Linear Programming II

HSPM J716

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2x+6y=36

5x+3y=30

8x+2y=40

400x+500y=3700

Coarse Stones (tons)

Fine

Sto

nes (

tons

)

Optimum

Isoprofit Line

Crawling along the simplex

• The simplex method moves from one corner to the next until the amount to be maximized stops rising.

Learning Objectives Competencies for linear programming

• Competency 1: Recognize problems that linear programming can handle:

• Linear programming lets you optimize an objective function subject to some constraints. The objective function and constraints are all linear.

Linear Programming Competencies

• Competency 2: Know the elements of a linear programming problem:

• an objective function that shows the cost or profit depending on what choices you make,

• constraint inequalities that show the limits of what you can do, and

• non-negativity restrictions, because you cannot turn outputs back into inputs.

Linear Programming Competencies

• Competency 2: Know the elements of a linear programming problem:

• an objective function that shows the cost or profit depending on what choices you make,

• constraint inequalities that show the limits of what you can do, and

• non-negativity restrictions, because you cannot turn outputs back into inputs.

Linear Programming Competencies

• Competency 3: Understand the principles that the computer uses to solve a linear programming problem.

• The computer uses the simplex method to systematically move along the edges of the feasible area (the simplex). It goes from one corner to the next and stops when the objective function stops getting better.

Linear Programming Competencies

• Competency 3: Understand the principles that the computer uses to solve a linear programming problem.

• The computer uses the simplex method to systematically move along the edges of the feasible area (the simplex). It goes from one corner to the next and stops when the objective function stops getting better.

Linear Programming Competencies

• Competency 4a: What linear programming problems have no solution?

• Those that have no feasible area. This means it's impossible to satisfy all the constraints at once.

Linear Programming Competencies

• Competency 4b: What difference does linearity make?

• If the constraints are not linear, the feasible area has curved edges. The simplex method doesn't work because you can't be sure that the solution is at a corner. The solution may be in the middle of a curved edge. If the objective function is not linear, the solution may not even be on an edge. It may be in the interior of the feasible area.

Minimization problem• Animals need:– 14 units of nutrient A,– 12 units of nutrient B, and– 18 units of nutrient C.

• A bag of X has 2 units of A, 1 unit of B, and 1 unit of C.

• A bag of Y has 1 unit of A, 1 unit of B, and 3 units of C.

• A bag of X costs $2. A bag of Y costs $4.

Minimization problem

• Constraints:• 2X + 1Y >= 14 nutrient A requirement• 1X + 1Y >= 12 nutrient B requirement• 1X + 3Y >= 18 nutrient C requirement– Read vertically to see how much of each nutrient

is in each grain.• Cost = 2X + 4Y – objective function to be minimized

Minimization problem

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Series1Series3Series5Iso-cost

Linear Programming Competencies

• Competency 5: Be able to solve small linear programming problems yourself.

• The tricky part is setting up the objective function and the constraints. Write them on paper. Then set up your spreadsheet and solve.

• Then comes the other tricky part – coaxing Excel to work!

Rear Admiral Grace Hopper

• 1906-1992• Navy Reserve Lt.(J.G.)

1943 (age 36)• 3rd programmer of

Mark I computer at Harvard

• Invented the compiler• “Mother” of COBOL• First female Admiral

Bug in Mark II (1947)

Linear Programming Competencies

• Competency 6: Understand shadow prices.• Shadow Prices are what-it’s-worth-to-you-for-

another-unit-of-input prices.• Other names for shadow prices:

• Lagrange Multiplier (if you don’t check Assume Linear Model)

• Opportunity Cost (other spreadsheets use this)• Reduced Cost (if you check Assume Linear Model)• Reduced Gradient (if you don’t)

Shadow prices for nutrient example

Multiple optima ifIso-cost line and constraint are parallel

Mixed constraintsif constraint is D9 >= E9

Shadow prices are below

Mixed constraintsif constraint is B2 >= 5

its shadow price is “Reduced Cost”

Primal and Dual

Primal Dual

Primal and Dual SolutionsPrimal Dual

Applications -- steps

• Identify the activities• Specify the constraints• Specify the objective function

• Solve! Get Sensitivity Report• Get results from spreadsheet• Get shadow prices from Sensitivity Report

Scheduling Application

Day Need for StaffMon 180Tue 160Wed 150Thu 160Fri 190Sat 140Sun 120

• Staff work 5 days straight, then get 2 off.

• Objective is to minimize the total cost.– Which is roughly

proportional to the number of hires.

Scheduling Application

• Identify the activities– Each hiring schedule is an activity

• Specify the constraints– How many people you need at different times

• Specify the objective function– Minimize the total number of people hired

Transportation Application

• Identify the activities– Each route from a distribution center to a customer

is an activity• Specify the constraints– Each distribution center starts with a limited amount– Each customer has a requirement

• Specify the objective function– Minimize total cost of all product movements