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Math 1300: Section 5- 3 Linear Programing in Two Dimensions: Geometric Approach

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  • 1. Linear Programming Problem Math 1300 Finite MathematicsSection 5.3 Linear Programming In Two Dimensions: Geometric Approach Jason Aubrey Department of Mathematics University of Missouri university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 2. Linear Programming ProblemLinear programming is a mathematical process that has beendeveloped to help management in decision making and it hasbecome one of the most widely used and best-known tools ofmanagement science. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 3. Linear Programming ProblemIn general, a linear programming problem is one that isconcerned with finding the optimal value of a linear objectivefunction of the form z = ax + by(where a and b are not both 0) Where the decision variables xand y are subject to the problem constraints. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 4. Linear Programming ProblemIn general, a linear programming problem is one that isconcerned with finding the optimal value of a linear objectivefunction of the form z = ax + by(where a and b are not both 0) Where the decision variables xand y are subject to the problem constraints.The problem constraints are various linear inequalities andequations. In all problems we consider, the decision variableswill also satisfy the nonnegative constraints, x ≥ 0, y ≥ 0. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 5. Linear Programming ProblemThe set of points satisfying both the problem constraints andthe nonnegative constraints is called the feasible region orfeasible set for the problem. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 6. Linear Programming ProblemThe set of points satisfying both the problem constraints andthe nonnegative constraints is called the feasible region orfeasible set for the problem.Any point in the feasible region that produces the optimal valueof the objective function over the feasible region is called anoptimal solution. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 7. Linear Programming ProblemExample: Maximize P = 30x + 40y subject to 2x + y ≤ 10 x +y ≤7 x + 2y ≤ 12 x, y ≥ 0 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 8. Linear Programming ProblemTheorem (Fundamental Theorem of Linear Programming)If the optimal value of the objective function in a linearprogramming problem exists, then that value must occur at one(or more) of the corner points of the feasible region. (A cornerpoint is a point in the feasible set where one or more boundarylines intersect.) university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 9. Linear Programming ProblemApplying the theorem... university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 10. Linear Programming ProblemApplying the theorem... 1 Graph the feasible region, as discussed in Section 5-2. Be sure to find all corner points. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 11. Linear Programming ProblemApplying the theorem... 1 Graph the feasible region, as discussed in Section 5-2. Be sure to find all corner points. 2 Construct a corner point table listing the value of the objective function at each corner point. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 12. Linear Programming ProblemApplying the theorem... 1 Graph the feasible region, as discussed in Section 5-2. Be sure to find all corner points. 2 Construct a corner point table listing the value of the objective function at each corner point. 3 Determine the optimal solution(s) from the table. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 13. Linear Programming ProblemApplying the theorem... 1 Graph the feasible region, as discussed in Section 5-2. Be sure to find all corner points. 2 Construct a corner point table listing the value of the objective function at each corner point. 3 Determine the optimal solution(s) from the table. 4 For an applied problem, interpret the optimal solution(s) in terms of the original problem. