The Super Sausage Company (SSC) has recently experienced drastic changes in raw material prices, and the manager has directed an analyst to reexamine the proportions in which SSC mixes ingredients to manufacture sausage. Sausage manufacture involves meeting two key product requirements. The percentage of protein, by weight, must be at least 15 percent: and the percentage of fat, be weight, cannot exceed 30 percent (remaining weight is filler). SSC has the following four raw materials available for mixing, with the following characteristics: (a) Formulate a linear program (LP) that would aid SSC in determining its most desirable mixing schedule, and solve for an optimal solution. (b) How many percentage of protein requirement can be increased so that the optimal solution found in (a) is kept unchanged? (c) Use the sensitivity analysis to find out how much the cost of needs to be reduced to ensure C is included in the optimal mixing schedule.
(Solved Homework): The Super Sausage Company (SSC) has recently experienced drastic changes in raw material prices,…
Expert Answer
The decision variable in Green
The constrains are in Blue
And the objective function is in Orange
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On solving:
| Decision var | |||||
| Ingredient | Percent Protein | Pecent Fat | Cost per pound | Share of ingredient | |
| A | 40% | 10% | 1.8 | 0 | |
| B | 20% | 15% | 0.75 | 0.666666667 | 66.67% |
| C | 10% | 35% | 0.4 | 0 | |
| D | 5% | 40% | 0.15 | 0.333333333 | 33.33% |
| Total Protein | Total Fat | Total cost | Total | ||
| 15.00% | 23.33% | 0.55 | 1 | ||
| >= | <= | Minimize | = | ||
| 15% | 30% | Objective func | 1 | ||
| Constrain Protein | Constrain Fat | Totality constrain |
B.
As the protein constrain is binding , that can not be changed.
C.
| Variable Cells | |||||
| Final | Reduced | ||||
| Cell | Name | Value | Gradient | ||
| $F$4 | A Share of ingredient | 0 | 0.24999996 | ||
| $F$5 | B Share of ingredient | 0.666666667 | 0 | ||
| $F$6 | C Share of ingredient | 0 | 0.050000002 | ||
| $F$7 | D Share of ingredient | 0.333333333 | 0 | ||
| Constraints | |||||
| Final | Lagrange | ||||
| Cell | Name | Value | Multiplier | ||
| $C$9 | Total Protein | 0.15 | 3.999999901 | ||
| $D$9 | Total Fat | 0.233333333 | 0 | ||
| $F$9 | Total | 1 | -0.049999992 | ||
As the gradient is of 0.05, so the cost needs to be reduced by
| 0.050000002
to be included in the optimal mix |
