Mt. Douglas Secondary

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1 Foundations of Math 11 Section 6.3 Linear Programming Linear Programming Linear inequalities can be used to solve optimization problems, problems in which we find the greatest or least value of functions. The method used to solve such problems is called linear programming. The two-variable linear programming, which we will work with, consists of two parts: 1. An objective function tells us the quantit we want to maimize or minimize.. The sstem of constraints consists of linear inequalities whose solution is called the feasible solution with area called the feasible region. If there is an optimal solution, it must be at the vertices of the feasible region. The solution will be found b testing the function at each verte. Steps to Follow for Solving a Linear Programming Problem Step 1: Sketch the region R determined b the sstem of constraints. Step : Find the vertices of R. Step 3: Calculate the value of the objective function C at each verte of R. Step : Find the maimum or minimum value(s) of C. Eample 1 Find the maimum and minimum values of the objective function given b C = subject to the following constraints: , 0 Solution: Graph the constraints equations using method from section 6.. The maimum and minimum values of C must occur at a verte of R. Verte Value of C = (0, 0) 7(0) + 3(0) = 0 (0, ) 7(0) + 3() = 1 (5, 0) 7(5) + 3(0) = 35 (, 6) 7() + 3(6) = 3 (,6) (0,) (0,0) (5,0) Hence, the minimum value of C is 0, which occurs at (0, 0); the maimum value C = 35 occurs at (5, 0). their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

2 80 Chapter 6 Linear Sstems Foundations of Math 11 Eample Maimize and minimize C = 3 + subject to: , 0 Solution: Verte C = 3 + (0, 0) 0 (0, 5) 10 (0,15) (5, 0) 15 (, 3) 18 (0,5) (,3) (0,0) (5,0) (10,0) Maimum C = 18 occurs at (, 3); minimum C = 0 occurs at (0, 0). Eample 3 Maimize and minimize C = 3 + subject to: , 0 Solution: Verte C = 3 + (, ) (, 0) 1 (10, 0) 30 (0,8) (0,5) (,) (,0) (10,0) Maimum C = 30 occurs at (10, 0); minimum C = 1 occurs at (, 0). their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

3 Foundations of Math 11 Section 6.3 Linear Programming 81 Eample Maimize and minimize C = + 6 subject to: , 0 Solution: Verte C = + 6 (0, ) (3, ) 18 (9, 0) 18 (0,) (3,) (9,0) Minimum C = 18 occurs at line segment + 3 = 9 with 3 9. Ever point on this line segment + 3 = 9 between = 3 to = 9 gives C = 18. Although C = is the maimum of these 3 vertices, there is no upper boundar to the region, e.g. (0, 10) gives C = 60, (0, 100) gives C = 600. Therefore, there is no maimum value for this problem. Summar of Linear Programming 1. If the linear programming problem has an optimal solution, either maimum or minimum of the objective function, then it must occur at the corner of the feasible region.. If the feasible region is closed and bounded like eamples 1, and 3, then the objective function has both a maimum and minimum value. 3. If the objective function has the same optimum value at two corners, then the optimum value is an point on the line segment connecting the two corner points.. When the feasible region is not closed like eample, the objective function ma have a maimum onl, a minimum onl, or neither. their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

4 8 Chapter 6 Linear Sstems Foundations of Math Eercise Set 1. Determine which of the ordered pairs given produces the maimum value. a) C = ; (0, 0), (7, 0), (5, 3) (0, 8.5) b) C = ; (0, 0), (0, 1), (15, 0) (7.5, 1.5) c) C = ; (1, ), (, 1), (0, ) (3, 0) d) C = ; (, 3), (1, 5), (7, 1) (5, ). Determine which of the ordered pairs given produces the minimum value. a) C = ; (0, 0), (35, 0), (5, 15) (1, 11) b) C = ; (0, 9), (10, 0), (, 5) (5, ) c) C = 3 10 ; (5, 1), (, 0), (10, 3) (8, ) d) C = 0.3 ; (10, 1), (0, ), (7, 0) (0, 11) their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

