Reference no: EM133293198

CASE 1. FABRICS AND FALL FASHIONS

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. Οn 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.

a) Ted is trying to convince Katherine not to produce any velvet shins 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 texti1e 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 shοιι1d 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 το 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 ?n unlimited quantity during the November sale. (The previously mentioned upper limits οn demand concern only the sales during September and October.) What should the new production plan be to maximize profit?

CASE 2. AUTO ASSEMBLY
Automobile Alliance, a large automobile manufacturing company, organizes the vehicles it manufactures into three families: a family of trucks, a family of small cars, and a family of midsized and luxury cars. One plant outside Detroit, ΜΙ, assembles two models from the family of midsized and luxury cars. The first model, the Family Thrillseeker, is a four-door sedan with vinyl seats, plastic interior, standard features, and excel1ent gas mileage. It is marketed as a smart buy for middle-class families with tight budgets, and each Family Thrillseeker sold generates a modest profit of $3,600 for the company. The second model, the Classy Cruiser, is a two-door luxury sedan with leather seats, wooden interior, custom features, and navigational capabilities. It is marketed as a privilege of affluence for upper-middle-class families, and each Classy Cruiser sold generates a healthy profit of $5,400 for the company.

a) Formulate and solve a linear programming problem to determine the number of Family Thrillseekers and the number of Classy Cruisers that should be assembled.

Before she makes her final production decisions, Rachel plans to explore the following questions independently except where otherwise indicated.

b) The marketing department knows that it can pursue a targeted $500,000 advertising campaign that will raise the demand for the of Classy Cruiser next month by 20 percent. Should the campaign be undertaken'?

c) Rachel knows that she can increase next month's plant capacity by using overtime labor. She can increase the plant's labor-hour capacity by 25 percent. With the new assembly plant capacity, how many Family Thrillseekers and how many Classy Cruisers should be assembled?

d) Rachel knows that overtime labor does not come without an extra cost. What is the maximum amount she should be willing to pay for all overtime labor beyond the cost of this labor at regular time rates? Express your answer as a lump sum.

e) Rachel explores the option of using both the targeted advertising campaign and the overtime labor-hours. The advertising campaign raises the demand for the Classy Cruiser by 20 percent, and the overtime labor increases the plant's labor-hour capacity by 25 percent. How many Family Thrillseekers and how many Classy Cruisers should be assembled using the advertising campaign and overtime labor-hours if the profit from each Classy Cruiser sold continues to be 50 percent more than for each Family Thrill- seeke sold?

f) Knowing that the advertising campaign costs $500,000 and the maximum usage of overtime labor-hours costs $1,600,000 beyond regular time rates, is the solution found in part (e) a wise decision compared to the solution found in part (a)?

g) Automobile Alliance has determined that dealerships are actual1y heavily discounting the price of the Family Thrillseekers το move them off the lot. Because of a profit-sharing agreement with its dealers, the company is therefore not making a profit of $3,600 on the Family Thrillseeker but is instead making a profit of $2,800. Determine the number of Family Thril1seekers and the number of Classy Cruisers that should be assembled given this new discounted price.

h) The company has discovered quality problems with the Family Thril1seeker by randomly testing Thril1seekers at the end of the assembly line. Inspectors have discovered that in over 60 percent of the cases, two of the four doors on a Thril1seeker do not seal properly. Because the percentage of defective Thri11seekers determined by the random testing is so high, the floor supervisor has decided to perform quality control tests on every Thril1seeker at the end of the line Because of the added tests, the time it takes to assemble one Family Thril1seeker has increased from 6 to 7.5 hours. Determine the number of units of each model that shou1d be assembled given the new assembly time for the Family Thril1seeker.

i) The board of director's of Automobile Alliance wishes to capture a larger share of the luxury sedan market and therefore would like to meet the ful1 demand for Classy Cruisers. They ask Rachel to determine by how much the profit of her assembly plant would decrease as compared to the profit found in part (a). They then ask her to meet the full demand for Classy Cruisers if the decrease in profit is not more than $2,000,000.

j) Rachel now makes her final decision by combining all the new considerations described in parts (j), (g), and (h). What are her final decisions on whether to undertake the advertising campaign, whether to use overtime labor, the number of Family Thril1seekers to assemble, and the number of Classy Cruisers to assemble?

