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$.$ Formulate and solve a linear programming problem to maximize profit given the production, resource, and demand constraints.

Before she makes her final decision, Katherine plans to explore the following questions independently, except where otherwise indicated.

c$.$ The textile wholesaler informs Katherine that the velvet cannot be sent back because the demand forecasts show that the demand for velvet will decrease in the future. Katherine can 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 $c?$

$c.$ 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?