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 14. Linear Programming ProblemBack to our original problem: university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 15. Linear Programming ProblemBack to our original problem:Example: Maximize P = 30x + 40y subject to 2x + y ≤ 10 x +y ≤7 x + 2y ≤ 12 x, y ≥ 0 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 16. Linear Programming Problem654321 0 1 2 3 4 5 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 17. Linear Programming Problem 2x + y ≤ 106 a Boundary line:5 (a) y = 10 − 2x x y4 0 10 5 03 Check: 2(0) + 0 ≤ 10 Yes!2 ?1 0 1 2 3 4 5 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 18. Linear Programming Problem x +y ≤76 b a Boundary line:5 (b) y = 7 − x x y4 0 7 7 03 Check: 0 + 0 ≤ 7 Yes!2 ?1 0 1 2 3 4 5 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 19. Linear Programming Problem x + 2y ≤ 126 c b a Boundary line:5 (c) y = 6 − 1 x 2 x y4 0 63 12 0 Check: 0 + 2(0) ≤ 12 Yes!2 ?1 0 1 2 3 4 5 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 20. Linear Programming Problem Now we mark the feasible set and6 c b a find the coordinates of the corner points.5432 FS1 0 1 2 3 4 5 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 21. Linear Programming Problem Line c and the y -axis intersect at 6 c b a (0, 6)(0, 6) 5 4 3 2 FS 1 0 1 2 3 4 5 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 22. Linear Programming Problem Line c (y = 7 − x) and Line b c b a 1 6 (y = 6 − 2 x) intersect at (2, 5):(0, 6) 5 (2, 5) 1 4 7−x =6− x 2 3 1 − x = −1 2 2 FS x =2 1 y =7−2=5 0 1 2 3 4 5 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 23. Linear Programming Problem Line a (y = 10 − 2x) and Line b 6 c b a (y = 7 − x) intersect at (3, 4):(0, 6) 5 (2, 5) 10 − 2x = 7 − x 4 (3, 4) −x = −3 3 x =3 2 FS y =7−3=4 1 0 1 2 3 4 5 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 24. Linear Programming Problem Line a intersects the x-axis at 6 c b a (5, 0).(0, 6) 5 (2, 5) 4 (3, 4) 3 2 FS 1 (5, 0) 0 1 2 3 4 5 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 25. Linear Programming Problem Don’t forget (0, 0)! 6 c b a(0, 6) 5 (2, 5) 4 (3, 4) 3 2 FS 1 (0, 0) (5, 0) 0 1 2 3 4 5 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 26. Linear Programming Problem Now we construct our corner point ta- 6 c b a ble, and find the maximum value of(0, 6) the objective function P = 30x + 40y : 5 Corner Point P = 30x + 40y (2, 5) 4 (0, 6) 240 (3, 4) (2, 5) 260 3 (3, 4) 250 (5, 0) 150 2 FS (0, 0) 0 1 Therefore the maximum value of P (0, 0) (5, 0) 0 1 2 3 4 5 occurs at the point (2, 5). university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 27. Linear Programming ProblemExample: Minimize and maximize P = 20x + 10y subject to 2x + 3y ≥ 30 2x + y ≤ 26 −2x + 5y ≤ 34 x, y ≥ 0 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 28. Linear Programming Problem (8, 10)10 (3, 8)864 (12, 2)2 0 2 4 6 8 10 12 14 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 29. Linear Programming Problem The feasible set has been drawn (8, 10) for you. We now construct our10 corner point table. (3, 8)864 (12, 2)2 0 2 4 6 8 10 12 14 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 30. Linear Programming Problem The feasible set has been drawn (8, 10) for you. We now construct our10 corner point table. (3, 8)8 Corner Point P = 20x + 10y (3, 8) 1406 (8, 10) 2604 (12, 2) 260 (12, 2)2 0 2 4 6 8 10 12 14 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 31. Linear Programming Problem The feasible set has been drawn (8, 10) for you. We now construct our10 corner point table. (3, 8)8 Corner Point P = 20x + 10y (3, 8) 1406 (8, 10) 2604 (12, 2) 260 Therefore the maximum value of (12, 2)2 P is at (8, 10) and (12, 2) and the minimum value of P is at (3, 8). 0 2 4 6 8 10 12 14 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 32. Linear Programming ProblemExample: A manufacturing plant makes two types of inflatableboats, a two-person boat and a four-person boat. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 33. Linear Programming ProblemExample: A manufacturing plant makes two types of inflatableboats, a two-person boat and a four-person boat. Eachtwo-person boat requires 0.9 labor hours from the cuttingdepartment and 0.8 labor-hours from the assembly department. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 34. Linear Programming ProblemExample: A manufacturing plant makes two types of inflatableboats, a two-person boat and a four-person boat. Eachtwo-person boat requires 0.9 labor hours from the cuttingdepartment and 0.8 labor-hours from the assembly department.Each four person boat requires 1.8 labor-hours from the cuttingdepartment and 1.2 labor-hours from the assembly department. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 35. Linear Programming ProblemExample: A manufacturing plant makes two types of inflatableboats, a two-person boat and a four-person boat. Eachtwo-person boat requires 0.9 labor hours from the cuttingdepartment and 0.8 labor-hours from the assembly department.Each four person boat requires 1.8 labor-hours from the cuttingdepartment and 1.2 labor-hours from the assembly department.The maximum labor-hours available per month in the cuttingdepartment and the assembly department are 864 and 672,respectively. