The Really Big Shoe Company Is a Manufacturer of Basketball

Essay

The Really Big Shoe Company is a manufacturer of basketball shoes and football shoes.Ed Sullivan,the manager of marketing,must decide the best way to spend advertising resources.Each football team sponsored requires 120 pairs of shoes.Each basketball team requires 32 pairs of shoes.Football coaches receive $300,000 for shoe sponsorship and basketball coaches receive $1,000,000.Ed's promotional budget is $30,000,000.The Really Big Shoe Company has a very limited supply (4 liters or 4,000cc)of flubber,a rare and costly raw material used only in promotional athletic shoes.Each pair of basketball shoes requires 3cc of flubber,and each pair of football shoes requires 1cc of flubber.Ed desires to sponsor as many basketball and football teams as resources allow.However,he has already committed to sponsoring 19 football teams and wants to keep his promises.
a.Give a linear programming formulation for Ed.Make the variable definitions and constraints line up with the computer output appended to this exam.
b.Solve the problem graphically,showing constraints,feasible region,and isoprofit lines.Circle the optimal solution,making sure that the isoprofit lines drawn make clear why you chose this point.(Show all your calculations for plotting the constraints and isoprofit line on the left to get credit.)

c.Solve algebraically for the corner point on the feasible region.
d.Part of Ed's computer output is shown following.Give a full explanation of the meaning of the three numbers listed at the end.Based on your graphical and algebraic analysis,explain why these numbers make sense.(Hint: He formulated the budget constraint in terms of $000.)
See the computer printout that follows.
Solver-Linear Programming
Solution
$\begin{array} { r r r r } \text { Variable } & \begin{array} { r } \text { Variable } \\ \text { Label } \end{array} & \begin{array} { r } \text { Original } \\ \text { Value } \end{array} & \begin{array} { r } \text { Coefficient } \\ \text { Sensitivity } \end{array} \\ \text { Var1 } & 19.000 C & 1.000 C & 0 \\ \text { Var2 } & 17.9167 & 1.000 C & 0 \\ \text { Constraint } & \text { Original } & \text { Slack or } & \text { Shadow } \\ \text { Label } & \text { RHV } & \text { Surplus } & \text { Price } \\ \text { Const1 } & & & \\ \text { Const2 } & 19 & 0 & 0 \\ \text { Const3 } & 3000 \mathrm { C } & 6383 & 0.0104 \\ & 400 \mathrm { C } & 0 & 0 \end{array}$
Objective Function
Value:                                                                 $36.91666667$
Sensitivity Analysis and Ranges
Objective Function Coefficients
$$\begin{array}{l}\begin{array} { r r r r } \begin{array} { r } \text { Variable } \\\text { Label }\end{array} & \begin{array} { r r r } \text { Lower } & \text { Original } & \text { Upper } \\\text { Limit }\end{array} & \text { Coefficient } & \text { Limit } \\\text { Var1 } & \text { No Limit } & 1 & 1.25 \\\text { Var2 } & 0.8 & 1 & \text { No Limit }\end{array}\\\text { Right-Hand-Side Values }\\\begin{array} { r r r r } \text { Constraint } & \begin{array} { r } \text { Lower } \\\text { Label }\end{array} & \begin{array} { r } \text { Original } \\\text { Value }\end{array} & \begin{array} { r } \text { Upper } \\\text { Limit }\end{array} \\\text { Const1 } & 12.2807 & 19 & 33.33333333 \\\text { Const2 } & 23616.67 & 3000 \mathrm { C } & \text { No Limit } \\\text { Const3 } & 228 \mathrm { C } & 400 \mathrm { C } & 4612.8\end{array}\end{array}$$
First Number: The shadow price of 0.0104 for the "Const3" constraint.
Second Number: The slack or surplus of 6383 for the "Const1" constraint.
Third Number: The lower limit of 12.2807 for the "Const1" constraint.