5 Foundations of Math 11 Section 6.3 Linear Programming Find the maimum and minimum values of the given function in the indicated region and where the occur. a) b) C = 3 (1,8) (,6) C = + 3 (7,3) Ma (7,5) Ma (1,) Min (3,1) Min c) (0,9) C = 3 d) C = + (,) Ma (,0) Ma (0,0) (3,0) Min (,1) (5,0) Min e) C = 3 + f) C = (,6) (,6) Ma (1,) (5,5) Ma (0,0) (,0) Min (,1) Min their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

6 8 Chapter 6 Linear Sstems Foundations of Math 11. What are the maimum and minimum values of C = 3 + subject to the following constraints: , 0 5. What are the maimum and minimum values of C = subject to the following constraints: , 0 6. What are the maimum and minimum values of C = 0.3 subject to the following constraints: What are the maimum and minimum values of C = subject to the following constraints: their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

7 Foundations of Math 11 Section 6.3 Linear Programming What are the maimum and minimum values of C = subject to the following constraints: What are the maimum and minimum values of C = 8 + subject to the following constraints: Solve the following linear programming problems. 10. Maimize C = 6 + Subject to , Maimize C = Subject to , 0 their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

8 86 Chapter 6 Linear Sstems Foundations of Math Minimize C = Subject to , Minimize C = Subject to , 0 1. Maimize C = 9 +1 Subject to , Maimize C = Subject to , 0 their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

9 Foundations of Math 11 Section 6.3 Linear Programming Minimize C = 0 + Subject to , Minimize C = Subject to , Minimize and maimize C = Subject to , Minimize and maimize C = Subject to , 0 their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

10 88 Chapter 6 Linear Sstems Foundations of Math Minimize and maimize C = Subject to , 0 1. Minimize and maimize C = Subject to , 0 their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

11 Foundations of Math 11 Section 6. Application of Linear Programming Application of Linear Programming An compan that is developing a business plan for epansion most likel will include a detailed optimization stud of the chances of success. This section will look at the application of linear programming to solve optimization problems. Eample 1 Solution: Eample Solution: A parkade can fit at most 100 cars and trucks on its lot. A car covers 100 sq ft, and a truck 00 sq ft of lot space of sq ft. It charges $0 per car and $35 per truck to park these vehicles each week. How man of each vehicle will bring in maimum revenue? Maimize M = 0C + 35T 100C + 00T C + T 10 C + T 100 C 0, T 0 Verte 0C + 35T (0, 0) 0 (0, 60) 100 (80, 0) 300 (100, 0) 000 Should stock 80 cars and 0 trucks. Fertilizer for a lawn comes in brands as follows: Nitrogen Phosphoric acid Potash Brand A (kg per bag) 30 1 (0,100) (0,60) Brand B (kg per bag) 0 C (80,0) T (0,0) (100,0) (10,0) A lawn needs at least 10 kg nitrogen, at least 16 kg of phosphoric acid, and at least 1 kg of potash. Brand A costs $ a bag and brand B $18 per bag. How man bags of each brand should be used to minimize the cost? What is the minimum cost? M = A + 18B 30A + B 10 3A + B 1 A + B 16 A + B 8 A + B 1 Verte A + 18B (0, 6) 108 (, 3) 98 (, ) 1 (1, 0) 6 B (0,6) (0,) (0,3) (,3) (,) (0,0) (,0) (8,0) (1,0) A Bu bags of brand A and 3 bags of brand B for a cost of $98. their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

