A food factory is making a beverage for a customer from mixing two different existing

products A and B. The compositions of A and B and prices ($/L) are given as follows,

Amount (L) in /100 L of A and B

Lime Orange Mango Cost ($/L)

A 3 6 4 5

B 8 4 6 6

The customer requires that there must be at least 4.5 Litres (L) Orange and at least

5 Litres of Mango concentrate per 100 Litres of the beverage respectively, but no more

than 6 Litres of Lime concentrate per 100 Litres of beverage. The customer needs at

least 100 Litres of the beverage per week.

a) Explain why a linear programming model would be suitable for this case study.

[5 marks]

b) Formulate a Linear Programming (LP) model for the factory that minimises the total

cost of producing the beverage while satisfying all constraints.

[10 marks]

c) Use the graphical method to find the optimal solution. Show the feasible region and

the optimal solution on the graph. Annotate all lines on your graph. What is the minimal cost for the product?

[10 marks]

Note: you can use graphical solvers available online but make sure that your graph is

clear, all variables involved are clearly represented and annotated, and each line is clearly

marked and related to the corresponding equation.

d) Is there a range for the cost ($) of A that can be changed without affecting the optimum solution obtained above?

[5 marks]

2. A factory makes three products called Spring, Autumn, and Winter, from three materials

containing Cotton, Wool and Silk. The following table provides details on the sales price,

production cost and purchase cost per ton of products and materials respectively.

Sales price Production cost Purchase price

Spring $60 $5 Cotton $30

Autumn $55 $4 Wool $45

Winter $60 $5 Silk $50

The maximal demand (in tons) for each product, the minimum cotton and wool proportion in each product is as follows:

Demand min Cotton proportion min Wool proportion

Spring 4800 55% 30%

Autumn 3000 45% 40%

Winter 3500 30% 50%

a) Formulate an LP model for the factory that maximises the profit, while satisfying the

demand and the cotton and wool proportion constraints.

[10 Marks]

b) Solve the model using R/R Studio. Find the optimal profit and optimal values of the

decision variables.

[10 Marks]

Hints:

1. Let xij ≥ 0 be a decision variable that denotes the number of tons of products

j for j ∈ {1 = Spring, 2 = Autumn, 3 = W inter} to be produced from Materials

i ∈ {C=Cotton, W=Wool, S=Silk}.

2. The proportion of a particular type of Material in a particular type of Product can be

calculated as:

e.g., the proportion of Cotton in product Spring is given by: xC1

xC1 + xW1 + xS1

.

3. Helen and David are playing a game by putting chips in two piles (each player has two

piles P1 and P2), respectively. Helen has 6 chips and David has 4 chips. Each player

places his/her chips in his/her two piles, then compare the number of chips in his/her

two piles with that of the other player’s two piles. Note that once a chip is placed in one

pile it cannot be moved to another pile. There are four comparisons including Helen’s

P1 vs David’s P1, Helen’s P1 vs David’s P2, Helen’s P2 vs David’s P1, and Helen’s P2

vs David’s P2. For each comparison, the player with more chips in the pile will score 1

point (the opponent will lose 1 point). If the number of chips is the same in the two piles,

then nobody will score any points from this comparison. The final score of the game is

the sum score over the four comparisons. For example, if Helen puts 5 and 1 chips in her

P1 and P2, David puts 3 and 1 chips in his P1 and P2, respectively. Then Helen will get

1 (5 vs 3) + 1 (5 vs 1) – 1 (1 vs 3) + 0 (1 vs 1) = 1 as her final score, and David will get

his final score of -1.

(a) Give reasons why/how this game can be described as a two-players-zero-sum game.

[5 Marks]

(b) Formulate the payoff matrix for the game.

[5 Marks]

(c) Explain what is a saddle point. Verify: does the game have a saddle point?

[5 Marks]

(d) Construct a linear programming model for each player in this game;

[5 Marks]

(e) Produce an appropriate code to solve the linear programming model in part (c).

[5 Marks]

(f) Solve the game for David using the linear programming model you constructed in

part (d). Interpret your solution.

[5 Marks]

[Hint: To record the number of chips in each pile for each player you may use the notation

(i, j), where i is the number of chips in P1 and j is the number of chips in P2, for example

(2,4) means two chips in P1 and four chips in P2.]

4. Supposing there are three players, each player is given a bag and asked to contribute in

his own money with one of the three amount {$0, $3, $6}. A referee collects all the money

from the three bags and then doubles the amount using additional money. Finally, each

player share the whole money equally. For example, if both Players 1 and 2 put $0 and

Player 3 puts $3, then the referee adds another $3 so that the total becomes $6. After

that, each player will obtain $2 at the end. Every player want to maximise his profit,

but he does not know the amount contributed from other players. [Hint: profit = money

he obtained – money he contributed.]

(a) Compute the profits of each player under all strategy combinations and make the

payoff matrix for the three players. [Hint: you can create multiple payoff tables to

demonstrate the strategy combinations. The referee is not a player and should not be in

the payoff table.]

[10 Marks]

(b) Find the Nash equilibrium of this game. What are the profits at this equilibrium?

Explain your reason clearly.