Question

Consider the following LP formulation of the modified Dog Food example you learned in class (by adding gruel 5). The objective coefficients represent unit costs ($ per 16 oz) for the gruels. The RHS’s represent the nutrition requirements (measured in oz). MIN 4 G1 + 6 G2 + 3 G3 + 2 G4 + 5 G5 SUBJECT TO 2) 3 G1 + 5 G2 + 2 G3 + 3 G4 + 4 G5 >= 3 (Protein Req.) 3) 7 G1 + 4 G2 + 2 G3 + 8 G4 + 2 G5 >= 5 (Carbohydrate Req.) 4) 5 G1 + 6 G2 + 6 G3 + 2 G4 + 4 G5 >= 4 (Fat Req.) 5) G1 + G2 + G3 + G4 + G5 = 1 (Total %) END Below is the LINDO output of the problem. OBJECTIVE FUNCTION VALUE 1) 3.000000 VARIABLE VALUE REDUCED COST G1 0.000000 0.500000 G2 0.166667 0.000000 G3 0.333333 0.000000 G4 0.500000 0.000000 G5 0.000000 1.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 -1.000000 3) 0.333333 0.000000 4) 0.000000 -0.500000 5) 0.000000 2.000000 RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE G1 4.000000 INFINITY 0.500000 G2 6.000000 2.000000 3.000000 G3 3.000000 1.000000 3.000000 G4 2.000000 2.000000 INFINITY G5 5.000000 INFINITY 1.000000 RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE 2 3.000000 1.000000 0.500000 3 5.000000 0.333333 INFINITY 4 4.000000 0.250000 2.000000 5 1.000000 0.142857 0.038462 Consider yourself as the dog food producer. Answer the following questions based on the LP output:

1. (1pts) What is the minimum cost to make such dog food?

2. (3pts) The dog food buyer is willing to pay $0.5 more if you increase the fat by 1 oz in the 16 oz can of dog food and keep other nutrients unchanged. Do you want to take the offer? Explain.

3. (2pts) Among all gruels, which is least attractive to you? Explain.

4. (3pts) If the gruel 1 seller wants to offer you 10% discount on gruel 1, will you be interested in buying the gruel 1? If not, what would be the attractive price for gruel 1? Explain.

5. (4pts) If the unit price (cost) of gruel 2, gruel 3, and gruel 5 increase by $0.5, respectively, and that of gruel 4 decreases by $0.5, do you want to change the current dog food mix? Will the minimum cost for the dog food change? If so, by how much? Show your calculations.

6. (3pts) Would you be interested in buying gruel 6 that sells for $4 per 16 oz, which contains 3 oz of protein, 6 oz of carbohydrate, and 7 oz of fat? Show your calculations.

7. (4pts) If the dog food buyer requests to increase the protein level by 0.6 oz in exchange for reducing the fat level by 0.7 oz, would you take this offer? Explain and show your calculation.

Answer 1

1.

Minimum cost = Objective function value at the optimality condition = $3

2.

The dual price is -$0.5. Increasing the fat more than the allowable increase which is 0.25 oz will cause more cost increase than the dual price. So, the $0.5 additional profit will not be sufficient. So, the offer may not be considered.

3.

Protein is the most cost-affecting binding resource. If we compute the unit profit per unit protein usage for all the gruels, we will find that the highest is for G3 (1.5). So, G3 will be the most profitable.

4.

The reduced cost is 0.500, so the reduction should at least be $0.50. Here, the reduction proposed in only 10$ of $4 which is $0.40. So, this is not an offer worth considering.

5.

The total % change is less than 100%. So, the optimal solution i.e. the current mix remains unchanged.

The total minimum cost also changes according to the increase and decrease of these coefficients.

So, the new minimum cost = 4*0 + (6+0.5)*0.16667 + (3+0.5)*0.3333 + (2 - 0.5)*0.50 + (5 + 0.5)*0.000 = 3.00

6.

3 * Dual price of protein + 6 * Dual price of carbs + 7 * Dual price of fat = 3*(-1) + 6*(0) + 7*(-1) = -$10

The cost per unit is $4. Since the reduction is cost per resource saved will be more than the cost of purchase (i.e. 10 > 4), gruel-6 is worth buying.

7.

Since the total % change is less than 100%, the dual prices of these two constraints will remain intact i.e. will be -1.0 and -0.50 respectively.

The profit will further reduce by 0.6*(-1) + (-0.7)*(-0.5) = -0.25, which is negative reduction i.e. increase So, no, this is an offer not worth considering.