State whether the following statement is True or False: Corner point method is most suitable method for solving the LPP graphically - Mathematics and Statistics

True or False

State whether the following statement is True or False:

Corner point method is most suitable method for solving the LPP graphically


  • True

  • False




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Chapter 2.6: Linear Programming - Q.2 (B)


The postmaster of a local post office wishes to hire extra helpers during the Deepawali season, because of a large increase in the volume of mail handling and delivery. Because of the limited office space and the budgetary conditions, the number of temporary helpers must not exceed 10. According to past experience, a man can handle 300 letters and 80 packages per day, on the average, and a woman can handle 400 letters and 50 packets per day. The postmaster believes that the daily volume of extra mail and packages will be no less than 3400 and 680 respectively. A man receives Rs 225 a day and a woman receives Rs 200 a day. How many men and women helpers should be hired to keep the pay-roll at a minimum ? Formulate an LPP and solve it graphically.

A company produces two types of goods A and B, that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of golds while that of type B requires 1 g of silver and 2 g of gold. The company can procure a maximum of 9 g of silver and 8 g of gold. If each unit of type A brings a profit of Rs 40 and that of type B Rs 50, formulate LPP to maximize profit.

A small manufacturing firm produces two types of gadgets A and B, which are first processed in the foundry, then sent to the machine shop for finishing. The number of man-hours of labour required in each shop for the production of each unit of and B, and the number of man-hours the firm has available per week are as follows:

Gadget Foundry Machine-shop
A 10 5
B 6 4
 Firm's capacity per week 1000 600

The profit on the sale of A is Rs 30 per unit as compared with Rs 20 per unit of B. The problem is to determine the weekly production of gadgets A and B, so that the total profit is maximized. Formulate this problem as a LPP.



A firm manufactures 3 products AB and C. The profits are Rs 3, Rs 2 and Rs 4 respectively. The firm has 2 machines and below is the required processing time in minutes for each machine on each product : 

Machine Products
4 3 5
2 2 4

Machines M1 and M2 have 2000 and 2500 machine minutes respectively. The firm must manufacture 100 A's, 200 B's and 50 C's but not more than 150 A's. Set up a LPP to maximize the profit.

Amit's mathematics teacher has given him three very long lists of problems with the instruction to submit not more than 100 of them (correctly solved) for credit. The problem in the first set are worth 5 points each, those in the second set are worth 4 points each, and those in the third set are worth 6 points each. Amit knows from experience that he requires on the average 3 minutes to solve a 5 point problem, 2 minutes to solve a 4 point problem, and 4 minutes to solve a 6 point problem. Because he has other subjects to worry about, he can not afford to devote more than

\[3\frac{1}{2}\] hours altogether to his mathematics assignment. Moreover, the first two sets of problems involve numerical calculations and he knows that he cannot stand more than 
\[2\frac{1}{2}\]  hours work on this type of problem. Under these circumstances, how many problems in each of these categories shall he do in order to get maximum possible credit for his efforts? Formulate this as a LPP.


A farmer has a 100 acre farm. He can sell the tomatoes, lettuce, or radishes he can raise. The price he can obtain is Rs 1 per kilogram for tomatoes, Rs 0.75 a head for lettuce and Rs 2 per kilogram for radishes. The average yield per acre is 2000 kgs for radishes, 3000 heads of lettuce and 1000 kilograms of radishes. Fertilizer is available at Rs 0.50 per kg and the amount required per acre is 100 kgs each for tomatoes and lettuce and 50 kilograms for radishes. Labour required for sowing, cultivating and harvesting per acre is 5 man-days for tomatoes and radishes and 6 man-days for lettuce. A total of 400 man-days of labour are available at Rs 20 per man-day. Formulate this problem as a LPP to maximize the farmer's total profit.

