduled. This trip}

consists of a start depot where the vehicle is initially loaded; one or more order locations that are visited; zero or more reload depots at which the vehicle reloads in-between the delivery of orders; and finally a end depot, as depicted in Figure1.2.

The trip already contains the depot stops, but the actual depot visited at these depot stops has not been determined yet. As described in_§1.2, the costs involved in loading at a depot have variable and fixed components.

After a depot is chosen for each vacant depot position, the resulting trip 1_{A vehicle is a combination of a truck, a trailer and a driver, as defined in}_§_1.7_.

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must be feasible. As described in_§2.2.5, an important aspect is the limit on total work time. In practice there are many more constraints in addition to the limit on working time. Although these make the problem more complex, the core of the problem remains the same. These extra constraints and their complexities will be discussed in§3.5.

As was already stated at the end of §2.2.5, the objective is that given the vehicle, the depots should be chosen in such a way that:

1. the resulting trip scheduled for the vehicle is feasible; 2. the total costs of the feasible trip are minimal.

Whether or not a feasible depot assignment exists is not known a priori.

3.2 Model

The context of the problem is, that a trip schedule is given. As the focus is on one vehicle in this schedule, there are no variables denoting vehicles. The orders to be delivered in the trip are fixed. The depots at which the vehicle loads still have to be chosen.

The problem of assigning depots is formulated in mathematical terms in Table3.1.2 _{We will show in} _§_3.5_{that various issues are not represented}

very well in this model, which makes this model too simple to be realistic for the problem of assigning depots to trips in practice. This model is intended only as start point for our analysis.

The parameters, decision variables, objective and constraints in the model will now be discussed.

Parameters

Let ns be the number of depot stops in the trip. A depot stop is a stop at which a depot is visited in order to load the truck, as defined in§1.7. Each depot visited is used as either a start depot, reload depot or end depot. A trip always contains at least two depot stops (a start depot and an end depot) therefore ns≥ 2.

Let there be n_d available depots. It is assumed that the set of depots is the same for every depot stop s. (In §3.5.1we will show that this is an assumption that can be made without loss of generality.) As the degenerate case is not interesting, it is assumed that there are at least 2 depots: n_d≥ 2.

Let T be the total time available for driving to depots, loading at them, and driving to the next stop after the depots; T > 0. This is the maximum

2_{As for notation, when a set contains n}

iitems, the items i are numbered from 1 to ni.

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3.2. MODEL 23

indices:

s depot stops s = 1 . . . ns

d depots d = 1 . . . n_d

decision variables:

a_sd a_sd = 1 ⇒ assignment of depot d to stop s a_sd∈ {0, 1} parameters:

c_sd costs of assigning depot d to stop s c_sd> 0 tsd driving time for depot d at depot stop s tsd> 0

T total available driving time T > 0

objective: minasd P s P d csd asd constraints:_P d asd = 1 ∀s P s P d tsd asd ≤ T

Table 3.1: An incomplete model for the problem of assigning depots to a trips.

work time of the driver minus the time spent on delivery at gasoline stations. As we focus here on the stops in the trip of one vehicle, T is a known constant.

For each possible stop-depot combination there are associated costs csd, that are incurred if depot d is chosen for stop s. (For example, c3,6 = 14

means that the costs of assigning depot 6 to stop 3 are 14.) Costs are assumed to be positive: csd> 0.

Each stop-depot combination has corresponding duration tsd, denoting the total time needed between departing from the last customer before stop

s and arriving at the first customer after stop s, where the truck is loaded at

depot d in the meantime. It includes the loading time at the depot d, which can differ between depots. These durations are assumed to be positive:

tsd> 0.

Decision variables

An assignment of depots to stops can be thought of as a mapping from the set of stops to the set of depots. For example, if there are 4 order stops (ns= 4) and 10 depots (nd= 10), the mapping {(1, 4); (2, 5); (3, 9); (4, 2)} corresponds to taking depot 4 as the start depot, depot 5 as the first reload

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Such a mapping is represented using binary variables: let a_sd denote the assignment of a depot to a stop: asd = 1 if depot d is chosen for stop s, and

asd = 0 otherwise. The example mapping from above is then equivalent to:

a1,4 = a2,5= a3,5 = a4,2= 1, and asd = 0 elsewhere.

