Class 5 Worksheet

docx

School

Bentley University *

*We aren’t endorsed by this school

Course

610

Subject

Mechanical Engineering

Date

Jan 9, 2024

Type

docx

Pages

Uploaded by ProfessorKnowledgeDolphin27

Class 5 Sensitivity Analysis Example 1: A pen manufacturer makes three items – a ballpoint pen, a mechanical pencil, and a fountain pen. All are available only in black and sell in specialty stores and online. The per-unit profit of the items is $3.00 for the ballpoint pen, $3.00 for the mechanical pencil, and $5.00 for the fountain pen. These values take into account labor, cost of materials, packing, quality control, and so on. The company is currently trying to plan its production mix for each week. They believe they can sell all the pens and pencils they produce, but production is limited by the available resources. They are able to get at most 1000 ounces of plastic, 1200 ounces of chrome and 2000 ounces of stainless steel each week. Each ballpoint pen requires 1.2 ounces of plastic, 0.8 ounces of chrome, and 2 ounces of stainless steel. Each mechanical pencil requires 1.7 ounces of plastic, no chrome, and 3 ounces of stainless steel. Each fountain pen requires 1.2 ounces of plastic, 2.3 ounces of chrome and 4.5 ounces of stainless steel. Other resources such as labor time available are not taken into consideration at this point. a) Formulate (algebraic formulation) the LP that finds the production plan that maximizes profit. b) Use Analytic Solver to find the solution and generate the sensitivity report. c) Due to some unexpected cost increases, the manufacturer realizes that the profit on the mechanical pencil may actually be less than $3.00. If the actual profit is $2.00 on the pencil, will they have to change the production schedule? d) If the profit on the ballpoint pens is $3.50 will the production schedule change?

e) If 500 ounces of additional stainless steel are available for 0.60 more than what is currently being paid, should the company buy additional steel at this price? f) Based on the allowable ranges, which unit profit is most sensitive? g) If additional plastic is available at $1.00 over the usual cost of $5.00 per ounce, if the manufacturer agrees to purchase 500 ounces. Should the plastic be purchased? Does the answer change if they can purchase less than 500 ounces at this price? h) If the manufacturer discovers that only 1,000 ounces of chrome are available this week, what happens to the profit? i) Due to some unexpected cost savings, the profits on the ballpoint and fountain pens may actually be larger than expected. If the profit on the ballpoint pens is $3.50 and the profit on the fountain pens is $5.50, will the production schedule change?

Example 2: Case 6-5 and 8-4 in textbook The Springfield School Board 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 the 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 to 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 Student s Percent in 6 th Grade Percent in 7 th Grade Percent in 8 th 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 1100 1000 The school board 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. a) Please find the Excel spreadsheet for the problem. b) Obtain the sensitivity report c) How much can the change in busing cost from area 6 to school 1 increase (assuming no change in the cost for other schools) before the current optimal solution will no longer be optimal?

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

d) Now assume the bussing cost from area 6 to all schools will increase by 1%. Will the solution still be optimal? What if the change is 10%? e) The school can add a portable classroom to increase the capacity of one or more of the middle schools for a few years. However, this is a costly move that the board would only consider if it would significantly decrease busing costs. Each portable classroom holds 20 students and has a leasing cost of $2,500 per year. Is it worthwhile to add portable classrooms to any school? f) Will this analysis be valid if we consider adding portable classrooms to all schools considered in part e?