12 90 Chapter 6 Linear Sstems Foundations of Math Eercise Set 1. A manufacturer makes two tpes of bike: downhill and all-terrain. Use the following table to determine the maimum profit.. A farmer has 10 acres of land for planting wheat and corn. The cost and time are listed below. Find the maimum profit. Downhill All-terrain Ma. time available Assembl hrs 1 hr 0 hrs Finish 1 hr 1 hr 3 hrs Profit $70 $50 Wheat Corn Ma. Preperation cost per acre $60 $30 $1800 Work per acre 3 10 Profit per acre $180 $ A small manufacturer makes to cars and boats. The table shows the maimum assembl and finish time allowed, plus the profit. How man to cars and boats should be made per da to maimize profit, and how much is the profit? Assembl Car 1 Truck Ma. hours per da hr hrs 8 hrs Finish 1 hr hrs 1 hrs Profit $0 $50. Two vitamin pills, A and B, have the following units of carbohdrates, protein and fats. The minimum units needed each da is listed. A B Min. units Carbs. 1 6 Protein 1 8 Fat 1 6 Cost per pill 0 30 How man of each pill should be taken to minimize cost? What is the minimum cost? their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

13 Foundations of Math 11 Section 6. Application of Linear Programming A compan produces three models of TVs at two production lines, A and B. The following table shows the minimum number of each TV needed to meet production quota with the cost. How man TVs should be produced on assembl lines A and B to minimize cost, and what is that cost? A B Min. 3 inch inch inch Cost per week $ $ A tea shop purchases mied qualities of teas. The need at least 80 kg of premium tea, and 00 kg of regular tea. Samples from wholesalers are shown below. Wholesaler Premium Regular Rejected A 0% 50% 30% B 0% 0% 0% Min. 80 kg 00 kg Wholesaler A charges $9 per kg and wholesaler B $1 per kg. How much should be purchased from each wholesaler to minimize cost? 7. A deli makes a profit of 15 cents on a sandwich, and 8 cents on a bun. To make a profit, the deli must sell between 300 and 800 sandwiches, and between 100 and 300 buns. The maimum number of sandwiches and buns produced each da is 800. How man of each should be made to maimize profit? 8. An investor has $ to invest in stocks and bonds. Her advisor tells her she should invest at least twice as much in stocks than in bonds. If the return is 10% on stocks, and 11% on bonds, what is the maimum profit she can make given her constraints? their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

14 9 Chapter 6 Linear Sstems Foundations of Math A small furniture manufacturer makes tables and chairs. A table takes hours of assembl and 1 hour of finishing. A chair takes 3 hours of assembl and half hour of finishing. The profit is $35 per table and $0 per chair. The can spend at most 108 hours on assembl, and 0 hours on finishing work each da. How man tables and chairs should be produced to maimize profit? 10. A manufacturer makes $1 on each tennis racket produced, and $10 on each badminton racket produced. For qualit control purposes, the number of tennis rackets produced should be between 0 and 60, and the number of badminton rackets produced should be between 10 and 30. The number of rackets in total cannot eceed 80. How man of each tpe should be produced dail to maimize profits? 11. A farmer raises no more than 5000 of two tpes of chickens. It costs 50 to raise the white chickens and 75 to raise the brown chickens, and the total cost cannot eceed $3000. At the end of 6 weeks, a white chicken will weigh 3 pounds, and a brown chicken pounds. How man chickens of each tpe should he raise to have the maimum number of pounds of chicken? 1. Farmer Brown raises potatoes and corn on 31 acres. Potatoes take 35 hours of labour per acre and corn 7 hours of labour per acre. There are 950 labour hours available. If the profit on potatoes is $180 per acre, and corn $160 per acre, how man acres of each should he plant to maimize profit? their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

