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The variables involved in LPP are called ______

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#### Solution

**decision variables**

#### APPEARS IN

#### RELATED QUESTIONS

Which of the following statements is correct?

**Find the feasible solution of the following inequation:**

3x + 2y ≤ 18, 2x + y ≤ 10, x ≥ 0, y ≥** **0

**Find the feasible solution of the following inequations:**

x - 2y ≤ 2, x + y ≥ 3, - 2x + y ≤ 4, x ≥ 0, y ≥ 0

A company produces two types of articles A and B which requires silver and gold. Each unit of A requires 3 gm of silver and 1 gm of gold, while each unit of B requires 2 gm of silver and 2 gm of gold. The company has 6 gm of silver and 4 gm of gold. Construct the inequations and find feasible solution graphically.

In a cattle breading firm, it is prescribed that the food ration for one animal must contain 14. 22 and 1 units of nutrients A, B, and C respectively. Two different kinds of fodder are available. Each unit of these two contains the following amounts of these three nutrients:

Fodder → |
Fodder 1 | Fodder 2 |

Nutrient ↓ |
||

Nutrients A | 2 | 1 |

Nutrients B | 2 | 3 |

Nutrients C | 1 | 1 |

The cost of fodder 1 is ₹ 3 per unit and that of fodder 2 ₹ 2. Formulate the LPP to minimize the cost.

A printing company prints two types of magazines A and B. The company earns ₹ 10 and ₹ 15 in magazines A and B per copy. These are processed on three Machines I, II, III. Magazine A requires 2 hours on Machine I, 5 hours on Machine II, and 2 hours on machine III. Magazine B requires 3 hours on machine I, 2 hours on machine II and 6 hours on Machine III. Machines I, II, III are available for 36, 50, and 60 hours per week respectively. Formulate the LPP to determine weekly production of magazines A and B, so that the total profit is maximum.

A company manufactures two types of fertilizers F_{1} and F_{2}. Each type of fertilizer requires two raw materials A and B. The number of units of A and B required to manufacture one unit of fertilizer F_{1} and F_{2} and availability of the raw materials A and B per day are given in the table below:

Fertilizers→ | F_{1} |
F_{2} |
Availability |

Raw Material ↓ | |||

A | 2 | 3 | 40 |

B | 1 | 4 | 70 |

By selling one unit of F_{1} and one unit of F_{2}, the company gets a profit of ₹ 500 and ₹ 750 respectively. Formulate the problem as LPP to maximize the profit.

If John drives a car at a speed of 60 km/hour, he has to spend ₹ 5 per km on petrol. If he drives at a faster speed of 90 km/hour, the cost of petrol increases ₹ 8 per km. He has ₹ 600 to spend on petrol and wishes to travel the maximum distance within an hour. Formulate the above problem as L.P.P.

**Solve the following LPP by graphical method:**

Minimize z = 8x + 10y, subject to 2x + y ≥ 7, 2x + 3y ≥ 15, y ≥ 2, x ≥ 0, y ≥ 0.

Which of the following is correct?

The maximum value of z = 10x + 6y subject to the constraints 3x + y ≤ 12, 2x + 5y ≤ 34, x, ≥ 0, y ≥ 0 is ______.

The point of which the maximum value of x + y subject to the constraints x + 2y ≤ 70, 2x + y ≤ 95, x, ≥ 0, y ≥ 0 is is obtained at ______.

Solution of LPP to minimize z = 2x + 3y, such that x ≥ 0, y ≥ 0, 1 ≤ x + 2y ≤ 10 is ______.

The corner points of the feasible solution are (0, 0), (2, 0), `(12/7, 3/7)`, (0, 1). Then z = 7x + y is maximum at ______.

The half-plane represented by 3x + 2y < 8 contains the point ______.

**Solve the following LPP:**

Maximize z = 5x_{1} + 6x_{2} subject to 2x_{1} + 3x_{2} ≤ 18, 2x_{1} + x_{2} ≤ 12, x_{1} ≥ 0, x_{2} ≥ 0.

**Solve the following LPP:**

Maximize z = 6x + 10y subject to 3x + 5y ≤ 10, 5x + 3y ≤ 15, x ≥ 0, y ≥ 0.

