Linear Programming. Linear Programming

Transcription

1 APPENDIX C Linear Programming C Appendi C Linear Programming C Linear Programming Linear Programming Application FIGURE C. 7 (, ) (, ) FIGURE C. Feasible solutions (, ) 7 NOTE In Eample, tr evaluating C at other feasible points in the graph of the constraints. No matter which point ou choose, the value of C will be greater than or equal to and less than or equal to. Linear Programming Sstems of linear inequalities are used etensivel in business and economics to solve optimization problems. The word optimize means to find the greatest or least. Man optimization problems can be solved using linear programming. A two-variable linear programming problem consists of the following.. An objective function that epresses the quantit to be maimized (or minimized).. A sstem of constraint linear inequalities whose solution set represents the set of feasible solutions. The solution of a linear programming problem is found b determining which point in the set of feasible solutions ields the optimal value of the objective function. For eample, consider a linear programming problem in which ou are asked to maimize the value of subject to a set of constraints that determine the region indicated in Figure C.. It can be shown that if there is an optimal solution, it must occur at one of the vertices of the region. In other words, ou can find the maimum value b testing C at each of the vertices, as illustrated in Eample. EXAMPLE Solving a Linear Programming Problem Find the minimum value and maimum value of C subject to the following constraints. C a b The graph of the constraint inequalities is shown in Figure C.. The three vertices are,,,, and,. To find the minimum and maimum values of C,evaluate C at each of the three vertices. At, : C Minimum value of C At, : At, : C C Maimum value of C The minimum value of C is. It occurs when and. The maimum value of C is. It occurs when and.

2 C APPENDIX C Linear Programming Guidelines for Solving a Linear Programming Problem To solve a linear programming problem involving two variables, use the following steps.. Sketch the region corresponding to the sstem of constraints. (The points inside or on the boundar of the region are called feasible solutions.). Find the vertices of the region.. Test the objective function at each of the vertices and select the values of the variables that optimize the objective function. For a bounded region, both a minimum and a maimum value will eist. (For an unbounded region, if an optimal solution eists, it will occur at a verte.) These guidelines will work whether the objective function is to be maimized or minimized. For instance, in Eample the same test was used to find the maimum value of C and the minimum value of C. 8 (, ) (, ) (, ) (, ) 8 FIGURE C. EXAMPLE Solving a Linear Programming Problem Find the minimum value and maimum value of C subject to the following constraints. The graph of the constraint inequalities is shown in Figure C.. The four vertices are,,,,,, and,. To find the minimum and maimum values of C, evaluate C at each of the four vertices. At, : C Minimum value of C At, : C ( At, : C 9 At, : C 9 Maimum value of C The minimum value of C is. It occurs when and. The maimum value of C is 9. It occurs when and.

3 APPENDIX C Linear Programming C 8 (, 8) (, ) (, ) 8 FIGURE C. EXAMPLE Solving a Linear Programming Problem Find the minimum value and maimum value of C 7 subject to the following constraints. The graph of the constraint inequalities is shown in Figure C.. The three vertices are, 8,,, and,. To find the minimum and maimum values of C, evaluate C 7 at each of the three vertices. At, 8: At, : C 78 C 7 Minimum value of C At, : C 7 9 The minimum value of C is. It occurs when and. There is no maimum value (the graph of the constraints is unbounded). EXAMPLE Solving a Linear Programming Problem (, ) (, ) (, ) (, ) Find the maimum value of C subject to the following constraints. FIGURE C. The constraints form the region shown in Figure C.. At the four vertices of this region, the objective function has the following values. At, : At, : At, : At, : C C C 8 C Maimum value of C So, the maimum value of C is 8, and this value occurs when and.

