Questions: The two liquid products that Tampa Chemical makes - TC1 and TC2 - yield excessive amounts of three different pollutants: P1, P2, and P3. The state government has ordered the company to install and to employ antipollution devices. The following table provides the current daily emissions of each of the three pollutants, in kg / 1000 liters of product (kilograms per 1000 liters), and the maximum of each pollutant allowed in kg. Pollutant TC1 TC2 MAXIMUM ALLOWED P1 20 35 40 P2 10 15 20 P3 80 60 50 The manager of the Production Department has approved the installation of two antipollution devices. The emissions from each product can be handled by either device in any proportion. (The emissions are sent through a device only once, that is, the output of one device cannot be the input to the other or back to itself.) The following table shows the percentage of each pollutant from each product that is removed by each device. Device1 Device2 POLLUTANT TC1 TC2 TC1 TC2 P1 40 35 30 20 P2 60 60 0 0 P3 50 50 60 85 For example, if the emission from TC1 is sent through Device 1, 40% of pollutant P1, 60% of pollutant P2, and 50% of pollutant P3 are removed. Manufacturing considerations dictate that TC1 and TC2 be produced in the ratio of 1 to 2. Formulate a single LP model to determine a plan that maximizes the total daily production (amount of TC1 plus amount of TC2) while meeting governmental requirements.

The two liquid products that Tampa Chemical makes - TC1 and TC2 - yield excessive amounts of three different pollutants: P1, P2, and P3. The state government has ordered the company to install and to employ antipollution devices. The following table provides the current daily emissions of each of the three pollutants, in kg / 1000 liters of product (kilograms per 1000 liters), and the maximum of each pollutant allowed in kg. Pollutant TC1 TC2 MAXIMUM ALLOWED P1 20 35 40 P2 10 15 20 P3 80 60 50 The manager of the Production Department has approved the installation of two antipollution devices. The emissions from each product can be handled by either device in any proportion. (The emissions are sent through a device only once, that is, the output of one device cannot be the input to the other or back to itself.) The following table shows the percentage of each pollutant from each product that is removed by each device. Device1 Device2 POLLUTANT TC1 TC2 TC1 TC2 P1 40 35 30 20 P2 60 60 0 0 P3 50 50 60 85 For example, if the emission from TC1 is sent through Device 1, 40% of pollutant P1, 60% of pollutant P2, and 50% of pollutant P3 are removed. Manufacturing considerations dictate that TC1 and TC2 be produced in the ratio of 1 to 2. Formulate a single LP model to determine a plan that maximizes the total daily production (amount of TC1 plus amount of TC2) while meeting governmental requirements.
Transcript text: The two liquid products that Tampa Chemical makes - TC1 and TC2 - yield excessive amounts of three different pollutants: P1, P2, and P3. The state government has ordered the company to install and to employ antipollution devices. The following table provides the current daily emissions of each of the three pollutants, in $\mathrm{kg} / 1000$ liters of product (kilograms per 1000 liters), and the maximum of each pollutant allowed in kg . \begin{tabular}{lcccc} & & \multicolumn{1}{c}{ Product } & & \\ Pollutant & TC1 & TC2 & MAXIMUM ALLOWED \\ P1 & 20 & 35 & 40 \\ P2 & 10 & 15 & 20 \\ P3 & 80 & 60 & 50 \\ \hline \end{tabular} The manager of the Production Department has approved the installation of two antipollution devices. The emissions from each product can be handled by either device in any proportion. (The emissions are sent through a device only once, that is, the output of one device cannot be the input to the other or back to itself.) The following table shows the percentage of each pollutant from each product that is removed by each device. \begin{tabular}{cccccc} & \multicolumn{2}{c}{ Device1 } & & \multicolumn{2}{c}{ Device2 } \\ \cline { 2 - 3 } \cline { 5 - 6 } POLLUTANT & TC1 & TC2 & & TC1 & TC2 \\ \hline P1 & 40 & 35 & & 30 & 20 \\ P2 & 60 & 60 & & 0 & 0 \\ P3 & 50 & 50 & & 60 & 85 \\ \hline \end{tabular} For example, if the emission from TC1 is sent through Device 1, $40 \%$ of pollutant P1, $60 \%$ of pollutant P2, and $50 \%$ of pollutant P3 are removed. Manufacturing considerations dictate that TC1 and TC2 be produced in the ratio of 1 to 2. Formulate a single LP model to determine a plan that maximizes the total daily production (amount of TC1 plus amount of TC2) while meeting governmental requirements.
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Solution

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Solution Steps

To solve this problem, we need to formulate a linear programming (LP) model. The objective is to maximize the total daily production of TC1 and TC2 while ensuring that the emissions of pollutants P1, P2, and P3 do not exceed the maximum allowed limits. The constraints will include the removal percentages of pollutants by the two devices and the production ratio requirement between TC1 and TC2.

  1. Define decision variables for the amount of TC1 and TC2 produced.
  2. Define decision variables for the proportion of each product's emissions treated by each device.
  3. Set up the objective function to maximize the sum of TC1 and TC2 production.
  4. Establish constraints for each pollutant to ensure emissions do not exceed the maximum allowed after treatment.
  5. Include the production ratio constraint between TC1 and TC2.
Step 1: Define the Objective Function

The objective is to maximize the total production of TC1 and TC2. Let \( x \) be the amount of TC1 produced and \( y \) be the amount of TC2 produced. The objective function is: \[ \text{Maximize } z = x + y \]

Step 2: Define the Constraints

The constraints are based on the maximum allowed emissions for each pollutant after treatment by the devices.

Pollutant P1

The emissions of P1 from TC1 and TC2 after treatment must not exceed 40 kg. The effective emissions are: \[ 20 \times (1 - 0.4) \times x + 35 \times (1 - 0.35) \times y \leq 40 \] Simplifying, we get: \[ 12x + 22.75y \leq 40 \]

Pollutant P2

The emissions of P2 from TC1 and TC2 after treatment must not exceed 20 kg. The effective emissions are: \[ 10 \times (1 - 0.6) \times x + 15 \times (1 - 0.6) \times y \leq 20 \] Simplifying, we get: \[ 4x + 6y \leq 20 \]

Pollutant P3

The emissions of P3 from TC1 and TC2 after treatment must not exceed 50 kg. The effective emissions are: \[ 80 \times (1 - 0.5) \times x + 60 \times (1 - 0.85) \times y \leq 50 \] Simplifying, we get: \[ 40x + 9y \leq 50 \]

Production Ratio

The production ratio of TC1 to TC2 must be 1:2, which implies: \[ x = \frac{1}{2}y \]

Step 3: Solve the Linear Programming Problem

Using the constraints and the objective function, we solve the linear programming problem to find the optimal values of \( x \) and \( y \).

Final Answer

The optimal production levels are:

  • TC1 production: \( x = 0.3703 \)
  • TC2 production: \( y = 0.5985 \)
  • Total production: \( x + y = 0.9688 \)

\[ \boxed{x = 0.3703, \, y = 0.5985, \, x + y = 0.9688} \]

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