1/17: DSCB Getting Started, Linear Programming Administrative Issues –Syllabus –Calendar –Get usernames, addresses, majors Linear Programming Small Groups Homework

Administrative Issues Syllabus: –Available at –Midterm exam: Thursday, February 21, due 2/23 –Final exam: Thursday, April 25, 6:30 – 9:00 p.m. Calendar: –Available at –PowerPoint viewer available at Grade page: –Available at

Administrative Issues Grade page: –Available at –Example at I need from you a sheet of paper with: –Your name (as you like to be called) –Your username for the grade page (don’t be obvious) –Your major, or what major(s) you are considering

Linear Programming What is it? –Synthesizing a problem in words into a series of equations. –A type of modeling tool –Optimizing a linear function subject to several constraints, expressed as inequalities.

LP - 4 Characteristics Objective Function –Something to be minimized or maximized –Usually maximizing profit, sometimes minimizing costs Constraints –Limitations on pursuit of the objective Alternative Courses of Action –Must have more than one possible course of action, or there is no need for LP Linear Equations (or inequalities)

EX: Toy Company A toy company makes 3 types of toys: wooden trucks, wooden dolls, and chess sets. Each requires some amount of hand labor, machine time, and wood. A wooden truck needs 10 min. hand time, 3 min. machine time, and 15 linear inches of wood. A wooden doll requires 8 min. hand time, 10 min. machine time, and 11 linear inches of wood. A chess set takes 3 min. hand time, 20 min. machine time, and 31 linear inches of wood. Per day, there are 8 hours of hand labor time, 8 hours on the machine, and 1000 linear feet of wood available. The profit margins for the truck, doll, and chess set are $7, $5, and $12, respectively.

Toy Company Formulate a linear program set to maximize the company's profit.

Terminology Z : variable to be optimized. x 1, x 2, x 3,… : decision variables. So we write Max Z ( profit ) = (some combo of x 1...x X ) S. T. ("subject to"): (the constraints)

Toy Company What are we supposed to maximize? What factors play a part in that? What constraints are there to the profit?

A toy company makes 3 types of toys: wooden trucks, wooden dolls, and chess sets. Each requires some amount of hand labor, machine time, and wood. A wooden truck needs 10 min. hand time, 3 min. machine time, and 15 linear inches of wood. A wooden doll requires 8 min. hand time, 10 min. machine time, and 11 linear inches of wood. A chess set takes 3 min. hand time, 20 min. machine time, and 31 linear inches of wood. Per day, there are 8 hours of hand labor time, 8 hours on the machine, and 1000 linear feet of wood available. The profit margins for the truck, doll, and chess set are $7, $5, and $12, respectively. Maximize the company’s profit.

A toy company makes 3 types of toys: wooden trucks, wooden dolls, and chess sets. Each requires some amount of hand labor, machine time, and wood. A wooden truck needs 10 min. hand time, 3 min. machine time, and 15 linear inches of wood. A wooden doll requires 8 min. hand time, 10 min. machine time, and 11 linear inches of wood. A chess set takes 3 min. hand time, 20 min. machine time, and 31 linear inches of wood. Per day, there are 8 hours of hand labor time, 8 hours on the machine, and 1000 linear feet of wood available. The profit margins for the truck, doll, and chess set are $7, $5, and $12, respectively. Maximize the company’s profit.

A toy company makes 3 types of toys: wooden trucks, wooden dolls, and chess sets. Each requires some amount of hand labor, machine time, and wood. A wooden truck needs 10 min. hand time, 3 min. machine time, and 15 linear inches of wood. A wooden doll requires 8 min. hand time, 10 min. machine time, and 11 linear inches of wood. A chess set takes 3 min. hand time, 20 min. machine time, and 31 linear inches of wood. Per day, there are 8 hours of hand labor time, 8 hours on the machine, and 1000 linear feet of wood available. The profit margins for the truck, doll, and chess set are $7, $5, and $12, respectively. Maximize the company’s profit.

A toy company makes 3 types of toys: wooden trucks, wooden dolls, and chess sets. Each requires some amount of hand labor, machine time, and wood. A wooden truck needs 10 min. hand time, 3 min. machine time, and 15 linear inches of wood. A wooden doll requires 8 min. hand time, 10 min. machine time, and 11 linear inches of wood. A chess set takes 3 min. hand time, 20 min. machine time, and 31 linear inches of wood. Per day, there are 8 hrs. of hand labor time, 8 hrs. machine time, and 1000 linear feet of wood available. The profit margins for the truck, doll, and chess set are $7, $5, and $12, respectively. Maximize the company’s profit.

