LINEAR PROGRAMMING
EXERCISE - A
1) Maximize Z= 5x+ 3y
Subject to: 3x+ 5y≤ 15 ; 5x+ 2y≤ 10 ; x, y ≥ 0. 20/19, 45/19, 235/19
2) Maximize Z= 9x+ 3y
Subject to: 2x+ 3y≤ 13 ; 3x+ y≤ 5 ; x, y ≥ 0. 2/7, 29/7, 15 or 5/3,0,15
3) Minimize Z= 18x+ 10y
Subject to: 4x+ y≥ 20 ; 2x+ 3y≥ 30 ; x, y ≥ 0. 3, 8, 134
4) Maximize Z= 50x+ 30y
Subject to: 2x+ y≤ 18 ; 3x+ 2y≤ 34 ; x, y ≥ 0. 10,2,560
5) Maximize Z= 4x+ 3y
Subject to: 3x+ 4y≤ 24 ; 8x+ 6y≤ 48; x≤ 5; y ≤ 6; x,y≥ 0. 24/7, 27/7, 24 or 5,4/3,24
6) Maximize Z= 15x+ 10y
Subject to: 3x+ 2y≤ 80 ; 2x+ 3y≤ 70 ; x, y ≥ 0. N 80/3, 0, 400 or 20,10,400
7) Maximize Z= 10x+ 6y
Subject to: 3x+ y≤ 12 ; 2x+ 5y≤ 34 ; x, y ≥ 0. 1,6,56
8) Maximize Z= 3x+ 4y
Subject to: 2x+ 2y≤ 80 ; 2x+ 4y≤ 120 ; x, y ≥ 0. 20, 20, 140
9) Maximize Z= 7x+ 10y
Subject to: x+ y≤ 30000 ; y≤ 12000; x≥ 6000; x≥ y ; x, y ≥ 0. 18000, 12000, 246000
10) Minimize Z= 2x+ 4y
Subject to: x+ y≥ 8 ; x+ 4y≥ 12 ; x≥ 3, y≥ 2; x, y ≥ 0. 4,2,16
11) Minimize Z= 5x+ 3y
Subject to: 2x+ y≥ 10 ; x+ 3y≥ 15 ; x≤ 10; y ≤ 8; x, y ≥ 0. 3, 4, 27
12) Minimize Z= 30x+ 20y
Subject to: x+ y≤ 8 ; x+ 4y≥ 12 ; 5x + 8y = 20; x, y ≥ 0. 4/7, 15/7 ,60
13) Maximize Z= 4x+ 3y
Subject to: 3x+ 4y ≤ 24 ; 8x+ 6y≤ 48 ; x≤3, y≤6; x, y ≥ 0. 24/7, 24/7,24
14) Minimize Z= x- 5y+ 20
Subject to: x- y≥ 0 ; - x+ 2y≥ 2 ; x≥ 3, y≤ 4; x, y ≥ 0. 4, 4, 4
15) Maximize Z= 3x+ 5y
Subject to: x+ 2y≤ 20 ; x+ y≤ 15 ; y≤ 5; x, y ≥ 0. 10,5,55
16) Minimize Z= 3x+ 5y
Subject to: x+ 3y≥ 3 ; x+ y≥ 2 ; x, y ≥ 0. 7
17) Maximize Z= 2x+ 3y
Subject to: x+ y≥ 1 ; 10x+ y≥ 5 ; x+ 10y ≥ 1; x, y ≥ 0. 2
18) Maximize Z= -x+ 2y
Subject to: -x+ 3y≤ 10 ; x+ y≤ 6 ; x - y ≤ 2; x, y ≥ 0. 20/3
19) Maximize Z= x+ y
Subject to: -2x+ y≤ 1 ; x≤ 2; x+ y≤ 3 ; x, y ≥ 0. 3
20) Maximize Z= 3x+ 4y, if possible,
Subject to constraints : x - y ≤ -1 ; - x+ y ≤ 0 ; x, y ≥ 0. Does not exist
21) Maximize Z= 3x+ 3y, if possible.
