Linear Programming, LP Problem, LP Function and Payoff Matrix in Game Theory

Assignment: Analytic 1
ASSIGNMENT 4
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Assignment: Analytic 2
Assignment 4
Question 1
(a) The basic requirement for a problem to be solved using linear programming method is
that the problem must be of maximization or minimization of linear objective function
subject to a set of linear constraints (Akpinar and Baykasoglu 2014; Higle and Sen
2013; Recht, Re and Bittorf 2012). The factory needs to minimize cost given the
linear combination of cost of each product needed to make the beverage. Therefore,
the fundamental requirement for a problem to be modelled by linear program is met
(minimization). In addition to the basic requirement there are other four more
assumptions that should be satisfied for the problem to be modelled using LP.
1) Proportionality – each products cost has a direct impact of the final cost of the
beverage thus the assumption is satisfied (Akpinar and Baykasoglu 2014). 2)
additivity - the decision on how many units of product A to use is independent of that
for product B. Thus, the contribution to the objective function for any product is
independent of the other. 3) divisibility- the variables being used in the decision
criteria are continuous for instance the customer requires a minimum of 4.5 liters of
orange. An indication that fractional units can be used. 4) certainty – the factory
cannot produce negative volumes of beverage since the customer requires a minimum
of 80 liters per week and it cannot use less than zero units of the products A and B.
(b) Suppose the factory use x = liters of product A and y liters of product B in making (x
+ y) liters of the beverage. Then, it implies that the total cost of producing (x + y)
liters of beverage is
C = 5x + 7y. (i)

Assignment: Analytic 3
Also, we know that liters of orange must not less than 4.5 liters in 100 liters of
beverage giving us equation (ii).
0.06 x +0.04 y ≥ 4.5 which simplifies to 3 x+ 2 y ≥ 225 (ii)
Next, mango used should not be less than 5 liters per 100liters of beverage
0.04 x +0.06 y ≥5 which simplifies to 2 x+3 y ≥ 250 (iii)
Similarly, lime used at most 6 liters per 100liters of beverage
0.03 x+ 0.08 y ≤ 6 which simplifies to 3x +8 y ≤ 600 (iv)
But the customer also has some demands as follows
x + y ≥ 80 (v)
Certainty constraints
x ≥ 0 and y≥ 0 form equation (vi) and (vii).
Therefore, the LP is to minimize equation (i) subject to constraints presented in
equation (ii) to (vii). That is
Min C = 5x + 7y.
Subject to:
3 x+ 2 y ≥ 225
2 x+3 y ≥ 250
3 x +8 y ≤ 600
x + y ≥ 80
x ≥ 0 and y≥ 0
(c) In plotting the inequalities, we input the signs in the reverse order (if ≥ we input ≤) in
the desmos.com function plotting platform (Desmos.com). The plotted graph is
shown below

Assignment: Analytic 4
The feasible region is shown in white and the conner points are indicated in brackets (x, y).
We substitute the values of the extreem points in the in equation (i) to obtain the costs as
follows:
In (x = 33.33, y = 62.5) impling TC = ( 5 x 33.33 ) + ( 7 x 62.5 )=604.15
In (x = 35, y = 60) implying TC=(5 x 35)+ ( 7 x 60 ) =595
In (x = 125, y = 0) implying TC = ( 5 x 125 ) + ( 7 x 0 )=625
In (x = 200, y = 0) implying TC= ( 5 x 200 ) + ( 7 x 0 ) =1000
The minimum cost combination is (x = 35, y = 60), an indication that the factory
should use 35 and 60 liters of product A and B respectively in making the beaverage.
(d) The Microsoft excel screenshot shows that the range of values that product A can take
without changing the optimal total cost is between 5 liters to 5.8 liters. Beyond 5.8