From the tenth floor of her office building, Katherine Rally watches the swarms of New Yorkers fight their way through the streets infested with yellow cabs and the sidewalks littered with hot dog stands. On this sweltering July day, she pays particular attention to the fashions worn by the various women and wonders what they will choose to wear in the fall. Her thoughts are not simply random musings; they are critical to her work since she owns and manages TrendLines, an elite women€™s clothing company.

The fall casual fashions include:

She knows that for the next month, she has ordered 45,000 yards of wool, 28,000 yards of acetate, 9,000 yards of cashmere, 18,000 yards of silk, 30,000 yards of rayon, 20,000 yards of velvet, and 30,000 yards of cotton for production. The prices of the materials are as follows:

(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)?

(e) The sewing staff encounters difficulties sewing the arms and lining into the wool blazers 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 Trend- Lines 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 concern only the sales during September and October.) What should the new production plan be to maximize profit?

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Labor and Machine Cost Clothing Item Materials Requirements Price Tailored wool slacks 3 yards of wool 2 yards of acetate for lining 1.5 yards of cashmere 1.5 yards of silk 0.5 yard of silk 2 yards of rayon 1.5 yards of acetate for lining 2.5 yards of wool 1.5 yards of acetate for lining $300 $160 Cashmere sweater Silk blouse Silk camisole Tailored skirt $450 $180 $120 $270 $150 $100 60 $120 Wool blazer 320 $140 Labor and Machine Cost Clothing Item Materials Requirements Price Velvet pants 350 3 yards of velvet 2 yards of acetate for lining 1.5 yards of cotton 0.5 yard of cotton 1.5 yards of velvet 1.5 yards of rayon $175 Cotton sweater Cotton miniskirt Velvet shirt Button-down blouse $130 S 75 $200 $120 S 60 S 40 $160 90 MaterialPrice per yarod Wool Acetate Cashmere Silk Rayon Velvet Cotton 9.00 1.50 60.00 $13.00 2.25 $12.00 2.50