CASE 3. AGRICULTURAL ALLOCATION MODEL

Α university has offered a rent subsidy alterative to the families living in the graduate student housing complex. Near the complex are 20 acres that can be farmed. The university wil1 allow the residents' organization to farm all or part of this land. Any derived profits can be apportioned to reducing the rent for the graduate families. Families may keep whatever produce they wish for their own use. The university has agreed to buy produce from the students, up to certain limits, for use within the dining services.

Formulate and solve the LP model that would enable the students to determine the number of acres they should allocate to each crop so as to maximize total profit. Assume that student demands are to be satisfied exactly. Profit is earned on crops sold to the university (over and above student demands) and on the soybean crop.
Include the following conditions:

a) Lettuce, potatoes, tomatoes, and soybeans require 5,000, 2,500, 6,500 and 3,500 gallons of water per acre, respectively, over and above natural rainfall during the growing season. Available water is limited to 40,000 gallons.

b) Because of required crop rotation, the number of acres planted in soybeans can be no more than 50 percent of the combined total acres planted in the other three crops.

c) Composted manure is to be used for fertilizer. The requirements are 2,500 lb. per acre for lettuce, 1,000 lb. per acre for potatoes, 3,000 lb. per acre for tomatoes, and 900 lb. per acre for soybeans; however only 40,000 lb. of fertilizer are available overall.

d) Much as the graduate students and their families enjoy working the farm, academic demands (and other pleasures) limit total available labor to 1,000 hours over the growing season. Lettuce requires 75 hours per acre, potatoes 35 hours per acre, tomatoes 60 hours per acre and soybeans 90 hours per acre.

You are additionally required to contact extended sensitivity analysis on the total available water, available manure, demand and on at least one of the objective function coefficients.

Excercise 4.
Fulgencio Batista led Cuba with a cold heart and iron fist-greedily stealing from poor citizens, capriciously ruling the Cuban population that looked to him for guidance, and violently murdering the innocent critics of his politics. In 1958, tired of watching his fellow Cubans suffer from corruption and tyranny, Fidel Castro led a guerrilla attack against the Batista regime and wrested power from Batista in January 1959. Cubans, along with members of the international community, believed that political and economic freedom had finally triumphed on the island. The next two years showed, however, that Castro was leading a Communist dictatorship-killing his political opponents and nationalizing all privately held assets. The United States responded to Castro's leadership in 1961 by invoking a trade embargo against Cuba. The embargo forbade any country from selling Cuban products in the United States and forbade businesses from selling American products to Cuba. Cubans did not feel the true impact of the embargo until 1989 when the Soviet economy collapsed. Prior to the disintegration of the Soviet Union, Cuba had received an average of $5 billion in annual economic assistance from the Soviet Union. With the disappearance of the economy that Cuba had almost exclusively depended upon for trade, Cubans had few avenues from which to purchase food, clothes, and medicine. The avenues narrowed even further when the United States passed the Torricelli Act in 1992 that forbade American subsidiaries in third world countries from doing business with Cuba that had been worth a total of $700 million annually.

a. How many basic, advanced, and supreme packages should Mr. Baker send to Cuba?

b. Mr. Baker reevaluates the levels of importance he places on each of the three goals. To sell his efforts to potential donors, he must show that his program is effective. Donors generally judge the effectiveness of a program on the number of people reached by aid packages. Mr. Baker therefore decides that he must put more importance on the goal of reaching at least 20 percent of the population. He decides to penalize his plan by 10 points for every half a percentage point below his 20 percent goal. The penalties for his other two goals remain the same. Under this scenario, how many basic, advanced and supreme packages should Mr. Baker send to Cuba? How sensitive is the plan to changes in the penalty weights?

c. Mr. Baker realizes that sending more doctors along with the supreme packages will improve the proper use and distribution of the packages' contents, which in turn will increase the effectiveness of the program. He therefore decides to send one doctor with every 75 supreme packages. The penalties for the goals remain the same as in part b. Under this scenario, how many basic, advanced, and supreme packages should Mr. Baker send to Cuba?

