Operations Research I

MID TERM

Instructions:  Answer each question.  Use Lindo to solve all problems.  Show all work by describing variables, stating assumptions, illustrating model and showing output solution to the problem.  The exam work should be completed in a word file and must be downloaded by the due date.  You must also upload answers to the excel answer template.  It is advisable that you put all work within the excel template by copy and pasting lindo output, using text, etc.

1. Controlling Air Polution

Nori & Leets Co. is one of the major producers of steel in its part of the world. It is located in the city of Steeltown and is the only large employer there. Steel­town has grown and prospered along with the company which now employs nearly 50,000 residents. Therefore, the attitude of the townspeople always has been "What's good for Nori & Leets is good for the town." However, this attitude is now changing; uncontrolled air pollution from the company's furnaces is ruin­ing the appearance of the city and endangering the health of its residents.

A recent stockholders' revolt resulted in the election of a new en­lightened board of directors for the company. These directors are determined to follow socially responsible policies, and they have been discussing with Steeltown city officials and citizens' groups what to do about the air pollution problem. Together they have worked out stringent air quality standards for the Steeltownairshed.

The three main types of pollutants in this airshed are particu­late matter, sulfur oxides, and hydrocarbons. The new standards require that the company reduce its annual emission of these pol­lutants by the amounts shown in the following table.

The board of directors has instructed management to have the engineering staff determine how to achieve these reductions in the most economical way. The steelworks have two primary sources of pollution, namely, the blast furnaces for making pig iron and the open-­hearth furnaces for changing iron into steel. In both cases, the engineers have decided that the most effective abatement meth­ods are (1) increasing the height of the smokestacks, (2) using filtler devices (including gas traps) in the smokestacks, and (3) including cleaner, high-grade materials among the fuels for the furnaces. Each of these methods has a technological limit on how heavily it can be used (e.g., a maximum feasible increase in the height of the smokestacks), but there also is considerable flexibility for using the method at a fraction of its technological limit.

The next table shows how much emissions (in millions of pounds per year) can be eliminated from each type of furnace by fully using any abatement method to its technological limit.

Reduction in Emission Rate from the Maximum Feasible Use of an Abatement Method

For purposes of analysis, it is assumed that each method also can be less fully used to achieve any fraction of the abatement ca­pacities shown in this table. Furthermore, the fractions can be dif­ferent for blast furnaces and open-hearth furnaces. For either type of furnace, the emission reduction achieved by each method is not substantially affected by whether or not the other methods also are used.

After these data were developed, it became clear that no single method by itself could achieve all the required reductions. On the other hand, combining all three methods at full capacity on both types of furnaces (which would be prohibitively expensive if the company's products are to remain competitively priced) is much more than adequate. Therefore, the engineers concluded that they would have to use somecombination of the methods, perhaps with fractional capacities, based on their relative costs. Furthermore, be­cause of the differences between the blast and the open-hearth fur­naces, the two types probably should not use the same combination.

An analysis was conducted to estimate the total annual cost that would be incurred by each abatement method. A method's an­nual cost includes increased operating and maintenance expenses, as well as reduced revenue due to any loss in the efficiency of the production process caused by using the method. The other major cost is the start-up cost (the initial capital outlay) required to in­stall the method. To make this one-time cost commensurable with the ongoing annual costs, the time value of money was used to cal­culate the annual expenditure that would be equivalent in value to this start-up cost.

This analysis led to the total annual cost estimates given in the following table for using the methods at their full abatement capacities.

Total Annual Cost from the Maximum Feasible Use of an Abatement Method

It also was determine that the cost of a method being used at a lower level is roughly proportional to the fraction of the abatement capacity given in the preceding table that is achieved. Thus, for any given fraction achieved, the total annual cost would be roughly that fraction of the corresponding quantity in the cost table.

The stage now is set to develop the general framework of the company's plan for pollution abatement. This plan needs to spec­ify which types of abatement methods will be used and at what fractions of their abatement capacities for (l) the blast furnaces and (2) the open-hearth furnaces.

Part a: You have been asked to head an operations research team to an­alyze this problem. Management wants you to begin by determining which plan would minimize the total annual cost of achieving the re­quired reductions in annual emission rates for the three pollutants.  Develop a LP model and solve by stating what methods should be used to attack the pollution problem

Sensitivity Analysis Requirement:

Management wants to conduct some what-if analysis.  Since the company does not have prior experience with the pollution abatement methods under consideration, the cost estimates provided for each technology could easily be off by as much as 10% in either direction. 

Part b:  Analyze the effect of an inaccuracy in estimating each cost parameter for the abatement methods.   If the true value is 10% less or 10% more than the estimated value, would this change the optimal solution.  Provide a table illustrating whether the optimal value will change as a result of a 10% increase or decrease.  Using this table, make a recommendation about where to focus further work in estimating the cost parameter more closely.

Part c:  Suppose the requirement for particles emission is increased to 65 million pounds.  What effect will this have on the company with regards to cost?