Objective

The objective is minimizing the total depot-related costs (which is the sum of the travel costs for truck and driver, and the loading costs), which is the sum of the costs of each individual depot stop.

Constraints

The minimizing of costs is subject to two contraints. In the first place, for every depot stop exactly one depot has to be chosen. In the second place, the total driving time should not exceed the time available for driving to the chosen depots, loading at the depots, and driving from each depots to the next stop.

3.3 Example Problem

An instance of the Depot Assignment Problem is shown in Figure3.1. The trip contains five depot stops and four order stops. For each of the first four depot stops, one depot must be chosen from the row of available depots.

For each chosable depot, the first number (in brackets) identifies the depot. It is followed by the required travel and loading time (the time needed to driver from the previous stop to this depot, load at this depot, and drive from this depot to the next stop; in hours and minutes) and the transport and product costs (scaled).3 In each row, the depots are sorted by increasing travel time.

We have chosen to display a trip in which a feasible assignment exists: when for each depot stop the depot with lowest travel and load time is chosen, the resulting assignment is feasible. Only the “feasible alternative depots” are shown for each depot stop, in the sense that when chosing a cer-tain depot for a cercer-tain depot stop always leads to an infeasible assignment, the depot is not displayed as possible depot for that depot stop. (Depot d is a “feasible alternative depot” for depot stop s, if choosing d for s, while chosing for all other depot stops the depot with lowest travel and load time, forms a feasible assignment.)

3_{The costs and travel and load time corresponding with chosing a depot for a certain}

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3.3. EXAMPLE PROBLEM 25 From the figure, we can derive that the total time available for driving and loading for the first four depot stops must be at least 6:43 hours (namely: 1:59 + 1:03 + 0:43 + 2:57). In the model from§3.2the (known) total time available for driving and loading is denoted by T ; we thus know that T must be at least 6:43.

The figure shows that depot (9) is a possible start depot, but the depot is not feasible as first reload depot. Possible explanations for this are:

 Depot (9) is located too far away from the location of order 2;  For order 2 a product must be loaded that is not available at depot

(9).

There is only one depot that can be chosen as end depot. This lack of choice can occur for several reasons:

The location of the last order and/or the end location are relatively remote, so that the other depots would require too much driving time; There can be multiple shift of a vehicle per day. When this trip is the first shift, the start location of the a second shift can be predefined to be this end depot;

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start location (12) 2:08 $ 231 (9) 1:59 $ 289 (s=1) order 1 feasible start depots:

(4) 3:07 $ 168 (11) 2:59 $ 150 (10) 1:03 $ 173 (s=2) order 2 feasible reload depots: (0) 3:03 $ 173 (4) 2:44 $ 216 (11) 2:38 $ 187 (13) 1:39 $ 163 (10) 0:43 $ 218 (s=3) order 3 feasible reload depots: (4) 4:30 $ 238 (11) 4:23 $ 216 (12) 4:06 $ 302 (10) 2:57 $ 314 (s=4) order 4 feasible reload depots: end depot 0:23 $ 183 (s=5) end location One feasible end depot:

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3.4. RELATION TO WELL-KNOWN PROBLEMS 27

3.4 Relation to Well-Known Problems

The Depot Assignment Problem as formulated in§3.2shares similarity with two problems often encountered in the literature: the Assignment Problem and the Knapsack Problem. A comparison with these problems will now be given.

3.4.1 Assignment Problem

The Depot Assignment Problem is somewhat similar to the classical Assign-ment Problem, which is formulated below.

Assignment Problem: There are n_j jobs and n_mmachines, n_m ≥ n_j > 0.

Job j can be executed by machine m at cost cjm> 0.

The problem is to find the assignment of jobs to machines that mini-mizes total cost, such that every job is assigned to one machine, but every machine handles at most one job:

min {ajm} X j X m cjmajm subject to X j ajm ≤ 1 ∀m X m a_jm = 1 ∀j ajm∈ {0, 1} ∀j ∀m £ The following is an attempt to map the Depot Assignment Problem onto the Assignment Problem: let job j stand for depot stops and machines m for a depot. Then:

 Assigning a job to a machine, ajm= 1, corresponds to choosing depot

m for depot stop j;

 The costs cjm represent the costs of using that depot for the stop. However, this mapping falls short for the following reasons:

The same depot (machine) may be chosen for several stops (jobs); Time aspects (the time needed for executing a job on a certain

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3.4.2 Knapsack Problem

The Binary Knapsack Problem is defined as follows:

Binary Knapsack Problem: There is a knapsack, whose weight capacity

c > 0 is given. There is a set of ni items, where item i has value vi> 0 and weight wi> 0.

