Regulations

• The deadline for the assessment is Tuesday 22 November 2022 at 12:00, UK time. You should submit the assessment on the STAT0025 Moodle page.

• There are three questions.  Answer all questions.  Question 1 is worth 19 marks. Question 2 is worth 14 marks. Question 3 is worth 7 marks.

• You may refer to your notes and other resources.  You may use a calculator or a computer for routine calculations, including matrix manipulations.  You may not use linear programming software.

• Late submission will incur a penalty unless there are extenuating circumstances supported by appropriate documentation.  Penalties are set out in the latest edi- tions of the Statistical Science Department student handbooks, available from the departmental web pages.

• Failure to submit this in-course assessment will mean that your overall examination mark is recorded as “non-complete”, i.e., you will not obtain a pass for the course.

• Your work will be marked and returned to you for feedback. The grade you receive is a provisional grade, and will be confirmed at the Statistics Examiners’ Meeting.

Formatting your solutions for submission

• You should submit ONE document that contains your solutions for all questions/part-     questions. Please follow UCL’s guidance on combining text and photographed/scanned   work, available here: https://www.ucl.ac.uk/news/2020/apr/seven-simple-steps-  -                     -answers-           -exams-or-submithandwrittenmoodleassessments

• In handwritten solutions (on paper or via tablet) please use blue or black pen.

Submitting your document

• Only upload ONE file that contains ALL your work.

• You will be allowed to submit only once.   Ensure that you submit the correct document!

• It is safer to submit a PDF document as it will be stable in terms of formatting across platforms.

• Name your file using your student number and the course code. For example, if your student number is 18002119 then you should name your file 18002119-STAT0025. This should also be input as the submission title on Moodle.

• Please write your student number on the first page of your submission. Do not write your name on your submission.

Plagiarism and collusion

• You must work alone.  You are encouraged to read the Department of Statistical Science’s advice on collusion and plagiarism, which you can find at                        https://www.ucl.ac.uk/statistics/sites/statistics/files/shbpc.pdf

• If there is any doubt as to whether the solutions you submit are entirely your own work, this will be reported to the Department of Statistical Science, who may take further action.

1. A company makes baby food out of applesauce, carrot puree, and crushed peas. They sell product 1, First Taste, and product 2, Second Choice.  Each product is made by mixing together the component foods in different proportions, but there is some flexibility in exactly how much of each component is in each product. The allowable proportions, and the price for which each product is sold for (in £ per litre), are given in this table:

Each component has a cost to purchase it, and a maximum amount that is available to the company (i.e., an upper bound).  These costs (in £ per litre) and upper bounds (in litres) are given in this table:

(a) With the goal of maximising the profit  (the difference between total sales revenue and cost of components), write down a linear programming problem (LPP) to determine the optimal composition of each product (in % of each component), and how much of each product to produce (in litres).  Explain your objective function and constraints. Do not attempt to solve the LPP.

(Hint: if you struggle to find linear constraints, you may need to rethink your choice of control variables.)                                                          [10 marks]

In addition to mixing the components, the components themselves must be trans- ported from their manufacture sites to the mixing sites, and in the problem above the cost of transportation has been neglected. They are transported via two inter- mediate warehouses, and the road network between component manufacture sites, warehouses and product mixing sites looks like this:

A

3

2

W                           1

3

3

4

V                           2

2

P

In this diagram, the vertices A, C and P are the component manufacture sites of applesauce, carrot puree and crushed peas, respectively.  The vertices W and V are two intermediate warehouses through which the components must pass.  The vertices 1 and 2 are the product mixing sites at which products 1 and 2, respectively, are produced.  The annotation on each edge in the diagram is the cost (in £ per litre) of transporting any component along that edge.

(b) With the goal of maximising profit (the difference between total sales revenue and the cost of both components and transportation), write down a linear programming problem to determine the optimal composition of each product (in % of each component), how much of each product to manufacture (in litres) and how much of each component to transport along each edge of the network (in litres).  Explain your objective function and constraints.  Do not attempt to solve the LPP.                                                                           [9 marks]

2.   (a) A company makes products 1 and 2, which consume resources R and S. They represent the problem of profit maximisation subject to these resource con- straints  (among other constraints) using the following linear programming problem:

Maximise z = 8x1 + 2x2

subject to x1 + 5x2  ≤ 25       (resource R)                       (1)

x1 + 3x2    0                                                     (2)

3x1 + x2  ≤ 10       (resource S)                       (3)

x1 , x2    0.                                                    (5)

Solve this problem using the graphical method (not the graphical naive method.)

[6 marks]