LINEAR PROGRAMMING Learning Outcomes and Assessment Standards Learning Outcome 2: Functions and algebra Assessment Standard 12.2.8 Solve linear programming problems by optimising a function in two variables, subject to one or more linear constraints, by establishing optima by means of a search line, and further comparing the gradients of the objective functions and linear constraint boundary lines. Lessons 28 30 Overview In this lesson you will: learn the terminology of linear programming learn to translate words into linear inequalities (constraints draw these graphs indicate the feasible region formulate the objective function which we will call a search line use this search line to find a maximum or minimum so as to solve the problem Lesson Important Terminology 1. Implicit constraints: Inequalities that arise naturally 2. Constraints: Inequalities that you make from the given information 3. The feasible region: The area on the graph the satisfies the above constraints 4. The objective function (often called a search line): The straight line graph you draw to help you solve the problem DVD Making inequalities (constraints) Question 1 A party is to be arranged. Let x be the number of girls and y the number of boys that will attend this party. Make inequalities (constraints), if you are given the following information. There must be at least 200 people at the party. However, the hall can only fit 600 people. We need at least 150 girls and at the most 400 boys and the ratio of boys to girls must not be more than 2 : 1. The organisers must make at least R4 000 on entrance fees. Each boy pays R20 to attend and each girl pays R15. ³ 200 For attendance: Attendance Door Fees x + y 600 (max hall can take) Girls (x) 1 15 ³ 150 x + y ³ 200 (min they want to attend) Boys (y) 1 20 400 600 ³ 4 000 Door fees: 15x + 20y ³ 4 000 3x + 4y ³ 800 (simplified by 5) 1 LC G12 Linear prog LWB.indb 1 2008/09/03 03:44:44 PM
Ratio: boys to girls not more than two to 1 So: y : x 2 : 1 _ y x _ 2 1 y 2x Girls: x ³ 150 they want more than 150 girls Boys: 0 y 400 a max of 400 boys and min of 0 and x and y (we cannot allow for fractions or negative numbering) y : x 2 : 1 Implicit constraints _ y x _ 2 1 x; y y 2x ratio of boys to girls not more than two to one. x + y 200 more than 200 must attend x + y 600 hall can house a maximum 600 people x 150 more than 150 girls y 400 a maximum of 400 boys must attend 15x + 20y 4 000 door fees 3x + 4y ³ 800 (always simplify your equations) Question 2 A company has two types of airplanes, a 707 and a 747. Let the number of 707s be x and the number of 747s be y and make inequalities if given the following information. A 707 carries 250 passengers and a 747 carries 300 passengers. The company is required to transport at least 3 000 passengers per day. The company must use at least four 747s and at least two 707s each day. The company only has 16 pilots available each day. Passengers Pilots 707 (x) 250 1 ³ 2 747 (y) 300 1 ³ 4 ³ 3 000 16 Implicit constraints: x; y Constraints: 250x + 300y 3 000 x 4 x 2 x + y 16 2 Question 3 The dairy delivers milk and orange juice. A housewife can order a maximum of 28 bottles per week. She does not want more than 12 bottles of orange juice. She must have at least 2 bottles of milk. If the milk costs R8,00 per bottle and the orange juice R12,00 per bottle, what is the maximum that she spends per week. Let x be the number of bottles of milk Let y be the number of bottles of orange juice Implicit constraints: x; y N LC G12 Linear prog LWB.indb 2 2008/09/03 03:44:46 PM
Bottles Cost Milk (x) 1 8 ³ 2 Orange (y) 1 12 12 28 Constraints: a x + y 28 b y 12 c x 2 Now we will draw the graph and shade in the feasible region. Objective function (search line) C = 8x + 12y ABCD is the feasible region 12y = 8x + C y = _ 8 12 x + _ C 12 Maximum at B x = 16 y = 12 16 bottles of milk 12 bottles of orange juice Cost 8(16) + 12(12) = 128 + 144 = 272 Question 4 A manufacturer makes two types of soup, A and B. He cannot sell more than 12 cases of A and 8 cases of B per day. It takes half-an-hour to produce a case of A and one hour to produce a case of B per day. The plant operates for 9 hours per day and he has enough material to make not more than 14 cases of both together per day. Let x be the number of cases of A and y be the number of cases of B. a) Represent these inequalities and shade in the feasible region. b) If the profit on A is R20 per case and on B, R30 per case, determine the number of cases to give a maximum profit? Let number of cases of A be x Let number of case of B be y Implicit constraints: x; y Time (h) Materials Profit A (x) _ 1 2 1 20 12 B (y) 1 1 30 8 9 14 3 LC G12 Linear prog LWB.indb 3 2008/09/03 03:44:46 PM
