The Textile Mill produces five different fabrics. Each fabric can be woven on on

ID: 346599 • Letter: T

Question

The Textile Mill produces five different fabrics. Each fabric can be woven on one or more of the mill’s 38 looms. The sales department’s forecast of demand for the next month is shown below, along with data on the selling price per yard, variable cost per yard, and purchase price per yard. The mill operates 24 hours a day and is scheduled for 30 days during the coming month.

Fabric

Demand (yards)

Selling price ($/yard)

Variable Cost ($/yard)

Purchase Price ($/yard)

16,500

0.99

0.66

0.80

22,000

0.86

0.55

0.70

62,000

1.10

0.49

0.60

7,500

1.24

0.51

0.70

62,000

0.70

0.50

0.70

The mill has two types of looms: dobbie and regular. The dobbie looms are more versatile and can be used for all five fabrics. The regular looms can produce only three of the fabrics. The mill has a total of 38 looms: 8 are dobbie and 30 are regular. The rate of production for each fabric on each type of loom is given in the following table. The time required to change over from producing one fabric to another is negligible and does not have to be considered.

Loom Rate (yards/hour)

Fabric

Dobbie

Regular

4.63

—

4.63

—

5.23

4.17

The Textile Mill satisfies all demand with either its own fabric or fabric purchased from another mill. Fabrics that cannot be woven at the Scottsville Mill because of limited loom capacity will be purchased from another mill. We use following linear programming model to maximize the profit of the Textile Mill and to answer the management’s questions:

Let X3R = Yards of fabric 3 on regular looms

X4R = Yards of fabric 4 on regular looms

X5R = Yards of fabric 5 on regular looms

X1D = Yards of fabric 1 on dobbie looms

X2D = Yards of fabric 2 on dobbie looms

X3D = Yards of fabric 3 on dobbie looms

X4D = Yards of fabric 4 on dobbie looms

X5D = Yards of fabric 5 on dobbie looms

Y1 = Yards of fabric 1 purchased

Y2 = Yards of fabric 2 purchased

Y3 = Yards of fabric 3 purchased

Y4 = Yards of fabric 4 purchased

Y5 = Yards of fabric 5 purchased

Max 0.61X3R + 0.73X4R + 0.20X5R + 0.33X1D + 0.31X2D + 0.61X3D + 0.73X4D + 0.20X5D + 0.19Y1 + 0.16Y2 + 0.50Y3 + 0.54Y4

Subject to:

0.1912X3R + 0.1912X4R + 0.2398X5R £ 21600 (Regular Hours Available)

0.21598X1D + 0.21598X2D + 0.1912X3D + 0.1912X4D + 0.2398X5D £ 5760 (Dobbie Hrs Available)

X1D + Y1

= 16500

X2D + Y2

= 22000 (Demand Constraints)

X3R + X3D + Y3

= 62000

X4R + X4D + Y4

= 7500

X5R + X5D + Y5

= 62000

ALL variables >=0

OPTIMAL SOLUTION OBTAINED WITH LINGO:

Optimal Objective Value

62531.49090

Variable

Value

Reduced Cost

X3R

27707.80815

0.00000

X4R

7500.00000

0.00000

X5R

62000.00000

0.00000

X1D

4668.80000

0.00000

X2D

22000.00000

0.00000

X3D

0.00000

-0.01394

X4D

0.00000

-0.01394

X5D

0.00000

-0.01748

11831.20000

0.00000

-0.01000

34292.19185

0.00000

-0.08000

0.00000

-0.06204

Constraint

Slack/Surplus

Dual Value

0.00000

0.57530

0.00000

0.64820

0.00000

0.19000

0.00000

0.17000

0.00000

0.50000

0.00000

0.62000

0.00000

0.06204

Objective

Allowable

Coefficient

Increase

Decrease

0.61000

0.01394

0.11000

0.73000

Infinite

0.01394

0.20000

Infinite

0.01748

0.33000

0.01000

0.01575

0.31000

Infinite

0.01000

0.61000

0.01394

Infinite

0.73000

0.01394

Infinite

0.20000

0.01748

Infinite

0.19000

0.01575

0.01000

0.16000

0.01000

Infinite

0.50000

0.11000

0.01394

0.54000

0.08000

Infinite

0.00000

0.06204

Infinite

RHS

Allowable

Value

Increase

Decrease

21600.00000

6556.82444

5297.86007

5760.00000

2555.33477

1008.38013

16500.00000

Infinite

11831.20000

22000.00000

4668.80000

11831.20000

62000.00000

Infinite

34292.19185

7500.00000

27707.80815

7500.00000

62000.00000

22092.07648

27341.95794

What is the optimal production schedule and loom assignments for each fabric?

