Linear Programming II

1
Linear Programming II

• HSPM J716

2
Crawling along the simplex

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

3
Learning Objectives

• Learning Objective 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.

4
Learning Objectives

• Learning objective 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.

5
Learning Objectives

• Learning objective 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.

6
Learning Objectives

• Learning objective 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.

7
Learning Objectives

• Learning objective 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.

8
Learning Objectives

• Learning objective 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.

9
Learning Objectives

• Learning objective 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.

10
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.

11
Minimization problem

• Constraints
• 2X 1Y gt 14 nutrient A requirement
• 1X 1Y gt 12 nutrient B requirement
• 1X 3Y gt 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

12
Minimization problem
13
Learning Objectives

• Learning objective 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!

14
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

15
Bug in Mark II (1947)
16
Learning Objectives

• Learning objective 6 Understand shadow prices.
• Shadow Prices are what-its-worth-to-you-for-anoth
  er-unit-of-input prices.
• Other names for shadow prices
• Lagrange Multiplier (if you dont check Assume
  Linear Model)
• Opportunity Cost (other spreadsheets use this)
• Reduced Cost (if you check Assume Linear Model)
• Reduced Gradient (if you dont)

17
Shadow prices for nutrient example
18
Multiple optima ifIso-cost line and constraint
are parallel
19
Mixed constraintsif constraint is D9 gt
E9Shadow prices are below
20
Mixed constraintsif constraint is B2 gt 5its
shadow price is Reduced Cost
21
Primal and Dual

• Primal

• Dual

22
Primal and Dual Solutions

• Primal

• Dual

23
Applications framework

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

24
Scheduling Application

• Staff work 5 days, then get 2 off.

Day Need for Staff
Mon 180
Tue 160
Wed 150
Thu 160
Fri 190
Sat 140
Sun 120
25
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

26
Transportation Application
27
Transportation Application

• Identify the activities
• Each route from distribution center to customer
  is an activity
• Specify the constraints
• Only so much at each distribution center
• Each customers requirement
• Specify the objective function
• Minimize total cost of all product movements