LINEAR PROGRAMMING This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

ALGEBRA Variable “x” or “y” It can also be “x1” and “x2” This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

LINEAR EQUATION x+2y=16 x=0 y=8 x=2 y=7 x=6 y=5 x=14 y=1 x=16 y=0 This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

SERIES OF LINEAR EQUATIONS x+2y=16 x+ y=12 x+2y-2y=16-2y x=16-2y (16-2y)+y= y+y=12 16-y=12 16-y+y=12+y This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

SERIES OF LINEAR EQUATIONS x+2y=16 x+ y=12 16=12+y 16-12=16+y-12 4=y x+2(4)=16 x+8=16 x+8-8=16-8 x=8 This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

SERIES OF LINEAR EQUATIONS x+2y=16 x + y=12 x=8 y=4 (8)+2(4)=16 8+8=16 16=16 (8)+ (4)=12 12=12 This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

INEQUALITY x + y<12 x=0y=0 x=0y=1 x=1y=0 This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

GRAPH LINEAR EQUATION y-2x=1 x=0y=1 x=1y=3 x=2y=5 This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

GRAPH INEQUALITY y<x y=1x=0,-1,-2,-3….. y=0x=-1,-2,-3,-4….. y=5x=4,3,2,1….. This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

LINEAR PROGRAMMING For many manufactures the ability to maximize profits is limited, or constrained by their machine, production line, and assembly line capacity, as well as by the size of their workforce. This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

LINEAR PROGRAMMING Linear programming assists managers in making complex product-mix decisions in the presence of constraints. This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

LINEAR PROGRAMMING The founder of linear programming was American operations researcher George Dantzing. The first problem solved was a minimum-cost diet problem that involved the solution of 9 equations with 77 decision variables. This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

LINEAR PROGRAMMING Dantzing, working with The National Bureau of Standards, supervised the solution of the diet problem, which took 120 person-days using hand-operated desk calculators. This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

MCDONALD'S FRANCHISES This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

MCDONALD'S FRANCHISES This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

MCDONALD'S FRANCHISES This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

MCDONALD'S FRANCHISES This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

HIGH STEP SPORTS SHOE The High Step Shoe Corporation wants to maximize its profits. Airheads=$10 Groundeds=$8.50 This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

HIGH STEP SPORTS SHOE In developing a production plan, managers will often be constrained by limited resources such as number of workers, availability of raw materials, maximum demand for a product, and so forth Quantities that can change (vary) and that managers are able to control are called decision variables. This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

DECISION VARIABLE What are the decision variables that the managers at High Step Sports Shoe must consider? The main decision variable to consider are the number of Airheads (A) and Groundeds (G) to produce each week This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

OBJECTIVE FUNCTION The goal is to make the most money, or to maximize profits. P=10A + 8.5G This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

CONSTRAINTS The steps in manufacturing the shoes include cutting the materials on a machine and having workers assemble the pieces into a pair of shoes. This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

CONSTRAINTS There are 6 machines that are used to cut materials. Each pair of Airheads requires 3 minutes of cutting time, while each pair of Groundeds requires 2 minutes. This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

CONSTRAINTS There are 850 workers who assemble the shoes. It takes a single worker 7 hours to assemble a pair of Airheads and 8 hours to assemble a pair of Groundeds. This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

CONSTRAINTS The assembly plan operates 40-hours per 5-day work week. Also, each cutting machine is operated only 50 minutes per hour to allow for routine maintenance. This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

HIGH STEP SPORTS SHOE How many minutes of work can 6 machines do in 40-hour work week? Now use the decision variables to write an inequality to represent a constraint that is based on the limited time the cutting machines operate each week. 6 X 50 X 40 = minutes 3A + 2G < This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

HIGH STEP SPORTS SHOE How many hours of work can 850 assembly workers do in a week? Write an inequality to model a constraint based upon the limited number of worker hours available for shoe assembly each week. 850 X 40 = hours 7A + 8G < This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

HIGH STEP SPORTS SHOE Could the number of pairs of shoes of each style that are produced each week ever be negative? Could it be zero? Why or why not? Then, are there more constraints? Which? This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

HIGH STEP SPORTS SHOE Graphing the system of inequalities shows the feasible region of the graph. This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

HIGH STEP SPORTS SHOE What does each pair of coordinates represent? Which of the six intersection points satisfy all the constraints? What does the shaded region of the graph represent? This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

HIGH STEP SPORTS SHOE Of all the feasible points, one will give the maximum profit. The process of determining this best solution is called optimization, and the solution itself is called optimal solution. This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

HIGH STEP SPORTS SHOE How many feasible points are there? Pick three points in the interior of the feasible region. List the corresponding values of A and G in a table like this one, and evaluate the profit P for each point selected. This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

HIGH STEP SPORTS SHOE Compare your answers with those of other students, and see which point has the most profit. Now test each feasible corner point. Enter these values of A, G, and P in another table This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

HIGH STEP SPORTS SHOE Which point from either table yields the largest profit? What do the coordinates of this point represent? This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

HIGH STEP SPORTS SHOE This problem is an example of the corner principle. This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

