A9.1 Linear programming

Transcription

1 pplications 9. Linear programming 9. Linear programming efore ou start You should be able to: show b shading a region defined b one or more linear inequalities. Wh do this? Linear programming is an eample of optimisation which is ver important in manufacturing. bjectives You will be able to find the maimum and minimum value of a linear function within a region in the plane. You will be able to formulate and solve a linear programming problem in two variables. Get Read raw the lines with equations: a b Show b shading on our graph the regions: a b Show b shading the region of points which satisf all of these inequalities:, and Ke Points set of linear inequalities of the form a b c can define an enclosed region, R, known as the feasible region. The coordinates of all the points within and on the boundaries of the feasible region satisf all the inequalities. linear function P is of the form P a b c where a, b and c are numbers. Within an enclosed region R, the maimum and minimum values of an linear function are attained at one of the corners of the region or along an edge of the region. Note: it is easier to show the region which satisfies all the inequalities as unshaded. Eample The feasible region shown (unshaded) below satisfies these linear inequalities:,, and The feasible region is the interior (and edges) of the quadrilateral Find the maimum and minimum values of the following linear functions: a P b P 09-hapter 09_ indd 9// 09::

2 hapter 9 Linear programming a So the maimum value of P (0) is at and the minimum value () is at. b t each of the corners, P takes the values,, 0, 0 So the maimum value occurs (0) at the Points and (and in fact anwhere along the edge ). The minimum value () occurs at. (a) t the corner (, ), P takes the value. t the corner (, ), P takes the value. t the other corners P takes the values (), and (), Eercise 9 F E Find the value of each of these linear functions at the point,,,, E and F as shown in the diagram. a b c d e f raw the region which satisfies all of the following inequalities:,, Find the value of each of these linear functions at the vertices of the region and at the point with coordinates (, ): a b c d e The diagram shows the finite region. Write down the equations of each of the boundar lines of. Find the inequalities that points within or on the boundar of must satisf. Work out the maimum and minimum values of the following functions at points within or on the boundar of : a b c d 09-hapter 09_ indd 9// 09::

3 pplications 9. Linear programming Here is a sketch. Find the maimum and minimum values taken b each of these functions within or on the boundar of the finite region. a b c d The region satisfies the following inequalities:. Show, b shading, the region. Find the maimum and minimum values of the following functions which satisf all of the above inequalities. a b c d Show, b shading, the region which satisfies all of the following inequalities: Find the maimum and minimum values of each of the following functions for the set of points which satisf all of the above inequalities. a b c 0 d 0 Eample radio broadcast compan has two stations: Hot Hits and ool lassics. The compan spends dail at least twice as much on Hot Hits as on ool lassics. The compan spends dail at least 00 on ool lassics and at least 000 on Hot Hits. The compan can afford to spend dail no more than a total of 000. Let be the mone the compan spends dail on Hot Hits. Let be the mone the compan spends dail on ool lassics. a Write down constraints that and/or must satisf. b raw a suitable diagram and identif the region that satisfies all of the constraints. The dail profit on ool lassics is epected to be 0 per pound spent and the dail profit on Hot Hits is epected to be per pound spent. c Write down an epression for the total dail profit P in terms of and. d Use our diagram to find the maimum dail profit and the values of and at which it occurs. 09-hapter 09_ indd 9// 09::

4 hapter 9 Linear programming a b constraint is a mathematical condition that must be satisfied. In linear programming the constraints can be written as linear inequalities Eaminer s Tip andidates often write the wrong wa round as _. c P 0 d Maimum profit occurs at (000, 000) and is Eercise This is the feasible region it satisfies all four of the constraints. farmer puts fertiliser on his fields. He knows he must put on at least 00 kg of phosphate and at least 00 kg of nitrate. The maimum total amount of fertiliser he will put on his fields is 000 kg. Let kg be the mass of phosphate. Let kg be the mass of nitrate. a Write down constraints that and/or must satisf. b raw a graph and indicate on the graph the region which satisfies all constraints. The cost of one kg of phosphate is 0p. The cost of one kg of nitrate is p. c Write down an epression for the cost pence of kg of phosphate and kg of nitrate. d Find the minimum and maimum cost which satisfies all the constraints. compan makes shirts and vests. Each da the compan must make at least 00 shirts and must make at least 0 vests. The compan makes at least as man shirts as vests each da. The compan can make a maimum of 00 of these garments each da. Let be the number of shirts. Let be the number of vests. a Write down the inequalities that and/or must satisf. b n graph paper, show b shading, the region which satisfies all of the inequalities. The cost of making a shirt is. The cost of making a vest is 0. c Write an epression for the total cost of making shirts and vests. d Work out the minimum cost that satisfies all the constraints. e The profit on a shirt is and the profit on a vest is. ssuming that the compan sells all the articles it makes, work out the maimum profit. 09-hapter 09_ indd 9// 09::

