STAT1016 (2017/18)  Mathematics of Decision Making  Faculty Header ID: 300273
Banner Header ID for handin: 239317 
Contribution: 50% of course 
Course Leader:
Professor Vitaly Strusevich 
Coursework 1  Deadline Date: Thursday 09/11/2017  
This coursework will be marked anonymously
YOU MUST NOT PUT ANY INDICATION OF YOUR IDENTITY IN YOUR SUBMISSION 

This coursework should take an average student who is uptodate with tutorial work approximately 25 hours
Feedback and grades are normally made available within 15 working days of the coursework deadline 

Learning Outcomes:
A Facilitate a multistage decision making by means of dynamic programming B Handle multiobjective s decision making by means of goal programming C Formulate and solve matrix, bimatrix and nperson games; 
Coursework Submission Requirements
 A paper copy of your work for this coursework must be submitted to the Faculty Reception Desk before the office closes on the Deadline Date of Thursday 09/11/2017.
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 Detailed Specification
This is an individual piece of work
 Deliverables
Worked solutions and printouts as answers to 4 questions
 Assessment Criteria
See the question paper
Mathematics of Decision Making Coursework 1
2017/2018
In all cases that a software output is asked to be delivered make sure that the corresponding printouts identify the question and your name, e.g., SmithQ1b
Clearly number all pages in your answer booklet. Consider the questions in the order of their numbering. Start the next question on a new page.
Question 1 [20 marks]
A company produces two products. Relevant information for 1 unit of each product is shown in the table below.
Product 1  Product 2  
Labour required  2 hours  4 hours 
Contribution to profit  £4  £3 
A total of 40 hours is available for production. Marketing considerations recommend at least 4 units of Product 1 should be produced and at least 8 units of Product 2 should be produced.
The company wants to generate at least £48 of profit and to achieve the goals listed below (in the order of importance):
Goal 1: Avoid underutilisation of labour Goal 2: Meet demand for Product 1 Goal 3: Meet demand for Product 2 Goal 4: Do not use any overtime
 Formulate and solve byhand a preemptive goal programming problem using the graphical Clearly explain the decision and deviation variables and all constraints. Formulate a linear programming problem for each level and its solution. Explicitly state the answer and give its meaningful interpretation.
[14 marks]
 Assume that a £2 penalty for each hour of overtime is incurred, and a £1 penalty is incurred for each hour of available labour that is not For each unit of Product 1 and Product 2 by which production falls short of demand, a penalty of £Z and of £5 pounds is introduced, respectively. Using Microsoft Excel, set up a nonpreemptive goal programming model. Running the model for various values of Z of your choice determine a value of Z such that in the obtained solution demand is satisfied for both products. Explicitly state the answer. Supply the printouts of the problem formulation, Solver setup and the solution sheet.
[6 marks]
Question 2 [25 marks]
A farm owns trucks of four types, their parameters are shown in the table below.
Type of truck  Quantity  Daily cost of running  Capacity 
T1  2  £20  400 kg 
T2  3  £40  700 kg 
T3  2  £80  1000 kg 
T4  1  £100  1500 kg 
The farm needs to deliver 2000 kg of grain to a mill. It takes one day for a truck to go to a mill and return back. The farm’s management is not going to spend more than £120 on that delivery.
 Formulate the problem of maximising total weight of delivered grain, provided that the budget of £120 is respected as a problem of integer linear programming. Give explanations regarding the decision
[7 marks]
 Solve the problem by using a dynamic programming algorithm. If the last digit in your registration number (before the dash) is odd use the matrix approach; otherwise, use the graphical approach. Clearly explain the stages and states, present the main recursion. Give a meaningful interpretation of the obtained solution; in particular, state whether it is possible to deliver 2000
[10 marks]
 Verify your solution by Microsoft Excel. Submit printouts of the input data, the settings for Solver and the found
[8 marks]
Question 3 [30 marks]
 Consider a 4×2 matrix game with the payoff table
æ a b ö

ç ÷
ç ÷ .
ç a d ÷

ç ÷
è ø
By performing an appropriate case analysis, demonstrate that for any values a, b, c and d the game admits a solution in pure strategies. Verify that the value of the game is min{max{a,c}, max{b,d}}.
[15 marks]
 Let A = (a₁,…,a_{m}), B = (b₁,…,b_{n}), C = (c₁,…,c_{m}) and D = (d₁,…,d_{n}) be four arrays ofnumbers, all numbers in arrays B and C being For an argument t, ∞ < t < +∞, define the functions
f(t) = max{a_{i} – c_{i} t  1≤ i ≤ m}; g(t) = max{ b_{j} + d_{j} t 1 ≤ j ≤ n}.
Notice that both functions f(t) and g(t) are continuous. As t changes from ∞ to +∞, function f(t) decreases from +∞ to ∞, while function g(t) increases from ∞ to +∞.
Therefore, there exists a value t⁰ such that
f(t⁰) = g(t⁰);
in other words, the graphs of these functions intersect. More precisely, if


f (t0 )= a – c t0
for some index i⁰, 1 ≤ i⁰ ≤ m, and

g (t0 )= –b

 d t0
for some index j⁰, 1 ≤ j⁰ ≤ n, then





a – c t0 = –b + d t0
Use this information to show that an m×n matrix game with a payoff matrix W = (w_{ij}) m×n
such that
wij
= ai + bj
ci + d j
admits a solution in pure strategies and t⁰ is the value of the game.
[15 marks]
Question 4 [25 marks]
Two manufacturers, A and B, are competing for sales in the same area. Both make and sell Product 1 and Product 2. The other producers of the same products are minor and can be neglected. If for a manufacturer its share of sales of Product 1 is a1 and that of Product 2 is a2 then it controls (a1 + a2)/2 of the market. At the moment Manufacturer B has sales volume three times as large as Manufacturer A for each of the two products.
Because of the recent technological breakthrough both companies will be making major improvements in both products. Each of them can develop both products improvements simultaneously and will have them ready for sale in 12 months. Alternatively, each of them can have a ‘crash programme’ for one of the products followed by a similar programme for the other product. If one product is developed first, it can be marketed ahead of the competitor. The crash programme for the first product (either Product 1 or Product 2) will take 9 months for Manufacturer B and 10 months for Manufacturer A. The subsequent crash programme for the other product will require 9 months for each of the manufacturers.
If both manufacturers market their improved models simultaneously then Manufacturer A will increase its share for that product by 2%. If Manufacturer A is ahead of its competitor by 2, 6 or 8 months, then its share for the corresponding product will increase by 20, 30 and 40 percent, respectively. Similarly, if Manufacturer B is ahead of its competitor by 1, 3, 7 or 10 months, then Manufacturer A will lose 6, 10, 12 and 14 percent of its share for the corresponding product, respectively.
 Verify that the situation in which each manufacturer is aimed at maximising its market share can be formulated as a matrix games with the
B1  B2  B3  
A1  27  35  35 
A2  29  19  38 
A3  29  38  19 
Clearly state the players, the strategies of each player and explain how the entries of the payoff matrix are derived.
[7 marks]
 Verify that the above game is not solvable in pure strategies. Formulate it as a linear programming problem and solve by using Microsoft Provide the printouts of the input, answer sheets and solver settings. State the answers accurate up to 4 d.p. Give interpretation and explanation of the found solution.
[9 marks]
 Assume that the board of Manufacturer B due to technical reasons has decided not to develop both products simultaneously, which is equivalent to the removal of strategy B1 from the table in (i). Solve the obtained game geometrically. State the full range of options for Manufacturer B. Give interpretation and explanation of the found