Model production in a furniture facture Suppose a company produces only tables and chairs A table is made of 2 large and 2 small pieces A chair is made of 1 large and 2 small pieces Profit: Table - $16 ; Chair - $10 Determine product mix that maximizes the company’s profits using the available resources ACTIVITY #3: LEGO FURNITURE This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

ACTIVITY #3: SOLUTION This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

THE MEAT INDUSTRY IN NEW ZEALAND This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

THE MEAT INDUSTRY IN NEW ZEALAND This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

THE MEAT INDUSTRY IN NEW ZEALAND This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

TRIM LOSS AT THE “CUTTING TIMES” Products such as paper, sheet metal, and photographic film are manufactured in very wide rolls that need to be cut into sets of smaller strips to meet specific product needs. This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

TRIM LOSS AT THE “CUTTING TIMES” The Cutting Times is a small local newspaper with its own printing press. The paper receives large 48-inch-wide rolls of newsprint, which must to be cut into 25-inch-wide rolls for the ordinary pages in the paper and 21-inch-wide rolls for smaller inserts This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

TRIM LOSS AT THE “CUTTING TIMES” Lets draw some models of each possible cutting pattern. This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

TRIM LOSS AT THE “CUTTING TIMES” For the Sunday edition of the Times right before Thanksgiving, the Times needs twenty 25-inch rolls for the ordinary part of the edition and fifty 21-inch rolls for the extra advertising inserts. This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

TRIM LOSS AT THE “CUTTING TIMES” When the times decides how many of each pattern to cut, what should be considered? How can the newspaper control the amount of waste? Why would the newspaper want to control the amount of waste? This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

DECISION VARIABLES Let x= the number of large rolls cut into two 21-inch rolls Let y= the number of large rolls cut into one 25-inch and one 21-inch roll This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

DECISION VARIABLES Now write a function representing the amount of waste, w, in terms of the decision variables x and y. Min w= 6x + 2y This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

CONSTRAINTS When a large roll is cut into two 21-inch rolls, how many 25-inch rolls and how many 21-inch rolls are produced? When a large roll is cut into one 25-inch roll and one 21-inch roll, how many 25- inch roll and how many 21-inch roll are produced? This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

CONSTRAINTS If 5 rolls are cut using the first pattern and 10 rolls are cut using the second pattern, how many 21-inch rolls and how many 25-inch rolls are produced? Ten 25-inch rolls and twenty 21-inch rolls This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

CONSTRAINTS If x rolls are cut using the first pattern and y rolls are cut using the second pattern. How many 25-inch rolls and how many 21-inch rolls will be produced? y= the number of 25-inch rolls 2x + y= the number of 21-inch rolls This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

CONSTRAINTS Now lets use the cut paper requirements to write two inequalities based on the number of 25-inch and 21-inch rolls that must be produced, in terms of x and y. 25-inch rolls requirement: y> inch rolls requirement: 2x + y > 50 This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

THE FEASIBLE REGION One of the decision variables is restricting the other decision variable, which one and why? y, and because the 25-inch rolls are produced using only one of the two patterns This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

THE OPTIMAL SOLUTION Choose two points in the feasible region and use the objective function to calculate the amount of waste (w 1 and w 2 ) produced for the points you selected. This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

THE OPTIMAL SOLUTION At what point do we have the minimum amount of waste? What does this point mean in terms of the original problem? This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

ACTIVITY #4: CUTTING STOCK Solve the given problem in teams of 4 One member of each team will be chosen to solve the problem on the board This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

ACTIVITY #4: SOLUTION A health care corporation wants to cut large 55-inch wide beams of gauze into 5- inch and 4-inch widths.  List all of the possible cutting patterns. 5” Strips4” StripsWaste This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

ACTIVITY #4: SOLUTION A health care corporation wants to cut large 55-inch wide beams of gauze into 5- inch and 4-inch widths.  What variable names could be assigned to represent the number of beams of gauze cut using each of the patterns? Use the first 12 letters of the alphabet. A better solution would be the use of subscripts. i.e. x1,x2,x3…x10,x11,x12.  The company also wants to cut some of the 5-inch rolls of gauze into 3/4-inch rolls to be used in the manufacture of adhesive strips and 3-inch rolls to be used in the manufacture of square gauze pads. The company needs 120 thousand 3/4-inch rolls and ten thousand 3-inch rolls. How should the 5-inch rolls be cut to minimize waste?  Both patterns minimize the waste, however, only the first pattern can give both cuts. ¾”(.75)3”Waste This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13

ACTIVITY #4: SOLUTION Decision variables: Let x = the number of 5” strips cut into 6 ¾” rolls Let y = the number of 5” strips cut into 2 ¾” and 1 3” roll Objective Function: Minimize w = 6x + 2y Constraints: subject to 2x + y ≥ 120,000 y ≥ 10,000 This opportunity if funded by the United States Department of Education. Award # DAY 7 : 6/26/2012 This opportunity is funded by the United States Department of Education. Award # WEEK 3 – 6/24/13 – 6/28/13