The Nori & Leets Co. is one of the major producers of steel in its part of the world. It is located in the city of Steel-town and is the only large employer there. Steel-town 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 Steel-town city officials and citizens' groups what to do about the air pollution problem. Together they have worked out stringent air quality standards for the Steel-town air-shed.

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.

Pollutant ______________________ Required Reduction in Annual Emission Rate

___________________________________________ (Million pounds)

Particulates ......................................................... 60

Sulfur oxides ...................................................... 150

Hydrocarbons ..................................................... 125

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, 1 (2) using filter 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.

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 some combination 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 next table for using the methods at their full abatement capacities.

It also was determined 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 (1) the blast furnaces and (2) the open-hearth furnaces.
You have been asked to head a management science 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.
a. Display the model on a spreadsheet.
b. Obtain an optimal solution and generate the sensitivity report.
Management now wants to conduct some what-if analysis with your help. Since the company does not have much prior experience with the pollution abatement methods under consideration, the cost estimates given in the third table are fairly rough, and each one could easily be off by as much as 10 percent in either direction. There also is some uncertainty about the values given in the second table, but less so than for the third table. By contrast, the values in the first table are policy standards and so are prescribed constants.
However, there still is considerable debate about where to set these policy standards on the required reductions in the emission rates of the various pollutants. The numbers in the first table actually are preliminary values tentatively agreed upon before learning what the total cost would be to meet these standards. Both the city and company officials agree that the final decision on these policy standards should be based on the tradeoff between costs and benefits. With this in mind, the city has concluded that each 10 percent increase in the policy standards over the current values (all the numbers in the first table) would be worth $3.5 million to the city. Therefore, the city has agreed to reduce the company's tax payments to the city by $3.5 million for each 10 percent increase in the policy standards (up to 50 percent) that is accepted by the company.
Finally, there has been some debate about the relative values of the policy standards for the three pollutants. As indicated in the first table, the required reduction for particulates now is less than half of that for either sulfur oxides or hydrocarbons. Some have argued for decreasing this disparity. Others contend that an even greater disparity is justified because sulfur oxides and hydrocarbons cause considerably more damage than particulates. Agreement has been reached that this issue will be reexamined after information is obtained about which trade-offs in policy standards (increasing one while decreasing another) are available without increasing the total cost.
c. Letting q denote the percentage increase in all the policy standards given in the first table, use a parameter analysis report to systematically find an optimal solution and the total cost for the revised linear programming problem for each q = 10, 20, 30, 40, 50. Considering the tax incentive offered by the city, use these results to determine which value of u (including the option of q = 0) should be chosen by the company to minimize its total cost of both pollution abatement and taxes.
d. For the value of q chosen in part h, generate the sensitivity report and repeat parts f and g so that the decision makers can make a final decision on the relative values of the policy standards for the three pollutants?

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Reduction in Emission Rate from the Maximum Feasible Use of an Abatement Method Taller Smokestacks Open-Hearth Filters Blast Furnaces Better Fuels Blast Blast Furnaces Open-Hearth Furnaces Open-Hearth Furnaces Furnaces 12 35 37 Pollutant Particulates Sulfur oxides Hydrocarbons Furnaces 25 18 28 20 31 24 17 56 29 13 49 20 42 53 Total Annual Cost from the Maximum Feasible Use of an Abatement Method Abatement Open-Hearth Blast Furnaces $8 million Method Furnaces Taller smokestacks Filters Better fuels $10 million 6 million 9 million 7 million 11 million