I am doing problem #14 in chapter 2 of the introduction to management science. H

Question

I am doing problem #14 in chapter 2 of the introduction to management science. How do you come up with the numbers on the two points after graphing?

I have (0,25) and (35,0) as two of the extreme points but the other two do not fall right on the graph so when I checked my homework I had the wrong numbers. Is there a way to calculate those two points? If so how?

RMC, Inc., is a small firm that produces a variety of chemical products. In a particular pro- duction process, three raw materials are blended (mixed together) to produce two products a fuel additive and a solvent base. Each ton of fuel additive is a mixture of ton of material 1 and of material 3. A ton of solvent base is a mixture of ½ ton of material 1, ton of material 2, and ío ton of material 3. After deducting relevant costs, the profit contribution is $40 for every ton of fuel additive produced and $30 for every ton of solvent base produced. RMC's production is constrained by a limited availability of the three raw materials. For the current production period, RMC has available the following quantities of each raw material: Raw Material Material 1 Material 2 Material 3 Amount Available for Production 20 tons 5 tons 21 tons Assuming that RMC is interested in maximizing the total profit contribution, answer the following a. What is the linear programming model for this problem? b. Find the optimal solution using the graphical solution procedure. How many tons of each product should be produced, and what is the projected total profit contribution? c. Is there any unused material? If so, how much'? d. Are any of the constraints redundant? If so, which ones?

Explanation / Answer

a) Let the ton of fuel additive produced be x1

Let the ton of solvent base produced be x2

Maximize Z = 40x1 + 30x2

Constraints:

2/5x1 + 1/2x2 <= 20 (Constraint for material x1)

1/5x2 <= 5 (Constraint for material x2)

3/5x1 + 3/10x2 <= 21 (constraints for material x3)

Now to answet your problem, we need to solve all the constraints to pick up all the intersection points

Example: Start with constraint for material 2

1/5 x2 <= 5

x2 <= 25

Now from the first material x1, we get

2/5x1 + 1/2 * 25 <= 20

2/5x1 + 12.5 <= 20

2/5x1 <= 7.5

x1 <= 18.75

So one point will be (18.75,25)

Now putting in the third equation

3/5x1 + 75/10 = 21

3/5x1 = 13.5

x1 = 13.5 * 5/3 = 22.5

Hence the point will be (22.5,25)

Now solving both 1 and 3 together we get

Equation 1: 2/5x1 + 1/2x2 = 20

Equation 3: 3/5x1 + 3/10x2 = 21

so converting into decimal form we get

0.4x1 + 0.5x2 = 20

0.6x1 + 0.3x2 = 21

Multiply the first equation by 3 and second by 5 we get

1.2x1 + 1.5x2 = 60

3x1 + 1.5x2 = 105

subtracting first from second we get

1.8x1 = 45

x1 = 25

Similarly, putting this value in the first equation we get

0.5x2 = 20 - 0.4x1 = 20 - 0.4(25) = 10

x2 = 20

x1 = 25 and x2=20

Since this satisfies the equation 2 so this is also a valid point

Now we can compute the values of maxima at all exterior points

The maxima will exists at this point

Z(max) = 40(25) + 20(30) = 1600$

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