Lesson Topics. B.3 Integer Programming Review Questions

Transcription

1 Lesson Topics Rounding Off (5) solutions in continuous variables to the nearest integer (like 2.67 rounded off to 3) is an unreliable way to solve a linear programming problem when decision variables should be integers. Sensitivity Analysis with Integer Variables is more important than with continuous variables because a small change in a constraint coefficient can cause a relatively large change in the optimal solution. Assignment Problems with Valuable Time (2) minimize the total time of assigning workers to jobs. Minimizing total time is appropriate when each worker has the same value of time. Assignment Problems with Supply and Demand (2) are Transportation Problems of suppliers to demanders except that each demand is assigned to exactly one supplier. 1

2 Rounding Off Question. Consider the following all-integer linear program: Max 1x 1 + 1x 2 s.t. 4x 1 + 6x x 1 + 5x x 1 + 1x 2 9 x 1, x 2 0 and integer a. Graph the constraints for this problem. Use dots to indicate all feasible integer solutions. b. Find the optimal integer solution. c. Solve the LP Relaxation of this problem (that is, the problem without requiring decision variables to be integers). 2

3 Answer to Question: a. b. The optimal solution to the LP Relaxation is shown on the above graph to be x1 = 4, x2 = 1. Its value is 5. c. The optimal integer solution is the same as the optimal solution to the LP Relaxation. This is always the case whenever all the variables take on integer values in the optimal solution to the LP Relaxation. 3

4 Rounding Off Question. Muir Manufacturing produces two popular grades of commercial carpeting among its many other products. In the coming production period, Muir needs to decide how many rolls of each grade should be produced in order to maximize profit. Each roll of Grade X carpet uses 3 units of synthetic fiber, requires 1 hour of production time, and needs 1 unit of foam backing. Each roll of Grade Y carpet uses 1 unit of synthetic fiber, requires 3 hours of production time, and needs 1 units of foam backing. The profit per roll of Grade X carpet is $3 and the profit per roll of Grade Y carpet is $2. In the coming production period, Muir has 9 units of synthetic fiber available for use. Workers have been scheduled to provide up to 7 hours of production time. The company has 10 units of foam backing available for use. a. Develop a linear programming model for this problem to determine how many rolls of each carpet should be produced. b. Graphically solve the linear-programming problem from Part a if you require that the number of rolls of each carpet be integers. c. Graphically solve the linear-programming problem from Part a if you do not require that the number of rolls of each carpet be integers (instead, the number of rolls of each carpet are continuous variables). d. Compare your solutions in Parts b and c. Tip: Your written answer should define the decision variables, formulate the objective and constraints, and solve for the optimum. --- You will not earn full credit if you just solve for the optimum; you must also define the decision variables, and formulate the objective and constraints. 4

5 Answer to Question: Part a: Let X = the number of rolls of Grade X carpet to make Let Y = the number of rolls of Grade Y carpet to make Part c: Max 3X + 2Y s.t. 3X + Y < 9 X + 3Y < 7 X + Y < 10 X, Y > A graph of the feasible set reveals that the third costraint is redundant --- it does not affect the feasible set. Adding a graph of isovalue lines reveals the optimum occurs where the first and second constraints bind (the third constraint is redundant). Solving the binding form of those two constraints yields the optimal solution: X = 2.5, Y = 1.5 5

6 Part b: A graph of the feasible set reveals that the third costraint is redundant --- it does not affect the feasible set. Use dots to indicate all feasible integer solutions. Adding a graph of isovalue lines reveals the optimum occurs at X = 3, Y = 0 Part d: Comparing solutions, the optimal integer solution in Part b is not the result of rounding off the optimal continuous solution in Part c. 6

7 Rounding Off Question. The Iron Works, Inc. seeks to maximize profit by making two products from steel and labor. It just received this day's allocation of 9 pounds of steel, and there is 8 hours of labor available. It takes 3 pounds of steel to make a unit of Product 1, and 1 pound of steel to make a unit of Product 2. It also takes 2 hours of labor to make a unit of Product 1, and 5 hours of labor to make a unit of Product 2. The physical plant has the capacity to make up to 5 units of total product (Product 1 plus Product 2). Product 1 has unit profit 3 dollars, and Product 2 has 9 dollars. a. Develop a linear programming model for this problem to determine how much should be produced. b. Graphically solve the linear-programming problem from Part a you require that production units be integers. c. Graphically solve the linear-programming problem from Part a if you do not require that production units be integers (instead, production units are continuous variables). d. Compare your solutions in Parts b and c. Tip: Your written answer should define the decision variables, formulate the objective and constraints, and solve for the optimum. --- You will not earn full credit if you just solve for the optimum; you must also define the decision variables, and formulate the objective and constraints. 7