4 C APPENDIX C Linear Programming Application EXAMPLE Finding the Maimum Profit You own a biccle manufacturing plant and can assemble biccles using two processes. The hours of unskilled labor, machine time, and skilled labor per biccle are shown below. You can use up to hours of unskilled labor and up to hours each of machine time and skilled labor. Process A earns a profit of $ per bike, and process B earns a profit of $ per bike. How man biccles should ou assemble b each process to obtain a maimum profit? Unskilled Machine Skilled labor time labor Hours for process A Hours for process B b (, ) (, ) 8 (, ) (, ) (, ) 8 FIGURE C. a Let a and b represent the numbers of biccles assembled b the two processes. Because ou want a maimum profit P, the objective function is P a b. The constraints are as follows. a b a b a b a b Profit: $ per bike for process A $ per bike for process B Unskilled labor: Up to hours Machine time: Up to hours Skilled labor: Up to hours Cannot produce a negative amount Cannot produce a negative amount The region that represents the feasible solutions is shown in Figure C.. The profits at the vertices of the region are as follows. At, : At, : At, : At, : At, : P $, P $8, P $, P $, P $ Maimum profit The maimum profit is obtained b making biccles b process A and biccles b process B.

5 APPENDIX C Linear Programming C C Eercises In Eercises, find the minimum and maimum values of the objective function subject to the constraints. (For each eercise, the graph of the region determined b the constraints is provided.). :. : C C 8 : :. :. : C C 7 : : (See Eercise.) (See Eercise.). :. : C C : : (, ) (, ) (, ) (,) (, ) (, ) (, ) ) (, (, ) (, ) (, ) (, ) (, ) 9 (, ) 7. : 8. : C. C : : (See Eercise.) (See Eercise.) 9. :. : C 7 C : : (, ) (, ) (, ). :. : C C : : (See Eercise 9.) (See Eercise.) In Eercises, sketch the region determined b the constraints. Then find the minimum and maimum values of the objective function subject to the constraints.. :. : C C : : (, ) (, ) (, 8) (, ) (7,) 7 ( 9, )

6 C APPENDIX C Linear Programming. :. : C C : : 7. : 8. : C C 7 8 : : 9. :. : C 9 C 7 : : (See Eercise 7.) (See Eercise 8.) In Eercises, use a graphing utilit to graph the region determined b the constraints. Then find the minimum and maimum values of the objective function subject to the constraints.. :. : C C : : 7 8. :. : C 7 C : : (See Eercise.) (See Eercise.). Maimum Profit A fruit grower has acres of land available to raise two crops, A and B. It takes da to trim an acre of crop A and das to trim an acre of crop B, and there are das per ear available for trimming. It takes. da to pick an acre of crop A and. da to pick an acre of crop B, and there are das per ear available for picking. Find the number of acres of each fruit that should be planted to maimize profit, assuming that the profit is $ per acre for crop A and $ per acre for crop B.. Investments An investor has up to $, to invest in two tpes of investments. Tpe A investments pa % annuall and tpe B investments pa % annuall. To have a well-balanced portfolio, the investor imposes the following conditions. At least one-half of the total portfolio is to be allocated to tpe A investments and at least one-fourth of the total portfolio to tpe B investments. How much should be allocated to each tpe of investment to obtain a maimum return? 7. Minimum Cost A farming cooperative mies two brands of cattle feed. Brand X costs $ per bag and contains two units of nutrient A, two units of nutrient B, and two units of nutrient C. Brand Y costs $ per bag and contains one unit of nutrient A, nine units of nutrient B, and three units of nutrient C. The minimum requirements of nutrients A, B, and C are units, units, and units, respectivel. Find the number of bags of each brand that should be mied to produce the required miture having a minimum cost. 8. Minimum Cost A pet suppl compan mies two brands of dr dog food. Brand X costs $ per bag and contains eight units of nutrient A, one unit of nutrient B, and two units of nutrient C. Brand Y costs $ per bag and contains tow units of nutrient A, one unit of nutrient B, and seven units of nutrient C. Each bad of dog food must contain at least units, units, and units of nutrients A, B, and C, respectivel. Find the number of bags of brands X and Y that should be mied to produce a miture meeting the minimum nutritional requirements and having a minimum cost. 9. Maimum Profit A manufacturer produces two models of biccles. The amounts of time (in hours) required for assembling, painting, and packaging each model are as follows. Assembl Painting Packaging Model A Model B..7 The total amounts of time available for assembl, painting, and packaging are hours, 8 hours, and hours, respectivel. The profits per unit for the two models are $ for model A and $ for model B. How man of each model should be produced to obtain a maimum profit?