Constraints: a x 12 b y 8 c _ 1 x + y 9 2 x + 2y 18 d x + y 14 Now we will draw the graph and shade in the feasible region. Objective function (search line) P = 20x + 30y OABCDE is the feasible region 30y = 20x + P y = _ 2 3 x + _ P 30 Maximum at C x = 10 y = 4 Question 5 The school mountaineering club wants to organise a hike to the Drakensberg. They need to compile an emergency pack which will provide sufficient nourishment for a pupil for one day. They decide on a combination of tins of canned beef and packets of biscuits. The contents of such a pack are subject to certain limitations of the amount of protein and kilojoules, as well as the number of each that can be carried. You, as a member of the mathematics club, must help them solve their problem. They supply the following information: 1. No more than 5 tins of canned beef and 8 packets of biscuits, but at least one of each must be included. 2. The protein content of a tin of canned beef is 30 g, and that of a packet of biscuits if 20 g. The minimum amount of protein needed per day is 140 g. 3. A tin of canned beef provides 1 500 kj, while a packet of biscuits provides 2 000 kj. A maximum of 15 000 kj may be consumed per day. 4 Let x be the number of tins of beef Let y be the number of packets of biscuits Implicit constraints: x; y Protein (g) Kilojoules 1 Beef (x) 30 1 500 5 1 Biscuits (y) 20 2 000 8 ³ 140 15 000 LC G12 Linear prog LWB.indb 4 2008/09/03 03:44:48 PM
Constraints: a x 5 b y 8 c x 1 d y 1 e 30x + 20y 140 3x + 2y 14 f 1 500x + 2 000y 15 000 15x + 20y 150 3x + 4y 30 Objective function(search line) M = x + y y = x + M m = _ 1 1 ABCDE is the feasible region Minimum at E y = 1 x = 4 Question 6 A carpenter makes two types of tables, A and B. He has 400 m 2 floor space available. Table A requires 30 m 2 and table B 40 m 2 of floor space. He does not have enough skilled labourers to make more than 6 tables of type B and 8 tables of type A. According to public demand, the ratio of table A to table B must not be more than 2 : 1. Let x be the number of type A tables and y the number of type B tables. a) Write down the constraints, graph them and indicate the feasible region. b) If the profit on each table is the same, determine graphically how many tables of each type must be made for maximum profit. Implicit constraints: x; y 5 LC G12 Linear prog LWB.indb 5 2008/09/03 03:44:50 PM
6 Floor space (m 2 ) Profit table A : table B 2 : 1 A (x) 30 1 8 x : y 2 : 1 B (y) 40 1 6 x_ y _ 2 1 400 x 2y y ³ _ x 2 30x + 40y 400 3x + 4y 40 0 x 8 0 y 6 x : y Ratio x : y 2 : 1 Constraints a 30x + 40y 400 3x + 4y 40 b y 6 c x 8 d y _ 1 2 x Objective function (search line) P = x + y OABC is the feasible region y = x + P m = 1 Maximum at C x = 8 y = 4 y 14 13 12 11 10 9 8 7 A 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Question 7 A farmer has 80 hectares that can be used for fruit farming. The annual cost of maintaining an orchard of peach trees is R12 000 per hectare, while an orchard of plum trees cost R8 000 per hectare. The farmer has an annual budget of R720 000. He has contracts that require him to cultivate at least 10 hectares of peaches. The local demand from the fruit market is such that the number of hectares of peaches must not be more than twice the number of hectares of plums. If the profit from peaches and plums are R6 000 and R5 000 per hectare respectively. a) Let the number of hectares of peach trees be x and the number of hectares of plum trees be y, list the constraints and shade the feasible region. b) Write down the objective function. B C x LC G12 Linear prog LWB.indb 6 2008/09/03 03:44:51 PM
c) Determine the number of hectares of peaches and plums that should be cultivated in order to maximise profit. Implicit constraints: x; y Hectares Maintenance Costs Profit Peach (x) 1 12 000 6 000 ³ 10 Plum (y) 1 8 000 5 000 80 720 000 Ratio x : y 2 : 1 Constraints a y _ 1 2 x b x + y 80 c 12x + 8y 720 3x + 2y 180 d x 10 Objective function (search line) P = 6 000x + 5 000y 8 000y = 6 000x + P m = _ 6 8 = _ 3 4 Maximum at B x = 10 y = 70 Activity 1. A local health board is producing a guide for healthy living. The guide should provide advice on health education, healthy lifestyles and the like. The board intends to produce the guide in two formats: one will be in the form of a short video; the other as a printed binder. The board is currently trying to decide how many of each type to produce for sale. It has estimated that it is likely to sell no more than 10 000 copies, of both items together. At least 4 000 copies of the video and at least 2 000 copies of the binder could be sold, although sales of the binder are not expected to exceed 4 000 copies. Let x be the number of videos sold and y the number of printed binders sold. a) Write down the constraint inequalities that can be deduced from the given information. b) Use graph paper to represent these inequalities graphically and indicate the feasible region clearly. c) The board is seeking to maximise the income, I, earned from the sales of the two products. Each video will sell for R50 and each binder for R30. Write down the objective function for the income. 7 LC G12 Linear prog LWB.indb 7 2008/09/03 03:44:52 PM