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 36. Linear Programming ProblemExample: A manufacturing plant makes two types of inflatableboats, a two-person boat and a four-person boat. Eachtwo-person boat requires 0.9 labor hours from the cuttingdepartment and 0.8 labor-hours from the assembly department.Each four person boat requires 1.8 labor-hours from the cuttingdepartment and 1.2 labor-hours from the assembly department.The maximum labor-hours available per month in the cuttingdepartment and the assembly department are 864 and 672,respectively. The company makes a profit of $25 on each2-person boat and $40 on each four-person boat. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 37. Linear Programming ProblemExample: A manufacturing plant makes two types of inflatableboats, a two-person boat and a four-person boat. Eachtwo-person boat requires 0.9 labor hours from the cuttingdepartment and 0.8 labor-hours from the assembly department.Each four person boat requires 1.8 labor-hours from the cuttingdepartment and 1.2 labor-hours from the assembly department.The maximum labor-hours available per month in the cuttingdepartment and the assembly department are 864 and 672,respectively. The company makes a profit of $25 on each2-person boat and $40 on each four-person boat.How many of each type should be manufactured each month tomaximize profit? What is the maximum profit? university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 38. Linear Programming ProblemTo solve such a problem involves: university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 39. Linear Programming ProblemTo solve such a problem involves: 1 Constructing a mathematical model of the problem, and university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 40. Linear Programming ProblemTo solve such a problem involves: 1 Constructing a mathematical model of the problem, and 2 using the mathematical model to find the solution. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 41. Linear Programming ProblemTo solve such a problem involves: 1 Constructing a mathematical model of the problem, and 2 using the mathematical model to find the solution.So far we have studied some of the techniques we will use forfinding the solution. Here, we introduce a 5 step procedure forconstructing a mathematical model of the problem. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 42. Linear Programming ProblemTo solve such a problem involves: 1 Constructing a mathematical model of the problem, and 2 using the mathematical model to find the solution.So far we have studied some of the techniques we will use forfinding the solution. Here, we introduce a 5 step procedure forconstructing a mathematical model of the problem.Step 1: Identify the decision variables. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 43. Linear Programming ProblemTo solve such a problem involves: 1 Constructing a mathematical model of the problem, and 2 using the mathematical model to find the solution.So far we have studied some of the techniques we will use forfinding the solution. Here, we introduce a 5 step procedure forconstructing a mathematical model of the problem.Step 1: Identify the decision variables.In this problem, we have x = number of 2-person boats to manufacture y = number of 4-person boats to manufacture university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 44. Linear Programming ProblemStep 2: Summarize relevant information in table form, relatingthe decision variables with the rows in the table, if possible. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 45. Linear Programming ProblemStep 2: Summarize relevant information in table form, relatingthe decision variables with the rows in the table, if possible.Recall: Each two-person boat requires 0.9 labor hours from thecutting department and 0.8 labor-hours from the assembly de-partment. Each four person boat requires 1.8 labor-hours fromthe cutting department and 1.2 labor-hours from the assembly de-partment. The maximum labor-hours available per month in thecutting department and the assembly department are 864 and672, respectively. The company makes a profit of $25 on each2-person boat and $40 on each four-person boat. