Operations Research Answers

Use the simplex method to obtain the optimal solution of the following linear programming model

𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑍 = 35𝑥1 + 50𝑥2

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜

3𝑥1 + 𝑥2 ≤ 30

𝑥1 + 2𝑥2 ≤ 15

4𝑥1 + 4𝑥2 ≤ 40 𝑥1,

𝑥2 ≥ 0

Solve the following linear programming problem using the simplex method:

𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑧 = 14𝑥 + 15𝑦

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜

13𝑥 + 15𝑦 ≤ 80

−12𝑥 − 17𝑦 ≥ −120

𝑥 ≥ 0, 𝑦 ≥ 0

) Find the dual program of the following linear programming problem.

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑃 = 16𝑥 − 2𝑦 − 5𝑧

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜

𝑥 + 4𝑦 − 𝑧 ≥ 120

𝑥 + 𝑦 + 3𝑧 ≤ 130

𝑥 ≥ 0, 𝑦 ≥ 0, 𝑧 𝑖𝑠 𝑢𝑛𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑒𝑑.

Find the dual program of the following linear programming problem.

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑍 = 30𝑥1 − 50𝑥2 + 10𝑥3

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜

3𝑥1 + 2𝑥2 − 𝑥3 ≥ 44

𝑥1 − 𝑥2 + 𝑥3 = 7

𝑥1 𝑖𝑠 𝑢𝑛𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑒𝑑, 𝑥2 ≥ 0, 𝑥3 ≥ 0,

Find the dual program of the following linear programming problem.

𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑧 = 5𝑥1 − 2𝑥2

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜

3𝑥1 + 2𝑥2 ≥ 16

𝑥1 − 𝑥2 ≤ 4

𝑥1 ≥ 5

𝑥1 ≥0,𝑥2 𝑖𝑠 𝑢𝑛𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑒𝑑 

 i) Define the term operations research.

ii) Give a detailed description of the origin of operations research .

iii) Explain the methodology of operations research. iv) Discuss the applications of operations research

v) Define the following terms as used in operations research

Model, Objective function, Constraints , Model formulation, Feasible solution, Transportation problems, Allocation problems

State and explain the operations research techniques.

i) Discuss the significance of operations research.

ii) Identify the limitations of operations research.

iii) Outline and briefly explain the five principle phases of operations research.

iv) Define the term linear programming and outline the four steps followed when formulating a linear programming model mathematically.

List the basic properties of linear programming models indicating also the assumptions of linear programming problems

Arrivals at a telephone booth are considered to be Poisson with an average time of 10 minutes between one arrival and the next. The length of a phone call is assumed to be distributed exponentially with mean of 3 minutes.

i) What is the probability that a person arriving at the booth will have to wait?

ii) What is the average length of the queues that form time to time?

The telephone department will install a second booth when convinced that an arrival would expect to have to wait at least three minutes for the phone. By how much must the flow of arrivals be increased in order to justify a second booth?