5 pplications 9. Linear programming compan makes chairs and settees. Ever da the compan can make a maimum of 00 pieces of furniture. The compan makes at most twice as man chairs as settees. The compan makes at least 0 chairs and at most 00 settees each da. The cost of making a chair is 0 and the cost of making a settee is 0. Let be the number of chairs and be the number of settees. a Epress each of the constraints as inequalities. b Epress the total cost in terms of and. c raw the feasible region on a grid of squares. d Find the minimum and maimum costs and the number of chairs and the number of settees at which the minimum cost is attained. market gardener grows cabbages and carrots. She has a maimum of 0 hectares for growing. She grows carrots on at least 0% more land than she grows cabbages. She must use at least hectares for carrots and at least hectares for cabbages. Let hectares be the area used for growing carrots and hectares be the area used for growing cabbages. a Write down the inequalities. The revenue from a hectare of carrots is 00 and the revenue from a hectare of cabbages is 00. b Write down an epression in terms of and for the total revenue, R. c Find the maimum value of the revenue R. newspaper runs a lotter in which there are prizes and prizes. There must be at least prizes. There must be at least prizes. The number of prizes must be not be more than more than the number of prizes. The total amount of mone available for the prize fund must not be greater than 00. Let be the number of prizes and be the number of prizes. a i Eplain wh 0 ii Write the other constraints as inequalities. b raw these inequalities on graph paper and identif the region that satisfies all the inequalities. c What is the maimum total number of prizes that can be given? Tickets at a concert cost either or 0. The number of 0 tickets must be no more than 0 more than the number of tickets. There must be at least 00 0 tickets and at most 00 tickets. The total number of tickets must not be more than. Let be the number of 0 tickets and let be the number of tickets. a Write these constraints as inequalities. b raw these inequalities on a suitable grid. The profit from each 0 ticket is and the profit from each ticket is. c Write down an epression for the total profit, P. d Find the maimum profit. 09-hapter 09_ indd 9// 09::

6 hapter 9 Linear programming 7 ill takes lots of eercise. Each week he covers between 0 miles and 0 miles b a combination of walking and jogging. He walks at most half as far as he jogs. He walks a minimum of miles, and jogs a minimum of miles. Let miles be the distance he walks and let miles be the distance he jogs. a Write down relevant constraints that and must satisf. b raw a graph to show the region of points satisfied b the constraints. ill uses up 0 calories per mile when he walks and 0 calories per mile when he jogs. c Use our graph to find the smallest number and the largest number of calories that ill can use up each week through this eercise. compan makes two tpes of phones, and. Each da it must make at least 0 tpe and at least 00 tpe. The number of tpe must be at most 0% more than tpe. The total number of phones made each da must not be more than 00 and must not be less than 00. Let be the number of tpe phones. Let be the number of tpe phones. a Write down all the constraints as inequalities. b Show b shading the region which satisfies all the constraints. The profit from making a tpe is. The profit from making a tpe is 7.0. c ssuming that the compan sells all the phones it makes, work out the maimum profit from the da. Review The solution to a linear programming problem requires the evaluation of a linear function at the corners of the feasible region. The maimum (minimum) value of a linear function in an enclosed region defined b a set of linear inequalities occurs at one of the corners of the region or at all the points along the edge of the region. 09-hapter 09_ indd 9// 09::

7 nswers nswers 9. Get Read answers 7 a Eercise 9 E F b 7 7 (, ) Through Through Through Through Inequalities satisfied are (min) (ma) (min) 0 (ma) (min) (ma) (min) (min) 0 (ma) 7 09-hapter 09_ indd 7 9// 09::7

8 hapter 9 Linear programming (min) 7 (ma) (min) (ma) (min) 0 (ma) 0 (ma) (min) (ma) (ma) (min) (ma) (min) (ma) (min) (ma) 0 (min) 9 (ma) 9 (ma) (min) 0 (ma) 0 (ma) 0 (min) 0 0 (ma) 0 0 (min) 0 0 (ma) 0 0 (min) 90 0 (ma) 00 Eercise 9 answers a 00, 00, 000 b c 0 d min, ma a 00, 0,, c 0 d 000 e 9000 a 0, 00, 00, _ b hapter 09_ indd 9// 09::

9 nswers d min 000 at 0, 0, ma at 00, 00 a 0,.,, b R c R ma 7 0 at (,) a i Total prize fund, so 00 b ii, c Ma value of is 0 a, 00, 00, 0 b a b c Maimum and minimum values of 0 0 are 7 00 and 000 a 0, 00, 00, 00, _ b c c P d P ma hapter 09_ indd 9 9// 09::