The problem is, to find the selection of items that fits and maximizes the total value:

max {xi} X i vixi subject to X i wixi ≤ c with xi ∈ {0, 1} £

The mapping from Depot Assignment Problem to Knapsack Problem can be made as follows:

Each item represents a depot–stop combination;

 An item’s value is the negation of its costs (vi= −c_ds) and the item’s weight denotes the required driving time (w_i = t_ds) for some depot– stop combination;

The capacity of the knapsack denotes the total time available for driving (c = T );

Maximizing the total value of the items then means minimizing the total depot costs;

Driving times and costs are assumed to be constant and known, which is also the case for the values and weights of items in the Knapsack Problem.

However, a problematic aspect in the mapping is that the Depot As-signment Problem has a notion of “groups”: for each depot stop, exactly one depot must be chosen from the group of available depots. In terms of knapsack items, this means that after chosing an item that corresponds to assigning some depot to some stop, none of the items that correspond to assigning another depot to that same stop may be chosen.

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3.5. PRACTICAL COMPLEXITIES 29 Multiple-Choice Knapsack Problem:

There is a knapsack with capacity c > 0. There are ng ≥ 1 groups, with each group containing ni ≥ 1 items. (For simplicity, the number of items is assumed to be the same in each group.) Item i of group g has value vgi≥ 0 and weight wgi≥ 0.

The problem is to find the selection of items, consisting of exactly one item from each group, that fits and maximizes the total value:

max {xgi} X i X g vgixgi subject to X i xgi= 1 ∀g X i X g w_gix_gi≤ c with xgi∈ {0, 1} £

In our case, a group of items g denotes the set of depots that can be chosen for a certain depot stop. Selecting exactly one item from each group corresponds to assigning exactly one depot to each depot stop.

In short, the Depot Assignment Problem as formulated in §3.2 can be regarded as a Multiple-Choice Knapsack Problem.

A treatment of the Multiple-Choice Knapsack Problem, including heuris-tics and algorithms to solve it (mc kp algorithms for short), are given in §5.2.

3.5 Practical Complexities

Compared to the problem as it occurs in practice, the model of the Depot Assignment Problem introduced in _§3.2 is lacking in some aspects. The relevant practical issues will now be discussed.

It will turn out, that some these issues do not influence the model, as the model has abstracted away from it. Other aspects can be taken into account by applying a suitable transformation to the problem parameters. Finally, some practical issues do have implications for the applicability of the mc kp algorithms, in that they can not be taken into account given the current problem formulation. It will be explicitly mentioned when this is the case.

3.5.1 Depot Infeasibilities

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When a depot is an infeasible choice for all order stops in the trip, then it can simply be removed from the set of depots. If it is an infeasible choice for only some of the order stops, this can be catched in the existing problem formulation by setting the corresponding t_sd to a sufficiently large value (for example: tsd = T +1) to ensure the depot will never be chosen.

3.5.2 Flexible Order Volumes

In§1.5it was stated, that an order contains the exact volume of each product demanded. For example, the order in Table 1.3 demands 10,000 liters of euro.

In practice, however, it is usually acceptable to deviate slightly from the fixed volume, as there is always a margin between the amount that is to be delivered and the amount that can be stored (allowing more volume), and also a margin between the strictly minimum needed volume and the calculated order volume (allowing less volume).

For example, one oil company allows the delivered volume to be up to 5 % above or 10 % below the given order volume. With those margins, the sample order with fixed quantities from Table1.3would look as in Table3.2.

product min volume max volume (old fixed volume)

euro 9,000 10,500 (10,000)

plus 3,600 4,200 (4,000)

diesel 17,100 19,950 (19,000)

Table 3.2: Sample order with minimum and maximum volumes.