Subject to constraints : x- y≤ 1 ; x+ y≥ 3; x, y ≥ 0. Max value is infinite i.e., the solution is unbounded
22) Show that the solution zone of the following inequalities on a graph paper.
5x+ y≥ 10 ; x+ y≥ 6 ; x+ 4y≥ 12; x, y ≥ 0. 1, 5, 13
23) Find the maximum and minimum value of 2x+ y Subject to constraints : x+ 3y≥ 6 ; x - 3y ≤ 3 ; 3x+ 4y ≤ 24; -3x +2y≤ 6; 5x + y ≥ 5, x, y≥ 0. 84/13,15/3,43/3
24) Find the minimum value 3x+ 5y subject to constraints : -2x+ y≤ 4 ; x+ y≥ 3 ; x - 2y ≤ 2, y≥ 0;
25) Solved the following linear programming problem graphically:
Maximize Z= 60x+ 15y
Subject to constraints: x+ y ≤ 50 ; 3x+ y≤ 90 ; x, y ≥ 0.
26) Find graphically, the maximum value of Z= 2x+ 4y,
Subject to constraints: 2x+ 4y≤ 8 ; 3x+ y≤ 6 ; x+ y ≤ 4; x, y ≥ 0.
EXERCISE - B
1) A Small manufacturing firm produces two types of gadgets A and B , which are first proceeds in the foundary, 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 A and B, and the number of man-hours the firm has available per weak as follows:
Gadget Foundary Machine-shop
A 10 5
B 6 4
Firm's cap 1000 600
The profit on the sale of A is Rs30 per unit as compared with Rs20 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 LPP.
2) A company is making two products Aand B. The cost of producing one unit of products A and B are Rs60 and Rs80 respectively. As per the agreement, the company has to supply atleast 200 units of products B to its regular consumers. One unit of product A requires one machine hour whereas product B has machine hours available abundantly within the company. Total machine hours available for product A are 400 hours . One unit of each product A and B requires one labour hour each and total of 500 labour hours are available. The company wants to minimise the cost of production by satisfying the given requirements. Formulate the problem as a LPP.
3) A firm manufacturers 3 products A, B, C. The profits areo Rs3, Rs2 and Rs4 respectively. The firm has 2 machine and below is the required processing time in minutes for each machine on each product:
Machine Products
A B C
M₁ 4 3 5
M₂ 2 2 4
Machines M₁ and M₂ have 2000 and 2500 machine minutes respr. 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.
4) A firm manufacturers two of products A and B and sells them at a profit of Rs2 on type A and Rs3 on type B. Each product processed on two machines M₁ and M₂ . Type A requires 1 minute of processing time on M₁ and 2 minutes of M₂; type B requires one minute on M₁ and one minute on M₂. The machine M₁ is available for not more than 6 hours 40 minutes while machine M₂ is available for 10 hour during any working day. Formulate the problem as a LPP.
5) A rubber company is engaged in producing three types of tyres A, B and C. Each type requires processing in two plants, Plant I and Plant II . The capacities of the two plants, in number of tyres per day, are as follows:
Plant A B C
I 50 100 100
II 60 60 200
The monthly demand for tyre A, B and C is 2500, 3000 and 7000 respectively. If plant I costs Rs2500 per day p, and plant II costs Rs3500 per day to operate, how many days should each be run per months to minimise cost while meeting the demand ? formulate the problem as LPP .
6) A company sells two different products A and B . The two products are produced in a common production process and are sold in two different markets. The production process has a total capacity of 45000 man-hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The market has been surveyed and company officials feel that the maximum number of units of A that can be sold is 7000 and that of B is 10000. If the profit is Rs60 per unit for the product A and Rs40 per unit for the product B, how many units of each product should be sold to maximize profit? formulate the problem as LPP .
