For a, c, d explain in detail using proper English. Justify your answers with necessary mathematical operations

CASE 4.1 Fabrics and Fall Fashions She turns back to her sprawling glass desk and looks at From the tenth floor of her office building. Katherine Rally the numerous papers covering it. Her eyes roam across the watches the swarms of New Yorkers fight their way through clothing patterns designed almost six months ago, the lists the streets infested with yellow cabs and the sidewalks lit- of materials requirements for each pattern, and the lists of tered with hot dog stands. On this sweltering July day, she demand forecasts for each pattern determined by customer pays particular attention to the fashions wom by the various surveys at fashion shows. She remembers the hectic and women and wonders what they will choose to wear in the sometimes nightmarish days of designing the fall line and fall. Her thoughts are not simply random musings; they are presenting it at fashion shows in New York, Milan, and Paris. critical to her work since she owns and manages TrendLines, Ultimately, she paid her team of six designers a total of an elite women's clothing company. $860,000 for their work on her fall line. With the cost of hir- Today is an especially important day because she must ing runway models, hair stylists, and makeup artists, sewing meet with Ted Lawson, the production manager, to decide and fitting clothes, building the set, choreographing and reupon next month's production plan for the fall line. Specif- hearsing the show, and renting the conference hall, each of ically, she must determine the quantity of each clothing the three fashion shows cost her an additional $2,700,000. item she should produce given the plant's production ca- She studies the clothing patterns and material requirepacity, limited resources, and demand forecasts. Accurate ments. Her fall line consists of both professional and casual planning for next month's production is critical to fall fashions. She determined the prices for each clothing item sales since the items produced next month will appear in by taking into account the quality and cost of material, the stores during September, and women generally buy the cost of labor and machining, the demand for the item, and majority of the fall fashions when they first appear in the prestige of the TrendLines brand name. September. The fall casual fashions include: She knows that for the next month, she has ordered shirts. Trendli ines does not want to produce more than the 45,000 yards of wool, 28,000 yards of acetate, 9,000 yards forecasied demand because once the pants and shirts go out of cashamere, 18,000 yards of silk. 30,000 yards of rayon, of style, the company cannot sell them. Trendl ines can pro20,000 yards of velvet, and 30,000 yards of cotion for pro-. doce less than the forecasted demand, however, since the duction. The prices of the materials are as follows: company is aot 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 hand 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 blawers have a great demand because they are basie itens needed in cvery professional wardrobe. Specifically, the demand for wool slacks is 7,000 pairs of slacks, and the demund for wool blarers is 5,000 blazers. Katherine wants to meet at least 60 persent of the Any material that is not used in production can be sent back demand for these two items in crder to maintain her loyal to the textile wholesaler for a full refund, although scrap ma- customer base and not lose business in the future. Although terial cannot be seat back to the wholesaler. the demand for tailored skins could not be estimated, She knows that the production of both the silk blouse Katherine feels she should make at leat 2,800of them. and cotten sweater leaves leftover scraps of material. Specifically, for the production of one silk blouse or one cothon (a) Ted is trying to convince Katherine not to produce any velvet sweater, 2 yards of silk and cotton, respectively, are needed. whirs since the demand for thas fachioe fad is quite low. He From these 2 yards, 1.5 yards are ased for the silk blouse argues that this fashion fad alooe acceunts for 5500000 of the or the coton sweater and 0.5 yard is left as scrap material. Fised design and other cons. The net contritution (price of She does not want to waste the material, so she plans to use. clothing incen - materials cost - laber cove from selling the the rectangular scrap of silk of cotton to produce a silk_. fashion fad shold coser these fived costs. Fach velvet shint camisole or cotton miniskirt, respectively. Therefore, when-_ generates a net contritution of 522 . He argues thut given the duced. Likewise, whenever a cotton sweater is prodoced, a yicld a prodit. What do yoe think of Ted x argumen? cotton miniskirt is also produced. Nose that it is possible to (b) Fomulate and solve a lineur programaning poothem to maximize produce a silk camisole without producing a silk blouse and profil given the production, mescerse, and demual constrainte. a cotton miniskirt without producing a cotton sweater. Before she makes her final decision, Katherine plans to ex - The demand forecasts indicate that some items have phore the following questions independently except where limited demand. Specifically, because the velvet pants and otherwise indicated. velvet shirts are fashion fads, Trendl.ines has forecasted that (c) The textile wholesaler informs Katherine that the velvet sanit can sell only 5,500 pairs of velvet pants and 6,000 velvet not be sent back because the demand forecaiss show that the demand for velvet will decrease in the future. Katherine can (f) The textile wholesaler informs Katherine that since another textherefore get no refund for the velvet. How does this fact change trile customer canceled his order, she can obtain an extra 10,000 the production plan? yards of acetate. How many of each clothing item should Trend- (d) What is an intuitive economic explanation for the difference be- Lines now produce to maximize profit? tween the solutions found in parts (b) and (c) ? (g) Trendl.ines assumes that it can sell every item that was not sold (e) The sewing staff encounters difficulties sewing the arms and during September and October in a big sale in November at 60 lining into the wool blazers since the blazer pattern has an awk-_ percent of the original price. Therefore, it can sell all items in ward shape and the heavy wool material is difficult to cut and unlimited quantity during the November sale. (The previously sew. The increased labor time to sew a wool blazer increases mentioned opper limits on demand concern only the sales durthe labor and machine cost for each blazer by S80, Given this ing September and October) What should the new production new cost, how many of each clothing item should Trendl ines plan be to maximize profit? produce to maximize profit? PREVIEWS OF ADDED CASES ON OUR WEBSITE (www.mhhe.com/hillier) CASE 4.2 New Frontiers of the students will be bused, so minimizing the total busing AmeriBank will soon begin offering Web banking to its cus- cost is one objective. Another is to minimize the inconvetomers. To guide its planning for the services to provide over nience and safety concerns for the students who will walk the Internet, a survey will be conducted with four different or bicycle to school. Given the capacities of the three age groups in three types of communities. AmeriBank is im- schools, as well as the need to roughly balance the number posing a number of constraints on how extensively each age of students in the three grades at each school, how can lingroup and each community should be surveyed. Linear pro- ear programming be used to determine how many students gramming is needed to develop a plan for the survey that from cach of the city's six residential areas should be aswill minimize its total cost while meeting all the survey con- signed to each school? What would happen if each entire straints under several different scenarios. residential area must be assigned to the same school? (This case will be continued in Cases 7.3 and 12.4.) CASE 4.3 Assigning Students to Schools After deciding to close one of its middle schools, the Springfield school board needs to reassign all of next year's middle school students to the three remaining middle schools. Many