# Question

Idaho Natural Resources (INR) has two mines with different production capabilities for producing the same type of ore. After mining and crushing, the ore is graded into three classes: high, medium, and low. The company has contracted to provide local smelters with 24 tons of high-grade ore, 16 tons of medium-grade ore, and 48 tons of low-grade ore each week. It costs INR $10,000 per day to operate mine A and $5,000 per day to run mine B. In a day's time, mine A produces 6 tons of high-grade ore, 2 tons of medium-grade ore, and 4 tons of low-grade ore. Mine B produces 2, 2, and 12 tons per day of each grade, respectively. Management's short-run problem is to determine how many days per week to operate each mine under current conditions. In the long run, management wishes to know how sensitive these decisions will be to changing economic conditions.

A report prepared for the company by an independent management consultant addressed the company's short-run operating concerns. The consultant claimed that the operating problem could be solved using linear programming techniques by which the firm would seek to minimize the total cost of meeting contractual requirements. Specifically, the consultant recommended that INR do the following:

Minimize Total Cost = $10,000A + $5,000B

subject to

6A + 2B ( 24 (high-grade ore constraint)

2A + 2B ( 16 (medium-grade ore constraint)

4A + 12B ( 48 (low-grade ore constraint)

A ( 7 (Mine A operating days in a week constraint)

B( 7 (Mine B operating days in a week constraint)

or, in their equality form,

6A + 2B - SH = 24

2A + 2B - SM = 16

4A + 12B - SL = 48

A + SA=7

B + SB = 7

Where

A, B, SH, SM, SL, SA, and SB ( 0

Here, A and B represent the days of operation per week for each mine; SH, SM, and SL represent excess production of high-, medium-, and low-grade ore, respectively; and SA and SB are days per week that each mine is not operated.

A graphic representation of the linear programming problem was also provided. The graph suggests an optimal solution at point X, where constraints 1 and 2 are binding. Thus, SH = SM = 0 and

6A + 2B – 0 = 24

minus 2A + 2B – 0 =16

4A = 8

A = 2 days per week

Substitute A = 2 into the high-grade ore constraint:

6(2) + 2B = 24

12 + 2B = 24

2B = 12

B = 6 days per week

A minimum total operating cost per week of $50,000 is suggested, because

Total Cost = $10,000A + $5,000B

= $10,000(2) + $5,000(6)

=$50,000

The consultant's report did not discuss a variety of important long-run planning issues. Specifically, INR wishes to know the following, holding all else equal:

A. How much, if any, excess production would result if the consultant's operating recommendation were followed?

B. What would be the cost effect of increasing low-grade ore sales by 50 percent?

C. What is INR's minimum acceptable price per ton if it is to renew a current contract to provide one of its customers with 6 tons of high-grade ore per week?

D. With current output requirements, how much would the cost of operating mine A have to rise before INR would change its operating decision?

E. What increase in the cost of operating mine B would cause INR to change its current operatingdecision?

A report prepared for the company by an independent management consultant addressed the company's short-run operating concerns. The consultant claimed that the operating problem could be solved using linear programming techniques by which the firm would seek to minimize the total cost of meeting contractual requirements. Specifically, the consultant recommended that INR do the following:

Minimize Total Cost = $10,000A + $5,000B

subject to

6A + 2B ( 24 (high-grade ore constraint)

2A + 2B ( 16 (medium-grade ore constraint)

4A + 12B ( 48 (low-grade ore constraint)

A ( 7 (Mine A operating days in a week constraint)

B( 7 (Mine B operating days in a week constraint)

or, in their equality form,

6A + 2B - SH = 24

2A + 2B - SM = 16

4A + 12B - SL = 48

A + SA=7

B + SB = 7

Where

A, B, SH, SM, SL, SA, and SB ( 0

Here, A and B represent the days of operation per week for each mine; SH, SM, and SL represent excess production of high-, medium-, and low-grade ore, respectively; and SA and SB are days per week that each mine is not operated.

A graphic representation of the linear programming problem was also provided. The graph suggests an optimal solution at point X, where constraints 1 and 2 are binding. Thus, SH = SM = 0 and

6A + 2B – 0 = 24

minus 2A + 2B – 0 =16

4A = 8

A = 2 days per week

Substitute A = 2 into the high-grade ore constraint:

6(2) + 2B = 24

12 + 2B = 24

2B = 12

B = 6 days per week

A minimum total operating cost per week of $50,000 is suggested, because

Total Cost = $10,000A + $5,000B

= $10,000(2) + $5,000(6)

=$50,000

The consultant's report did not discuss a variety of important long-run planning issues. Specifically, INR wishes to know the following, holding all else equal:

A. How much, if any, excess production would result if the consultant's operating recommendation were followed?

B. What would be the cost effect of increasing low-grade ore sales by 50 percent?

C. What is INR's minimum acceptable price per ton if it is to renew a current contract to provide one of its customers with 6 tons of high-grade ore per week?

D. With current output requirements, how much would the cost of operating mine A have to rise before INR would change its operating decision?

E. What increase in the cost of operating mine B would cause INR to change its current operatingdecision?

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