SCM 312 Final Exam (Chapter 10)

Changes in RHS Values of Constraints

1. A key question analysts want to answer is if adding more resources is worth investment - Cost versus Benefit - RHS values of constraints often represent resources available - If additional resources were available, a higher total profit could be realized 2. Sensitivity analysis about RHS values can determine how much should be paid for additional resources & how much more of a resource would be useful 3. If the right-hand side of a constraint is changed, the feasible region will change (unless the constraint is redundant) 4. Often the optimal solution will change 5. The amount of change in the objective function value that results from a unit change in the RHS of a constraint is called the dual price (or shadow price in max) - Improvement in objective function if constraint is "relaxed" one unit - One of most important values in LP results

4 Properties of an LP Problem

1. All problems seek to maximize or minimize some quantity (known as the objective function) 2. Restrictions or constraints that limit the degree to which we can pursue our objective are present 3. There must be alternative courses of action from which to choose 4. The objective and constraints in problems must be expressed in terms of linear equations or inequalities

Feasible Region of Constraints

1. Any point on or below the constraint plot will not violate the restriction 2. Any point above the plot will violate the restriction

5 Basic Assumptions of an LP Problem

1. Certainty - numbers in the objective and constraints are known with certainty and do not change during the period being studied 2. Proportionality - relationships in the objective and constraints are proportional 3. Additivity - total of all activities equals the sum of the individual activities 4. Divisibility - solutions need not be whole numbers 5. Non-negativity - all answers or variables are nonnegative

Changes in Technological Coefficients

1. Changing coefficients in the constraint equations can produce a significant change in the shape of the feasible region - This may change the optimal solution 2. Constraints in production problems are often based on technology available to make products 3. Changes in the technological coefficients often reflect changes in the state of technology 4. If the amount of resources needed to produce a product changes, coefficients in the constraint equations will change

Unboundness

1. Condition when an LP does not have a finite solution - Feasible region will be open ended - In a maximization problem, 1 or more decision variables (and the profit) can be made infinitely large without violating any constraints 2. This usually means the problem has been formulated improperly

Product Mix Problem Example

1. Consider a chocolate manufacturing company which produces only two types of chocolate - A and B - Both require milk and cocoa only 2. The company wishes to maximize its profit Key decision is how many units of A and B should it produce respectively 3. The following quantities are required to manufacture each unit of A and B - Each unit of A requires 1 unit of milk and 3 units of cocoa - Each unit of B requires 1 unit of milk and 2 units of cocoa 4. The company warehouse has a total of 5 units of milk and 12 units of cocoa 5. On each sale, the company makes a profit of $6 per unit A sold and $5 per unit B sold

Changes in the Objective Function Coefficient

1. Contribution rates in the objective functions often fluctuate 2. Graphically this means: - Feasible region remains exactly the same - Slope of the isoprofit or isocost line changes 3. Can often make modest increases or decreases in the objective function coefficient of any variable without changing the optimal corner point 4. Want to test how much an objective function coefficient can change before the optimal solution would be at a different corner point 5. See slide 8-10 on the third powerpoint for chapter 10 to see an excel solution

LP Problems in the Supply Chain

1. Devising an inventory and warehousing strategy for an e-tailer can be very complex - Millions of SKUs with different popularity in different regions to be delivered in defined time and resources - Would be impossible to make the best decision without finding optimal solution mathematically 2. Think about how complexity of deliveries for FedEx or UPS - FedEx delivers 14M packages/day and they are the small one - Not optimizing this means extra employees, late packages, more fuel use, etc.

Steps for using Excel Solver in an LP Problem

1. Enter the variable names, the coefficients for the objective function and constraints, and the right-hand-side values for each of the constraints 2. Designate specific cells for the values of the decision variables 3. Write a formula to calculate the value of the objective function 4. Write a formula to compute the left-hand sides of each of the constraints 6. See slide 53-62 on the second powerpoint for chapter 10

FedEx Example

1. FedEx delivery man has 6 packages to deliver in a day 2. The warehouse is located at point A 3. The 6 delivery destinations are given by U, V, W, X, Y and Z 4. The numbers on the lines indicate the distance between the cities 5. FedEx wants the delivery person to take the shortest route to save on fuel and time

Sensitivity Analysis

1. Involves a series of what-if? questions concerning how sensitive the optimal solution is to changes in model parameters - Test coefficients in constraints, RHS of constraints, and the objective function - Done after optimal solution is found (post-optimality) 2. Goal is to determine a range of changes in problem parameters that will not affect the optimal solution or change the variables in the solution 3. As important as optimal solution because supply chains are dynamic

Corner Point Solution Method

1. Involves finding the profit at every corner point of the feasible region 2. The mathematical theory behind LP is that the optimal solution must lie at one of the corner points (extreme points) in the feasible region 3. Must find the optimal point - It is the point where the two constraints intersect - Solve for one variable and plug back into the equations

Minimization Problems

1. Many LP problems involve minimizing an objective such as cost instead of maximizing a profit function 2. Minimization problems can be solved graphically like maximization problems - Isocost versus isoprofit - Corner points 3. They can also be solved in Excel and other software just as easily as maximization problems 4. See slide 64-71 on the second powerpoint for chapter 10