**Solve the following LPP:**

Maximize z = 2x + 3y subject to x - y ≥ 3, x ≥ 0, y ≥ 0.

**Solve each of the following inequations graphically using XY-plane:**

4x - 18 ≥ 0

**Solve the following LPP:**

Maximize z = 4x_{1} + 3x_{2} subject to

3x_{1} + x_{2} ≤ 15, 3x_{1} + 4x_{2} ≤ 24, x_{1} ≥ 0, x_{2} ≥ 0.

**Solve the following LPP:**

Maximize z =60x + 50y_{ } subject to

x + 2y ≤ 40, 3x + 2y ≤ 60, x ≥ 0, y ≥ 0.

A carpenter makes chairs and tables. Profits are ₹ 140 per chair and ₹ 210 per table. Both products are processed on three machines: Assembling, Finishing and Polishing. The time required for each product in hours and availability of each machine is given by the following table:

Product → | Chair (x) |
Table (y) |
Available time (hours) |

Machine ↓ | |||

Assembling | 3 | 3 | 36 |

Finishing | 5 | 2 | 50 |

Polishing | 2 | 6 | 60 |

Formulate the above problem as LPP. Solve it graphically

A company produces mixers and food processors. Profit on selling one mixer and one food processor is Rs 2,000 and Rs 3,000 respectively. Both the products are processed through three machines A, B, C. The time required in hours for each product and total time available in hours per week on each machine arc as follows:

Machine |
Mixer |
Food Processor |
Available time |

A | 3 | 3 | 36 |

B | 5 | 2 | 50 |

C | 2 | 6 | 60 |

How many mixers and food processors should be produced in order to maximize the profit?

A chemical company produces a chemical containing three basic elements A, B, C, so that it has at least 16 litres of A, 24 litres of B and 18 litres of C. This chemical is made by mixing two compounds I and II. Each unit of compound I has 4 litres of A, 12 litres of B and 2 litres of C. Each unit of compound II has 2 litres of A, 2 litres of B and 6 litres of C. The cost per unit of compound I is ₹ 800 and that of compound II is ₹ 640. Formulate the problems as LPP and solve it to minimize the cost.

A firm manufactures two products A and B on which profit earned per unit ₹ 3 and ₹ 4 respectively. Each product is processed on two machines M_{1} and M_{2}. The product A requires one minute of processing time on M_{1} and two minutes of processing time on M_{2}, B requires one minute of processing time on M_{1} and one minute of processing time on M_{2}. Machine M_{1} is available for use for 450 minutes while M_{2} is available for 600 minutes during any working day. Find the number of units of product A and B to be manufactured to get the maximum profit.

A firm manufacturing two types of electrical items A and B, can make a profit of ₹ 20 per unit of A and ₹ 30 per unit of B. Both A and B make use of two essential components a motor and a transformer. Each unit of A requires 3 motors and 2 transformers and each units of B requires 2 motors and 4 transformers. The total supply of components per month is restricted to 210 motors and 300 transformers. How many units of A and B should be manufactured per month to maximize profit? How much is the maximum profit?

In a cattle breeding firm, it is prescribed that the food ration for one animal must contain 14, 22, and 1 unit of nutrients A, B, and C respectively. Two different kinds of fodder are available. Each unit weight of these two contains the following amounts of these three nutrients:

Nutrient\Fodder |
Fodder 1 |
Fodder2 |

Nutrient A | 2 | 1 |

Nutrient B | 2 | 3 |

Nutrient C | 1 | 1 |

The cost of fodder 1 is ₹ 3 per unit and that of fodder ₹ 2 per unit. Formulate the L.P.P. to minimize the cost.

A printing company prints two types of magazines A and B. The company earns ₹ 10 and ₹ 15 on magazines A and B per copy. These are processed on three machines I, II, III. Magazine A requires 2 hours on Machine I, 5 hours on Machine II and 2 hours on Machine III. Magazine B requires 3 hours on Machine I, 2 hours on Machine II and 6 hours on Machine III. Machines I, II, III are available for 36, 50, 60 hours per week respectively. Formulate the Linear programming problem to maximize the profit.

