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)               (1)

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           (2)             1.000000

       ROW   SLACK OR SURPLUS     DUAL PRICES

        2)         0.000000         -1.000000

        3)           (3)              (4)

        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           (5)                  (6)

       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           (7)

        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 partial LP output:

1. (7pts) Complete the LP output by filling in the missing numbers. Provide either calculations or explanations for your answers:

(1), (2), (3), (4), (5), (6), and (7)

2. (3pts) The dog food buyer asks for a 20% discount (of the minimum dog food cost in LINDO solution, i.e. (1) ) in exchange for a reduction of protein by 0.6oz. Do you want to take the offer? Why?
3. (2pts) What would the attractive price for you to use Gruel 5? Explain
4. (4pts) If the unit price (cost) of 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? With the price changes, what will be the minimum cost for the dog food? Show your calculations.
5. (2pts) A new gruel supplier wants to sell you its new gruel mix which contains, in a 16 oz of gruel, 3 oz of protein, 4 oz of carbohydrate, and 5 oz of fat. What is the maximum price you are willing to pay for the new gruel? Show your calculations.
6. (3pts) If a dog food buyer requests to increase the protein level by 0.5 oz in exchange for a reduction of the fat level. What would you recommend the minimum reduction to keep the same dog food price? Explain and show your calculation.

Answer 1

LP Output

1. An objective function is $3 i.e. to have the food prepared in right quantity $3 is required which will have zero amount of G1 & G5 but G2, G3 & G4 are in the ratio of 0.167, 0.33 & 0.5 respectively.

2. G5 will have no contribution to the final food preparation. Therefore, the value of G5 is 0

3. Protein will have a surplus of 0.33. Therefore, the value in 3 will be 5.33

4. Dual Prices for protein is zero i.e. there is no improvement in the objective function even if we reduce the constraint by 1 unit.

5. Allowable Increase is Infinity as the increased price will have no effect on the output. therefore, the answer is infinite.

6. The allowable decrease is 0.5 i.e. if the prices of G1 decreases by 0.5$ then it will have some share in the final food preparation.

7. The allowable decrease is Infinity i,e, with the given configuration the carbohydrate requirement will have 5.33 as the least number

(3pts)If we reduce the RHS of Protein equation from 3 to 2.4 we will get the optimum price and in this case, we get 2.5as the least price which is higher than the 20% reduction in the original price. Current price is 3 and 20% reduction should bring down the price to 2.4 however, with protein requirement going down by 0.6oz the price will be $2.5. Therefore, I will not take the order under the given situation.

(2pts)If we reduce the price of G5 by greater than $1 then it will be attractive to me. It can be seen from the allowable decrease for G5. Then the G5 will be a part of the solution and only G1 will not be required to make the solution.

(4pts) Under the new price system, the current mix will not change however, the optimal price will decrease from $3 to $2.92. This can be seen by changing the equation in Lindo. Objective function will change to 4G1+6G2+3.5G3+1.5G4+5.5G5 keeping other things constant.

(2pts) In this case, the maximum price is $3.625. This simulation can be done by changing the protein, carbohydrate and fat constraint equation RHS to 3,4 & 5 respectively.