Question: CASE 4.1 Fabrics and Fall Fashions,hillier_lieberman_2005, introduction to opearation research, i wantto solve it on ampl but how ?
From the tenth floor of her office building, Katherine Rallywatches the swarms of New Yorkers fight their way through thestreets infested with yellow cabs and the sidewalks littered withhot dog stands, On this sweltering July day, she pays particularattention to the fashions worn by the various women and wonderswhat they will choose to wear in the fall. Her thoughts are notsimply random musings; they are critical to her work since she ownsand manages TrendLines, an elite women's clothing company.
Today is an especially important day because she must meet with TedLawson, the production manager, to decide upon next month'sproduction plan for the fall line. Specifically, she must determinethe quantity of each clothing item she should produce given theplant's production capacity, limited resources, and demandforecasts. Accurate planning for next month's production iscritical to fall sales since the items produced next month willappear in stores during September, and women generally buy themajority of the fall fashions when they first appear inSeptember.
She turns back to her sprawling glass desk and looks at thenumerous papers covering it. Her eyes roam across the clothingpatterns designed almost six months ago, the lists of materialsrequire3ments for each pattern, and the lists of demand forecastsfor each pattern determined by customer surveys at fashion shows.She remembers the hectic and sometimes nightmarish days ofdesigning the fall line and presenting it at fashion show in NewYork, Milan, and Paris. Ultimately, she paid her team of sixdesigners a total of $860,000 for their work on her fall line. Withthe cost of hiring runway models, hair stylists, and makeupartists, sewing and fitting clothes, building the set,choreographing and rehearsing the show, and renting the conferencehall, each of the three fashion shows cost her an additional$2,700,000.
She studies the clothing patterns and material requirements. Herfall line consists of both professional and casual fashions. Shedetermined the prices for each clothing item by taking into accountthe quality and cost of material, the cost of labor and machining,the demand for the item, and the prestige of the TrendLines brandname.
The fall professional fashions include:
Clothing Item | Materials Requirements | Price | Labor and Machine Cost |
Tailored Wool Slacks | 3 yards of wool, 2 yards of acetate for lining | $ 300 | $ 160 |
Cashmere Sweater | 1.5 yards of cashmere | $ 450 | $ 150 |
Silk Blouse | 1.5 yards of silk | $ 180 | $ 100 |
Silk Camisole | 0.5 yard of silk | $ 120 | $ 60 |
Tailored Skirts | 2 yards of rayon, 1.5 yards of acetate for lining | $ 270 | $ 120 |
Wool Blazer | 2.5 yards of wool, 1.5 yards of acetate for lining | $ 320 | $ 140 |
The fall casual fashions include:
Clothing Item | Materials Requirements | Price | Labor and Machine Cost |
Velvet Pants | 3 yards of velvet, 2yards of acetate for lining | $ 350 | $ 174 |
Cotton Sweater | 1.5 yards of cotton | $ 130 | $ 60 |
Cotton Miniskirt | 0.5 yards of cotton | $ 75 | $ 40 |
Velvet Shirt | 1.5 yards of velvet | $ 200 | $ 160 |
Button-Down Blouse | 1.5 yards of rayon | $ 120 | $ 90 |
She knows that for the next month, she has ordered 45,000 yards ofwool, 28,000 yards of acetate, 9,000 yards of cashmere, 18,000yards of silk, 30,000 yards of rayon, 20,000 yards of velvet, and30,000 yards of cotton for production. The prices of the materialsare as follows:
Material | Price per yard |
Wool | $ 9.00 |
Acetate | $ 1.50 |
Cashmere | $ 60.00 |
Silk | $ 13.00 |
Rayon | $ 2.25 |
Velvet | $ 12.00 |
Cotton | $ 2.50 |
Any material that is not used in production can be sent back tothe textile wholesaler for a full refund, although scrap materialcannot be sent back to the wholesaler.
She knows that the production of both the silk blouse and cottonsweater leaves leftover scraps of material. Specifically, for theproduction of one silk blouse or one cotton sweater, 2 yards ofsilk and cotton, respectively, are needed. From these 2 yards, 1.5yards are used for the silk blouse or the cotton sweater and 0.5yard is left as scrap material. She does not want to waste thematerial , so she plans to use the rectangular scrap of silk orcotton to produce a silk camisole or cotton miniskirt,respectively. Therefore, whenever a silk blouse is produced, a silkcamisole is also produced. Likewise, whenever a cotton sweater isproduced, a cotton miniskirt is also produced. Note that it ispossible to produce a silk camisole without producing a silk blouseand a cotton miniskirt without producing a cotton sweater.
The demand forecasts indicate that some items have limited demand.Specifically, because the velvet pants and velvet shirts arefashion fads, TrendLines ahs forecasted that it can sell only 5,500pairs of velvet pants and 6,000 velvet shirts. TrendLines does notwant to produce more than the forecasted demand because once thepants and shirts go out of style, the company cannot sell them.TrendLines can produce less than the forecasted demand, however,since the company is not required to meet the demand. The cashmeresweater also has limited demand because it is quite expensive, andTrendLines knows it can sell at most 4,000 cashmere sweaters. Thesilk blouses and camisoles have limited demand because many womenthink silk is too hard to care for, and TrendLines projects that itcan sell at most 12,000 silk blouses and 15,000 silk camisoles.
The demand forecasts also indicate that the wool slacks, tailoredskirts, and wool blazers have a great demand because they are basicitems needed in every professional wardrobe. Specifically, thedemand for wool slacks is 7,000 pairs of slacks, and the demand forwool blazers is 5,000 blazers. Katherine wants to meet at least 60percent of the demand for these two items in order to maintain herloyal customer base and not lose business in the future. Althoughthe demand for tailored skirts could not be estimated, Katherinefeels she should make at least 2,800 of them.
Q :Formulate and solve a linear programming problem to maximizeprofit given the production, resource, and demand constraints.using ampl
