1. MCCARL AND SPREEN TEXT CH. 2 HTTP://AGECON2.TAMU.EDU/PEOPLE/FACUL T Y/MCCARL-BRUCE/BOOKS.HTM Lecture 2: Basic LP Formulation

2. Elements of an LP Problem

4. Linear Programming Problem

5. Matrix Notation Max C T X 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

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  X fancy, X fine

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 X fancy + 1700 X fine

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 X fancy + 20 X fine ≤ 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  X fancy + X fine ≤ 12 Non-Negativity  Joe can only convert a non-negative number of vans  X fancy, X fine ≥ 0

11. Basic LP Application: Joe’s Van Conversion Shop In mathematical formulation, Joe’s LP problem is: Maximize 2000 X fancy + 1700X fine s.t. X fancy + X fine ≤ 12 25 X fancy + 20 X fine ≤280 X fancy, X fine ≥ 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  X fancy = 8  X fine = 4 Have we satisfied our 3 constraints  X fancy + X fine = 8 + 4 = 12 ≤ 12  25 X fancy + 20 X fine = 25 *8 + 20*4 = 280 ≤ 280  X fancy, X fine ≥ 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 i th RHS parameter by one unit  Tells us how much the i th 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: c j is the return per unit of X j  If X j =1, the net return from X j is c j  If X j =100, the net return from X j is 100*c j  Also applies to resource usage (a ij ) 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 X fancy + 1700X fine  Labor constraint: 25 X fancy + 20 X fine  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 (c j, b i, and a ij ) 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)