How many yards of each fabric must be purchased from another mill?

What is the maximum profit attainable with the suggested production schedule?

If the purchase price of fabric 3 is decreased by $0.10, would the optimal solution change?

If the mill increased the selling price of fabric 2 on dobbie looms to $1.00, would the production schedule change? How much profit change would you expect?

How much is it worth for the company to have an extra regular hour available?

How much is it worth for the company to have an extra dobbie hour available?

What is the maximum value of the 9th Dobbie Loom; i.e., how much they should be willing to pay for the additional dobbie loom?

Management would like to understand the effects of different demand levels for different fabrics on the optimal solution and the total profit. Discuss the range of feasibility and the value of extra demand for each fabric.

If the company has to choose only one fabric to promote by additional advertisement, which fabric they should choose and why?

If they increase the selling price for fabric 1 and 4 by $0.10 simultaneously, would the optimal solution change? What would be the optimal total cost?

After implementing lean strategies, they plan to increase available regular hours to 24700 and available dobbie hours to 3500. Will there be any savings or total cost increase?

Fabric

Demand (yards)

Selling price ($/yard)

Variable Cost ($/yard)

Purchase Price ($/yard)

16,500

0.99

0.66

0.80

22,000

0.86

0.55

0.70

62,000

1.10

0.49

0.60

7,500

1.24

0.51

0.70

62,000

0.70

0.50

0.70

Explanation / Answer

What is the optimal production schedule and loom assignments for each fabric?

How many yards of each fabric must be purchased from another mill?

What is the maximum profit attainable with the suggested production schedule?

Max. profit = 62531.49090

If the purchase price of fabric 3 is decreased by $0.10, would the optimal solution change?

The present objective coefficient of Y3 is $0.50, with a range of optimality [0.5 - 0.01394, 0.5+0.11] i.e. [0.486, 0.61]. So, when decrease by $0.10, it will be $0.40 and comes out of range. So, the optimal solution will change.

If the mill increased the selling price of fabric 2 on Dobbie looms to $1.00, would the production schedule change? How much profit change would you expect?

The relevant variable is X2D. The objective coefficient is 0.31 and the allowable increase in infinity. So, whatever be the increase in objective coefficient (or the selling price), the optimal solution will not change. The profit will, however, change as -

62531.4909 + (1 - 0.31)*22000 = 77711.4909

How much is it worth for the company to have an extra regular hour available?

The dual price of the regular hour constraint (i.e. the first one) is 0.57530. So, for one additional hour, a $0.57530 additional profit will be realized. So, the company can pay a price less than 0.57530 for one unit of an additional regular hour.

How much is it worth for the company to have an extra Dobbie hour available?

The dual price of the Dobbie hour constraint (i.e. the second one) is 0.64820. So, for one additional hour, a $0.64820 additional profit will be realized. So, the company can pay a price less than 0.64820 for one unit of an additional Dobbie hour.

What is the maximum value of the 9th Dobbie Loom; i.e., how much they should be willing to pay for the additional Dobbie loom?

One extra room = 1*30*24 = 720 extra hours. From the dual price of this constraint, the company should be willing to pay 720 x $0.64820 = $466.7

Constraints 3,4,5,6, and 7 are relevant to answer this question.

The allowable increase and decrease represent the range of the demand levels at which the dual prices remain intact.

For example, for fabric 4, the present demand level is 7500. if we increase the level of the range [7500-7500, 7500+27707], the present dual price of 0.62 remains the same. In other words, within this range of feasibility, the objective value will change by $0.62 for one unit change in demand. This applies to all others.

If the company has to choose only one fabric to promote by additional advertisement, which fabric they should choose and why?

Fabric 4 because it has the highest dual price of 0.62. One unit additional demand will give the highest impact in the profit.

Fabric Production in Dobbie Production in Regular Purchase Total 1 4669 - 11831 16500 2 22000 - 0 22000 3 0 27708 34292 62000 4 0 7500 0 7500 5 0 62000 0 62000

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