Lecture 2: Basic LP Formulation

1. Lecture 2: Basic LP Formulation McCarl and Spreen Text Ch. 2 http://agecon2.tamu.edu/people/facult y/mccarl-bruce/books.htm

2. Elements of an LP Problem • Decision Variables, • Level denotes amount undertaken of the respective unknowns • “n” decision variables where j = 1, 2, …, n • Objective Function, F(X) • Linear in form • Total objective value, Z • is the contribution of each to the objective function.

3. Elements of an LP Problem • Constraints • “m” different constraints • ith constraint: • i= (1,…,m) or • is the upper limit or right hand side imposed by the ith constraint • is the per unit use of the items in the ith constraint by a unit of • Exogenous Parameters • , , and • These are the “data” of an LP problem

4. Linear Programming Problem The LP problem is to choose so as to … … … … … …

5. Linear Programming Problem Matrix Notation Max CTX Subject to AX ≤ b X ≥ 0 X: Vector of decision variables C: Vector of decision variable coefficients A: Matrix of coefficients giving constraint usage by each variable b: Vector of right hand sides of the constraints

6. Elements of an LP Solution • Optimal value for the objective function • Optimal values for the decision variables • Shadow price or Lagrange Multiplier • Estimates of the change in the objective function value induced by changing the ith RHS parameter by one unit  Marginal value of the change in the ith RHS parameter (Marginal Value Product of RHS) • Reduced Cost • Reduction in the objective function value when one unit of ith decision variable that is not in the solution is forced into the solution

7. Basic LP Application: Joe’s Van Conversion Shop • Joe wishes to maximize profits • His shop converts plain vans into custom vans • Custom Vans are produced as two grades • Fine Vans • Fancy Vans • Joe must decide how many of each van type to convert • Decision Variables: the number to convert by van type • Xfancy , Xfine

8. Basic LP Application: Joe’s Van Conversion Shop • Both types require a $25,000 plain van • Fancy vans: conversion cost $10,000 • sell for $37,000 • profit margin $2,000 • Fine vans: conversion cost $6,000 • sell for $32,700 • profit margin $1,700 • Mathematically: • Maximize Z = 2000 Xfancy + 1700 Xfine

9. Basic LP Application: Joe’s Van Conversion Shop • Labor is a limited resource for Joe • Joe’s shop employs 7 people to convert vans • Joe’s shop operates 8 hours per day, 5 days a week • Therefore, Joe has at most 280 labor hours available a week • A Fancy van takes 25 labor hours • A Fine van takes 20 labor hours • 25 Xfancy + 20 Xfine≤ 280

10. Basic LP Application: Joe’s Van Conversion Shop • Capacity is another limiting resource • Joe’s shop can work on no more than 12 vans in a week • Xfancy + Xfine≤ 12 • Non-Negativity • Joe can only convert a non-negative number of vans • Xfancy , X fine≥ 0

11. Basic LP Application: Joe’s Van Conversion Shop In mathematical formulation, Joe’s LP problem is: Maximize 2000 Xfancy + 1700Xfine s.t.Xfancy+ Xfine≤ 12 25 Xfancy + 20 Xfine≤280 Xfancy , Xfine≥ 0

12. Basic LP Application: Joe’s Van Conversion Shop • Such a problem can be solved a number of ways • The answer is • Z=profits= $22,800 • Xfancy = 8 • Xfine = 4 • Have we satisfied our 3 constraints • Xfancy + Xfine = 8 + 4 = 12 ≤ 12 • 25 Xfancy + 20 Xfine = 25 *8 + 20*4 = 280 ≤ 280 • Xfancy , Xfine≥ 0 • Uses all of our labor and capacity

13. Basic LP Application: Joe’s Van Conversion Shop • Shadow prices (aka Lagrange Multiplier) • Capacity value = $500 per van • Labor value = $60 per hour • Definition of shadow price: • The change in the objective function value induced by changing the ith RHS parameter by one unit • Tells us how much the ith resource is worth in its current application • Binding vs. Non-Binding Constraints • Why is this important?

14. Assumptions of LP • Attributes of the Model • Objective Function Appropriateness • Decision Variable Appropriateness • Constraint Appropriateness • Mathematical Relationship within the Model • Proportionality • Additivity • Divisibility • Certainty

15. Assumptions: Attributes of the Model • Objective function Appropriateness • Sole criteria for choosing among the feasible values of the decision variables • Decision Variable Appropriateness • The decision variables (DV) are fully manipulatable within the feasible region and under the control of the decision maker • All appropriate DVs have been included in the model

16. Assumptions: Attributes of the Model • Constraint Appropriateness • Constraints fully identify the bounds placed on the decision variables • Resource availability, technology, etc. • Resources used and/or supplied within any single constraint are homogenous items • Constraints do not improperly eliminate admissible values of the decision variables • Constraints are inviolate

17. Assumptions: Mathematical Relationships • Proportionality • The contribution per unit of each variable in the objective function and constraints is assumed a constant times the variable level • No economics or diseconomies of scale • Example: cjis the return per unit of Xj • If Xj=1, the net return from Xj is cj • If Xj=100, the net return from Xj is 100*cj • Also applies to resource usage (aij) in a constraint • Potential Problems

18. Assumptions: Mathematical Relationships • Additivity • Contributions of variables to an equation are additive • The objective function value is the sum of the individual contributions of each variable • Total resource use is the sum of the resource use of each variable across all variables • Examples from Joe’s Van shop: • Objective function: 2000 Xfancy + 1700Xfine • Labor constraint: 25 Xfancy + 20 Xfine • Rules out the possibility of interaction terms in the objective function or the constraints

19. Assumptions: Mathematical Relationships • Divisibility • Decision variables can take on any non-neg. value • Decision variables are continuous • Assumption is violated when non-integer values make little sense • Decision to construct a building • Use Integer Programming instead • Certainty • All parameters (cj, bi, and aij) are known constants • This assumption implies that LP is a “non-stochastic” model • Potential problems  Sensitivity analysis

20. What We Know So Far… Matrix and mathematical notations Fundamental uses for LP Basic model formulation Key elements of an LP problem 7 Assumptions of LP Shadow prices and reduced costs

21. Where We are Going Next • Applying what we know to setting up and solving an LP problem • In Excel (covered in first lab) • Setting up the tableau (Overheads 3)