Chapter 4: Modeling and Analysis Chapter 4 in “DECISION SUPPORT AND BUSINESS INTELLIGENCE SYSTEMS” Chapter 17 part5 in “OPERATION MANAGEMENT. HEIZER, RENDER”

The Structure of MSS Mathematical Models Include financial and engineering models Link decision variables, uncontrollable variables, intermediate variables, and result variables together. The modeling process involves identifying the variables and the relationships among them. Solving a model determines the values of these variables.

The Components of Mathematical Models

The Components of Mathematical Models (cont.) Result (Outcome) Variables They indicate how well the system performs or attains its goal. They considered dependent variables Result variables are dependent on decision variables and uncontrollable variables. Decision variables: Decision variables describe alternative choices. The decision maker controls the decision variables.

The Components of Mathematical Models (cont.) Uncontrollable variables: They affect the result variables but are outside decision-maker’s control. These factors can be fixed factors, called parameters. They can vary, called variables Intermediate Result variables: intermediate outcomes produce intermediate result variables.

Examples of Components of Models Ex: Financial investment Result variable: total profit Decision variable: investment amounts. Uncontrollable variables: competition, tax Intermediate Result variable: cost, income

The Structure of MSS Mathematical Models The components are linked together by mathematical expression: P=R-C P =profit, R= revenue (income) , C=cost

Mathematical Programming Optimization A family of tools designed to help solve managerial problems in which the decision maker must allocate scarce resources among competing activities to optimize a measurable goal  Linear programming (LP) LP is best-known technique in the mathematical programming family. It is used to determine a level of operational activity in order to achieve an objective, subject to restrictions called constraints It used to find the optimal solution of resource allocation problems. All the relationships among the variables in this type of model are linear

Common Elements to LP Decision variables Should completely describe the decisions to be made by the decision maker Objective Function (OF) A linear mathematical function that relates the decision variables to the goal, measures goal attainment, and to be optimized Most frequent objective of business firms is to maximize profit Most frequent objective of individual operational units (such as a production or packaging department) is to minimize cost Objective function coefficients unit profit or cost coefficients indicating the contribution to the objective of one unit of decision variable

Common Elements to LP Constraints Restrictions on resources such as time, money, labor, etc. Expressed in a form of linear inequalities There are three types of constraints Upper limits where the amount used is ≤ the amount of a resource Lower limits where the amount used is ≥ the amount of the resource Equalities where the amount used is = the amount of the resource Capacities Upper or lower limits of constraints

LP Assumptions Objective function and constraints must be linear Certainty All coefficients are known with certainty We are dealing with a deterministic world The return of any allocation is independent of the others

Examples of LP Problems and Applications SCM: transportation models Product Mix Scheduling (Production and Labor)

Formulating LP Problems The product-mix problem at Shader Electronics The Shader Electronics Company produces two products, (1) the Shader Walkman, and (2) the Shader Watch-TV. Each Walkman takes four hours of electronic work and two hours in the assembly shop. Each Watch-TV requires three hours of electronics and one hour in assembly. During the current production period, 240 hours of electronic time and 100 hours of assembly department time are available. Each Walkman sold yields a profit of $7; each Watch-TV produced may be sold for a $5 profit. How many of each should be produced?

Formulating LP Problems (con.) Decision variables: Let X1 = ---------------- Let X2 = ----------------   2. Subject to: First Constraint: ---------------- Second Constraint: ---------------- Third Constraint: ---------------- Objective function coefficients: For x1 ------------- For x2--------------  4. Objective Function: Maximize Profit = ----------------

Excel Solver Solution Hours Required to Produce 1 Unit Walkman Watch-TVs Available Hours Department (X1) (X2) This Week Hours Required to Produce 1 Unit Electronic 4 3 240 Assembly 2 1 100 Profit per unit $7 $5 Table B.2 Decision Variables: X1 = number of Walkmans to be produced X2 = number of Watch-TVs to be produced

Simplex Solution Before the simplex algorithm can be used to solve an LP, the LP must be converted into a LP problem in a standard form where: All the constraints (inequalities) are converted to equations (equalities) All variables are nonnegative. 

Standard Maximization Problems in Standard Form A linear programming problem is said to be a standard maximization problem in standard form if its mathematical model is of the following form: Maximize the objective function Subject to problem constraints of the form With non-negative constraints

Slack Variables In real life problems, it’s unlikely that all resources will be used completely, so there usually are unused resources. Slack variables represent the unused resources between the left-hand side and right-hand side of each inequality. Used to express constraints inequalities as equalities

Sımplex Method Simplex algorithm for standard maximization problems Step-1 Write the standard maximization problem in standard form, introduce slack variables to form the initial system, and write the initial tableau. Step-3 Select the pivot column Step-5 Select the pivot element and perform the pivot operation STOP The optimal solution has been found. The linear programming problem has no optimal solution Step 2 Are there any negative indicators in the bottom row? Step 4 Are there any positive elements in the pivot column above the dashed line? Simplex algorithm for standard maximization problems

To solve a LP problem in standard form 1- Convert each inequality in the set of constraints to an equation by adding slack variables. Create the initial simplex tableau. 2- Check are there any negative indicators in the bottom row 3- Select the pivot column. ( The column with the “most negative value” element in the last row.) 4- Select the pivot row. (The row with the smallest non-negative result when the last element in the row is divided by the corresponding in the pivot column.) 5-Use elementary row operations calculate new values for the pivot row so that the pivot is 1 (Divide every number in the row by the pivot number.) 6- Use elementary row operations to make all numbers in the pivot column equal to 0 except for the pivot number. If all entries in the bottom row are zero or positive, this the final tableau. If not, go back to step 2. 7- If you obtain a final tableau, then the linear programming problem has a maximum solution, which is given by the entry in the lower-right corner of the tableau.

