Question 5. (20 points) The Textile Mill produces five different fabrics. Each fabric can be woven on one or more of the mills 38 looms. The sales departments 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)
1	16,500	0.99	0.66	0.80
2	22,000	0.86	0.55	0.70
3	62,000	1.10	0.49	0.60
4	7,500	1.24	0.51	0.70
5	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
1	4.63
2	4.63
3	5.23	5.23
4	5.23	5.23
5	4.17	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 managements 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
Y1	11831.20000	0.00000
Y2	0.00000	-0.01000
Y3	34292.19185	0.00000
Y4	0.00000	-0.08000
Y5	0.00000	-0.06204
Constraint	Slack/Surplus	Dual Value
1	0.00000	0.57530
2	0.00000	0.64820
3	0.00000	0.19000
4	0.00000	0.17000
5	0.00000	0.50000
6	0.00000	0.62000
7	0.00000	0.06204
Objective	Allowable	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	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

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

1 dobbie loom for fabric one, 6 dobbie looms for fabric two, 7 regular looms for fabric 3, 2 regular looms for fabric four, 20 regular looms for fabric 5 the whole time (one month); one doobie looms for fabric 1 for 4/10 of month and the rest of the time used for fabric two; one regular loom to make fabric 3 for 4/10 month and to make fabric 5 the rest of the month

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

Fabric 1 11813.2 purchased from another mill

Fabric 2 none

Fabric 3 34,292.18 purchased from somewhere else

Fabric 4 none

Fabric 5 - none

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

The profit would be $62,531.48

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

e) 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?

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

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

An extra doomie loom would be best to be used to make fabric 1 all month and an additional profit of $466.70 and 3333.6 yard of fabric one would be produced. This is because fabric 1 and 3 are being brought more expense than to produce them.

h) 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?

i) 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.

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

k) 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?

l) After implementing lean strategies, they plan to increase available regular hours to 25000 and available dobbie hours to 4000. Will there be any savings or total cost increase?