The store will carry the goods of 4 well-known designers. John has decided that his inventory should not exceed $75,000.

John must decide how to allocate the floor space to the 4 brands he has decided to carry. Each

manufacturer requires that a minimum amount of floor space be allocated to its goods. The total surface of John’s store is 3,360 square feet.

The space requirements and some budgeted figures for the first year of operation are as follows:

Manufacturer                                                #1                    #2                    #3                    #4

Minimum floor space (square feet)                500                  700                  600                  900

Expected annual sales ($ per square foot)      300                  480                  450                  600

Average gross margin                                     40%                 45%                 38%                 35%

The following are annual costs:

Salaries           $100,000

Rent                 $40 per square foot

Utilities           $4,500

Salaries, rent, and utilities are fixed costs.

To solve his problem, John has set up the following linear program:

Maximize: π (M1, M2, M3, M4) = 108M1 + 187.2M2 + 153M3 + 192M4

Subject to:

M1 + M2 + M3 +M4 <= 3,360                                   Total floor space constraint

M1 > = 500                                                                 Minimum floor space for Manufacturer #1

M2 > = 700                                                                 Minimum floor space for Manufacturer #2

M3 > = 600                                                                 Minimum floor space for Manufacturer #3

M4 > = 900                                                                 Minimum floor space for Manufacturer #4

15M1 + 22M2 + 23.25M3 + 32.50M4 < = 75,000     Maximum inventory

Where:

M1 represents the number of square feet allocated to goods of manufacturer #1

M2 represents the number of square feet allocated to goods of manufacturer #2

M3 represents the number of square feet allocated to goods of manufacturer #3

M4 represents the number of square feet allocated to goods of manufacturer #4

The linear program was solved using Solver. The results are included in Exhibit 7-1.

EXHIBIT 7-1

Answer Report

Target Cell (Max)

Cell                 Name                                                  Original Value           Final Value

$F$10              Total contribution margin                   449 640                       525 371

Adjustable Cells

Cell                 Name                                                  Original Value           Final Value

$B$9               Square feet allocated to M #1             500                             500

$C$9               Square feet allocated to M #2             700                             1,105

$D$9               Square feet allocated to M #3               600                            600

$E$9                Square feet allocated to M #4               900                            900

Constraints

Cell                 Name                                      Cell Value      Formula         Status             Slack

$F$9                Total surface available            3,105               $F$9<=3360 Not binding     255

$G$9               Maximum inventory               75 000.00        $G$9<=75000 Binding           0

$D$9               Square feet allocated to M#3     600              $D$9>=600     Binding           0

$E$9                Square feet allocated to M#4     900             $E$9>=900     Binding           0

$B$9               Square feet allocated to M#1      500              $B$9>=500     Binding          0

$C$9               Square feet allocated to M#2    1,105            $C$9>=700     Not binding    405

Sensitivity Report

Adjustable Cells

           

Final              Reduced         Objective        Allowable

Cell                 Name                          Value              Cost                Coefficient    Increase

$B$9               Sq.ft allocated to M#1   500               19.64               108                  20

$C$9               Sq.ft.allocated to M#2   1,105            0.00                187.2               1E+30

$D$9               Sq.ft allocated to M#3    600              44.84               153                  45

$E$9                Sq.ft allocated to M #4 900              84.55             192                  85

Constraints

Final               Shadow           Constraint      Allowable

Cell                 Name                          Value              Price               R.H. Side        Increase

$F$9                Total surface available 3,105               0                    3,360               1E+30

$G$9               Maximum inventory     75,000           8.51                75 000             5,620

Required:

1. What is the expected contribution margin and operating income for the first year of operations? 

2. By how much per unit and in total would the contribution margin change if the constraint on the level of inventories were set to $80,000? Explain your answer. 

3. Would the optimal solution be the same if there was not a constraint on the level of inventories? Explain your answer, and calculate the change in contribution margin if any. 

4. Explain and define what is “binding” and “slack”.