1) The production of three goods requires using two machines. Machine 1 can be u

Question

1) The production of three goods requires using two machines. Machine 1 can be utilized for 100 hours, and machine 2 can be utilized for 100 hours, too. The time spent for the production of one unit of each good is given by the following table:

Machine 1 Machine 2

Good 1: 3 2

Good 2: 1 2

Good 3: 4 1

The profits per unit produced of the three goods are 6, 3, and 4, respectively.

(a) Write down the linear programming problem this leads to. (2 points)

(b) Solve this problem geometrically. (3 points)

(c) If machine 1 increases its capacity to 101 (from 100), while the capacity of the second machine remains the same (100), what is the new maximal profit? (2 points)

Explanation / Answer

This is an example of product-mix problem in which same resources are meant for production of different products having different technological coefficients as well as different profits per unit.

Let G1, G2, G3 be the quantities of Good1, Good2, and Good3 respectively to be produced.

Objective is to have maximum Profit Z = 6G1 + 3G2 + 4G3

Constraints are availability of machine times: 3G1+G2+4G3 <=100 (machine1), 2G1+2G2+G3 <= 100 (machine2)

Lastly G1, G2, G3 >= 0 being quantities

Formulation and solution is as follows:

Solution

Solution in case machine1 capacity is 101

Details

G1 G2 G3 RHS Maximize 6 3 4 Max 6G1 + 3G2 + 4G3 Machine1 3 1 4 <= 100 3G1 + G2 + 4G3 <= 100 Machine2 2 2 1 <= 100 2G1 + 2G2 + G3 <= 100

Related Questions

Navigate

• Browse (All)
• Browse 1
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.