The corner points of the feasible region determined by the following system of linear inequalities:
2x + y ≤ 10, x + 3y ≤ 15, xy ≥ 0 are (0, 0), (5, 0), (3, 4) and (0, 5). Let Z = px + qy, where p, q > 0. Condition on p and q so that the maximum of Z occurs at both (3, 4) and (0, 5) is 

A firm manufactures two products, each of which must be processed through two departments, 1 and 2. The hourly requirements per unit for each product in each department, the weekly capacities in each department, selling price per unit, labour cost per unit, and raw material cost per unit are summarized as follows:

  Product A Product B Weekly capacity
Department 1 3 2 130
Department 2 4 6 260
Selling price per unit ₹ 25 ₹ 30  
Labour cost per unit ₹ 16 ₹ 20  
Raw material cost per unit ₹ 4 ₹ 4  

The problem is to determine the number of units to produce each product so as to maximize total contribution to profit. Formulate this as a LPP.

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

Maximize : Z = 10x + 25y subject to 0 ≤ x ≤ 3, 0 ≤ y ≤ 3, x + y ≤ 5 also find maximum value of z.

Fill in the blank :

Graphical solution set of the in equations x ≥ 0, y ≥ 0 is in _______ quadrant

The region represented by the inequality y ≤ 0 lies in _______ quadrants.

Solve the following problem :

Maximize Z = 5x1 + 6x2 Subject to 2x1 + 3x2 ≤ 18, 2x1 + x2 ≤ 12, x ≥ 0, x2 ≥ 0

Solve the following problem :

Maximize Z = 4x1 + 3x2 Subject to 3x1 + x2 ≤ 15, 3x1 + 4x2 ≤ 24, x1 ≥ 0, x2 ≥ 0

Solve the following problem :

A company manufactures bicyles and tricycles, each of which must be processed through two machines A and B Maximum availability of machine A and B is respectively 120 and 180 hours. Manufacturing a bicycle requires 6 hours on machine A and 3 hours on machine B. Manufacturing a tricycle requires 4 hours on machine A and 10 hours on machine B. If profits are ₹ 180 for a bicycle and ₹ 220 on a tricycle, determine the number of bicycles and tricycles that should be manufacturing in order to maximize the profit.

Solve the following problem :

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 unit 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 manufacture per month to maximize profit? How much is the maximum profit?

Choose the correct alternative:

If LPP has optimal solution at two point, then

Choose the correct alternative:

The corner points of feasible region for the inequations, x + y ≤ 5, x + 2y ≤ 6, x ≥ 0, y ≥ 0 are

Choose the correct alternative:

The corner points of the feasible region are (0, 3), (3, 0), (8, 0), `(12/5, 38/5)` and (0, 10), then the point of maximum Z = 6x + 4y = 48 is at

State whether the following statement is True or False:

The maximum value of Z = 5x + 3y subjected to constraints 3x + y ≤ 12, 2x + 3y ≤ 18, 0 ≤ x, y is 20

State whether the following statement is True or False:

If LPP has two optimal solutions, then the LPP has infinitely many solutions

State whether the following statement is True or False:

The point (6, 4) does not belong to the feasible region bounded by 8x + 5y ≤ 60, 4x + 5y ≤ 40, 0 ≤ x, y

A set of values of variables satisfying all the constraints of LPP is known as ______

If the feasible region is bounded by the inequations 2x + 3y ≤ 12, 2x + y ≤ 8, 0 ≤ x, 0 ≤ y, then point (5, 4) is a ______ of the feasible region

A company manufactures 2 types of goods P and Q that requires copper and brass. Each unit of type P requires 2 grams of brass and 1 gram of copper while one unit of type Q requires 1 gram of brass and 2 grams of copper. The company has only 90 grams of brass and 80 grams of copper. Each unit of types P and Q brings profit of ₹ 400 and ₹ 500 respectively. Find the number of units of each type the company should produce to maximize its profit

A dealer deals in two products X and Y. He has ₹ 1,00,000/- to invest and space to store 80 pieces. Product X costs ₹ 2500/- and product Y costs ₹ 1000/- per unit. He can sell the items X and Y at respective profits of ₹ 300 and ₹ 90. Construct the LPP and find the number of units of each product to be purchased to maximize its profit

A company manufactures two types of ladies dresses C and D. The raw material and labour available per day is given in the table.