Pivot Pivot Column: The column of the tableau representing the variable to be entered into the solution mix. Pivot Row: The row of the tableau representing the variable to be replaced in the solution mix. Pivot Number: The element in both the pivot column and the pivot row.

Step 1 The first step of the simplex method requires that each inequality be converted into an equation. ”less than or equal to” inequalities are converted to equations by including slack variables. Suppose S1 electronic hours and S2 assembly hours remain unused in a week. The constraints become; 4X1 + 3X2 + s1 = 240 2X1 + 1X2 + s2 = 100

Step 1(con.) The problem can now be considered as solving a system of 3 linear equations involving the 5 variables in such a way that P has the maximum value; Now, the system of linear equations can be written in 3x6 matrix. 4X1 + 3X2 + s1 = 240 2X1 + 1X2 + s2 = 100 P - 7X1 + 5X2 = 0

Step 1(con.) Basic Variables x1 x2 S1 S2 P Right Hand Side 4 3 1 240 2 The initial tableau is Basic Variables x1 x2 S1 S2 P Right Hand Side 4 3 1 240 2 100 -7 -5 The tableau represents the initial solution; The slack variables S1 and S2 form the initial solution mix. The initial solution assumes that all avaliable hours are unused. i.e. The slack variables take the largest possible values.

Step 1(con.) Variables in the solution mix are called basic variables. Each basic variables has a column consisting of all 0’s except for a single 1. all variables not in the solution mix take the value 0. The simplex process, a basic variable in the solution mix is replaced by another variable previously not in the solution mix. The value of the replaced variable is set to 0.

Yes , There are negative values in the last raw We can go on step 3

Step 3 Select the pivot column (determine which variable to enter into the solution mix). Choose the column with the “most negative” element in the objective function row. Basic Variables x1 x2 S1 S2 P Right hand side 4 3 1 240 2 100 -7 -5 Pivot column x1 should enter into the solution mix because each unit of x1 (a Walkmans ) contributes a profit of $7 compared with only $5 for each unit of x2 (a TV)

Step 4 No, There aren’t any positive elements in the pivot column above the dashed line. We can go on step 5

Step 5 Basic Variables x1 x2 S1 S2 P Right hand side 4 3 1 240 2 100 Select the pivot row (determine which variable to replace in the solution mix). Divide the last element in each row by the corresponding element in the pivot column. The pivot row is the row with the smallest non-negative result. Enter Basic Variables x1 x2 S1 S2 P Right hand side 4 3 1 240 2 100 -7 -5 Exit Pivot row Pivot column Pivot number

Step 5 (con) Basic Variables x1 x2 S1 S2 P Right hand side 4 3 1 240 S2 Should be replaced by x1 in the solution mix. Now calculate new values for the pivot row. Divide every number in the row by the pivot number. Basic Variables x1 x2 S1 S2 P Right hand side 4 3 1 240 1/2 50 -7 -5

Step 5 Basic Variables x1 x2 S1 S2 P Right hand side 1 -2 40 1/2 50 Use row operations to make all numbers in the pivot column equal to 0 except for the pivot number which remains as 1. Basic Variables x1 x2 S1 S2 P Right hand side 1 -2 40 1/2 50 - 3/2 7/2 350 7R2 + R3 If 50 Walkmans are made, then the unused electronic hours are reduced by 200 hours (4 h/Walkmans multiplied by 50 Walkmans ); the value changes from 240 hours to 40 hours. Making 50 Walkmans results in the profit being increased by $350; the value changes from $0 to $350. Back to step 2

Multiple Goals, Sensitivity Analysis, What-If Analysis, and Goal Seeking Refers to a decision situation in which alternatives are evaluated with several and simultaneous, sometimes conflicting, goals. Determining single measure of effectiveness is difficult. Handling methods: Utility theory Goal programming Linear programming with goals as constraints Point system

Multiple Goals, Sensitivity Analysis, What-If Analysis, and Goal Seeking Assesses impact of change in inputs or parameters on solutions Allows for adaptability and flexibility to changing condition. Sensitivity analysis Can be: automatic trial and error

Multiple Goals, Sensitivity Analysis, What-If Analysis, and Goal Seeking Sensitivity analysis Can be: automatic: Performed in standard quantitative model implementation such as LP. It reports the range within which a certain input variable can vary without having any significant impact on the proposed solution. It usually limited to one change at a time and only for certain value. Lindo is used to do that trial and error: It determines the impact of changes in any variable or in several variables. You change some input data and solve the problem again. Two approaches to do it: what-is analysis and goal seeking.

Multiple Goals, Sensitivity Analysis, What-If Analysis, and Goal Seeking A process that involves asking a computer what the effect of changing some of the input data or parameters would be Goal seeking Backwards approach, starts with goal Determines values of inputs needed to achieve goal Example is break-even point determination: Involves determining the value of the decision variables that generate zero profit

Problem-Solving Search Methods

Problem-Solving Search Methods Analytical techniques Analytical techniques use mathematical formulas to derive an optimal solution directly or to predict a certain result They are used for structured problems Algorithms: An algorithm is a step-by-step search process for obtaining an optimal solution Solutions are generated and tested for possible improvements.

Problem-Solving Search Methods Blind search Blind search techniques are arbitrary search approaches that are not guided. There are two types: In a complete enumeration all the alternatives are considered and therefore an optimal solution is discovered In an incomplete enumeration (partial search) continues until a good-enough solution is found (a form of suboptimization)

Problem-Solving Search Methods Heuristic searching Informal, judgmental knowledge of an application area that constitutes the rules of good judgment in the field. Heuristics also encompasses the knowledge of how to solve problems efficiently and effectively, how to plan steps in solving a complex problem, how to improve performance, and so forth