8 Answer to Question: Part a: Let X = units of Product 1 produced. Let Y = units of Product 1 produced. Max 3X + 9Y s.t. 3X + Y < 9 (steel) 2X + 5Y < 8 (labor) X + Y < 5 (capacity) X, Y 0 Part c: X on horizontal axis Y on vertical axis First constraint, through (3,0) and (0,9) Second constraint, through (4,0) and (0,1.6) Third constraint, (5,0) and (0,5) Feasible set of solutions is the region bounded by first, second, and non-negativity constraints A graph of the feasible set and isovalue lines reveals the optimum occurs where the non-negativity of X and the second constraint binds. Solving the binding form of those two constraints yields the optimal solution: X = 0, Y = 8/5 = 1.6 8

9 Part b: A graph of the feasible set with dots to indicate all feasible integer solutions and isovalue lines reveals the optimum occurs at X = 1, Y = 1 Part d: Comparing solutions, the optimal integer solution in Part b is not the result of rounding off the optimal continuous solution in Part c. 9

10 Rounding Off Question. Exxon Mobil Corporation seeks to maximize profit by making two grades of gasoline from crude oil and additives. It just received this day's allocation of 4 thousand gallons of crude oil, and 2 thousand gallons of additives. It takes 0.8 gallons of crude oil to make a gallon of Premium gasoline, and 0.4 gallons of crude oil to make a gallon of Regular gasoline. It also takes 0.2 gallons of additives to make a gallon of Premium gasoline, and 0.6 gallons of additives to make a gallon of Regular gasoline. Premium gasoline has unit profit 3 of dollars, and Regular gasoline has 1 dollar. a. Develop a linear programming model for this problem to determine how much should be produced. b. Graphically solve the linear-programming problem from Part a if you require that production units be integers. c. Graphically solve the linear-programming problem from Part a if you do not require that production units be integers (instead, production units are continuous variables). d. Compare your solutions in Parts b and c. Tip: Your written answer should define the decision variables, formulate the objective and constraints, and solve for the optimum. --- You will not earn full credit if you just solve for the optimum; you must also define the decision variables, and formulate the objective and constraints. 10

11 Answer to Question: Part a: Let X = thousands of units of Premium gasoline produced. Let Y = thousands of units of Regular gasoline produced. Part c: Max 3X + 1Y s.t. 0.8X + 0.4Y < 4 (crude oil) 0.2X + 0.6Y < 2 (additives) X, Y A graph of the feasible set and isovalue lines (dashed lines above) reveals the optimum occurs where the non-negativity of Y and the first constraint binds. Solving the binding form of those two constraints yields the optimal solution: X = 5, Y = 0 Part b: Since the continuous solution is discrete, it is also the solution to the discrete case. (Since (X,Y) are in thousands of units, (X,Y) would still be an integer solution as long as 1000X and 1000Y were integers. 11

12 Rounding Off Question. Exxon Mobil Corporation seeks to maximize profit by making two grades of gasoline from crude oil and additives. It just received this day's allocation of 1.5 thousand gallons of crude oil, and 0.8 thousand gallons of additives. It takes 0.6 gallons of crude oil to make a gallon of Premium gasoline, and 0.3 gallons of crude oil to make a gallon of Regular gasoline. It also takes 0.2 gallons of additives to make a gallon of Premium gasoline, and 0.3 gallons of additives to make a gallon of Regular gasoline. Premium gasoline has unit profit 3 of dollars, and Regular gasoline has 4 dollars. a. Develop a linear programming model for this problem to determine how much should be produced. b. Graphically solve the linear-programming problem from Part a if you require that production units be integers. c. Graphically solve the linear-programming problem from Part a if you do not require that production units be integers (instead, production units are continuous variables). d. Compare your solutions in Parts b and c. Tip: Your written answer should define the decision variables, formulate the objective and constraints, and solve for the optimum. --- You will not earn full credit if you just solve for the optimum; you must also define the decision variables, and formulate the objective and constraints. 12

13 Answer to Question: Part a: Let X = thousands of units of Premium gasoline produced. Let Y = thousands of units of Regular gasoline produced. Part c: Max 3X + 4Y s.t. 0.6X + 0.3Y < 1.5 (crude oil) 0.2X + 0.3Y < 0.8 (additives) X, Y A graph of the feasible set and isovalue lines (dashed lines above) reveals the isovalue lines are steeper than the second constraint, and the optimum occurs where the first and the second constraint bind. Solving the binding form of those two constraints yields the optimal solution: X = 1.75, Y =