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 46. Linear Programming ProblemStep 2: Summarize relevant information in table form, relatingthe decision variables with the rows in the table, if possible.Recall: Each two-person boat requires 0.9 labor hours from thecutting department and 0.8 labor-hours from the assembly de-partment. Each four person boat requires 1.8 labor-hours fromthe cutting department and 1.2 labor-hours from the assembly de-partment. The maximum labor-hours available per month in thecutting department and the assembly department are 864 and672, respectively. The company makes a profit of $25 on each2-person boat and $40 on each four-person boat. 2-person 4-person Available Cutting 0.9 1.8 864 Assembly 0.8 1.2 672 Profit $25 $40 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 47. Linear Programming ProblemStep 2: Summarize relevant information in table form, relatingthe decision variables with the rows in the table, if possible. 2-person 4-person Available Cutting 0.9 1.8 864 Assembly 0.8 1.2 672 Profit $25 $40Step 3: Determine the objective and write a linear objectivefunction. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 48. Linear Programming ProblemStep 2: Summarize relevant information in table form, relatingthe decision variables with the rows in the table, if possible. 2-person 4-person Available Cutting 0.9 1.8 864 Assembly 0.8 1.2 672 Profit $25 $40Step 3: Determine the objective and write a linear objectivefunction. P = 25x + 40y university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 49. Linear Programming ProblemStep 2: Summarize relevant information in table form, relatingthe decision variables with the rows in the table, if possible. 2-person 4-person Available Cutting 0.9 1.8 864 Assembly 0.8 1.2 672 Profit $25 $40Step 3: Determine the objective and write a linear objectivefunction. P = 25x + 40y Step 4: Write problem constraints using linear equationsand/or inequalities. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 50. Linear Programming ProblemStep 2: Summarize relevant information in table form, relatingthe decision variables with the rows in the table, if possible. 2-person 4-person Available Cutting 0.9 1.8 864 Assembly 0.8 1.2 672 Profit $25 $40Step 3: Determine the objective and write a linear objectivefunction. P = 25x + 40y Step 4: Write problem constraints using linear equationsand/or inequalities. 0.9x + 1.8y ≤ 864 0.8x + 1.2y ≤ 672 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 51. Linear Programming Problem P = 25x+40y 0.9x + 1.8y ≤ 864 0.8x + 1.2y ≤ 672Step 5: Write nonnegative constraints. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 52. Linear Programming Problem P = 25x+40y 0.9x + 1.8y ≤ 864 0.8x + 1.2y ≤ 672Step 5: Write nonnegative constraints. x ≥0 y ≥0 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 53. Linear Programming Problem P = 25x+40y 0.9x + 1.8y ≤ 864 0.8x + 1.2y ≤ 672Step 5: Write nonnegative constraints. x ≥0 y ≥0 We now have a mathematical model of the given problem. Weneed to find the production schedule which results in maximumprofit for the company and to find that maximum profit. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 54. Linear Programming ProblemNow we follow the procedure for geometrically solving a linearprogramming problem with two decision variables.Applying the Fundamental Theorem of Linear Programming university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 55. Linear Programming ProblemNow we follow the procedure for geometrically solving a linearprogramming problem with two decision variables.Applying the Fundamental Theorem of Linear Programming 1 Graph the feasible region, as discussed in Section 5-2. Be sure to find all corner points. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 56. Linear Programming ProblemNow we follow the procedure for geometrically solving a linearprogramming problem with two decision variables.Applying the Fundamental Theorem of Linear Programming 1 Graph the feasible region, as discussed in Section 5-2. Be sure to find all corner points. 2 Construct a corner point table listing the value of the objective function at each corner point. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 57. Linear Programming ProblemNow we follow the procedure for geometrically solving a linearprogramming problem with two decision variables.Applying the Fundamental Theorem of Linear Programming 1 Graph the feasible region, as discussed in Section 5-2. Be sure to find all corner points. 2 Construct a corner point table listing the value of the objective function at each corner point. 3 Determine the optimal solution(s) from the table. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 58. Linear Programming ProblemNow we follow the procedure for geometrically solving a linearprogramming problem with two decision variables.Applying the Fundamental Theorem of Linear Programming 1 Graph the feasible region, as discussed in Section 5-2. Be sure to find all corner points. 