d. The aid budget is cut, and Mr. Baker learns that he definitely cannot allocate more than
$20 million in aid to Cuba. Due to the budget cut, Mr. Baker decides to stay with his original policy of sending one doctor with every 100 supreme packages. How many basic, advanced, and supreme packages should Mr. Baker send to Cuba, assuming that the penalties for not meeting the other two goals remain the same as in part a?

e. Now that the aid budget has been cut, Mr. Baker feels that the levels of importance of his three goals differ so much that it is difficult to assign meaningful penalty weights to deviations from these goals. Therefore, he decides that it would be more appropriate to apply a preemptive goal-programming approach (which will ensure that his budget goal is fully met if possible), while retaining his original policy of sending one doctor with every 100 supreme packages. How many basic, advanced, and supreme packages should Mr. Baker send to Cuba according to this approach?

Excercise 5.

The Oakdale County School Board was meeting in special session. A federal judge had ordered the board to present an acceptable busing plan for racially balancing the four high schools in Oakdale County within a week. The judge had previously given the school board several opportunities to informally present a plan, but the members had been unable to agree among themselves. Every time they met and started to develop a plan to bus students from one high school district to another, an argument would arise before they got past the first busing move, and they would adjourn the meeting. This time, however, they knew the judge had lost patience and they had to agree on something.

Formulate and solve a goal programming model to help the board with its dilemma.

Excercise 6.
Broderick Crawford is the district commander for the Catawba Valley highway patrol district in western Pennsylvania. He is attempting to assign highway patrol cars to different road segments in his district. The primary function of the highway patrol force is to patrol roads outside of incorporated city and town limits in the district to deter traffic violators and accidents. This objective is typically achieved by maintaining a visible presence-letting motorists see patrol units on a regular basis and giving out warnings, citations, and so forth. Secondary activities of a patrol unit include providing assistance to motorists, answering distress calls, handling emergencies and accidents when called to the scene, and occasionally apprehending criminals.

Formulate and solve a goal programming model to determine the number of patrol units to assign to each road segment to achieve the commander's goal.

Excercise 7.
Adeline Jonasson lost two close friends in the collapse of the World Trade Center on September 11, 2001. Both had been vibrant young women who left grieving husbands and children behind. What terrible losses. Not a day goes by that she doesn't think of these friends and feel the anger yet again over those senseless deaths. Now she feels a real sense of mission to do something about it. What a relief it had been to be offered a top managerial position in the newly formed Transportation Security Administration. After being told that the job would involve heading a task force on airport security, Adeline had not hesitated a moment in accepting the position. She had greatly enjoyed her career as a management science consultant in the airline industry. It was very satisfying to help several airline companies save many millions of dollars. However, she now felt a greater calling. She would be able to use her expertise in management science to help save lives. There was no way to bring her friends back, but at least she could do everything possible to prevent this from happening again.

Now that it has obtained all the needed managerial input, the task force is ready to begin its analysis.

a. Identify the two decisions to be made and define a decision variable for each one.

b. Describe why this problem is a preemptive goal-programming problem by giving quantitative expressions for each of the goals in terms of the decision variables defined in part a.

c. Draw a single two-dimensional graph where the two axes correspond to the decision variables defined in part a. Consider each of the goals in order of priority and use the quantitative expression obtained in part b for this goal to draw a plot on the graph that displays the values of the decision variables that fully satisfy this goal. After completing this for all the goals, use the graph to determine the optimal solution for this preemptive goal-programming problem.

d. Use preemptive goal programming to formulate and solve this problem on computer software.

e. If it is possible to fully satisfy all the goals except the lowest priority goal, one can quickly solve a preemptive goal-programming problem by formulating and solving a linear programming model that includes all the goals except the last one as constraints and then uses the objective function to strive toward the lowest priority goal. Formulate and solve such a linear programming model for this problem on computer software. What would be the interpretation for the preemptive goal-programming problem if this linear programming model had no feasible solutions?

Note: Complete cases in attachment file.

Attachment:- MCD Homework Cases.rar