Introduction to Quantitative Business Methods (Do I REALLY Have to Know This Stuff?)

Management Science …is the study and development of techniques for the formulation and analysis of management and related business problems. Operations research models are often helpful in this process.

Operations Research …is the application of techniques developed in mathematics, statistics, engineering and the physical sciences to the solution of problems in business, government, industry, economics and the social sciences.

Quantitative Methods …employ mathematical models to reach a wide variety of business decisions. They give modern managers a competitive edge Managers do not need to have great mathematical skills Familiarity allows one to: Ask the right questions Recognize when additional analysis is necessary Evaluate potential solutions Make informed decisions

Introduction to Linear Programming

Mathematical Programming …is the development of modeling and solution procedures which employ mathematical techniques to optimize the goals and objectives of the decision- maker. Programming problems determine the optimal allocation of scarce resources to meet certain objectives.

Linear Programming Problems …are mathematical programming problems where all of the relationships amongst the variables are linear.

Components of a LP Formulation 1) Decision Variables 2) Objective Function 3) Constraints 4) Non-negativity Conditions

Decision Variables …represent unknown quantities. The solution for these terms are what we would like to optimize.

Objective Function …states the goal of the decision-maker. There are two types of objectives: Maximization, or Minimization

Constraints …put limitations on the possible solutions of the problem. The availability of scarce resources may be expressed as equations or inequalities which rule out certain combinations of variable values as feasible solutions.

Non-negativity Conditions …are special constraints which require all variables to be either zero or positive.

Special Terms 1) Parameters 2) RHS 3) Objective Coefficients 4) Technological Coefficients 5) Canonical Form 6) Standard Form

Parameters …are the constant terms. These are neither variables, nor their coefficients. In canonical form the parameters always appear on the right-hand side of the constraints.

Right-Hand Side (RHS) …are the numbers (parameters) located on the right-hand side of the constraints. In a production problem these parameters typically indicate the amount, or quantity, of resources available. In the conventional literature these are known as the “b”s.

Objective Coefficients …are the coefficients of the variables in the objective function. In a production problem these typically represent unit profit or unit cost. In the conventional literature these are known as the “c”s.

Technological Coefficients …also known as “exchange coefficients,” these are the coefficients of the variables in the constraints. In a production problem these typically represent the unit resource requirements. In the conventional literature these are known as the “a”s.

Canonical Form …refers to an LP problem with an objective function, all of the variables are non-negative and where all of the variables and their coefficients are on the left-hand sides of the constraints, and all of the parameters are on the right-hand sides of the constraints.

Standard Form …refers to an LP problem in canonical form. In addition, all of the constraints are expressed as equalities and every variable is represented in the same order of sequence on every line of the linear programming problem.

Redwood Furniture Company ResourceUnit RequirementsAmount Available TableChair Wood Labor Unit Profit 68

Graphical LP Solution Procedure 1) Formulate the LP problem 2) Plot the constraints on a graph 3) Identify the feasible solution region 4) Plot two objective function lines 5) Determine the direction of improvement 6) Find the most attractive corner 7) Determine the coordinates of the MAC 8) Find the value of the objective function

Redwood Furniture Problem X T = 4 tables X C = 9 chairs P = 6(4) + 8(9) = 96 dollars

Exercises: Use the graphical solution procedure to determine the optimal solutions for the following linear programming problems. For each make sure to show the feasible solution region, the direction of improvement, the most attractive corner, and solve for the decision variables and the objective function.

Problem #1 MIN Z = 3A – 2B s.t.5A + 5B > 25 3A < 30 6B < 18 3A + 9B < 36 where: A, B > 0 A = 2 B = 3 Z = 0

Problem #2 MAX Z = 6X – 3Y s.t.2X + 2Y < 20 6X > 12 4Y > 4 4X + Y < 20 where: X, Y > 0 X = 19/4 Y = 1 Z = 51/2

Problem #3 MAX Z = 5S – 5T s.t.3T < 18 4S + 4T < 40 2S < 14 6S - 15T < 30 3S > 9 where: S, T > 0 S = 7 T = 4/5 Z = 31

Special LP Cases For each of the following problems use the graphical solution procedure to try to determine the optimal solutions. You may find it difficult to proceed in some cases, and in all cases the results are interesting. In each case proceed as far as you can.

Special Case #1 MAX Z = 4X 1 + 3X 2 s.t.5X 1 + 5X 2 < 25 X 2 > 6 X 2 < 8 where: X 1, X 2 > 0 INFEASIBLE Problem

Special Case #2 MAX Z = 4X 1 + 3X 2 s.t.5X 1 + 5X 2 > 25 X 2 < 6 X 2 < 8 where: X 1, X 2 > 0 UNBOUNDED Problem Redundant Constraint

Special Case #3 MAX Z = 4X 1 + 4X 2 s.t.5X 1 + 5X 2 < 25 X 2 < 4 X 1 < 3 where: X 1, X 2 > 0 Multiple Optimal Solutions X 1 = 3 X 2 = 2 Z = 20 X 1 = 1 X 2 = 4 Z = 20