Stan’s Vans Modelling a business decision for a van hire company;

Project Introduction

Stanley Baker is a van driver and an entrepreneur. He owns a van hire company called Stan’s Van,

based in London, which has a single depot in Lewisham. Business is booming and Stan wants to expand

the business. He has purchased three more depots in Southampton, Cambridge and Skegness and needs

to decide how many new vans to buy. Not being very good with figures, Stan has commissioned an

operational research study to decide how many new vans to buy. The capital cost of buying the vans

is not an issue for Stan, as it is a one-off cost and the business has plenty of money in the bank. He is

more concerned about the cost of maintaining the vans, which he estimates is £22 per week for each

van. This cost covers storage, serving, tax and so on. Of course, these costs are balanced by the profit

the business makes from hiring out the vans.

Stan would like to offer his customers enough flexibility so that they can hire vans from one depot

and return them to another depot. This will make his service attractive to, for example, people moving

house who do not want to have to return vans to the same depot they hire them. However, Stan is

worried that there may be a build-up of vans at particular depots or a shortage of vans at others.

Therefore he proposes to hire employees to transport the vans from depot to depot as necessary. This

will be another cost to the business due to the employees’ wages and the cost of petrol.

Your task is to model and solve the problem of how many vans the business should own to maximise

profit. You are required to start by building a basic ‘simplified’ model of Stan’s business (described

in Section 1) before refining it in Section 2 into a more precise model. Stan is also considering taking

a loan out to expand some of his depots to as so increase their repair capacity. The details of this are

given in Section 3, and you are required to further extend your model according to these specifications

to help Stan to decide what to do.

Stan has also identified some important external factors that might influence his business in the

future. The government is planning two high speed rail construction projects, but due to economic

difficulties the government may be forced to abandon one of these projects. Whether the projects

go ahead or not will influence Stan’s business and in particular influence the future demand for van

hire. More details are given in Section 4, and you should build a stochastic program to model these

considerations.

1

Basic model

In this section a simplified version of Stan’s business is described. To aid you in your study, Stan

has done some market research to produce some data concerning the demand for van hire in the four

cities. The table below lists the demand for each day of the week. The business shuts on Sunday so

no vans will be hired on this day.

Demand

Skegness

Cambridge

London

Southampton

Monday

150

380

150

240

Tuesday

230

220

300

150

Wednesday

210

120

360

90

2

Thursday

130

330

170

150

Friday

180

320

110

180

Saturday

350

150

190

120

Vans can be hired at any point during the day between 9am and 6pm and must be returned the

following morning before 9am. For example, if a van is hired on Wednesday then it is returned first

thing in the morning on Thursday before 9am, so that it is ready to be hired by someone else on

Thursday. Sundays are not counted, so if a van is hired on Saturday it is returned on Monday.

The company charges £60 for the hire of a van, and the cost to the company (for ‘wear and tear’,

administration, etc.) is £30 every time a van is hired.

Stan also has estimates (which you can assume are accurate) for the proportion of the vans hired

from each depot that are returned to each other given depot. These are given below. So, for example,

of the vans hired from Cambridge, 0.15 of them are returned to Skegness.

Proportion returned from

Skegness

Cambridge

London

Southampton

To

Cambridge London

0.2

0.1

0.55

0.25

0.2

0.54

0.12

0.27

Skegness

0.6

0.15

0.15

0.08

Southampton

0.1

0.05

0.11

0.53

Every day the business can transfer vans from one depot to another. The costs of doing so are

listed below. You can assume that if a van is transported from one city to another it leaves at 9am

and arrives at 6pm, so that it cannot be hired out on that day, but can be hired out on the following

day. Stan needs to decide how many vans to transfer between the depots each day. Some of the vans

can be left at a depot during the day and then overnight.

Transport cost (£)

Skegness

Cambridge

London

Southampton

Skegness

0

30

50

80

Cambridge

30

0

30

60

London

50

30

0

40

Southampton

80

60

40

0

Each morning, at every depot the vans will be counted. Each of the vans must be accounted for,

and each comes into one of the following three categories:

• Vans that have stayed at the depot for the whole of the previous day

• Vans returned that morning by customers

• Vans transferred to that depot from other depots the previous evening

The total number of vans that come into these three categories is called the In-Number.

