# 2. Case Problem – Assigning Student to Schools (Note that the professor will offer the model for question (a); you must build on this to answer all

2. Case Problem – Assigning Student to Schools (Note that the professor will offer the model for question (a); you must build on this to answer all remaining questions b to g) Assigning Students to Schools The Springfield School Board (SSB) has made the decision to close one of its middle schools (sixth, seventh, and eighth grades) at the end of this school year and reassign all of next year’s middle school students to the three remaining middle schools. The school district provides busing for all middle school students who must travel more than approximately a mile, so the school board wants a plan for reassigning the students that will minimize the total busing cost. The annual cost per student for busing from each of the six residential areas of the city to each of the schools is shown in the following table (along with other basic data for next year), where 0 indicates that busing is not needed and a dash indicates an infeasible assignment.
Busing Cost per Student
Area
Number of Students
Percentage in 8th Grade School 1 School 2 School 3 1 450 32 38 30 \$300 \$ 0 \$700 2 600 37 28 35 — 400 500 3 550 30 32 38 600 300 200 4 350 28 40 32 200 500 — 5 500 39 34 27 0 — 400 6 450 34 28 38 500 300 0
School capacity: 900 1,100 1,000
The School board also has imposed the restriction that each grade must constitute between 30 and 36 percent of each school’s population. The above table shows the percentage of each area’s middle school population for next year that falls into each of the three grades. The school attendance zone boundaries can be drawn so as to split any given area among more than one school, but assume that the percentages shown in the table will continue to hold for any partial assignment of an area to a school.
You have been hired as a decision analyst/management science consultant to assist the SSB in determining how many students in each area should be assigned to each school. Please work on this problem to answer the following
questions and prepare a managerial report to the SSB. Please do provide all your model formulations and solutions.
a. Formulate and solve a linear programming model for this problem. b. What is resulting recommendation to the school board?
After seeing your recommendation, the school board expresses concern about all the splitting of residential areas among multiple schools. They indicate that they “would like to keep each neighborhood together.”
c. Adjust your recommendation as well as you can to enable each area to be assigned to just one school. (Adding this restriction may force you to fudge on some other constraints.) How much does this increase the total busing cost? (This line of analysis will be pursued more rigorously by formulating an Integer Programming model (Module 9, Chapter 10), which won’t be pursue in the assignment.)
The school board is considering eliminating some busing to reduce costs. Option 1 is to only eliminate busing for students traveling 1 to 1.5 miles, where the cost per student is given in the table as \$200. Option 2 is to also eliminate busing for students traveling 1.5 to 2 miles, where the estimated cost per student is \$300.
d. Revise the model from part a to fit Option 1, and solve. Compare these results with those from part b, including the reduction in total busing cost. e. Repeat part d for Option 2. The school board now needs to choose among the three alternative busing plans (the current one or Option 1 or Option 2). One important factor is busing costs. However, the school board also wants to place equal weight on a second factor: The inconvenience and safety problems caused by forcing students to travel by foot or bicycle a substantial distance (more than a mile, and especially more than 1.5 miles). Therefore, they want to choose a plan that provides the best trade-off between these two factors. f. Use your results from parts b, d, and e to summarize the key information related to these two factors that the school board needs to make this decision. g. Which decision do you think should be made? Why? Note: The SSB can consider many additional questions which will be better answered with an Integer Programming Model discussed in Chapter 10 (Module 9)).
3. Case Problem – Assigning Doctors to Medical Research Projects (The professor will offer the first Excel Model to answer the first question. Your job is to build from there and answer the remaining questions.)
Project Pickings Tazer, a pharmaceutical manufacturing company, entered the pharmaceutical market 12 years ago with the introduction of six new drugs. Five of the six drugs were simply permutations of existing drugs and therefore did not sell very heavily. The sixth drug, however, addressed hypertension and was a huge success. Since Tazer had a patent on the hypertension drug, it experienced no competition, and profits from the hypertension drug alone kept Tazer in business.
During the past 12 years, Tazer continued a moderate amount of research and development, but it never stumbled upon a drug as successful as the hypertension drug. One reason is that the company never had the motivation to invest heavily in innovative research and development. The company was riding the profit wave generated by its hypertension drug and did not feel the need to commit significant resources to finding new drug breakthroughs.
Now Tazer is beginning to fear the pressure of competition. The patent for the hypertension drug expires in five years, and Tazer knows that once the patent expires, generic drug manufacturing companies will swarm into the market like vultures. Historical trends show that generic drugs decrease sales of branded drugs by 75 percent.
Tazer is therefore looking to invest significant amounts of money in research and development this year to begin the search for a new breakthrough drug that will offer the company the same success as the hypertension drug. Tazer believes that if the company begins extensive research and development now, the probability of finding a successful drug shortly after the expiration of the hypertension patent will be high.
As head of research and development at Tazer, you are responsible for choosing potential projects and assigning project directors to lead each of the projects. After researching the needs of the market, analyzing the shortcomings of current drugs, and interviewing numerous scientists concerning the promising areas of medical research, you have decided that your department will pursue five separate projects, which are listed below.
Project Up: Develop a more effective antidepressant that does not cause serious mood swings Project Stable: Develop a drug that addresses manic-depression. Project Choice: Develop a less intrusive birth control method for women.
Project Hope: Develop a vaccine to prevent HIV infection. Project Release: Develop a more effective drug to lower blood pressure.
For each of the five projects, you are only able to specify the medical ailment the research should address since you do not know what compounds will exist and be effective without research.
You also have five senior scientists to lead the five projects. You know that scientist are very temperamental people and will only work well if they are challenged and motivating, you have established a bidding system for the projects. You have given each of the five scientists 1,000 bid points. They assigned bids to each project, giving a higher number of bid points to projects they most prefer to lead.
