Wk# Linear Programming

Transcription

1 Alg2H 4-11 Linear Programming Wk#10 The Graphical method to solve linear programming problems: Copy the complete problem on graph paper Number the constraints Graph and label (with numbers) the constraints Shade the feasible region Determine and label (with letters) the corner points Evaluate the objective function at the corner points Determine the OPTIMAL VALUE (MAX and/or MIN) of the objective function and at what value(s) of x and y it occurs Use the graphical method outlined above to solve the following. (Be sure to use graph paper.) #1 MAX and MIN: z = 6x + 3y Subject to: x + 2y > 0-2x 4y < -8 10x + 2y > 10 0<x<6 -y < 3 y>0 #2 MAX and MIN: z = 0.9x + 0.2y Subject to: 2x + 3y < 12-2x 3y < 6 3x + y < 4 -x y < 0 y<2 x>0 y>0

2 Complete on graph paper: WK #10 #3 NUCLEAR vs. COAL Problem The City of Scorpion Gulch is planning a new power generating station. They can use either a nuclear power plant, a coal-fired plant, or both. It costs $60 per mw each day to dispose of nuclear waste and only $20 per mw each day to dispose of the ashes from the coal. You have been hired to optimize the numbers of megawatts (mw) of each kind of plant built in order to comply with a new Environmental Protection Agency (EPA) rule which states that the cost of disposing of wastes must be a minimum. Identify the variables: Let x = the # of mw generated by the nuclear plant y = D = a. Write the objective function expressing D in terms of x and y. b. Write inequalities expressing the following constraints: i. The maximum amount of nuclear power available is 24 mw. ii. The two plants must be capable of generating at least 25 mw, total. iii. Nuclear fuel costs $100 per mw each day, and coal costs $200 per mw each day. The daily fuel cost can be no more than $4200. iv. Constructing a nuclear plant costs 5 million dollars per mw, and constructing coal-fired plants costs 4 million dollars per mw. The total cost of construction must not exceed 140 million dollars. v. For political reasons, the number of mw in the coal plant must be more than half the number of mw in the nuclear plant. c. On graph paper, graph and label (with numbers) the constraints. (Remember to label the axis!)

3 d. Shade the feasible region. e. Based on these constraints, is it feasible to use no nuclear power? Justify your answer. f. Determine and label (with letters) the corner points. If it is not possible to read the coordinates of a point clearly from the graph, solve the system of equations to determine the exact point of intersection. g. Using a different color pencil or a pen, shade the portion of the feasible region in which the cost of disposal, D, is at most $1200 a day. h. Evaluate the objective function at each of the corner points. i. Determine the number of mw generated by each plant to minimize the disposal cost and determine the minimum disposal cost.

4 Complete on graph paper: Wk #10 #4 Corn Chip Problem Joe s Corn Chips, Inc. is divided into two departments which put out two types of corn chips, Extra Larges and Really Smalls. The company makes a profit of $200 per kilobag of Extra Larges and $150 per kilobag of Really Smalls. (A kilobag is 1000 bags.) Each department has separate regulations concerning the number of bags produced per day. The company s goal is to maximize its profit while following these regulations. Identify the variables: Let x = # of kilobags of Extra Larges y = P = a. Write the objective function expressing P in terms of x and y. b. Write inequalities expressing the following constraints: i. No more than 20 kilobags of Extra Larges and no more than 30 kilobags of Really Smalls can be put out per day. ii. No more than 45 kilobags, total, can be manufactured each day. iii. The number of Extra Larges must be no less than ¾ the number of Really Smalls produced per day. iv. More than 300 hours of labor must be used each day to meet union requirements. It takes 10 hours to make a kilobag of Extra Larges and 15 hours to make a kilobag of Really Smalls. c. On graph paper, graph and label (with numbers) the constraints. (Remember to label the axis!)

5 d. Shade the feasible region. e. Determine and label (with letters) the corner points. If it is not possible to read the coordinates of a point clearly from the graph, solve the system of equations to determine the exact point of intersection. f. Using a different color pencil or a pen, shade the portion of the feasible region in which the daily profit would be at least $6000. g. Evaluate the objective function at each of the corner points. h. How many kilobags of each kind should be produced each day to give the greatest feasible profit? What is this profit?