When several orders with flexible volumes are scheduled together into one vehicle, it might be possible to completely fill the truck capacity by in-creasing or dein-creasing each of the volumes by a small amount. The flexibility can thus result in more efficient use of the available truck capacity.

Whether or not there is flexibility in the order volumes does not play a role in the Depot Assignment Problem, as the orders themselves do not play an active role in the problem (the volumes of orders influence the costs csd, but those costs are external to the model.)

3.5.3 Variable Load Rates at Depots

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3.5. PRACTICAL COMPLEXITIES 31 The variable loading costs can be included directly in c_sdand the variable loading times in tsd, this issue does not invalidate the model – not even in the presence of flexible order volumes (see §3.5.2).

3.5.4 Variable Unload Rates at Gasoline Stations

Similar to varying load rates at depots, the unload rates at gasoline stations are not the same everywhere. In most cases, the rate is 1,000 liters per minute. The duration of unloading depends on this rate and the exact volume delivered.

As the unloading durations are not directly part of the model (they are included in tsd), the model supports this aspect already.

3.5.5 Time Windows

Orders can have time windows associated with them, restricting the moment in time a delivery can take place. More exactly, an order time window denotes either (a) the interval in which the start of the delivery must take place, so the moment the delivery has finished may lie after the end of the time window; or (b) the interval in which the whole delivery must take place, so the moment the delivery has finished must also lie in the interval. Depots can have time windows, too, indicating at which points in time a truck can (start to) load at the depot. 4

When orders or depots have time windows, this introduces a new depen-dency between the depots chosen for different depot stops with respect to feasibility (in addition to the dependency caused by the restriction on total time spent on driving and loading at depots). This is shown by the following example.

Assume there are two depot stops, s1 and s2, and that choosing some

depot d_afor s₁and depot d_bfor s₂is a feasible assignment, but that choosing

dc for s1 would make db an infeasible choice for s2 because the time window

of depot db would then be violated (the arrival at db would be too late). Then the feasibility of d_b at s₂ depends on the choice made at s₁, in a way not caused by the restriction on total time.

Time spent waiting at a customer location before the actual delivery can take place, is called waiting time.

In the model, we have made a simplifying assumption, by stating that the only costs that depend on the choice of depots are the traveling and loading cost of the depot that were denoted by csd. In practice, there are variable costs of a vehicle that depend on the total time the vehicle is used. In particular, the costs of waiting time are usually the standard costs of the truck and driver per hour, which increase linearly with the time spent 4_{In shortrec, the exact meaning of time windows is determined by a configuration}

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waiting. And as the amount of waiting time at an order stop depends on the depots chosen for all depot stops visited before the order stop in the trip, the assumption that all costs corresponding to choosing a depot (csd) is fixed does not hold.

3.5.6 Truck Preloading

Each trip starts by the vehicle loading at the start depot, according to the description in §1.7. However, at some oil companies it is customary to preload the vehicle at the end of the day, so that the next day it can skip the start depot by directly driving to the first customer. This does not change the problem, except that depot stop 1 (s = 1) does not correspond to the stop at the start depot anymore, but now denotes the first reload depot.

A vehicle may have to preload for the next day. This happens at the end

depot. This aspect is already included in the model, because the preloading

time and costs can be included in csd and tsd.

3.5.7 Lunch

If the working time of a driver covers lunch time, usually some time should be reserved for to driver to have his lunch. A driver always has his lunch when he is at a stop (either an order stop or a depot stop).

If shortrec calculates a schedule and has to take drivers’ lunches into account, the first stop that lies in the lunch time interval is taken as the

lunch stop. As the lunch time interval is usually from 12 am to 2 pm, this

effectively means the first stop visited after 12 am determines the location and time for the driver’s lunch.

The duration of the lunch is usually fixed to half an hour. This is added to the regular time spent at the stop. For example, in case of a depot stop, the duration of the lunch is added to the time spent on loading at the depot. The model does not take the requirement for the drive to have lunch into account. (It could be taken into account somewhat, by reducing T with the duration of the lunch break. That does indeed lead to less work time. However, the moment the driver has his lunch is not controlled, and there are also interactions with Overtime, see_§3.5.8.)

