Question 4 (15 Marks)

With 7 years of experience as a salesperson for a national retailer, John has decided to start his own business. He plans to sell designer clothing and has already found a convenient location in a large suburban shopping centre. 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: (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 1

5M1 + 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 10

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

Cell Name FinalValue ReducedCost ObjectiveCoefficient AllowableIncrease

$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

Cell Name FinalValue ShadowPrice ConstraintR.H. Side AllowableIncrease

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

$G$9 Maximum inventory 75,000 8.51 75 000 5,620 11

Required: a) What is the expected contribution margin and operating income for the first year of operations? (4 Marks)

b) 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. (4 Marks)

c) 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 marks)

d) Explain and define "binding" and "slack". (3 Marks)