A construction project requires at least 8,000 yd^3 of aggregate. The aggregate

Question

A construction project requires at least 8,000 yd^3 of aggregate. The aggregate must contain no less than 45% sand and no more than 50% gravel. Materials may be obtained from two sites Material from each site has a different mixture of sand and gravel, and comes at a delivery cost. These are described below. Find the least-cost mixture of material from the two sources. Formulate the problem as a linear program, with a linear objective function and linear constraints. Identify and define the decision variables. Give the reason behind each constraint. Graph feasible region and solve the problem graphically. Solve the problem using Lagrange multipliers. You may use the graphical solution to help reduce the number of cases you examine with the Lagrange multiplier method What are the values of the Lagrangian multipliers? Do your solutions from the graphical and Lagrangian method agree?

Explanation / Answer

a> The objective is t minimize the cost of the mixture from the two sites.

Let x be the mixture in yd^3 from site 1

Let y be the mixture in yd^3 from site 2

hence minimize Z = 5x + 7y

subject to the constraints :

total aggregate form the two sites needs to be atleast 8000 yd^3

=> x + y >= 8000 ---------->(1)

the total aggregate has atlease 45% of sand and atmost 50% of gravel

=> .3x + .6y >= .45(x+y) -------->(2)

=> .7x + .4y <= .5(x+y) ------->(3)

x >= 0 ----------->(4)

y >= 0 ---------->(5)

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