The following table provides the bids from the five senior scientists for the five individual projects:
Project Dr. Kvaal Dr. Zuner Dr. Tsai Dr. Mickey Dr. Rollins Project Up 100 0 100 267 100 Project Stable 400 200 100 153 33 Project Choice 200 800 100 99 33 Project Hope 200 0 100 451 34 Project Release 100 0 600 30 800
You decide to evaluate a variety of scenarios you think are likely. 1In general, patents protect inventions for 17 years. In 1995, GATT legislation extending the protection given by new pharmaceutical patents to 20 years became effective. The patent for Tazer’s hypertension drug was issued prior to the GATT legislation, however. Thus, the patent only protects the drug for 17 years.
a. Given the bids, you need to assign one senior scientist to each of the five projects to maximize the preferences of the scientists. What are the assignments? b. Dr. Rollins is being courted by Harvard Medical School to accept a teaching position. You are fighting desperately to keep her at Tazer, but the prestige of Harvard may lure her away. If this were to happen, the company would give up the project with the least enthusiasm. Which project would not be done? c. You do not want to sacrifice any project since researching only four projects decreases the probability of finding a breakthrough new drug. You decide that either Dr. Zuner or Dr. Mickey could lead two projects. Under these new conditions with just four senior scientists, which scientists, will lead which projects to maximize preferences?
Project Dr. Mickey Dr. Kvaal Dr. Rollins Project Up 300 86 Can’t lead Project Stable Can’t lead 343 50 Project Choice 125 171 50 Project Hope Can’t lead Can’t lead 100 Project Release 175 Can’t lead 600
Which scientists should lead which projects to maximize preferences?
g. You decide that Project Hope and Project Release are too complex to be led by only one scientist. Therefore, each of these projects will be assigned two scientists as project leaders. You decide to hire two more scientists in or der to staff all projects: Dr. Arriaga and Dr. Santos. Because of religious reasons, neither of them want to lead Project Choice and so they assign 0 bid points to this project. The following table lists all projects, scientists, and their bids.
Project Kvaal Zuner Tsai Mickey Rollins Arriaga Santos Up 86 0 100 300 Can’t lead 250 111 Stable 343 200 100 Can’t lead 50 250 1 Choice 171 800 100 125 50 0 0 Hope Can’t lead 0 100 Can’t lead 100 250 333 Release Can’t lead 0 600 175 600 250 555
h. Do you think it is wise to base your decision in part g only on an optimal solution for a variant of an assignment problem?
4. Case Problem – EZ trailers Inc. (Professor will lead the way on how to begin this problem)
Case Problem EZ TRAILERS, INC. EZ Trailers, Inc., manufactures a variety of general purpose trailers, including a complete line of boar trailers. Two of their best-selling boar trailers are the EZ-190 and the EZ250. The EZ-190 is designed for boats up to 19 feet in length, and the EZ-250 can be used for boats up to 25 feet in length. EZ Trailer would like to schedule production for the next two months for these two models. Each unit of the EZ-190 requires four hours of production time, and each unit of the EQ-250 uses six hours of production time. The following orders have been received for March and April.
Model March April EZ-190 800 600 EZ-250 1100 1200 The ending inventory from February was 200 units of the EZ-190 and 300 units of the EQ-250. The total number of hours of production time used in February was 6300 hours.
The management of EZ Trailers in concerned about being able to satisfy existing orders for the EZ-250 for both March and April. In fact, it believes that this goal is the most important one that a production schedule should meet. Next in importance is satisfying existing orders fo the EZ190. In addition, management doesn’t want to implement any production schedule that would involve significant labor fluctuations from month to month. In this regard, its goal is to develop a production schedule that would limit fluctuations in labor hours used to a maximum of 1000 hours from one month to the next.
Managerial Report
Perform an analysis of EZ Trailers’s production scheduling problem, and prepare a report for EZ’s president that summarizes your findings. Your report must include your detailed models and a discussion and analysis of the following items.
1. The production schedule that best achieves the goals as specified by management. 2. Suppose that EZ Trailers’s storage facilities would accommodate only a maximum of 2. Case Problem – Assigning Student to Schools (Note that the professor will offer the model for question (a); you must build on this to answer all remaining questions b to g) Assigning Students to Schools The Springfield School Board (SSB) has made the decision to close one of its middle schools (sixth, seventh, and eighth grades) at the end of this school year and reassign all of next year’s middle school students to the three remaining middle schools. The school district provides busing for all middle school students who must travel more than approximately a mile, so the school board wants a plan for reassigning the students that will minimize the total busing cost. The annual cost per student for busing from each of the six residential areas of the city to each of the schools is shown in the following table (along with other basic data for next year), where 0 indicates that busing is not needed and a dash indicates an infeasible assignment.
1. The production schedule that best achieves the goals as specified by management. 2. Suppose that EZ Trailers’s storage facilities would accommodate only a maximum of 300 trailers in any one month. What effect would this have on the production schedule? 3. Suppose that EZ Trailers can store only a maximum of 300 trailers in any one month. In addition, suppose management would like to have an ending inventory in April of at least 100 units of each model. What effect would both changes have on the production schedule? 4. What changes would occur in the production schedule if the labor fluctuation goal was the highest priority goal?
trailers in any one month. What effect would this have on the production schedule? 3. Suppose that EZ Trailers can store only a maximum of 300 trailers in any one month. In addition, suppose management would like to have an ending inventory in April of at least 100 units of each model. What effect would both changes have on the production schedule? 4. What changes would occur in the production schedule if the labor fluctuation goal was the highest priority goal?

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