Assignment:
Q: 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)
|
|
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 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:
a) What is the optimal production schedule and loom assignments for each fabric?
|
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
|
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?