**LINEAR PROGRAMMING**

ASSIGNMENT #2: LINEAR PROGRAMMING

Due in marked boxes outside SBE2201, by 2:00 p.m. on July 23rd 2015

**Objectives:**

(1)To develop linear programming modeling skills

(2)To be able to solve linear programming problems using the Solver add-in to Excel;

(3)To practice and develop report writing skills, creative thinking, and a team approach to problem solving.

**Four Questions:**

**Problem 1:** 50 marks [35 marks for the mathematical formulation, 6 marks for Excel files, 9 marks for the report (you should consider all the scenarios described in the last paragraph of the question)

**Problem 2** : 50 marks [20 marks for the mathematical formulation, 6 marks for excel, 24 marks for the answers to questions a to h]

**Problem 3:** 50 marks [29 marks for formulation, 6 marks for Excel Report, 15 marks for report which should contain the solution and other aspects of the problem as well]

**Problem 4:** 100 marks [60 marks for the mathematical formulation, 20 marks for the Excel files, 20 marks for the report(solution should be described and some what-ifanalysis should be done in the report)]

**Administrative Notes**

This assignment is to be completed by three 3 person groups. The only exception is if the class is not divisible by three. In that case there can be two groups of 2 or one group of 2.

**Two submissions:**

Submission 1(in the dropbox outside SBE2201): Submit your work in word-processed report format, double-spaced using the cover page which is posted online. Binding is not necessary – stapling is adequate. Colour is optional. Double-sided printing is fine.

Submission 2(in the electronic dropbox on MyLS): Submit 4 excel files – one for each question into the appropriately titled electronic dropbox. Your excel file should have the last names of all your group members and the questions number.

The report format should be as follows for each problem:

**Part 1: **State all the assumptions you have made, define the decision variables and their units, type out the mathematical linear programming problem (includes objective function and all constraints). An excel print out of your formulation is not enough.

**Part 2: **For each problem, you should have a report, which, must be convincing to a general manager (so provide your reasoning in non-technical language).

**Part 3: **Include any computer output as an Appendix to each question.. The computer printout per se does not constitute the report.

**Problem 1**

A problem frequently encountered by managers of banks, mutual funds, investment services, and insurance companies is the selection of specific investments from among a wide variety of alternatives. The manager’s overall objective is usually to maximize expected return on investment, given a set of legal, policy, or risk restraints.

For example, a provincial credit union invests in short-term trade credits, corporate bonds, gold stocks, and construction loans. To encourage a diversified portfolio, the board of directors has placed limits on the amount that can be committed to any one type of investment. The credit union has $6 million available for immediate investment and wishes to do three things: (1) to maximize the interest earned on the investments made over the next six months, (2) The credit

Union’s financial advisor’s would like the Credit Union to invest at least 90% in corporate bonds, gold stocks and construction loans and (3) satisfy the diversification requirements as set by the board of directors.

**The specifics of the investment possibilities are as follows:**

**Table 1.**

Investment | Interest Earned (%) | Maximum investment ($ millions) |

Trade credit | 7.5 | 1.0 |

Corporate bonds | 10.5 | 2.5 |

Gold stocks | 19.7 | 1.5 |

Construction loans | 13.8 | 1.8 |

In addition, the board specifies that at least 60% of the funds invested be in gold stocks and construction loans, and that no less than 20% be invested in trade credit.

Formulate the problem, and find the optimal investment plan for the manager. If there is a conflict in meeting requirements of the Board of Directors and the Financial advisors, the Board’s recommendations are to be met at the expense of not meeting the financial advisors recommendations. Write an investment report to the board based on your analysis. (In the report, you should state the objective of the plan, the decisions that should be made, the real situation/constraints you have to consider, the final recommendation and the annual yield you anticipate).

The interest rate for each investment are subjective to possible adjustment, the interest rate of Trade credit may be anywhere between [6%,8%], and the interest rate of Gold stocks may change in the range of [19%, 20%]. However, we know that the change of the two interest rates will be positively correlated. Provide your recommendation to the board in case of the possible interest rate changes.

**Problem 2**

Flin Flon Fashions Inc., a Manitoba manufacturer of men’s wear, produces four varieties of ties.

One is an expensive, all-silk tie, one is an all-polyester tie, and two are blends of polyester and cotton. The following table illustrates the cost and availability (per monthly production planning period) of the three materials used in the production process:

**Table 2.**

Material | Cost per meters ($) | Material available per month (meters) |

Silk | 21 | 800 |

Polyester | 6 | 3000 |

Cotton | 9 | 1600 |

The firm has fixed contracts with several major department store chains to supply ties. The contracts require that Flin Flon Fashions supply a minimum quantity of each tie but allow for a larger demand if Flin Flon chooses to meet that demand. (Most of the ties are not shipped with the name Flin Flon on their label, incidentally, but with “private stock” labels supplied by the stores.) Table 3 summarizes the contract demand for each of the four styles of ties, the selling price per tie, and the fabric requirement of each variety.