7) To maintain his health a person must fulfil certain minimum daily requirements for several kinds of nutrients. Assuming that there are only three kinds of nutrients-- calcium, protein and calories and the person's diet consists of only two food items, I and II, whose price and nutrient content are shown in the table below:
Food(I) Food(Ii) Daily req.
Calcium 10 5 20
Protein 5 4 20
Calories. 2 6 13
Price (Rs) 60 100
What combination of two food items will satisfy the daily requirement and entail the least cost ? Formulate this as a LPP.
8) A Manufacturer can produce two products A and B, during a given time period. Each of these products requires four different manufacturing operations : grinding, turning, assembly and testing. The manufacturing requirements in hours per unit of products A and B are given below:
A B
grinding 1 2
Turning 3 1
Assemblo 6 3
Testing 5 4
The available capacities of these operations in hours for the given time period are grinding 30; turning 60, assembling 200; testing 200. The contribution to profit is Rs20 for each unit of A and Rs30 for each unit of B. The firm can sell all that it produces at the prevailing market price. Determine the optimum amount of A and B to produce during given time period. Formulate this as a LPP.
9) Vitamin A and B are found in two different fruits F₁ and F₂. One unit of food F₁ contains 2 units of vitamin A and 3 units of vitamin B. One units of food F₂ contains 4 units of vitamin A and 2 units of vitamin B. One unit of food M₁ and M₂ cost Rs50 and Rs25 respectively. The minimum daily requirements for a personmof vitamin A and B is 40 and 50 units respectively. Assuming that any thing in excess of daily minimum requirement of vitamin A and B is not harmful, find out the optimum mixture of food F₁ and F₂ at the minimum cost which meets the daily minimum requirement of vitamin A and B. Formulate this as a LPP.
10) An automobile manufacturer make an automobiles and trucks in a factory that is divided into two shops. Shop A, which performs the basic assembly operation, must work 5 man-dats on each truck but only two men-days on each automobile. Shop B, which performs finishing operations , must work 3 men-days for each automobile or truck that it produces. Because of men and machine limitations, shopA has 180 man-days per week available while shop B has 135 man-days per week. If the manufacturers makes a profit of Rs30000 On each truck and Rs2000 on each automobile. How many of each should he produce to maximize his profit ? Formulate this as LPP.
11) Two tailor A and B earns Rs150 and Rs200 per day respectively. A can stitch 6 shirts and 4 pants per day while B can stitch 10 shirts and 4 pants per day. Form a linear programming problem to minimise the labour cost to produce atleast 60 shirts and 32 pants.
12) An airline agrees to charter planes for a group. The group needs atleast 160 first class seats and atleast 300 tourist class seats. The airline must use atleast two of its model 314 planes which have 20 first class and 30 tourist classe seats . The airline will also use some of its model 535 planes which have 20 first class seats and 60 tourist class seats. Each flight of a model of 314 plane costs the company Rs100000 and each flight of a model 535 plane costs Rs150000. How many of each type of plane should be used to minimise the flight cost ? Formulate this as a LPP
13) Amit's mathematics teachers has given him 3 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, these 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 four point problem, and 4 minute to solve as a 6 point problem. Because he has other subjects to worry about, he cannot afford to devote more than 7/2 hours to altogether to the mathematics assignment. Moreover , the first two sets of problems involves numerical calculation and he knows that he cannot stand more than 5/2 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.