During the course of the day the fate of the vans falls into three categories:

• Vans that are hired out from that depot on that day

• Vans transferred to one of the other depots

• Vans left at the depot for the whole day

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The total number of vans that come into these three categories is called the Out-Number. Evidently,

the In-Number and the Out-Number must be the same for any particular depot on any given day, so

that all vans are accounted for.

Your task is to decide how many vans the company should own and how many vans should be

transported from each depot to each other depot every day. The company also wishes to know the

number of vans that will be hired out from each depot each day and the number of vans that will be

left in each depot each day.

You can assume that for any given depot and any given day of the week these numbers will be the

same each week. You must ensure that

• At each depot on each day the In-Number and the Out-Number are equal.

• The number of vans hired at any depot on any day does not exceed demand.

2

Refined model

The main simplification made in the last section was that it was assumed that none of the vans were

returned damaged. In reality, 10% of the vans returned are damaged. In this case, customers are

charged a fine of £150, regardless of the amount and nature of the damage. There is no direct cost

to the company as this is covered by their insurance. However, vans cannot be hired out again until

they are repaired. Damaged vans can be repaired at either the Cambridge or London depots. It

takes a day to repair a van, and the maximum number of vans that can be repaired in a day at

these depots are 18 and 30 respectively. A damaged van must be transported to either Cambridge

or London before it can be repaired. When it has been repaired it can be hired out again the next

day (or transported somewhere else). For example a van could be returned on Wednesday morning

to Skegness, transported to Cambridge that day, repaired at Cambridge on Thursday, and hired out

from Cambridge on Friday (or it could be transported to a different depot on Friday and hired out on

Saturday).

This complicates the modelling process, as you now need to specify

• the number of vans the company should own

• the number of damaged and undamaged vans that will be transported from each depot to each

other depot every day

• the number of vans that will be repaired in each depot each day

• the number of vans that will be hired out from each depot each day

• the number of damaged and undamaged vans that will be left in each depot each day

The In-Number now breaks down into Damaged In-Number and the Undamaged In-Number. The

Damaged In-Number for a given depot on a given day is the number of vans falling into the following

categories:

• Damaged vans that have stayed at the depot for the whole of the previous day

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• Vans returned damaged that morning by customers

• Damaged vans transferred to that depot from other depots the previous evening

The Undamaged In-Number is given by the number of vans falling into the following categories:

• Undamaged vans that have stayed at the depot for the whole of the previous day

• Vans returned undamaged that morning by customers

• Undamaged vans transferred to that depot from other depots the previous evening

• Vans that were repaired by that depot the previous day and are now undamaged

Similarly, the Out-Number breaks down into the Damaged Out-Number and the Undamaged OutNumber. The Damaged Out-Number for a given depot on a given day is the number of

vans falling

into the following categories:

• Damaged vans transferred to one of the other depots that day

• Damaged vans left at the depot for the whole day

• Damaged vans that are repaired at that depot during that day

The Undamaged Out-Number at a given depot is the number of vans falling into the following

categories:

• Vans that are hired out from that depot on that day

• Undamaged vans transferred to one of the other depots that day

• Undamaged vans left at the depot for the whole day

Analogously to the last section, the Damaged In- and Out-Numbers must be equal and the Undamaged In- and Out-Numbers must be equal.

Another simplification made by the previous model was that it did not include the ‘Saturday

discount’. Stan offers a discount of £10 for any van that is hired on a Saturday.

Stan would like to build a more sophisticated model that incorporates the added complications

of damaged vans and the Saturday discount to find help him to choose how many vans the company

should own.

