The LP Model and LINGO Output Represent a Problem Whose

question 24

Essay

The LP model and LINGO output represent a problem whose solution will tell a specialty retailer how many of four different styles of umbrellas to stock in order to maximize profit.It is assumed that every one stocked will be sold.The variables measure the number of women's,golf,men's,and folding umbrellas,respectively.The constraints measure storage space in units,special display racks,demand,and a marketing restriction,respectively.
MAX 4 X1 + 6 X2 + 5 X3 + 3.5 X4
SUBJECT TO
2)2 X1 + 3 X2 + 3 X3 + X4 <= 120
3)1.5 X1 + 2 X2 <= 54
4)2 X2 + X3 + X4 <= 72
5)X2 + X3 >= 12
END
OBJECTIVE FUNCTION VALUE
1)318.00000 $$\begin{array} { c c c } \text { VARIABLE } & \text { VALUE } & \text { REDUCED COST } \\\text { X1 } & 12.000000 & .000000 \\\text { X2 } & .000000 & .500000 \\\text { X3 } & 12.000000 & .000000 \\\text { X4 } & 60.000000 & .000000\end{array}$$ $$\begin{array} { c c r } { \text { ROW } } & \text { SLACK OR SURPLUS } & \text { DUAL PRICE } \\\hline2) & .000000 & 2.000000 \\\text { 3) } & 36.000000 & .000000 \\\text { 4) } & .000000 & 1.500000 \\\text { 5) } & .000000 & -2.500000\end{array}$$ RANGES IN WHICH THE BASIS IS UNCHANGED: $\begin{array}{c}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text { OBJ. COEFFICIENT RANGES }\\\begin{array}{c}VARIABLE \\X 1 \\X 2 \\X 3 \\X 4\end{array}\begin{array}{ccc}\frac{\text { CURRENT }}{\text { COEFFICIENT }} & \frac{\text { ALLOWABLE }}{\text { INCREASE }} & \frac{\text { ALLOWABLE }}{\text { DECREASE }}\\4.000000 & 1.000000 & 2.500000 \\6.000000 & .500000 & \text { INFINITY } \\5.000000 & 2.500000 & .500000 \\3.500000 & \text { INFINITY } & .500000\end{array}\end{array}$ $$\begin{array}{l}\begin{array} { c c c c } &&\quad\quad\quad\quad\quad\quad\quad\quad\text { RIGHTHAND SIDE RANGES }\\& \text { CURRENT } & \text { ALLOWABLE } & \text { ALLOWABLE } \\\hline\text { ROW } & \text { RHS } & \text { INCREASE } & \text { DECREASE } \\2 & 120.000000 & 48.000000 & 24.000000 \\3 & 54.000000 & \text { INFINITY } & 36.000000 \\4 & 72.000000 & 24.000000 & 48.000000 \\5 & 12.000000 & 12.000000 & 12.000000\end{array}\end{array}$$ Use the output to answer the questions.
a.How many women's umbrellas should be stocked?
b.How many golf umbrellas should be stocked?
c.How many men's umbrellas should be stocked?
d.How many folding umbrellas should be stocked?
e.How much space is left unused?f. How many racks are used?g. By how much is the marketing restriction exceeded?h. What is the total profit?i. By how much can the profit on women's umbrellas increase before the solution would change?j. To what value can the profit on golf umbrellas increase before the solution would change?k. By how much can the amount of space increase before there is a change in the dual price?l. You are offered an advertisement that should increase the demand constraint from 72 to 86 for a total cost of $20. Would you say yes or no?

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Definitions:

Variable Costs

Charges that adjust in direct proportion to the amount of goods produced or sold.

Fixed Costs

Costs that do not change with the level of output or sales, such as rent, salaries, and loan payments.

Tax Rate

The percentage at which an individual or corporation is taxed, which can vary depending on income levels or profit margins.

Variable Cost

Expenditures that fluctuate based on production or sales volume, for instance, materials and manpower.