2 Construct a corner point table listing the value of the objective function at each corner point. 3 Determine the optimal solution(s) from the table. 4 For an applied problem, interpret the optimal solution(s) in terms of the original problem. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 59. Linear Programming Problem500400300200100 100 200 300 400 500 600 700 800 900 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 60. Linear Programming Problem First we plot the boundary line 0.9x + 1.8y = 864:500 x y 0 480400 960 0300200100 100 200 300 400 500 600 700 800 900 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 61. Linear Programming Problem Test: 0.9(0) + 1.8(0) ≤ 0 Yes!500 ? So we choose the lower half-400 plane.300200100 100 200 300 400 500 600 700 800 900 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 62. Linear Programming Problem Now plot boundary line 0.8x + 1.2y = 672:500 x y 0 560400 840 0300200100 100 200 300 400 500 600 700 800 900 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 63. Linear Programming Problem Test: 0.8(0) + 1.2(0) ≤ 672 Yes!500 ? So we choose the lower half-400 plane.300200100 100 200 300 400 500 600 700 800 900 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 64. Linear Programming Problem Next, we find the intersection point of the lines:500 1 2 480 − x = 560 − x400 2 3 1 x = 80300 6 (480, 240) x = 480200 y = 480 − (1/2)(480) = 240100 100 200 300 400 500 600 700 800 900 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 65. Linear Programming Problem500400300 (480, 240)200 FS100 100 200 300 400 500 600 700 800 900 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 66. Linear Programming ProblemNow we solve the problem: Corner Point P = 25x + 40y (0, 0) 0 (0, 480) 19,200 (480, 240) 21,600 (860, 0) 21,500We conclude that the company can make a maximum profit of$21,600 by producing 480 two-person boats and 240four-person boats. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 67. Linear Programming ProblemRemember, there are two parts to solving an applied linearprogramming problem: constructing the mathematical modeland using the geometric method to solve it. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 68. Linear Programming ProblemExample: A chicken farmer can buy a special food mix A at 20cents per pound and a special food mix B at 40 cents perpound. Each pound of mix A contains 3,000 units of nutrient N1and 1,000 units of nutrient N2 ; each pound of mix B contains4,000 units of nutrient N1 and 4,000 units of nutrient N2 . If theminimum daily requirements for the chickens collectively are36,000 units of nutrient N1 and 20,000 units of nutrient N2 , howmany pounds of each food mix should be used each day tominimize daily food costs while meeting (or exceeding) theminimum daily nutrient requirements? What is the minimumdaily cost? Construct a mathematical model and solve usingthe geometric method. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 69. Linear Programming ProblemStep 1: Identify the decision variables. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 70. Linear Programming ProblemStep 1: Identify the decision variables.In this problem, we are asked, how many pounds of each food mix should be used each day to minimize daily food costs while meeting (or exceeding) the minimum daily nutrient requirements?So, x = number of pounds of mix A to use each day y = number of pounds of mix B to use each day university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 71. Linear Programming ProblemStep 2: Summarize relevant information in table form, relatingthe decision variables with the rows in the table, if possible. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 72. Linear Programming ProblemStep 2: Summarize relevant information in table form, relatingthe decision variables with the rows in the table, if possible.A chicken farmer can buy a special food mix A at 20 cents perpound and a special food mix B at 40 cents per pound. Eachpound of mix A contains 3,000 units of nutrient N1 and 1,000 unitsof nutrient N2 ; each pound of mix B contains 4,000 units of nutrientN1 and 4,000 units of nutrient N2 . If the minimum daily require-ments for the chickens collectively are 36,000 units of nutrient N1and 20,000 units of nutrient N2 . . . university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 73. Linear Programming ProblemStep 2: Summarize relevant information in table form, relatingthe decision variables with the rows in the table, if possible.A chicken