Question

A company produces regular and premium gasoline for independent service stations. The company manufactures the products by blending three components: 1, 2, and 3. The gasolines are sold at different prices while the petroleum components have different costs. Data show that regular gasoline can be sold for $2.90 per gallon and premium gasoline for $3.00 per gallon. For the current production period, the company can obtain the three components at the cost per gallon and in the quantities shown in Table 1. Table 2 shows the product specifications that restrict the amount of each component used in each product. Current commitments to distributors require the company to produce at least 10,000 gallons of regular gasoline. The company wants to determine how to mix the three components into the two gasoline products and maximize total profit. Formulate a model to determine how many gallons of each component should be used in the regular gasoline and the premium gasoline. [Note: Xij = no. of gallons of component i used in gasoline j, where i = 1, 2, 3 and j = R (Regular) and P (Premium).] Table 1 -------------------------------------------------------------------- Component Cost/gallon Maximum Available 1: $2.50 14,000 gallons 2: $2.60 10,000 gallons 3: $2.84 10,000 gallons -------------------------------------------------------------------- Table 2 -------------------------------------------------------------------- Product Specifications Regular gas: At most 23% component 1 Regular gas: At least 27% component 2 Premium gas: At most 45% component 2 Premium gas: At least 30% component 3 -------------------------------------------------------------------- 1) Which of the following constraints is correct? X1P + X2P = 14,000 X1P + X1P > 14,000 X1R + X1P ≤ 14,000 2) Which of the following constraints is correct? X2R ≥ 0.27(X1R + X2R + X3R) X2R < 0.27(X1R + X2R + X3R) X2R = 0.27(X1P + X2P + X3P) 3) Which of the following constraints is correct? X1R > 0.23(X1R + X2P + X3R) X2R = 0.23(X1P + X2P + X3P) X1R ≤ 0.23(X1R + X2R + X3R)

Answer 1

A linear programming model is used to determine the optimal blend of different components for regular and premium gasoline to maximize profit. The correct constraints are established to ensure component usage is within allowable limits and product specifications are met.

Step-by-step explanation:

To address the problem of maximizing profit for a company producing regular and premium gasoline, we need to first establish the correct constraints based on the given data and variables.

We can formulate a linear programming model for this problem where Xij represents the quantity of component i in gasoline j, with i being 1, 2, or 3 and j being R (Regular) or P (Premium).

Constraint Verification

The correct constraint that ensures no more than 14,000 gallons of component 1 are used is X1R + X1P ≤ 14,000.

1. For regular gasoline to have at least 27% of component 2, the correct constraint is X2R ≥ 0.27(X1R + X2R + X3R).
2. To ensure that regular gasoline has at most 23% of component 1, the appropriate constraint is X1R ≤ 0.23(X1R + X2R + X3R).

All these constraints work to ensure that the production of gasoline meets the company's costs and resources, and the product specifications for regular and premium gasoline.