15 Foundations of Math 11 Section 6. Application of Linear Programming A school graduation class wants to hire buses and vans for a trip to Jasper National Park. Each bus hold 0 students and 3 teachers and costs $100 to rent. Each van holds 8 students and 1 teacher and cost $100 to rent. The school has at least 00 students wanting to go, but at most 36 teachers. What is the minimum transportation cost? 1. An electronics compan assembles two tpes of TVs: plasma and LCD. A plasma TV costs $00 to assemble and takes 0 hours of labour. The LCD TV costs $50 and requires 30 hours of labour. The compan has $0 000 in capital, and 160 hours of labour available for assembl. What is the maimum number of TVs the electronics compan can assemble? 15. Ace Machiner produces two products, A and B, using three machines, I, II and III. Item A makes a profit of $0 and item B makes a profit of $30. Machine I can be used up to 100 hours and produce item A s and 1 item B s each hour. Machine II can be used up to 10 hours and produce 8 item A s and 8 item B s each hour. Machine III can be used up to 8 hours and can produce 6 item A s onl. Find the number of each item the compan should produce to maimize profit. 16. Acme Manufacturers makes two products, X and Y, with 3 machines, A, B and C. Product X takes 3 hours on A, 1 hour on B and 1 hour on C to produce. Product Y takes hours on A, hours on B and 1 hour on C to produce. Machine A can be used for onl hours. Machine B can be used for onl 16 hours. Machine C can be used for onl 9 hours. Find the maimum profit each da if product X makes a profit of $1, and product Y makes a profit of $16 on each item. their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

16 9 Chapter 6 Linear Sstems Foundations of Math Chapter Review Section Graph the inequalit. a) b) < their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

17 Foundations of Math 11 Section 6.5 Chapter Review 95. Solve the sstem of inequalities. a) + + < 0 6 b) < c) d) > + < e) + < 6 f) < 10 > their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

18 96 Chapter 6 Linear Sstems Foundations of Math Derive a set of inequalities to describe the region. a) b) c) rectangle: vertices at ( 3, ), ( 3, ), (1, ) (1, ) d) parallelogram: vertices at (, 0), (0, 3), (6, 3) (, 0) e) triangle: vertices at (, 0), (0, ), (, ) f) trapezoid: vertices at ( 3, ), (, ), ( 3, 1) (1, 1) their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

19 Foundations of Math 11 Section 6.5 Chapter Review 97. A furniture manufacturer makes tables and chairs. Each table takes 1 hr to assemble and hrs to finish. Each chair takes 1 1 hours to assemble and 1 hr to finish. The plant assembl line is available 1 hrs per da, and the finishing line 16 hrs per da. Graph the sstem, and label the corner points. 5. A compan sells two size of kaaks. Model A sells for $800, and model B for $100. The compan does not want more than $0 000 in inventor at an one time, and wants at least of each model on displa. Graph the sstem, and label the corner points. Section Find the maimum and minimum values of the objective function, and when the occur. a) C = + Constraints: ; 0 b) C = + Constraints: ; 0 their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

20 98 Chapter 6 Linear Sstems Foundations of Math Find the maimum and minimum values of the objective function, and when the occur. a) C = Constraints: ; 0 b) C = Constraints: ; 0 Section A kennel mies two brands of dog food. Brand X costs $5 per bag, and brand Y $0 per bag. Carbs Fat Protein Brand X Brand Y The minimum units of carbohdrate, fat and protein are 1, 36 and units respectivel. How man bags of each brand should be mied to minimize cost, and what is that cost? 9. A shoe manufacturer produces two tpes of hiking boots. Model A makes a profit of $18, and model B a profit of $. The boots go through a three-stage process. 1 3 Model A 1 1 Model B 1 1 Hours available How man of each model should be made to maimize profits, and what is that profit? their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

21 Foundations of Math 11 Section 6.5 Chapter Review You have $0 000 to invest in stocks and bonds. Your financial advisor tells ou to invest between $6000 and $ 000 in stocks, and no more than $ in bonds. If stocks pa 8%, and bonds 7 1 %, what is our maimum profit? 11. A moving compan wants to purchase a minimum of 15 trucks with a load capacit of at least 36 tons. Model A holds tons and costs $ Model B holds 3 tons and costs $ 000. Find the number of each model to minimize cost. What is that cost? their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.

22 300 Chapter 6 Linear Sstems Foundations of Math 11 their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.