B. There would be a zero cost impact of an increase in low-grade ore sales from 48 to 72 tons (= 1.5 X 48). With A = 2 and B = 6, 80 tons of low-grade ore are produced. A 50% increase in low-grade ore sales would simply reduce excess production from SL = 32 to SL = 8, because

Graphically, the effect of a 50% increase in low-grade ore sales would be to cause a rightward shift in the low-grade ore constraint to a new constraint line with endpoints (0B, 18A) and (6B, 0A). Although such a shift would reduce the feasible space, it would not affect the optimal operating decision of A = 2 and B = 6 (at point X). C. If INR did not renew a contract to provide one of its current customers with 6 tons of highgrade ore per week, the high-grade ore constraint would fall from 24 to 18 tons per week. The new high-grade ore constraint, reflecting a parallel leftward shift, is written

and has endpoints (0B, 3A) and (9B, 0A). With such a reduction in required high-grade ore sales, the high-grade ore constraint would no longer be binding and the optimal production point would shift to point W, and A = 1 and B = 7 (because SM = SB = 0). At this point, high-grade ore production would equal 20 tons, or 2 tons more than the new high-grade ore requirement:

with operating costs of

Total Cost = $10,000A + $5,000B = $10,000(1) + $5,000(7) = $45,000

Therefore, renewing a contract to provide one of its current customers with 6 tons of highgrade ore per week would result in our earlier operating decision of A = 2 and B = 6 and total costs of $50,000, rather than the A = 1 and B = 7 and total costs of $45,000 that would otherwise be possible. The marginal cost of renewing the 6-ton contract is $5,000, or $833 per ton.

Marginal Cost = Change in Operating Costs Number of Tons

D. In general, the isocost relation for this problem is

where Cq is any weekly cost level, and CA and CB are the daily operating costs for mines A and B, respectively. In terms of the graph, A is on the vertical axis and B is on the horizontal axis. From the isocost formula we find the following:

with an intercept of Cq/Ca and a slope equal to -(CB/Ca). The isocost line will become steeper as CB increases relative to CA. The isocost line will become flatter (slope will approach zero) as CB falls relative to CA.

If CA increases to slightly more than $15,000, the optimal feasible point will shift from point X (6B, 2A) to point V (7B, 1.67A), because the isocost line slope will then be less than -1/3, the slope of the high-grade ore constraint (A = 4 - (1/3)B). Thus, an increase in CA from $10,000 to at least $15,000, or an increase of at least $5,000, is necessary before the optimal operating decision will change.

E. An increase in CB of at least $5,000 to slightly more than $10,000 will shift the optimal point from point X to point Y (2B, 6A), because the isocost line slope will then be steeper than -1, the slope of the medium-grade ore constraint (A = 8 - B).

An increase in CB to slightly more than $30,000 will be necessary before point Z (1.67B, 7A) becomes optimal. With CB > $30,000 and CA = $10,000, the isocost line slope will be steeper than -3, the slope of the low-grade ore constraint, A = 12 - 3B.

As seems reasonable, the greater CB is relative to CA, the more mine A will tend to be employed. The greater CA is relative to CB, the more mine B will tend to be employed.

ST9.2 Profit Maximization. Interstate Bakeries, Inc., is an Atlanta-based manufacturer and distributor of branded bread products. Two leading products, Low Calorie, QA, and High Fiber, QB, bread, are produced using the same baking facility and staff. Low Calorie bread requires 0.3 hours of worker time per case, whereas High Fiber bread requires 0.4 hours of worker time per case. During any given week, a maximum of 15,000 worker hours are available for these two products. To meet grocery retailer demands for a full product line of branded bread products, Interstate must produce a minimum of 25,000 cases of Low Calorie bread and 7,500 cases of High Fiber bread per week. Given the popularity of low-calorie products in general, Interstate must also ensure that weekly production of Low Calorie bread is at least twice that of High Fiber bread.

Low Calorie bread is sold to groceries at a price of $42 per case; the price of High Fiber bread is $40 per case. Despite its lower price, the markup on High Fiber bread substantially exceeds that on Low Calorie bread. Variable costs are $30.50 per case for Low Calorie bread, but only $17 per case for High Fiber bread.

A. Set up the linear programming problem that the firm would use to determine the profit-maximizing output levels for Low Calorie and High Fiber bread. Show both the inequality and equality forms of the constraint conditions.

B. Completely solve the linear programming problem.

C. Interpret the solution values for the linear programming problem.

D. Holding all else equal, how much would variable costs per unit on High Fiber bread have to fall before the production level indicated in part B would change?

ST9.2 Solution

A. First, the profit contribution for Low Calorie bread, QA, and High Fiber bread, QB, must be calculated.

Thus,

Profit contribution _ . Variable costs = Price -per unit per unit nA = $42 - $30.50 = $11.50 per case of QA nB = $40 - $17 = $23 per case of QB

This problem requires maximization of profits, subject to limitations on the amount of each product produced, the acceptable ratio of production, and available worker hours. The linear programming problem is

Maximize Subject to

n |
= |
$11.50QA |

QA |
— |
25,000 |

QB |
— |
7,500 |

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