Suppose the mining problem in Question-4 is slightly different. There are different sites within a mine where high, medium, or low grade minerals are extracted. For instance, if extraction is carried on for one day at the high-grade site in mine X, the production yield is 6 tons of high-grade mineral. Likewise, if extraction is carried on for one day at the medium-grade site in mine Y, the production yield is 1 ton of medium-grade mineral. Additional production yield data is provided in Table-Q5-1 below.

Mine	Production (Tons/Day)
	High	Medium	Low
X	6	3	4
Y	1	1	6

Table-Q5-1. Production yield of extraction at each site of each mine.

The only way to obtain high, medium, and low grade minerals is to extract at the designated high, medium, and low sites, respectively, within each mine.

Additional information has become available about the market for the products.

The company has the option the sell the high, medium, low-grade oresminerals extracted from any of the sites to other manufacturers other than the smelting plant. However, the unit revenue ($/ton) earned from the contract with smelting plant is higher than what the other manufacturers will pay. The revenues are provided below.

Product type	High grade	Medium grade	Low grade
Revenue ($/ton) when sold to the smelting plant	$10	$12	$8
Revenue ($/ton) when sold in the market to other manufacturers	$10	$9	$7

Table Q5-2. Revenue per ton from different customers

The demand of the smelting plant must be met exactly because of the contract. The demand for other manufacturers is higher than the companys production capacity. The company has the market power to determine the amount of products to sell to other manufacturers (and can chose to not sell at all). The demand information is provided below.

Product type	High grade	Medium grade	Low grade	Demand fulfillment
Demand of smelting plant	12 tons/week	8 tons/week	24 tons/week	Must be met exactly
Demand from other manufacturers	unlimited	unlimited	unlimited	Optional

Table Q5-3. Demand from different customers

The mining company is interested in maximizing revenues. You will use linear programming to determine a weekly mining plan. The weekly plan will show how the time at each mine will be allocated to different product grades sites. Specifically, you have to determine the number of days per week at each site mine that will be dedicated to producingextracting each grade (at the corresponding site). Keep in mind that each mine site operates no more than 5 days per week.

NOTE: SHOW ALL YOUR WORK.

a) (3 points) What are the decision variables? How many decision variables are there? Clearly explain the decision variables in English and provide the mathematical notation used in the algebraic formulation of the decision problem.

b) (2 points) What is the objective function? Clearly explain the objective function in English and provide the mathematical formulation.

c) (2 points) What are the constraints of the problem? Clearly explain each constraint in English and provide the mathematical formulation.

d) (1 points) Are there any sign or type restrictions in your formulation? Why or why not? Clearly explain the sign and type restrictions (if any).

e) Prepare a spreadsheet to determine the optimal production mix of using Excel Solver.

i. (2 points) Provide a screenshot of your spreadsheet model. The screenshot should show all the cells involved in formulating the optimization problem. NOTE: Do not copy and paste your spreadsheet to your answer report as an Excel object. Instead, capture a screenshot in Excel and paste this as an image.

ii. (2 points) Provide a screenshot of your spreadsheet with the formula view. You can switch from normal view to formula view in Excel by clicking the CTRL and ~ keys on your keyboard simultaneously.

iii. (2 points) Prepare the Solver model by logging all the information to the Solver Parameters window in Excel. Provide a screenshot of the Solver Parameters window.

iv. (2 points) Solve the optimization problem using Excel Solver. What is the optimal solution, i.e., the optimal values of the objective function and the decision variables?

NOTE: Do not forget to change the options in the Solver. Unclick ignore integer constraints and set the integer optimality (%) to 0.001. Use Simplex LP as your solving method.

a) Suppose the contract with the smelting plant has ended and is not going to be renewed. The unit revenue ($/ton) from the smelting plant is higher from that of other manufacturers for each grade. The executives are unhappy that the contract has ended with the smelting plant and state that the revenues will decline because of that. You can test whether this statement is true by solving a new linear program. Keep in mind that the company will sell its products to other manufacturers only. You should determine the optimal product mix that maximizes revenue, without any obligations to meet the demand of a particular customer. Adjust your linear programming formulation and resolve the problem to determine the new optimal solution.

i. (2 points) What is the new optimal revenue? What is the new weekly mining plan?

ii. (2 points) Is the optimal revenue higher or lower without the contract with the smelting plant? Why or why not? Clearly explain.