14) a farmer as 100 acre forms he can save the tomatoes letus varieties he can raise the pricken obtained is one per kilogram per tomato 0.75 head for letting him to per kilogram for reduce average for redmi1000 kilogram of radius fertilizers is available at 0.5 0 per kg and required for acre is 100 kg switch off all tomatoes and latives and 50 kilograms for reduces labourer required for showing cultivating and harvesting for acre is fine days for tomatoes and disease and 6 men days for letting the total of 4 men days of labours are available at 20% a farm is a transported list 100 package daily using large Van switch carry 200 packages and small bhains which can take it to packages each the cost of engaging which large when is 400 in which Mall ban is 200 not more than 3000 to be spend Delhi on the job in the number of large banks cannot be exceed the number of small Bank permanent this problem as lpt given that object is minimum cost A farm manufacture to product daburly requirement for unit for each product in which department the weekly capacities in East department selling price for unit labour cost per unit and raw materials for units are samurai does follows the problem is to determine the number of universe to produce each product so as to maximize the total country to profit formula
factory produces two problems each of products required to hours for moulding 3 hours for grinding and 4 hours with policy and each of the product required for hours for moulding 2 hours for grinding 2 hours for policy in the factory has molding machine available for 20 hours grinding machine for 24 hours and polishing machine available for 13 hours the profit is five per unit and cheaper units and the factory can sale all that it produces a toy company manufacturing 2 types of double a and a Deluxe doll which doll of type b text twice as long as produce as one of IP and the company would have time to make a maximum 2000 per day to produces only the basic version the supply of plastic is sufficient to produce 1500 does her month both and be combine that fancy dress if the company makes profit 3 and 5 per door respectively on 12:00 a.m. Dolby how many of these should be produced party in order to maximize prof a farm can produce three types of clothes says the three kinds of polar required for its Erode old greenhole and blue whale 1 units of length needs to metres of red wine units of 3 metres upgrade to metres of women in 2 ML and 1 units of clothes 5 M Greenwood 4 m sub the form as only stop of 16 mold 20 m Green it is associum that the income obtained from one unit of length per blade the problem has a linear programming problem to maximize the income at furniture palm manufacture chairs and tables which requires the use of three machines A B and C production of one share it cost in the same one hour and machine is tablets one hour it on the senior see the propri realised by selling one share is 30 oil for a table is 16 the total time available for weekend how many chairs and tables should be made for weakestro to maximise broker developer mathematical formulation a manufacturer of a line a patent medicine is preparing a product plan on machine in be there a sufficient ingredient available to make 2000 bottle of UN 40000 but the but there only 45000 into which either of the machine can be put for the more it take 3 hours to bottles of the endless 66 hours available and 7 portal for burn the problem as a linear programming problem resourceful home decorator manufacture two types of Reliance says a and be both lands go through to the scenes for a cutter second a finisher library awards for the cut-off's time and 1 hour for the finisher times lambi request one hour of cutters and 2 hours of finisher 10 the cutter as 104 hours and finish our 76 hour of the time available is month profit and the language in on the land these 11 that you can all the produces how many a company makes two kinds of leather belts can be built a is highly quality belt and b is a lower quality the respective profit after 10:30 per build its belt of type a require twice as much as time as a bell type b and all the breadth were type be the company could make 1000 best birthday the supply of laser is sufficient for 8000 base per day both and be combined required same fancy drone only 400 buckles for they are available there only 7 bubbles available for built the what should be the daily production of which type of formula of the problemaditation wishes to mix two types of food in such a way that the vitamin contents of the mixture contain atlist 8 units of vitamin and 10 minutes a vitamin C food One contest to units per KGF vitamin A and 1 minutes for vitamin C food contance one unit per KGN to units per kg vitamin C its cos 50 per kg to purchase food and 7 produced formality the above linear programming problem to minimise the cost of mixture a diet is to contain at least 400 units of carbohydrate 500 m subtit and 300 units of proteins to foods are available which cost to per unit and which cost 4 in needs a unit of food contains 10 minutes of carbohydrates 20 minutes a fat and 15 minutes supporting unit of food contents 25 minutes of our 10 minutes of fat 20 minutes supporting find the minimum cost for a diet that consist the mixture of this and also the objective of a diet problem is to a set on the quantities of certain food that should be eaten to meet certain nutrient requirement at the minimum cause of consideration is limited the number of milligram of each of these vitamins content within a unit food is given what is the linear programming formulation for this problem
Comments
Post a Comment