14 Part b: You would earn full credit if you noted that since (X,Y) are in thousands of units, X = 1.75 thousand and Y = 1.50 thousand are integer solutions. You would also earn full credit by solving the problem if you require that X and Y are integers. Max 3X + 4Y s.t. 0.6X + 0.3Y < 1.5 (crude oil) 0.2X + 0.3Y < 0.8 (additives) X, Y 0 A graph of the feasible set with dots to indicate all feasible integer solutions and isovalue lines reveals the optimum occurs at X = 1, Y = Part d: The integer solution in Part b is not the result of rounding off the continuous solution in Part c. 14

15 Rounding Off Question. Jacuzzi produces two types of hot tubs: Fuzion and Torino. There are 5 pumps, 36 hours of labor, and 60 feet of tubing available to make the tubs per week. Here are the input requirements, and unit profits: Fuzion Torino Pumps 1 1 Labor 9 hours 4 hours Tubing 8 feet 15 feet Unit Profit $6 $9 a. Develop a linear programming model for this problem to determine how much should be produced. b. Graphically solve the linear-programming problem from Part a if you require that production units be integers. c. Graphically solve the linear-programming problem from Part a if you do not require that production units be integers (instead, production units are continuous variables). d. Compare your solutions in Parts b and c. Tip: Your written answer should define the decision variables, formulate the objective and constraints, and solve for the optimum. --- You will not earn full credit if you just solve for the optimum; you must also define the decision variables, and formulate the objective and constraints. 15

16 Answer to Question: Part a: Let F = Fuzion units produced per week. Let Y = Torino units produced per week. Part c: Max 6 F + 9 T s.t. F + T < 5 (pumps) 9 F + 4 T < 36 (labor hours) 8 F + 15 T < 60 (tubing) F, T A graph of the feasible set and isovalue lines (dashed lines above) reveals the optimum occurs where the first and the third constraint bind. Solving the binding form of those two constraints yields the optimal solution: F = 2.143, T =

17 Part b: Max 6 F + 9 T s.t. F + T < 5 (pumps) 9 F + 4 T < 36 (labor hours) 8 F + 15 T < 60 (tubing) F, T A graph of the feasible set dots and isovalue lines (dashed lines above) reveals the optimum occurs at either (0,4) or (1,3) or (3,2). It turns out there are two optima: (0,4) and (3,2). Part d: The integer solutions in Part b are not the result of rounding off the continuous solution in Part c. 17

18 Assignment with Valuable Time Question. Sibeau Custom Tailoring has five idle tailors and two custom garments to make. The estimated time (in hours) it would take each tailor to complete a garment is shown in the next slide. Tailor Garment Polyester Suit Silk Suit Formulate an appropriate binary-integer program for determining the tailorgarment assignments that minimize the total estimated time spent making the two garments. No tailor is to be assigned more than one garment, and each garment is to be worked on by exactly one tailor. Formulate the problem, but you need not solve for the optimum. Tip: Your written answer should define the decision variables, and formulate the objective and constraints. 18

19 Answer to Question: Define the decision variables x ij = 1 if garment i is assigned to tailor j, and = 0 otherwise. Minimize total time spent making garments: Min 9x x x x x x x x x x 25 Define the constraints of exactly one tailor per garment: 1) x 11 + x 12 + x 13 + x 14 + x 15 = 1 2) x 21 + x 22 + x 23 + x 24 + x 25 = 1 Define the constraints of no more than one garment per tailor: 3) x 11 + x 21 < 1 4) x 12 + x 22 < 1 5) x 13 + x 23 < 1 6) x 14 + x 24 < 1 7) x 15 + x 25 < 1 19

20 Assignment with Valuable Time Question. Sibeau Custom Tailoring has five idle tailors and two custom garments to make. The estimated time (in hours) it would take each tailor to complete a garment is shown in the next slide. (An 'X' in the table indicates an unacceptable tailor-garment assignment.) Tailor Garment Polyester Suit Silk Suit X 12 9 Formulate and solve a binary-integer program for determining the tailorgarment assignments that minimize the total estimated time spent making the two garments. No tailor is to be assigned more than one garment, and each garment is to be worked on by exactly one tailor. Tip: Your written answer should define the decision variables, formulate the objective and constraints, and solve for the optimum. --- You will not earn full credit if you just solve for the optimum; you must also define the decision variables, and formulate the objective and constraints. 20

