DEAR EXPERT I need to formulate a mathematical model for the business problem , we should define the business problem by explaining what the problem is ,constraints and variables. Definee what the objective is, what you want to learn for optimization, and what your limitations are , gives feasible solution , and need some sensitivity analysis.

Thank you

CASES 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 patter determined by customer pays particular attention to the fashions worn by the various surveys at fashion shows. She remembers the hectic and women and wonder 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 Ultimately, she paid her team of six designers a total of presenting it at fashion shows in New York, Milan, and Paris. critical to her work since she owns and manages TrendLines, $860.000 for their work on her fall line. With the cost of hir- an elite women's clothing company 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 re- upon next month's production plan for the fall line. Specifhearsing 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 require pacity, limited resources, and demand forecasts. Accurate fashions. She determined the prices for each clothing item ments. Her fall line consists of both professional and casual planning for next month's production is critical to full by taking into account the quality and cost of material , the sales since the items produced next month will appear in 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 professional fashions include: Price Labor and Machine Cost $160 $300 Clothing Item Tailored wool stacks Cashmere sweater Silk blouse Silk camisole Tailored skirt Materials Requirements 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 $450 $180 $120 $270 $150 $100 $ 60 $120 Wool blazer $320 $140 CASES 161 The fall casual fashions include: Price 5350 Labor and Machine Cost 5175 Clothing Item Velvet pants Cotton sweater Cotton miniskirt Velvet shirt Button-down blouse Materials Requirements 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 $130 5 75 5200 5120 5 60 $ 40 5160 5 90 Wool $ 1.50 $13.00 $ 2.25 $12.00 $ 2.50 Velvet She knows that for the next month, she has ordered shirts. TrendLines does not want to produce more than the 45,000 yards of wool, 28.000 yards of acetate. 9.000 yards forecasted demand because once the pants and shirts go out of cashmere. 18.000 yards of silk, 30.000 yards of rayon. of style, the company cannot sell them. TrendLines can pro- 20,000 yards of velvet, and 30,000 yards of cotton for pro- duce less than the forecasted demand, however, since the duction. The prices of the materials are as follows: company is not required to meet the demand. The cashmere sweater also has limited demand because it is quite expen- sive, and TrendLines knows it can sell at most 4,000 cash- Material Price per yard mere sweaters. The silk blouses and camisoles have limited $ 9.00 demand because many women think silk is too hard to care Acetate for, and TrendLines projects that it can sell at most 12,000 Cashmere $60.00 silk blouses and 15.000 silk camisoles. Silk The demand forecasts also indicate that the wool Rayon slacks, tailored skirts, and wool blazers have a great demand Cotton because they are basic items needed in every professional wardrobe. Specifically, the demand for wool slacks is 7,000 pairs of slacks, and the demand for wool blazers is 5.000 blazers. Katherine wants to meet at least 60 percent of the Any material that is not used in production can be sent back demand for these two items in order 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 sent back to the wholesaler. the demand for tailored skirts could not be estimated. She knows that the production of both the silk blouse Katherine feels she should make at least 2,800 of them. and cotton sweater leaves leftover scraps of material. Specif- ically, for the production of one silk blouse or one cotton fa) Ted is trying to convince Katherine not to produce any velvet sweater, 2 yards of silk and cotton, respectively, are needed. shirts since the demand for this fashion fad is quite low. He From these 2 yards, 1.5 yards are used for the silk blouse argues that this fashion fad alone accounts for $500,000 of the or the cotton sweater and 0.5 yard is left as scrap material. fixed design and other costs. The net contribution (price of She does not want to waste the material, so she plans to use clothing item - materials cost - labor cost) from selling the the rectangular scrap of silk or cotton to produce a silk fashion fad should cover these fixed costs. Each velvet shirt camisole or cotton miniskirt, respectively. Therefore, when generates a net contribution of $22. He argues that given the net contribution, even satisfying the maximum demand will not ever a silk blouse is produced, a silk camisole is also pro vield a profit. What do you think of Ted's argument? 162 CHAPTER 4 SOLVING LINEAR PROGRAMMING PROBLEMS: THE SIMPLEX METHOD demand for velvet will decrease in the future. Katherine can) The textile wholesaler informs Katherine that since another tex. therefore get no refund for the velvet. How does this fact change tile 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)? (2) TrendLines 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 pattem 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 upper limits on demand concem only the sales dur- the labor and machine cost for each blazer by $80. Given this ing September and October.) What should the new production new cost, how many of each clothing item should TrendLines plan be to maximize profit? produce to maximize profit