Q1. Recall the Reddy Mikks model: Reddy Mikks produces both interior and exterio
ID: 387618 • Letter: Q
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Q1. Recall the Reddy Mikks model: Reddy Mikks produces both interior and exterior paints from two raw materials M1 and M2. The following table provides the basic data of the problem: Tons of raw material needed per ton Raw msterial Maxinu daily availability (tons) 24 M2 A market survey indicates that the daily production for interior paint cannot exceed that for exterior paint by more than 1 ton Also, the maximum daily production for interior paint is 2 tons. Reddy Mikks wants to determine the optimum (best) product mix of interior and exterior paints that maximizes the total daily profit. (a) Write the linear programming formulation using decision variables x1 and x2 for the total daily tons produced of exterior and interior paints, respectively (b) Determine the best feasible solution among the following solutions of the LP from part (a) (ii)-,-2, z2= 2. (iv) 2,21 (c) For the feasible solution x1 3, x2 1, determine the unused amounts of raw materials M1 and M2. (d) Construct the following constraints and express them with a linear left-hand side and a constant right-hand side (i) The daily production for interior paint exceeds that of exterior paint by at least 1 ton (ii) The daily usage of raw material of M2 in tons is at most 6 and at least 3 ii) The production for interior paints cannot be less than the production for exterior paint. (iv) The minimum quantity of paint that should be produced is 3 tons (v) The proportion of interior paint to the total production of both interior and exterior paint must not exceed 0.5. (e) Using the formulation from part (a) (i) Graph the feasible region. Be sure to label each ineuality and each axis, and shade the feasible region. (ii) Draw and label the evel curve for10 (i.e., the level curve through (2 (2,0)) (iii) Calculate, draw, and label the gradient 2 at the point (2.0) (iv) Determine the optimal solution to the Reddy Mikks probl rate the optimal solution on your graph. (v) What is the optimal value'. Draw the corresponding level curve for the optimal point (vi) Which constraints are binding at ? What value do you report to Reddy Mikks?Explanation / Answer
a) The linear programmng formulation -
Maximize Z = 5x1 + 4x2
where Z is profit
subject to
x2-x1<=1
x2<=2
b) Put values of x1 and x2 from each laternative solution into the above function to determine max value of Z.
1) x1 = 1, x2 = 5; Z = 5*1 + 4*5 = 5+20 = 25
2) Z = 5*2 + 4*2 = 10 + 8 = 18
3) Z = 5*3 + 4*1.5 = 15+6 = 21
4) Z=5*2 + 4*1 = 10+4 = 14
5) Z = 5*2 + 4(-1) = 10 - 4 = 6
From the above Z is maximum when x1 = 1 and x2=5. But it does not meet the criteria that x2 should not xceed x1 more than 1 ton and x2 cant be more than 2.
Other options meet the criteria. From other options Z is maximum when x1 = 3 and x2 = 1.5
hence option ii) is optimal.
c) When x1 = 3 then M1 will be used in 6*3 = 18 and M2 will be used in 3 tons resectively. and x2 = 1 then M! will be use in 4*1 = 4 and M2 will be use in 2*1 = 2 tons. Thus M1 used = 18+4 = 22 and M2 used = 3+2 = 5 tons.
Unused M1 = 24 - 22 = 2 tons and M2 = 6-5 = 1 tons.
d) i) x1 + 1 <= x2
ii) 6<=M>=3
iii) x2>=x1
iv) x1+x2 >= 3
v) x2/(x1+x2) <= 0.5
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