A manufacturer produces bulbs and tubes. Each of these must be processed through two machines M_{1} and M_{2}. A package of bulbs requires 1 hour of work on Machine M_{1} and 3 hours of work on M_{2}. A package of tubes requires 2 hours on Machine M_{1} and 4 hours on Machine M_{2}. He earns a profit of ₹ 13.5 per package of bulbs and ₹ 55 per package of tubes. If maximum availability of Machine M_{1} is 10 hours and that of Machine M_{2} is 12 hours, then formulate the L.P.P. to maximize the profit.

**Solve the following L.P.P. by graphical method :**

Maximize : Z = 4x + 6y subject to 3x + 2y ≤ 12, x + y ≥ 4, x, y ≥ 0.

**Choose the correct alternative :**

Of all the points of the feasible region the optimal value of z is obtained at a point

**Choose the correct alternative :**

The corner points of the feasible region given by the inequations x + y ≤ 4, 2x + y ≤ 7, x ≥ 0, y ≥ 0, are

If the corner points of the feasible region are (0, 0), (3, 0), (2, 1) and `(0, 7/3)` the maximum value of z = 4x + 5y is ______.

**Choose the correct alternative :**

The half plane represented by 3x + 2y ≤ 0 constraints the point.

**Fill in the blank :**

“A gorage employs eight men to work in its shownroom and repair shop. The constraints that there must be at least 3 men in showroom and at least 2 men in repair shop are ______ and _______ respectively.

A train carries at least twice as many first class passengers (y) as second class passengers (x) The constraint is given by_______

**Fill in the blank :**

A dish washing machine holds up to 40 pieces of large crockery (x) This constraint is given by_______.

**State whether the following is True or False :**

The point (1, 2) is not a vertex of the feasible region bounded by 2x + 3y ≤ 6, 5x + 3y ≤ 15, x ≥ 0, y ≥ 0.

The feasible region is the set of point which satisfy.

Which value of x is in the solution set of inequality − 2X + Y ≥ 17

Maximize z = 7x + 11y subject to 3x + 5y ≤ 26, 5x + 3y ≤ 30, x ≥ 0, y ≥ 0

Maximize z = 3x + 5y subject to x + 4y ≤ 24, 3x + y ≤ 21, x + y ≤ 9, x ≥ 0, y ≥ 0 also find the maximum value of z

Minimize z = 7x + y subjected to 5x + y ≥ 5, x + y ≥ 3, x ≥ 0, y ≥ 0

Minimize z = 2x + 4y is subjected to 2x + y ≥ 3, x + 2y ≥ 6, x ≥ 0, y ≥ 0 show that the minimum value of z occurs at more than two points

x − y ≤ 1, x − y ≥ 0, x ≥ 0, y ≥ 0 are the constant for the objective function z = x + y. It is solvable for finding optimum value of z? Justify?

**Choose the correct alternative:**

The feasible region is

**Choose the correct alternative:**

Z = 9x + 13y subjected to constraints 2x + 3y ≤ 18, 2x + y ≤ 10, 0 ≤ x, y was found to be maximum at the point

**State whether the following statement is True or False:**

Objective function of LPP is a relation between the decision variables

**State whether the following statement is True or False:**

LPP is related to efficient use of limited resources

Constraints are always in the form of ______ or ______.

The constraint that in a particular XII class, number of boys (y) are less than number of girls (x) is given by ______

A company produces two types of pens A and B. Pen A is of superior quality and pen B is of lower quality. Profits on pens A and B are ₹ 5 and ₹ 3 per pen respectively. Raw materials required for each pen A is twice as that of pen B. The supply of raw material is sufficient only for 1000 pens per day. Pen A requires a special clip and only 400 such clips are available per day. For pen B, only 700 clips are available per day. Formulate this problem as a linear programming problem.

**Solve the following linear programming problems by graphical method.**

Maximize Z = 6x_{1} + 8x_{2} subject to constraints 30x_{1} + 20x_{2 }≤ 300; 5x_{1} + 10x_{2} ≤ 110; and x_{1}, x_{2} ≥ 0.

**Solve the following linear programming problems by graphical method.**

Minimize Z = 20x_{1} + 40x_{2} subject to the constraints 36x_{1} + 6x_{2} ≥ 108; 3x_{1} + 12x_{2} ≥ 36; 20x_{1} + 10x_{2} ≥ 100 and x_{1}, x_{2} ≥ 0.