Resources Dress C(x) Dress D(y) Max. availability
Raw material 5 4 60
Labour 5 3 50

P is the profit, if P = 50x + 100y, solve this LPP to find x and y to get the maximum profit

Smita is a diet conscious house wife, wishes to ensure certain minimum intake of vitamins A, B and C for the family. The minimum daily needs of vitamins A, B, and C for the family are 30, 20, and 16 units respectively. For the supply of the minimum vitamin requirements Smita relies on 2 types of foods F1 and F2. F1 provides 7, 5 and 2 units of A, B, C vitamins per 10 grams and F2 provides 2, 4 and 8 units of A, B and C vitamins per 10 grams. F1 costs ₹ 3 and F2 costs ₹ 2 per 10 grams. How many grams of each F1 and F2 should buy every day to keep her food bill minimum

A wholesale dealer deals in two kinds of mixtures A and B of nuts. Each kg of mixture A contains 60 grams of almonds, 30 grams of cashew and 30 grams of hazel nuts. Each kg of mixture B contains 30 grams of almonds, 60 grams of cashew and 180 grams of hazel nuts. A dealer is contemplating to use mixtures A and B to make a bag which will contain at least 240 grams of almonds, 300 grams of cashew and 540 grams of hazel nuts. Mixture A costs ₹ 8 and B costs ₹ 12 per kg. How many kgs of each mixture should he use to minimize the cost of the kgs

Maximize Z = 5x + 10y subject to constraints

x + 2y ≤ 10, 3x + y ≤ 12, x ≥ 0, y ≥ 0

Maximize Z = 400x + 500y subject to constraints

x + 2y ≤ 80, 2x + y ≤ 90, x ≥ 0, y ≥ 0

Minimize Z = 24x + 40y subject to constraints

6x + 8y ≥ 96, 7x + 12y ≥ 168, x ≥ 0, y ≥ 0

Minimize Z = 2x + 3y subject to constraints

x + y ≥ 6, 2x + y ≥ 7, x + 4y ≥ 8, x ≥ 0, y ≥ 0

Solve the LPP graphically:
Minimize Z = 4x + 5y
Subject to the constraints 5x + y ≥ 10, x + y ≥ 6, x + 4y ≥ 12, x, y ≥ 0

Solution: Convert the constraints into equations and find the intercept made by each one of it.

Inequations Equations X intercept Y intercept Region
5x + y ≥ 10 5x + y = 10 ( ___, 0) (0, 10) Away from origin
x + y ≥ 6 x + y = 6 (6, 0) (0, ___ ) Away from origin
x + 4y ≥ 12 x + 4y = 12 (12, 0) (0, 3) Away from origin
x, y ≥ 0 x = 0, y = 0 x = 0 y = 0 1st quadrant

∵ Origin has not satisfied the inequations.

∴ Solution of the inequations is away from origin.

The feasible region is unbounded area which is satisfied by all constraints.

In the figure, ABCD represents

The set of the feasible solution where

A(12, 0), B( ___, ___ ), C ( ___, ___ ) and D(0, 10).

The coordinates of B are obtained by solving equations

x + 4y = 12 and x + y = 6

The coordinates of C are obtained by solving equations

5x + y = 10 and x + y = 6

Hence the optimum solution lies at the extreme points.

The optimal solution is in the following table:

Point Coordinates Z = 4x + 5y Values Remark
A (12, 0) 4(12) + 5(0) 48  
B ( ___, ___ ) 4( ___) + 5(___ ) ______ ______
C ( ___, ___ ) 4( ___) + 5(___ ) ______  
D (0, 10) 4(0) + 5(10) 50  

∴ Z is minimum at ___ ( ___, ___ ) with the value ___

A linear function z = ax + by, where a and b are constants, which has to be maximised or minimised according to a set of given condition is called a:-

Maximised value of z in z = 3x + 4y, subject to constraints : x + y ≤ 4, x ≥ 0. y ≥ 0

If z = 200x + 500y  .....(i)

Subject to the constraints:

x + 2y ≥ 10  .......(ii)

3x + 4y ≤ 24  ......(iii)

x, 0, y ≥ 0  ......(iv)

At which point minimum value of Z is attained.


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