**Table 3.**

Variety of | Selling price | Monthly | Monthly | Material | Material |

tie | per tie ($) | Contract | demand | required per tie | Requirements |

Minimum | (meters) | ||||

All silk | 6.70 | 6000 | 7000 | 0.125 | 100% silk |

All polyester | 3.55 | 10000 | 14000 | 0.08 | 100% polyester |

Poly-cotton- | 4.31 | 13000 | 16000 | 0.10 | 50% polyester- |

blend 1 | 50% cotton | ||||

Poly-cotton- | 4.81 | 6000 | 8500 | 0.10 | 30% polyester- |

blend 1 | 70% cotton |

a.Recommend to Flin Flon Fashions a production plan for each type of tie each month. What is the monthly profit you anticipate for your production recommendation?

b.Because of the uncertainty of global business environment, the raw material price for cotton goes up by $2, and total amount of cotton available per month reduces to 1400 meters. How would you change your recommendation? Why?

c.If the total silk available is decreased to 760 meters, what happens to optimal solution and objective function value? Can you use the rule of sensitivity analysis to solve the problem?

d.What happens if the total silk available is increases to 900 meters? What decision is sensitive to these changes? Why?

e.If the cost of Polyester is decreased to $4, what happens to optimal solutions and objective function value? Can you use the rule of sensitivity analysis to solve the problem? What happens if cost of Polyester is increased to $10? What decision is sensitive to these changes? Why?

f.One department advertises Flin Flon Fashion’s polyster tie on prime time TV which results in an increase in demand for this tie by 2000. How does this impact the profit and what are the optimal production quantities?

g.If there is a fall in demand of Poly cotton Blend 1 by 1000, how would this effect the production quantities and profit?

h.If the minimum requirement of blend 1 were to increase to 18,000, what would be the profit and production quantities? If the minimum requirement of blend 1 were to increase to 15,000, what would be the profit and production quantities?

**Problem 3**

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 next year’s middle school students. Students who must travel more than approximately a mile need to be bussed.

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.

**Table 8.**

Number | Percentage | Percentage | Percentage | Busing Cost per Student (dollars) | |||

Area | of | in 6th grade | in 7th grade | in 8th grade | School 1 | School 2 | School 3 |

students | |||||||

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 also has imposed the restriction that each grade must constitute between30-36 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 management science consultant to assist the school board in determining how many students in each area should be assigned to each school.

a.Formulate and solve a linear programming model for this problem.

b.What is your resulting recommendation to the school board? Do you have any concerns about the solution? If yes, what are they?

**Problem 4**

Maple Wood Goods is a company that is currently involved in the clear-felling of two forest tracts. The trees felled will supply the firm’s sawmill and chipboard plant. Some of the logging output is also available for export. The harvested trees are cut into sections referred as first cuts, second cuts and third cuts. The estimated daily yield for each cut for each forest is as follows.

**Forest**

Cut | 1 | 2 | |

1 | 53.76 | 88.32 | (100’s Cubic feet) |

2 | 51.20 | 78.72 | |

3 | 23.04 | 24.96 |

The log cuts (other than those for export) are transported to either the sawmill or the chipboard plant which are at different locations. Transportation costs are as follows.

Forest | Sawmill | Chipboard plant | |

1 | 8 | 14 | ($’s per unit) |

2 | 6 | 10 |

Handling costs ($’s per unit) at the two plants depend on the type of cut as shown.

Cut | Sawmill | Chipboard plant |

1 | 5.00 | 2.40 |

2 | 7.00 | 3.00 |

3 | 10.00 | 4.00 |

At the sawmill logs are cut into three grades of finished products: clear grade, dressing grade and construction grade. A fraction of the incoming wood ends up as scrap and sawdust. The following table shows the log conversion factors as well as the processing times in minutes per unit of input.

Cut | Timber | Scraps | Sawdust | Processing |

Time/unit | ||||

1 | 0.72 | 0.20 | 0.08 | 1.8 |

2 | 0.66 | 0.24 | 0.10 | 2.6 |

3 | 0.54 | 0.30 | 0.16 | 3.9 |

The productive capacity of the mill is 360 minutes per day.

Yields in percentages for each cut of cut timber at each forest are as follows.

Grade | ||||

Clear | Dressin g | Construction | ||

Cut 1 | Forest 1 | 35 | 48 | 17 |

Forest 2 | 28 | 42 | 30 | |

Cut 2 | Forest 1 | 10 | 18 | 72 |

Forest 2 | 3 | 9 | 88 | |

Cut 3 | Forest 1 | 0 | 5 | 95 |

Forest 2 | 0 | 0 | 100 |

The prices per unit of finished products are $300 for clear grade, $220 for dressing grade and $160 for construction grade. Scraps at the sawmill are transported to the chipboard plant at a cost of $8 per unit. Sawdust is used as fuel in the sawmill and saves $24 in other fuel costs per unit.

At the chipboard plant logs and scraps are chipped, mixed with additives and pressed into chipboard. Each unit of wood yields .76 units of chipboard. The plant can produce up to 112 units per day. Chipboard selling price is $210 per unit.

The minimum daily requirements have been specified by the management and these are 32 units of clear grade, 36 units of dressing grade, 48 units of construction grade and 96 units of chipboard. Export prices are $190 per unit for first cuts and $176 per unit for second cuts. Third cuts are not exported.

Maple Wood Goods wishes to identify the optimal daily operating policy. Formulate the problem as a linear programming problem and solve it. Do a sensitivity analysis once you obtain the solution. Write a report with your recommendations.