Toy Company What are we supposed to maximize? –THE PROFIT What factors play a part in that? –PROFIT FROM TRUCKS, DOLLS, and CHESS SETS What constraints are there to the profit? –HAND TIME, MACHINE TIME, and WOOD

Toy Company Let x 1 = toy trucks, x 2 = dolls, x 3 = chess sets.

Toy Company Let x 1 = toy trucks, w/ a $7 profit each x 2 = dolls, w/ a $5 profit each x 3 = chess sets w/ a $12 profit each

Toy Company Let x 1 = toy trucks, w/ a $7 profit each x 2 = dolls, w/ a $5 profit each x 3 = chess sets w/ a $12 profit each So Max Z (profit) = 7 x x x 3

Toy Company - constraints Hand Time: total of 8 hours. -- or 480 min. Truck - 10 min. Doll - 8 min. Chess Set - 3 min.

Toy Company - constraints Hand Time: total of 8 hours. -- or 480 min. Truck - 10 min. Doll - 8 min. Chess Set - 3 min. so 10 x x x 3 <= 480

Toy Company - constraints Machine Time: total of 8 hrs. -- or 480 min. Truck - 3 min. Doll - 10 min. Chess Set - 20 min.

Toy Company - constraints Machine Time: total of 8 hrs. -- or 480 min. Truck - 3 min. Doll - 10 min. Chess Set - 20 min. so 3 x x x 3 <= 480

Toy Company - constraints Wood: total of 1000 ft. -- or 12,000 in. Truck - 15 in. Doll - 11 in. Chess Set - 31 in.

Toy Company - constraints Wood: total of 1000 ft. -- or 12,000 in. Truck - 15 in. Doll - 11 in. Chess Set - 31 in. so 15 x x x 3 <= 12000

Toy Company - constraints Other constraints: Integers:x 1, x 2, x 3 must be integers. Positive: x 1, x 2, x 3 >= 0

Toy Company - total LP Max Z (profit) = 7 x x x 3 S. T.: 10 x x x 3 <= x x x 3 <= x x x 3 <= x 1, x 2, x 3 >= 0 x 1, x 2, x 3 must be integers.

EX: Camping Trip. PCF $/lb beef jerky dried potatoes granola mix NutriGrain bars Must have 30 g. protein, 60 g. carbohydrates, and 15 g. of fat. Minimize the cost.

Graphical Solutions for LP Sparky Electronics 2 products, WalkFM & WristTV profit: $7 $5 machine time 4 3 assembly time 2 1 Total machine time 240 Total assembly time 100

LP - Graphical Solution Limitation to the method: only TWO decision variables can exist.

LP - Graphical Solution Maximize ? S. T. :? ?

LP - Graphical Solution Maximize Z ( profit ) = 7 x x 2 S. T. :? ?

LP - Graphical Solution Maximize Z ( profit ) = 7 x x 2 S. T. :4 x x 2 <= 240

LP - Graphical Solution Maximize Z ( profit ) = 7 x x 2 S. T. :4 x x 2 <= x x 2 <= 100

LP - Graphical Solution Maximize Z ( profit ) = 7 x x 2 S. T. :4 x x 2 <= x x 2 <= 100 x 1, x 2 >= 0

LP - Graphical Solution 4 x x 2 = x x 2 = 100 x 1. x 2 >= 0

LP - Graphical Solution 4 x x 2 = 240 x1x1 x2x2

LP - Graphical Solution 4 x x 2 = x x 2 = 100 x1x1 x2x2

LP - Graphical Solution 4 x x 2 = x x 2 = 100 Feasible Solution Region x1x1 x2x2

LP - Graphical Solution 4 x x 2 = x x 2 = 100 x1x1 x2x2

LP - Graphical Solution 4 x x 2 = x x 2 = 100 Max Z = 7 x x 2 Z = $400 Z = $410 Z = $350 x1x1 x2x2

LP - Graphical Solution 4 x x 2 = x x 2 = 100 Max Z = 7 x x 2 Z = $400 Z = $410 Z = $350

Small Groups: Kelson Sporting Gds. Kelson Co. makes two types of baseball gloves: a standard model and a catcher’s mitt. There is 900 hours of production time available in the cutting department, 300 hours in finishing, & 100 hours in packaging. Write the LP model for this problem. Use the graphical solution method to find the optimal solution. What is the total profit possible? ModelCuttingFinishingPacking$ profit Regular11/21/8$5 Catcher’s 1 ½ 1/3 ¼ $8

Homework ch. 1 #11, 18 ch. 2 #11, 21 Reminder: should be legible, labeled with your name and the date, and make sense.