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Constraints are always in the form of ______ or ______.

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#### Solution

Constraints are always in the form of **equations** or **inequations**.

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#### RELATED QUESTIONS

Which of the following statements is correct?

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

2x + 3y ≤ 6, x + y ≥ 2, 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.

A furniture dealer deals in tables and chairs. He has ₹ 1,50,000 to invest and a space to store at most 60 pieces. A table costs him ₹ 1500 and a chair ₹ 750. Construct the inequations and find the feasible solution.

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 doctor has prescribed two different units of foods A and B to form a weekly diet for a sick person. The minimum requirements of fats, carbohydrates and proteins are 18, 28, 14 units respectively. One unit of food A has 4 units of fat, 14 units of carbohydrates and 8 units of protein. One unit of food B has 6 units of fat, 12 units of carbohydrates and 8 units of protein. The price of food A is ₹ 4.5 per unit and that of food B is ₹ 3.5 per unit. Form the LPP, so that the sick person’s diet meets the requirements at a minimum cost.

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.

The company makes concrete bricks made up of cement and sand. The weight of a concrete brick has to be at least 5 kg. Cement costs ₹ 20 per kg and sand costs of ₹ 6 per kg. Strength consideration dictates that a concrete brick should contain minimum 4 kg of cement and not more than 2 kg of sand. Form the L.P.P. for the cost to be minimum.

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

Maximize z = 11x + 8y, subject to x ≤ 4, y ≤ 6, x + y ≤ 6, x ≥ 0, y ≥ 0

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

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

**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.

**Select the appropriate alternatives for each of the following question:**

The value of objective function is maximum under linear constraints

Which of the following is correct?

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

**Solve the following LPP:**

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

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

4x - 18 ≥ 0

**Solve the following LPP:**

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

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

**Solve the following LPP:**

Minimize z = 4x + 2y

Subject to 3x + y ≥ 27, x + y ≥ 21, x + 2y ≥ 30, x ≥ 0, y ≥ 0

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 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 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:

Raw Material\Fertilizers |
F_{1} |
F_{2} |
Availability |

A | 2 | 3 | 40 |

B | 1 | 4 | 70 |

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

Objective function of LPP is ______.

**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

**Choose the correct alternative :**

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

**Fill in the blank :**

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

**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 :**

Saina wants to invest at most ₹ 24000 in bonds and fixed deposits. Mathematically this constraints is written as x + y ≤ 24000 where x is investment in bond and y is in fixed deposits.

**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.

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

The feasible solution of LPP belongs to only quadrant I.

The feasible region is the set of point which satisfy.

Maximize z = 10x + 25y subject to x + y ≤ 5, 0 ≤ x ≤ 3, 0 ≤ y ≤ 3

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

Maximize z = −x + 2y subjected to constraints x + y ≥ 5, x ≥ 3, x + 2y ≥ 6, y ≥ 0 is this LPP solvable? Justify your answer.

**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

The variables involved in LPP are called ______

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.

A company produces two types of products say type A and B. Profits on the two types of product are ₹ 30/- and ₹ 40/- per kg respectively. The data on resources required and availability of resources are given below.

Requirements |
Capacity available per month |
||

Product A |
Product B |
||

Raw material (kgs) | 60 | 120 | 12000 |

Machining hours/piece | 8 | 5 | 600 |

Assembling (man hours) | 3 | 4 | 500 |

Formulate this problem as a linear programming problem to maximize the profit.

A company manufactures two models of voltage stabilizers viz., ordinary and auto-cut. All components of the stabilizers are purchased from outside sources, assembly and testing is carried out at the company’s own works. The assembly and testing time required for the two models are 0.8 hours each for ordinary and 1.20 hours each for auto-cut. Manufacturing capacity 720 hours at present is available per week. The market for the two models has been surveyed which suggests a maximum weekly sale of 600 units of ordinary and 400 units of auto-cut. Profit per unit for ordinary and auto-cut models has been estimated at ₹ 100 and ₹ 150 respectively. Formulate the 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.**

Maximize Z = 22x_{1} + 18x_{2} subject to constraints 960x_{1} + 640x_{2} ≤ 15360; x_{1} + x_{2} ≤ 20 and x_{1}, x_{2} ≥ 0.

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

Given an L.P.P maximize Z = 2x_{1} + 3x_{2} subject to the constrains x_{1} + x_{2} ≤ 1, 5x_{1} + 5x_{2} ≥ 0 and x_{1} ≥ 0, x_{2} ≥ 0 using graphical method, we observe

A firm manufactures pills in two sizes A and B. Size A contains 2 mgs of aspirin, 5 mgs of bicarbonate and 1 mg of codeine. Size B contains 1 mg. of aspirin, 8 mgs. of bicarbonate and 6 mgs. of codeine. It is found by users that it requires at least 12 mgs. of aspirin, 74 mgs. of bicarbonate and 24 mgs. of codeine for providing immediate relief. It is required to determine the least number of pills a patient should take to get immediate relief. Formulate the problem as a standard LLP.

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

Maximize Z = 3x_{1} + 5x_{2} subject to the constraints: x_{1} + x_{2} ≤ 6, x_{1} ≤ 4; x_{2} ≤ 5, and x_{1}, x_{2} ≥ 0.

**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.

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:

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.

Food F_{1} contains 2, 6, 1 units and food F_{2} contains 1, 1, 3 units of proteins, carbohydrates, fats respectively per kg. 8, 12 and 9 units of proteins, carbohydrates and fats is the weekly minimum requirement for a person. The cost of food F_{1} is Rs. 85 and food F_{2} is Rs. 40 per kg. Formulate the L.P.P. to minimize the cost.