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# Linear programming

The "Mill" produces five different fabrics. Each fabric can be woven
on one or more of the mill's 38 looms. The sales department has
forecast demand for the next month. The demand data are shown in Table
1.0, 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.

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 Table 1.1. The time to
change over from producing one fabric to another is negligible and
does not have to be considered.

The Mill satisfies all demand with either its own fabric or fabric
purchased from another mill. That is, fabrics that cannot be woven at
The Mill because of limited loom capacity will be purchased from
another mill. The purchase price of each fabric is also shown in Table
1.0.

Question
Develop a model that can be used to schedule production for The Mill,
and at the same time, determine how many yards of each fabric must be
purchased from another mill. Include a discussion and analysis of the

1. The final production schedule and loom assignments for each fabric
2. The projected total contribution to profit
3. A discussion of the value of additional loom time (The Mill is
the monthly profit contribution of this additional loom?)
4. A discussion of the objective coefficient ranges
5. A discussion of how the objective of minimizing total costs would
provide a different model than the objective of maximizing total
profit contribution: How would the interpretation of the objective
coefficients ranges differ for these two models?

Table 1.0
Monthly Demand, Selling Price, Variable Cost, and Purchase Price Data
for The Mill

Demand Selling Price Variable Cost Purchase Price
Fabric (yards) (\$/yard) (\$/yard) (\$/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

Table 1.1
Loom Production Rates for The Mill

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

* Fabrics 1 and 2 can be manufactured only on the dobbie loom.