d) Determine graphically, by using a search line, the number of videos and binders that ought to be sold to maximise the income. e) What maximum income will be generated by the two guides. 2. A factory employs unskilled workers at R1 000 per month and skilled workers at R2 000 per month, whilst the pay-roll should not exceed R90 000 per month. To keep going, at least 50 operators are necessary. Furthermore, regulations determine that the number of skilled workers should be at least half that of the unskilled workers. a) If x represents the number of unskilled and y the number of skilled workers, write down all the restrictions in terms of x and y. b) Illustrate the restrictions in a neat diagram on graph paper. c) Determine from the diagram the minimum and maximum number of unskilled workers that could be employed. 3. (Extension) A manufacturer of a liquid detergent uses two basic chemical ingredients, Mix A and Mix B. The detergent is packaged and sold to two separate markets: Household market (H) and Commercial market (C). The products are sold in five litre bottles. For the household detergent each five litre bottle requires four litres of Mix A and one litre of Mix B, whereas the corresponding composition of the commercial detergent is two litres of Mix A and three litres of Mix B as shown in the table. Chemical composition of detergents per five litre bottle: Ingredient Household detergent (H) Commercial detergent (C) Mix A 4 litres 2 litres Mix B 1 litre 3 litres 8 On a weekly basis, the company has supplies of no more than 20 000 litres of Mix A and 15 000 litres of Mix B. Furthermore the company can buy no more than 4 000 containers a week for H and 4 500 containers a week for C. Let x be the number of five litre bottles for H Let y be the number of five litre bottles for C. a) Write down the inequalities representing the availability of Mix A and Mix B and the availability of the containers for H and C. b) Represent the inequalities on graph paper, using the scale 2 cm = 2 000 litres on both axes. c) Shade the feasible region on the graph. d) If H contributes R2,40 to the profit per five litre bottle produced and C contributes R2,00 per five litre bottle, write down the equation for the profit (P), and determine the gradient of the profit line. e) Draw the profit line as a dotted line on the graph. f) Determine how many five litre bottles of each type of detergent must be produced on a weekly basis to maximize profit. (You may read off the answer from the graph.) g) Determine the maximum profit per week. LC G12 Linear prog LWB.indb 8 2008/09/03 03:44:52 PM
4. (Extension) A vegetable farmer grows carrots (x kg) and beetroot (y kg) for the hotel trade. The load in the crate, used to send the produce to his customers, cannot exceed 100 kg. His clients insist on at least 10 kg of each vegetable and also at least three times as many carrots as beetroot in weight. The farmer makes a profit of R1 per kilogram on the carrots and R1,20 per kilogram on the beetroot. a) Write down a set of inequalities that will represent the restrictions. b) Use graph paper to illustrate the feasible region. c) Find graphically the quantity of carrots and beetroot that should be dispatched per crate to yield the maximum profit. d) Determine the maximum profit. 9 LC G12 Linear prog LWB.indb 9 2008/09/03 03:44:53 PM
ANSWERS AND ASSESSMENT Lesson 28 30 Question 1 1.1. x + y 10 000... a x 4 000... b 2 000 y 4 000... c d x; y 0 1.2. 1.3. I = 50x + 30y y = _ 5 3 x + _ 1 30 1.4. From 1.3 Maximum income at (8 000; 2 000) 1.5 Income = 50(8 000) = 30(2 000) = R460 000 Question 2 2.1 a x; y > 0 b 1 000x + 2 000y 90 000 x + 2y 90 c x + y 50 d y _ x 2 10 LC G12 Linear prog LWB.indb 10 2008/09/03 03:44:53 PM
TIPS FOR THE TEACHER Lesson 28-30 I have emphasized word sum examples because I believe that is the spirit of linear programming and the new curriculum. 11 LC G12 Linear prog LWB.indb 11 2008/09/03 03:44:54 PM