$118$ Chapter Three Linear Programming: Formulation and Applications

company cannot sell them. TrendLines can produce less than the

forecasted demand, however, since the company is not required

to meet the demand. The cashmere sweater also has limited

demand because it is quite expensive, and TrendLines knows it

can sell at most $4,000$ cashmere sweaters. The silk blouses and

camisoles have limited demand because many women think silk

is too hard to care for, and TrendLines projects that it can sell at

most $12,000$ silk blouses and $15,000$ silk camisoles.

The demand forecasts also indicate that the wool slacks,

tailored skirts, and wool blazers have a great demand because

they are basic items needed in every professional wardrobe.

Specifically, the demand is $7,000$ pairs of wool slacks and $5,000$

wool blazers. Katherine wants to meet at least $60$ percent of the

demand for these two items to maintain her loyal customer base

and not lose business in the future. Although the demand for

tailored skirts could not be estimated, Katherine feels she should

make at least $2,800$ of them.

a$.$ Ted is trying to convince Katherine not to produce any velvet

shirts since the demand for this fashion fad is quite low. He

argues that this fashion fad alone accounts for $$500,000$ of

the fixed design and other costs. The net contribution $($price

of clothing item $-$ materials cost $-$ labor cost$)$ from selling

the fashion fad should cover these fixed costs. Each velvet

shirt generates a net contribution of $$22.$ He argues that given

the net contribution, even satisfying the maximum demand

will not yield a profit. What do you think of Ted's argument?

b$.$ Formulatc and solve a linear programming problem to maximize

profit given the production, resource, and demand constraints.

Before she makes her final decision, Katherine plans $t_0$

explore the following questions independently, except where

otherwise indicated.

c$.$ The textile wholesaler informs Katherine that the velvet can$-$

not be sent back because the demand forecasts show that the

demand for velvet will decrease in the future. Katherine $ci{n}^2$

therefore get no refund for the velvet. How does this fact

change the production plan?

d$.$ What is an intuitive economic explanation for the difference

between the solutions found in parts $b$ and $c?$

$e.$ The sewing staff encounters difficulties sewing the arms and

lining into the wool blazer since the blazer pattern has an

awkward shape and the heavy wool material is difficult to

cut and sew. The increased labor time to sew a wool blazer

increases the labor and machine cost for each blazer by $$80.$

Given this new cost, how many of each clothing item should

TrendLines produce to maximize profit?

f$.$ The textile wholesaler informs Katherine that since another

textile customer canceled his order, she can obtain an extra

$10,000$ yards of acetate. How many of each clothing item

should TrendLines now produce to maximize profit?

g$.$ TrendLines assumes that it can sell every item that was not

sold during September and October in a big sale in November

at $60$ percent of the original price. Therefore, it can sell all

items in unlimited quantity during the November sale. $($The

previously mentioned upper limits on demand only concern

the sales during September and October.$)$ What should the

new production plan be to maximize profit?