2. Old McDonalds Farm

Old Macdonald's 200 acre farm sells wheat, alfalfa, and beef.   Wheat sells for $30 per bushel, alfalfa sells for $200 per bushel, and beef sells for $300 per ton.  Up to 1000 bushels of wheat and up to 1000 bushels of alfalfa can be sold, but demand for beef is unlimited.  If an acre of land is devoted to raising wheat, alfalfa or beef, the yield and required labor are given below:

CROP             YIELD/ACRE                 LABOR/ACRE

Wheat             50 bushels                  30 hours

Alfalfa              100 bushels                20 hours

Beef                10 tons                        50 hours

Up to 2000 hours of labor can be purchased at $15 per hour.  Each acre devoted to beef requires 5 bushels of alfalfa.  The LINDO output is attached and shows how to maximize profit.  The variables are as follows:

W  = acres devoted to wheat

AS = bushels of alfalfa sold

A  = acres devoted to alfalfa

B  = acres devoted to beef

AB = bushels of alfalfa devoted to beef

L = hours of labor purchased

Answer the following questions and justify your answers from anon-integer LP Lindo solution

a) What is the maximum profit of the operation?

b) How much must the price of a bushel of wheat increase before it becomes profitable to grow wheat?

c) Old Macdonald is offered 2000 bushels of alfalfa at a cost of $150 per bushel.  Should Old Macdonald accept this offer?  What affect would this have on profit?

d) What is the most Old Macdonald is willing to pay for an additional hour of  labor?

e) Old Macdonald is offered 25 additional acres of land at a cost of $60 per acre. Should Old Macdonald accept this offer?  What affect would this have on profit?

f) What would be the new solution if alfalfa sold for $20 per bushel

3. Market Research Firm

A market research firm has been retained to conduct focus group interviews for a real estate company.  Plans are made to classify participants to age: over 45 or 45 and under and as to whether or not they have recently bought and/or sold a house (within the last two years).  Specific guidelines have been developed:

• At least 100 people must be selected
• At least 75% of the participants must have bought or sold a house recently
• Of those who have not bought or sold recently, at least 40% must be over 45 years of age
• No more than 40% of participants should be 45 or under

There are preliminary screenings with associated costs.  The cost for each participant begins at $10.  Additional contact with those over 45 raises the cost by $12.  Additional time with the recent buyers and sellers adds $7 to the cost of including those participants.  Formulate a linear programming model to determine the amount of and type of participants that minimizes total cost (use lindo)

4. Steelco

Steco, a steel wholesaler, leases regional warehouses for its operation on a monthly basis.  Steco currently has a list of three potential warehouses that it can lease.  The cost per month to lease each warehouse is shown in table 1 below.   There are four sales districts in Stecos operation.  The monthly demand (in truck load requirements) is indicated in table 1 as well as the unit cost of sending a truck from a warehouse to a given district.  Steco wants to know which warehouses to lease and how many trucks to send from each warehouse to each district that will minimize overall cost.  Keep in mind that every warehouse does not have to be used.  Formulate an LP program and solve this problem.

Table 1

5. Hoggs Corn & Wheat Farm

The Hogg company are a group of Oklahoma agricultural producers.  The Hoggs want to derive a production process that will satisfy their goals. These farmers can produce corn, wheat, and/or hogs. The acreage they can cultivate for crops is 1100 acres, of which 600 acres is for corn production and 500 acres for wheat production. The Hogs have a maximum of 2000 hr of labor available for crop operations and 600 hr of labor for their hog operation.  Corn production requires 1.6 hr of labor per acre, wheat requires 2.0 hr of labor per acre, and the hog operation requires 20 hr of labor per hog unit. This farm can produce 120 bushel of corn per acre and 32 bushel of wheat per acre. Corn and wheat can be sold on the market at $2.50 per bushel and $2.35 per bushel, respectively, and/or can be used as feed in the hog operation. Also, corn and wheat may be purchased from the city feed store at $2.70 per bushel and $2.55 pr bushel, respectively, for feeding the hogs. The hog unit requires 169.1 bushel of feed during a single production period. The corn and wheat used as hog feed can be transformed into corn equivalent units; that is, 1.1 bushel of wheat equals 1 bushel of corn.

The cost per acre to produce corn and wheat are $146.40 per acre and $48.60 per acre.  The returns to the hog operation are $919.20 per hog unit.  A hog unit is a sow and her litter, which are sold on the market.  Returns to the hog operation are net values, for example, after maintenance and replacement stock cost.

The Hogg company is considering the following strategy:

As the goal with the highest priority, P1 they would like to see returns to the agriculture operation of at least $100,000.  As their second priority, P2, at least three fourths of the land available for corn production and at least one-half of the land available for wheat production must be planted. The producers also desire that the under-planting of corn be weighed three times greater than that for wheat.  The third and last priority goal P3, is that the hog operation beno greater than 40 hog units.