21 Answer to Question: Define the decision variables x ij = 1 if garment i is assigned to tailor j, and = 0 otherwise. Minimize total time spent making garments: Min 9x x x x x x x x x 25 Define the constraints of exactly one tailor per garment: 1) x 11 + x 12 + x 13 + x 14 + x 15 = 1 2) x 21 + x 22 + x 24 + x 25 = 1 Define the constraints of no more than one garment per tailor: 3) x 11 + x 21 < 1 4) x 12 + x 22 < 1 5) x 13 < 1 6) x 14 + x 24 < 1 7) x 15 + x 25 < 1 21

22 Interpretation: Assign tailor 1 to garment 3 and tailor 2 to garment 5. Time spent is 15 hours. 22

23 Assignment with Supply and Demand Question. Dow Chemical uses the chemical Rbase in production operations at two divisions. Only three suppliers of Rbase meet Dow s quality standards. The quantity of Rbase needed by each Dow division and the price per gallon charged by each supplier are as follows: Demand (1000s of gallons) Price per Gallon ($) Div 1 40 Sup Div 2 45 Sup The cost per gallon ($) for shipping from each supplier to each division are as follows: Cij Div 1 Div 2 Sup Sup Sup Sup Dow wants to diversify by spreading its business so that each division s demand is assigned to exactly one supplier. Formulate the optimal assignment of suppliers to divisions as a linearprogramming problem. Formulate the problem, but you need not solve the problem. 23

24 Answer to Question: Linear programming formulation (supply inequality, demand equality). Variables: Xij = 1 if Supplier i is assigned to Division j, else 0 Assignment Costs: The total cost is the sum of the purchase cost and the transportation cost. Supplier 1 assigned to Division 1 (cost in $1000s): o Purchase cost: (40 x $12.60) = $504 o Transportation Cost: (40 x $1) = $40 o Total Cost: $544 Assignment Costs: Cij = Cost of assigning Supplier i to Division j Cij Div 1 Div 2 Sup Sup Sup

25 Linear programming formulation (supply inequality, demand equality). Objective (minimize cost): Min 544X X X X X X32 Demand Constraints (since each division s demand is assigned to exactly one supplier): X11 + X21 + X31 = 1 X12 + X22 + X32 = 1 Optional: There is no mention of supply constraints, which are common in assignment problems. Here is what those common constraints would be in this problem. Supply Constraints (Each supplier can supply at most 1 Division): X11 + X12 < 1 X21 + X22 < 1 X31 + X32 < 1 25

26 Assignment with Supply and Demand Question. The Goodyear Tire and Rubber Company uses rubber in its American, Asian, and European tire manufacturing plants. Goodyear can buy rubber from either India, or Indonesia, or Malaysia, or Thailand. The number of tons of rubber needed daily by each tire plant and the price per ton charged by each supplier are as follows: Demand (tons) Price (dollars per ton) American 5 India 3 Asian 4 Indonesia 4 European 3 Malaysia 2 Thailand 5 The cost (dollars per ton) for shipping from each supplier to each manufacturing plant are as follows: American Asian European India Indonesia Malaysia Thailand To reduce fixed costs, Goodyear wants each manufacturing plant s demand to be assigned to exactly one rubber supplier. And because of capacity constraints, each rubber supplier can supply at most one tire plant. Formulate the optimal assignment of rubber suppliers to manufacturing plant s as a linear-programming problem. Formulate the problem, but you need not solve the problem. 26

27 Answer to Question: Linear programming formulation (supply inequality, demand equality). Variables: Xij = 1 if Supplier i is assigned to Plant j, else 0 Assignment Costs: The total cost is the sum of the purchase cost and the transportation cost. Supplier 1 (India) assigned to Plant 1 (American) (cost in dollars): o Purchase cost: (5 x $3) = $15 o Transportation Cost: (5 x $3) = $15 o Total Cost: $30 Assignment Costs: Cij = Cost of assigning Supplier i to Plant j Cij Plant 1 Plant 2 Plant 3 Sup 1 5(3+3)=30 4(3+6)=36 3(3+4)=21 Sup 2 5(4+5)=45 4(4+9)=52 3(4+8)=36 Sup 3 5(2+2)=20 4(2+6)=32 3(2+4)=18 Sup 4 5(5+7)=60 4(5+1)=24 3(5+2)=21 27

28 Linear programming formulation (supply inequality, demand equality). Objective (minimize cost): Min 30X X X X X X X X X X X X43 Subject to: Demand Constraints (each demand is assigned to exactly one supplier): X11 + X21 + X31 + X41 = 1 X12 + X22 + X32 + X42 = 1 X13 + X23 + X33 + X43 = 1 Supply Constraints (each supplier can supply at most one demander): X11 + X12 + X13 < 1 X21 + X22 + X23 < 1 X31 + X32 + X33 < 1 X41 + X42 + X43 < 1 28