3.5.8 Normal Work Time versus Overtime

Often a driver is paid a fixed wage for each hour worked. Yet when the working time is in excess of the standard number of hours, a higher overtime wage might have to be paid. This can not be incorporated in the model, because the costs csd are fixed (not depending on the total work time, which follows from the actual depot stops visited in the trip). Normally, the costs

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3.5. PRACTICAL COMPLEXITIES 33 This implies that the costs of choosing a depot for a certain depot stop are not independent of the depots chosen for the other depot stops, as will be explained below.

Assume a certain trip, for which choosing the nearest depots leads to a feasible trip. Now focus on one of the depot stops, s, holding fixed the depots chosen for the other depot stops (and ignoring time windows).

Let dn be the depot nearest to stop s. Assume dn is temporarily chosen for s, and that this choice leads to a total work time within the standard number of hours, i.e. no overtime.

Assume there exists a depot dc that would be cheaper to use than the nearest depot dn. Then one of the following is the case:

1. Choosing dc would exceed the standard number of hours. In other words, choosing dc leads to overtime while dn does not.

2. Choosing dc for s does not lead to overtime hours, as the standard number of hours for the driver is higher than in the previous case. Because overtime hours are more expensive than regular working hours, the decrease in costs following from choosing dc instead of dn in the first case is less than the decrease in the second case. Actually, in the first case, there could even be a cost increase.

This implies that the decrease in costs following from changing a depot for one depot stop s depends on what depots are chosen for the other depot stops, as that determines whether (more) overtime will occur.

This dependency violates the assumption that the travel times and costs for each depot-stop combination (t_sd and c_sd) do not depend on the depots chosen for the other stops.

Therefore, the model does not support the notion of more expensive overtime hours.

3.5.9 Varying Driving Times and Rush Hours

Certain trucks can have lower average speed than others. For example, this could be caused by different weights of the trucks, or the use of different engines. In the model, the deviating average driving speed can be directly included in the travel time of the truck, as all travel times are given with respect to one given vehicle.

On the other hand, as traffic is more dense at certain times (“rush hours”) of the day than at others, a constant travel time tsd (for a given vehicle) is not always realistic. Therefore, in shortrec, for a given road different travel times can be specified for different hours of the day.

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This dependency between the depots can not easily be included in the model.

3.5.10 Summary

Of the practical complexities discussed here, the following turn out to be irrelevant for the modeling of the Depot Assignment Problem as

Multiple-Choice Knapsack Problem:

Depot infeasibilities Flexible order volumes Variabel load rates at depots

Variable unload rates at gasoline stations Truck preloading

However, the following issues violate assumptions made in the MC

Knap-sack Problem:

Lunch

Time windows of depots Time windows of orders

Varying driving times and rush hours Normal work time versus overtime

3.6 Conclusion

A model for the Depot Assignment Problem was introduced. It was shown that the problem can be regarded as a variant of the Knapsack Problem.

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Chapter 4

Solution Approaches

In the previous chapter, the Depot Assignment Problem was formulated and described. Subsequently it was evaluated with respect to complication issues playing a role in practice. We now turn to approaches that can be used to solve the problem.

The two heuristics currently used in shortrec are described first, and the quality of their solutions will be evaluated. Based on this evaluation, we will propose three new heuristics derived from the existing two. In total, we then have five heuristics.

To measure the quality of these five heuristics, a set of benchmark scheduling problems is introduced, and quality criteria for schedules are defined. After analyzing the schedules calculated by each of the heuristics, conclusions are drawn regarding the quality of the different heuristics. We end the chapter with a recommendation regarding the heuristic to use.

4.1 Current shortrec Heuristics

The heuristics used by shortrec to tackle the Depot Assignment Problem will be described now. At the time this thesis project was started, in short-rec two mutually exclusive solution approaches for the Depot Assignment Problem were implemented: Locally-Cheapest and Locally-Nearest. (We will use ‘local’ to denote choices made per depot stop, irrespective of the effect of the choice on the (feasibility of the) total trip; and ‘global’ for choices that are made for all depots at the same time, taking into account the overall effect.)

4.1.1 Locally-Cheapest Heuristic

The Locally-Cheapest heuristic chooses a depot for every depot stop in the trip, one depot stop at a time. For every depot stop, the depot that is the cheapest (in terms of travel cost and loading cost) is chosen. The distance

The Choice of Depots in Oil Truck Scheduling