Maximize: z = 3x_{1} + 4x_{2} subject to 2x_{1} + x_{2} ≤ 40, 2x_{1} + 5x_{2} ≤ 180, x_{1}, x_{2} ≥ 0. In the LPP, which one of the following is feasible comer point?

A solution which maximizes or minimizes the given LPP is called

In the given graph the coordinates of M_{1} are

The maximum value of the objective function Z = 3x + 5y subject to the constraints x ≥ 0, y ≥ 0 and 2x + 5y ≤ 10 is

The minimum value of the objective function Z = x + 3y subject to the constraints 2x + y ≤ 20, x + 2y ≤ 20, x > 0 and y > 0 is

A firm manufactures two products A and B on which the profits earned per unit are ₹ 3 and ₹ 4 respectively. Each product is processed on two machines M_{1} and M_{2}. Product A requires one minute of processing time on M_{1} and two minutes on M_{2}, While B requires one minute on M_{1} and one minute on M_{2}. Machine M_{1} is available for not more than 7 hrs 30 minutes while M_{2} is available for 10 hrs during any working day. Formulate this problem as a linear programming problem to maximize the profit.

**Solve the following linear programming problem graphically.**

Maximize Z = 60x_{1} + 15x_{2} subject to the constraints: x_{1} + x_{2} ≤ 50; 3x_{1} + x_{2} ≤ 90 and x_{1}, x_{2} ≥ 0.

The maximum value of Z = 3x + 5y, subject to 3x + 2y ≤ 18, x ≤ a, y ≤ 6, x, y ≥ 0 is ______.

The LPP to maximize Z = x + y, subject to x + y ≤ 1, 2x + 2y ≥ 6, x ≥ 0, y ≥ 0 has ________.

The values of θ satisfying sin7θ = sin4θ - sinθ and 0 < θ < `pi/2` are ______

Which of the following can be considered as the objective function of a linear programming problem?

The minimum value of z = 5x + 13y subject to constraints 2x + 3y ≤ 18, x + y ≥ 10, x ≥ 0, y ≥ 2 is ______

The point which provides the solution of the linear programming problem, Max.(45x + 55y) subject to constraints x, y ≥ 0, 6x + 4y ≤ 120, 3x + 10y ≤ 180, is ______

Solve the following LP.P.

Maximize z = 13x + 9y,

Subject to 3x + 2y ≤ 12,

x + y ≥ 4,

x ≥ 0,

y ≥ 0.

The optimal value of the objective function is attained at the ______ of feasible region.

The set of feasible solutions of LPP is a ______.

Solution which satisfy all constraints is called ______ solution.

Shamli wants to invest ₹ 50, 000 in saving certificates and PPF. She wants to invest atleast ₹ 15,000 in saving certificates and at least ₹ 20,000 in PPF. The rate of interest on saving certificates is 8% p.a. and that on PPF is 9% p.a. Formulation of the above problem as LPP to determine maximum yearly income, is ______.

The maximum value of Z = 9x + 13y subject to constraints 2x + 3y ≤ 18, 2x + y ≤ 10, x ≥ 0, y ≥ 0 is ______.

For the following shaded region, the linear constraint are:

**Solve the following problems by graphical method:**

Maximize z = 4x + 2y subject to 3x + y ≥ 27, x + y ≥ 21, x ≥ 0 y ≥ 0

**Solve the following LPP by graphical method:**

Maximize: z = 3x + 5y Subject to: x + 4y ≤ 24, 3x + y ≤ 21, x + y ≤ 9, x ≥ 0, y ≥ 0

Two kinds of foods A and B are being considered to form a weekly diet. The minimum weekly requirements of fats, Carbohydrates and proteins are 12, 16 and 15 units respectively. One kg of food A has 2, 8 and 5 units respectively of these ingredients and one kg of food B has 6, 2 and 3 units respectively. The price of food A is Rs. 4 per kg and that of food B is Rs. 3 per kg. Formulate the L.P.P. and find the minimum cost.

**Find graphical solution for the following system of linear in equation:**

x + 2y ≥ 4, 2x - y ≤ 6