$1$ A construction project requires at least $8,000y{d}^3$ 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 different

delivery cost. These afe described below.

Find the least$-$cost mixture of material from the two sources.

a$.$ 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. $($Hint: There are five constraints$)$

b$.$ Graph the feasible region and solve the problem graphically. $($Hint: Corner points$)$

c$.$ 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.

d$.$ What are the values of the Lagrangian multipliers?

e$.$ Do your solutions from the graphical and Lagrangian method agree?

$2$ Solve this formulation using the Solver in MS$-$Excel $($found under the Tools bar$).$ Be sure to

specify the "Assume Linear Model" option.

a$.$ Provide a hard copy of the spreadsheet which makes the decision variables, objective

function, and constraints clear. This should fit on one page.

b$.$ Provide copies of the Answer Report, Sensitivity Report, and Limits Report from the

Solver solution.

c$.$ Interpret the results from Solver. What are the optimal values of the decision variables?

What is the minimum cost? What are the marginal values of the constraints? $($Hint:

Marginal values $=$ Lagrangian multipliers $=$ Shadow values