Formulate a linear programming model for the problem 3. Findthe optimal solution for the problem

Vale S.A. is a Brazilian multinational corporation engaged in metals and mining and the largest producer of iron ore and nickel in the World. One of the cities where Vale operates is Mariana. Mariana has grown and thrived along with Vale, which now is the largest employer in the city. Therefore, the population has always supported the company. Nevertheless, a new generation has emerged in the city. Youth is aware that Vale’s production process is posing a risk to the population’s health, damaging the environment, and impacting the looks of Mariana.

A recent students’ demonstration caused the election of a new board of directors for Vale in Mariana. The new board is more environmental concern than the previous one, and then willing to change some of the policies of the company so that some of the impacts that Vale is causing in Mariana are diminished. The new directors have had meetings with Mariana council officers and a youth’s group, to discuss about one of the worst environmental impacts caused by Vale in Mariana: air pollution. Together they have agreed upon setting air quality standards for Mariana.

Vale contributes to air pollution through mainly two sources: blast andpuddling furnaces. The types of air pollutants discharged by Vale in Mariana arenitrogen dioxide (NO2), sulphur dioxide (SO2) andfine particulate matter (PM2.5). According to the news air quality standards, Vale must reduce its annual pollutants emissions (NO2, SO2, PM2.5) by the extends presented in Table 1.


Table 1: New clean air standards for Vale S.A. in Mariana

Pollutant Required reduction in annual pollutant emission rate (millions of kilos)
PM2.5 60
SO2 150
NO2 125


Management has been requested by the board of directors to determine how to accomplish these reductions at minimum cost. Management found out that there are three types of reduction techniques: using filter devices, adopting cleaner products in the fuels for the furnaces and extending the height of the smokestacks. Each of these techniques has a technological limit but can be used at a fraction of its technological limit. Once one of these techniques is implemented on a furnace, the amount of pollutant emissions produced by a furnace decrease. Table 2 displays the pollutant emission reductionbased on the type of furnace and by adopting a reduction technique to its full technological limit.For purposes of analysis, it is assumed that each method also can be used less fully to achieve any fraction of the emission reductions shown in Table 2. Moreover, the fractions can be different for blast furnaces and for puddling furnaces. For either type of furnace, the emission reduction achieved by each method is not substantially affected by whether the other methods also are used.





Table 2: Reduction in emission rate (in millions of kilos per year) from the maximum feasible use of a reduction technique

Pollutant Filter devices Cleaner products Higher smokestacks
Blast furnaces Puddling furnaces Blast furnace Puddling furnaces Blast furnace Puddling furnaces
PM2.5 12 9 25 20 17 13
SO2 35 42 18 31 56 49
NO2 37 53 28 24 29 20


Once these data were created, management realized that no single reduction technique by itself could accomplish all the necessary reductions. On the other hand, implementing all the three techniques at full capacity (technological limit) on both blast andpuddling furnaces would be very expensive and much more than the adequate. Accordingly, management figured that the best approach is to use some combination of the reduction techniques, such as fractional capacities, based upon the relative costs. Besides, because of the differences between the blast andpuddling furnaces, the same combination of reduction techniques should not be adopted on both types of furnaces.

A study was performed to estimate the total annual cost of each reduction technique. The total annual cost of a reduction technique consists of operating and maintenance costs together with reduced profit due to any decrease in the performance of the production process caused by the reduction technique. Apart from operating and maintenance costs, the total annual cost also includes the initial capital cost required to install the technique. To change this one-time cost comparable with the ongoing annual costs, the time value of money was used to estimate the annual expenditure (over the expected life of the technique) that would be equivalent in value to the initial capital cost. The total annual costs for implementing each of the three reduction techniques at their full capacities is presented in Table 3. This study also determined that the total annual cost of a technique being adopted at a lower capacity level is proportional to the fraction of the reduction capacity displayed in Table 2 that is achieved. Therefore, for any given fraction accomplished, the total annual cost would be that fraction of the corresponding quantity in Table 3.

Table 3: Total annual cost from the maximum feasible use of a reduction technique for Vale S.A in Mariana ($ millions)

Reduction technique Blast furnaces Puddling furnaces
Filter devices 8 10
Cleaner products 7 6
Higher smokestacks 11 9

Now, Vale wants to create a plan for the reduction of pollutant emissions. This plan must specify which reduction techniques will be used and at what fractions of their capacities for the blast furnaces and puddling furnaces. Due to the nature of the problem, the linear programming technique must be used.

Since Vale S.A. does not have much priori experience with the presented pollution emission reduction techniques, the estimated total annual costs shown in Table 3 are fairly rough, and each could easily be off by as much as 10% in either direction. Vale then needs to understand the effect of an inaccuracy in estimating each cost parameter given in Table 3. Moreover, Mariana City Council decided to reduce Vale’s tax payment to the city by $3.5 million for a reduction of 20% in the air quality standard offine particulate matter.Vale S.A. thus wants to know if a decrement of 20% on the air quality standard forfine particulate matter yields a reduction plan more attractive.

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1. Define the business problem 2. Formulate a linear programming model for the problem 3. Findthe optimal solution for the problem 4. Discuss the sensitivity analysis

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