ACTIVITY 1: TRAVELING SALESMAN PROBLEM

Sandy’s grandmother lives in an old one-story house. There are many connecting doors between the rooms. One day, Sandy wanted to find a route that would take her though each door exactly once. Help Sandy find a route.

TSP SOLUTION

If a room has an odd number of doors, you must either begin in that room or end in that room

This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 6 : 6/25/2012

EXPAND THE CIRCUIT

Steve just added a visit to Columbus, Ohio.

This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 6 : 6/25/2012

THE NEAREST-NEIGHBOR ALGORITHM

I. Choose a node as your starting point

II. From the starting node, travel to the node for which the fare is the cheapest. We call this node the “nearest neighbor”. If there is a tie, choose one arbitrarily.

III. Repeat the process, one node at a time, traveling to nodes that have not yet been visited. Continue this process until all nodes have been visited.

IV. Complete the Hamiltonian circuit by returning to the starting point.

V. Calculate the cost of the circuit.This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 6 : 6/25/2012

THE NEAREST-NEIGHBOR ALGORITHM

This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 6 : 6/25/2012

What is the cost of the route? What is the new route?$531.00; WPCoCSAW

THE NEAREST-NEIGHBOR ALGORITHM

Why does using the nearest-neighbor algorithm make more sense than using the brute-force method in this case?

Will the nearest-neighbor algorithm always give a good route? Why or why not?

This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 6 : 6/25/2012

REPETITIVE NEAREST-NEIGHBOR ALGORITHM

I. Select any node as a starting point. Apply the nearest-neighbor algorithm from that node.

II. Calculate the cost of that circuit.

III.Repeat the process using each of the other nodes as the starting point.

IV.Choose the “best” Hamiltonian circuit.

This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 6 : 6/25/2012

REPETITIVE NEAREST-NEIGHBOR : SOLUTION1.Start at P: PWACSCoP = 74+76+104+65+110+79= $508

2.WACSCoPW; its cost is the same, $508. A circuit goes into and out of each city. Therefore, the starting point within a given circuit has not effect on the total cost.

3.Start at C: CSWPCoAC = 65+105+74+79+121+104 =$548

4.Start at A: AWPCoCSA = 76+74+79+88+65+149 = $531

5.Start at S: SCCoPWAS = 65+88+79+74+76+149 = $531

6.Start at Co: CoPWACSCo = 79+74+76+104+65+110 = $508. This is identical to the circuit found by starting the nearest –neighbor algorithm at Pittsburgh. The cheapest circuit found by starting the algorithm at either Pittsburgh or Columbus translates to WACSCoPW and costs $508.

This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 6 : 6/25/2012

ACTIVITY #2

Instructions:• Using the poster board and

the pins, place the different characteristics under the correct method

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Award # 2010-38422-19963 - DAY 6 : 6/25/2012

SOLUTION

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Award # 2010-38422-19963 - DAY 6 : 6/25/2012

LINEAR P

ROGRAMMING

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Award # 2010-38422-19963 - DAY 7 : 6/26/2012

ALGEBRA

Variable “x” or “y”

It can also be “x1” and “x2”

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Award # 2010-38422-19963 - DAY 7 : 6/26/2012

LINEAR EQUATION

x+2y=16

x=0 y=8x=2 y=7x=6 y=5x=14 y=1x=16 y=0

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Award # 2010-38422-19963 - DAY 7 : 6/26/2012

SERIES OF LINEAR EQUATIONS

x+2y=16

x+ y=12

x+2y-2y=16-2y

x=16-2y

(16-2y)+y=1216-2y+y=1216-y=12

16-y+y=12+y

This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 7 : 6/26/2012

SERIES OF LINEAR EQUATIONS

x+2y=16

x+ y=1216=12+y

16-12=16+y-124=y

x+2(4)=16

x+8=16

x+8-8=16-8x=8

This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

INEQUALITY

x + y<12

x=0 y=0x=0 y=1x=1 y=0

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Award # 2010-38422-19963 - DAY 7 : 6/26/2012

GRAPH LINEAR EQUATION

y-2x=1

x=0 y=1x=1 y=3x=2 y=5

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Award # 2010-38422-19963 - DAY 7 : 6/26/2012

GRAPH INEQUALITY

y<x

y=1 x=0,-1,-2,-3…..y=0 x=-1,-2,-3,-4…..y=5 x=4,3,2,1…..

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Award # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

MCDONALD'S FRANCHISES

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Award # 2010-38422-19963 - DAY 7 : 6/26/2012

MCDONALD'S FRANCHISES

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Award # 2010-38422-19963 - DAY 7 : 6/26/2012

MCDONALD'S FRANCHISES

This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 7 : 6/26/2012

MCDONALD'S FRANCHISES

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Award # 2010-38422-19963 - DAY 7 : 6/26/2012

HIGH STEP SPORTS SHOE

The High Step Shoe Corporation wants to maximize its profits.

Airheads=$10Groundeds=$8.50

This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

OBJECTIVE FUNCTION

The goal is to make the most money, or to maximize profits.

P=10A + 8.5G

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Award # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 = 12000 minutes

3A + 2G < 12000This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 7 : 6/26/2012

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 = 34000 hours

7A + 8G < 34000This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

• 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 # 2010-38422-19963 - DAY 7 : 6/26/2012

ACTIVITY #3: SOLUTION

 

 

This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 7 : 6/26/2012

THE MEAT INDUSTRY IN NEW ZEALAND

This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 7 : 6/26/2012

THE MEAT INDUSTRY IN NEW ZEALAND

This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 7 : 6/26/2012

THE MEAT INDUSTRY IN NEW ZEALAND

This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 insertsThis opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 rolls2x + y= the number of 21-inch rolls

This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 7 : 6/26/2012

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> 2021-inch rolls requirement: 2x + y > 50

This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

THE OPTIMAL SOLUTION

Choose two points in the feasible region and use the objective function to calculate the amount of waste (w1 and w2) produced for the points you selected.

This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012

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” Strips 4” Strips Waste11 0 010 1 19 2 28 3 37 5 06 6 15 7 24 8 33 10 02 11 11 12 20 13 3

This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 7 : 6/26/2012

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” Waste1 1 1.252 1 0.503 0 2.754 0 2.005 0 1.256 0 0.50

This opportunity if funded by the United States Department of Education.

Award # 2010-38422-19963 - DAY 7 : 6/26/2012

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 # 2010-38422-19963 - DAY 7 : 6/26/2012