farmer can buy a special food mix A at 20 cents perpound and a special food mix B at 40 cents per pound. Eachpound of mix A contains 3,000 units of nutrient N1 and 1,000 unitsof nutrient N2 ; each pound of mix B contains 4,000 units of nutrientN1 and 4,000 units of nutrient N2 . If the minimum daily require-ments for the chickens collectively are 36,000 units of nutrient N1and 20,000 units of nutrient N2 . . . mix A mix B Min daily req. units of N1 3,000 4,000 36,000 units of N2 1,000 4,000 20,000 Cost per pound $0.20 $0.40 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 74. Linear Programming ProblemStep 2: Summarize relevant information in table form, relatingthe decision variables with the rows in the table, if possible. mix A mix B Min daily req. units of N1 3,000 4,000 36,000 units of N2 1,000 4,000 20,000 Cost per pound $0.20 $0.40Step 3: Determine the objective and write a linear objectivefunction. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 75. Linear Programming ProblemStep 2: Summarize relevant information in table form, relatingthe decision variables with the rows in the table, if possible. mix A mix B Min daily req. units of N1 3,000 4,000 36,000 units of N2 1,000 4,000 20,000 Cost per pound $0.20 $0.40Step 3: Determine the objective and write a linear objectivefunction. We want to minimize daily food costs so the objectivefunction is C = 0.20x + 0.40y university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 76. Linear Programming ProblemStep 4: Write problem constraints using linear equations and/orinequalities. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 77. Linear Programming ProblemStep 4: Write problem constraints using linear equations and/orinequalities. 3, 000x + 4, 000y ≥ 36, 000 1, 000x + 4, 000y ≥ 20, 000 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 78. Linear Programming ProblemStep 4: Write problem constraints using linear equations and/orinequalities. 3, 000x + 4, 000y ≥ 36, 000 1, 000x + 4, 000y ≥ 20, 000Step 5: Write nonnegative constraints. x ≥0 y ≥0 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 79. Linear Programming ProblemWe now have a mathematical model of the given problem:Minimize C = 0.20x + 0.40ysubject to: 3, 000x + 4, 000y ≥ 36, 000 1, 000x + 4, 000y ≥ 20, 000 x ≥0 y ≥0We will now use the geometric method to determin how manypounds of each food mix should be used each day to minimizedaily food costs while meeting (or exceeding) the minimumdaily nutrient requirements. We can then determine theminimum daily cost as well. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 80. Linear Programming Problem 10 8 6 4 2−2 0 2 4 6 8 10 12 14 16 18 20 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 81. Linear Programming Problem 10 8 6 4 2−2 0 2 4 6 8 10 12 14 16 18 20Now plot boundary line 3, 000x + 4, 000y = 36000: x y 0 9 12 0 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 82. Linear Programming Problem 10 8 6 4 2−2 0 2 4 6 8 10 12 14 16 18 20Test:3, 000(0) + 4, 000(0) ≥ 36, 000 No! ?So we choose the upper half-plane. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 83. Linear Programming Problem 10 8 6 4 2−2 0 2 4 6 8 10 12 14 16 18 20Now plot boundary line 1, 000x + 4, 000y = 20, 000: x y 0 5 20 0 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 84. Linear Programming Problem 10 8 6 4 2−2 0 2 4 6 8 10 12 14 16 18 20Test:1, 000(0) + 4, 000(0) ≥ 20, 000 No! ?So we choose the upper half-plane. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 85. Linear Programming Problem 10 (0, 9) 8 6 4 2 (8, 3) (20, 0)−2 0 2 4 6 8 10 12 14 16 18 20Next, we find the intersection point of the lines: 3, 000x + 4, 000y = 36, 000 – 1, 000x + 4, 000y = 20, 000 2, 000x + 0y = 16, 000 x =8And so, 1, 000(8) + 4, 000y = 20, 000; this implies that y =3. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 86. Linear Programming Problem 10 (0, 9) 8 6 FS 4 2 (8, 3) (20, 0)−2 0 2 4 6 8 10 12 14 16 18 20So, we have the feasible set shown above. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 87. Linear Programming Problem 10 (0, 9) 8 6 FS 4 2 (8, 3) (20, 0)−2 0 2 4 6 8 10 12 14 16 18 20And now we plug the corner points into the objective functionto find the minimum cost: Point C = 0.20x + 0.40y (0, 9) $3.60 (8, 3) $2.80 (20, 0) $4.00 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 88. Linear Programming Problem Point C = 0.20x + 0.40y (0, 9) $3.60 (8, 3) $2.80 (20, 0) $4.00We therefore conclude that the chicken farmer can feed thechickens at a minimal cost of $2.80 per day using 8 pounds ofmix A and 3 pounds of mix B. university-logo Jason Aubrey Math 1300 Finite Mathematics