Operations Research Answers

A dispatcher for a City’s Taxi Association has five taxi cabs at different locations and four customers who have called for transportation service. The distance (in km) from each taxi’s present location to each customer is shown in the following table. (3 points) 

Cab 

Customer I II III IV A 7 2 4 10 B 5 1 5 6 C 8 7 6 5 D 2 5 2 4 E 3 3 5 8 

1. a. Determine the optimal assignment that will minimize the total distance travelled.
2. b. Compute the total minimum distance of the optimal assignment. 

Question 1

A train can accommodate at most 80 passengers, economy class (x) and first-

class (y) passengers. To avoid making a loss the train must carry at least ten

economy class passengers and at least twenty first class passengers. Due to the

number of complementary items the first-class passengers receive, the number

of first-class passengers should be at most three times the number of economy

class passengers.

First class tickets cost N$5000 while economy class tickets cost N$3000.

i) State/ describe what the variables x and y represent.

ii) State the objective function.

iii) List the constraints.

iv) Draw and clearly label the constraints on graph paper.

v) Clearly show and label the feasible region.

vi) From the graph deduce the number of economy and first-class tickets

that should be sold to make as much money as possible.

vii) Work out the maximum amount of money from ticket sales that the

train company will make.

A train can accommodate at most 80 passengers, economy class (x) and first-

class (y) passengers. To avoid making a loss the train must carry at least ten

economy class passengers and at least twenty first class passengers. Due to the 

number of complementary items the first-class passengers receive, the number 

of first-class passengers should be at most three times the number of economy 

class passengers. 

First class tickets cost N$5000 while economy class tickets cost N$3000. 

i) State/ describe what the variables x and y represent.

ii) State the objective function.

iii) List the constraints.

iv) Draw and clearly label the constraints on graph paper.

v) Clearly show and label the feasible region.

vi) From the graph deduce the number of economy and first-class tickets 

that should be sold to make as much money as possible.

vii) Work out the maximum amount of money from ticket sales that the 

train company will make.

This time, our immune system is the best defense. With this, a Melagail wishes to mix two types of foods in such a way that vitamin contents of the mixture contain at least 8 units of vitamin A and 10 units of vitamin C. Food A contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C. Food B contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs ₱50 per kg to purchase Food A and ₱70 per kg to purchase Food B. Formulate this problem as a linear programming problem to minimize the cost of such a mixture.

A. Define the decision variables

B. Write the LP Model

C. Determine the feasible region

D. Determine the optimal solution

E. Final Answer: (interpret the result)

assume you are given the following linear programming model. max z= 2x1 + 3x2, subject to x1+x2<20, 2x1+4x2<60, x1>0, x2>0. compute a unit profit allowable range(range of optimal) for chair and table interpret your answer

To state whether the following is true or false with a short proof or a counter example in support of the answer :

The following 4/3/F/F_ max problem can be reduced to a machine problem:

Job Processing time (in hours) on

M₁ M₂ M₃

1 8 6 10

2 5 2 13

3 4 11 11

4 6 7 10

To state if the statements are true or false with a short proof or a counter example in support of the answer.

1.) The optimal solution for the following LLP is Z* = 30 :

Maximise Z = x₁ - x₂ + 3x₃

Subject to : x₁ + x₂ + x₃ ≤ 10

x₁ , x₂ , x₃ ≥ 0

2.) For the mixed generator r₍ₙ₊₁₎=(5r+7)(mod 8), if r₀=4, then rₙ=0

3.) If the availabilities and requirements of a transportation problem are integers, the optimal solution to the problem will have integer values.

4.) The optimal solution of ILLP can be obtained by rounding off the optimal solution of its LP relaxation.

To use the simplex method to solve the following LLP :

Maximise z = 4x₁ + 3x₂

Subject to :

2x₁ + x₂ <= 1000

x₁ + x₂ <= 800

x₁ <= 400

x₂ <= 700

x₁ , x₂ >= 0

To write the dual of the following LLP :

Minimise Z = 16x₁ + 9x₂ + 21x₃

Subject to the constraints :

x₁ + x₂ + x₃ = 16

2x₁ + x₂ + x₃ >= 12

x₁ , x₂ >= 0

x₃ - unrestricted.

To solve the ILLP given below by the graphical method :

Maximum Z = 95x₁ + 100x₂

Subject to the constraints

5x₁ + 2x₂ <= 20

x₁ >= 3

x₂ <= 5

x₁ , x₂ are non-negative integers.