3

Possible expansion

Stan is considering expanding the depots at Cambridge, London or Southampton in order to increase

their repair capacities, which are currently 18, 30 and 0 vans per day respectively (as detailed in

Section 2). There are five options available. The first three are:

(a) Increase the repair capacity of the Cambridge depot by 10 vans per day at a cost of £18,000

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(b) Increase the repair capacity of the London depot by 10 vans per day at a cost of £20,000

(c) Increase the repair capacity of the Southampton depot by 10 vans per day at a cost of £16,000

These costs cover the interest payments on the loans he would need to take out to fund the

expansions. If Stan chooses option (a) he has the option of increasing the repair capacity of the

Cambridge depot by an additional 10 vans per day at a further cost of £6,000; similarly if he chooses

option (b) then he can increase the repair capacity of the London depot by an additional 8 vans per

day at a further cost of £8,000. Of these five options, at most three can be carried out.

Stan would like to know whether it would be cost effective to carry out these expansions and if so,

which ones he should choose. Extend your model from Section 2 to incorporate these options.

4

Stochastic demand

As mentioned in the Introduction, the government is planning to build high speed rail links to

Southampton and Cambridge. Both these projects will go ahead with probability 0.4, but due to

economic difficulties the government may abandon one of the projects (but not both). There is a

probability of 0.4 that they will abandon the rail link to Southampton and a probability of 0.2 they

will abandon the link to Cambridge.

Stan’s demand estimates in Section 1 are based on the assumption that both projects will go ahead

and that Southampton’s and Cambridge’s local economies will be improved leading to his optimistic

predicted demand for van hire. If the planned rail link to one of the towns is abandoned, you can

assume the demand for van hire in that town will be 50% lower than Stan’s predictions.

Stan’s Vans can wait until the government has made its decision before it decides how many vans

to own, but the company must decide very soon which of the five expansion options described in

Section 3 to choose. Adapt the model from Section 3 to maximise the expected profit of Stan’s Vans

over each of the three possible scenarios.

Stan would also like to know whether to try and delay the decision on which expansion options to

choose until he has more information about the government’s intentions. Advise Stan on whether it

would be worthwhile delaying.

6

Deliverables and Report Contents

Stan’s Vans has commissioned you to develop LP models to model the problem described in Sections

1 and 2 and MIP models to model the problem described in Section 3 and Section 4. The company

wants you to use AMPLDev to implement and solve these models. They also expect that you will

then analyse and discuss your findings on the basis of the questions outlined in the previous sections.

You must deliver a Project Report containing your analysis and suggestions but also detailing the

modelling that you have undertaken.

The Project report should consist of the following (explained below in more detail):

(a) An Executive Summary (as short as possible), discussing your main findings.

(b) A concise Management Report discussing all of your findings.

(c) A number of Technical Appendices, detailed below.

(d) Electronic copies of all of the AMPL files that you develop as well as any additional computer

files used (if any).

(a) The Executive Summary is intended for Stan. It should be completely free of any mathematical

terms and discussion of modelling technicalities. It should describe the main characteristics of the

solutions obtained and answer all the questions posed by Stan’s Vans.

(b) In the Management Report, which is intended for Stan’s local operational managers at the depots,

you should concisely discuss all of your findings and analysis in more detail. The Management

Report should be independent of the Executive Summary (i.e., self-contained). Ideally, the Management Report will avoid the use of unnecessary mathematics, technical

terms and discussion of

modelling technicalities.

7

Marking scheme

Modelling

Implementation

Analysis

Organisation/Presentation

Total

25

25

25

25

100

• Modelling. The basic ingredients of the model are appropriately and meaningfully defined. The

essential decision variables for the model are introduced with suitable types. Constraints are

formulated correctly.

• Implementation. All scenarios and the extension are implemented in a sound manner. Data

is provided in an appropriate form and separated from the model. Multiple optimal solutions

are considered. Further, demonstrating sufficient capacity to developing “good” MIP models:

avoiding unnecessary variables and constraints, compactness of data representation as well as

the use of more advanced AMPLDev features for defining model entities.

• Analysis. The report should sufficiently demonstrate the ability to recognise and report all key

results of the models and relate these to the real-world problem as well as demonstrating sufficient

understanding of these results with a discussion of requisite insight. Discussing limitations of

the model, the possible sources of these and how they may impact on the problem decision.

• Organisation/Presentation. A main report and technical appendices which are clear, requisitely

concise, well organised, well formatted and well presented with appropriate use of figures and

tables, as well as organised, well commented and easy to navigate AMPLDev files.

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