Introduction to Linear Programming

1. Many SCM decisions involve trying to make the most effective & efficient use of limited resources 2. Linear programming (LP) is a widely used mathematical modeling technique designed to help managers in planning and decision making relative to resource allocation - Involves solving a problem mathematically - Must develop a set of equations to describe problem

Four special cases in LP

1. No feasible solution 2. Unboundedness 3. Redundancy 4. Alternate Optimal Solutions

Reduced Cost

1. Only non-zero when the optimal value of a variable is zero 2. Represents how much the coefficient of the activity represented by the variable must be improved before any of that activity will be done (optimal solution will not be zero)

Excel Solution for a Basic LP Problem

1. See slide 5-7 on the first powerpoint for chapter 10 2. Copy the data into excel to practice

Integer Programming

1. Some problems require the optimal value of the decision variables to be integers - Do not want to purchase or use half a machine for a production schedule - Usually would not create an optimal schedule with half a worker 2. Need to add a constraint that all decision variables must be integers - Similar to non-negativity constraint (X≥0) - Can be easily added in Excel Solver

Isoprofit Line Method Steps

1. Start with a small but possible profit value and graph the objective function 2. Move the objective function line in the direction of increasing profit (while maintaining the slope) 3. The last point it touches in the feasible region is the optimal solution

How do we find the best integer solution?

1. The LP without the integer constraint is known as the relaxed problem - Relaxed solution provides upper bound of the optimal solution of the IP - Optimal integer solutions must be ≤ than optimal relaxed solution 2. Optimal solution is no longer always at a corner point - Can be hundreds or thousands of "candidate" solutions that must be checked - Algorithms developed to find these solutions until one is found within x% of optimal relaxed problem - This suboptimality tolerance must be specified in software (including Excel Sover)

Excel Solver for LP Problems

1. The Solver tool in Excel can be used to find solutions to - LP problems - Integer programming problems - Non-integer programming problems 2. Solver is limited to 200 variables and 100 constraints 3. To use Solver, it is necessary to enter formulas based on the initial model 4. Solver Add-Ins that solve larger problems are available commercially and are inexpensive

Slack

1. The amount of a resource that is not used in a less-than-or-equal constraint 2. Slack = Amount of resource available - amount of resource used

Changes in RHS Values are Bounded

1. The amount of possible increase in the right-hand side of a resource is limited - E.g. If the number of hours available increases beyond the upper bound, then the objective function would no longer increase by the dual price - There would simply be excess (slack) hours of a resource - The objective function may change different amount from the dual price 2. The dual price is relevant only within limits

Surplus

1. The amount over the constraint that is being produced or utilized in a greater-than-or-equal to constraint 2. Surplus = Actual amount - minimum amount

Changing the tolerance in Excel Solver

1. The default tolerance is 1% 2. Changing it to zero or close to zero will force Excel to try every feasible integer solution

Graphical Representation of a Constraint Process

1. The first step in solving the problem is to identify a set or region of feasible solutions - To do this we plot each constraint equation on a graph 2. We start by graphing the equality portion of the constraint equations 3. We solve for the axis intercepts and draw the line

Graphical Solution to an LP Problem

1. The graphical method only works when there are just two decision variables 2. When there are more than two variables, a more complex approach is needed 3. The graphical method provides valuable insight into how the more complex approaches work 4. Each axis represents one variable - tables and chairs

Product Mix Problem

1. Two or more products are produced using limited resources such as personnel, machines, and raw materials 2. The profit that the firm seeks to maximize is based on the profit contribution per unit of each product 3. The company would like to determine how many units of each product it should produce so as to maximize overall profit given its limited resources

Constraints

The constraints are the restrictions or limitations on the decision variables. They usually limit the value of the decision variables

Redundancy

1. A redundant constraint does not affect the feasible solution region 2. One or more constraints bind entire problem 3. Very common occurrence in the real world 4. It causes no particular problems, but eliminating redundant constraints simplifies the model

4 Steps in an LP Problem

1. Completely understand the managerial problem being faced 2. Identify the objective and the constraints 3. Define the decision variables 4. Use the decision variables to write mathematical expressions for the objective function and the constraints

No feasible solution

1. Exists when there is no feasible solution to the problem that satisfies all the constraint equations 2. This is a common occurrence in the real world 3. Generally one or more constraints are relaxed until a solution is found

Alternate Optimal Solutions

1. Two or more optimal solutions exist 2. When the objective function's isoprofit (or isocost) line runs perfectly parallel to one of the constraints 3. Allows management great flexibility in deciding which combination to select

Optimization

1. We all have finite resources & time - We want to make the most of them - We all implicitly think about how to optimize the use of that time to make it as productive as possible based on priorities 2. Managing supply chains involves the same types of decisions - E.g., labor, facilities, scheduling - Not optimizing these would be throwing away $$$$ - Most problems have too many potential solutions to count

First Two Questions We Always Ask in an LP problem

1. What is the goal? - Maximize profit 2. What decision do we need to make in order to reach our goal? - How many of each type of item should we produce - We need two decision variables to represent the number of each hot tub produced - X1 and X2

Non-Negativity Restriction

For all linear programs, the decision variables should always take non-negative values

Shadow Price

How much the optimal solution can be increased or decreased if we change the RHS values (resources available) one unit

Objective Function

It is defined as the objective of making decision

Decision Variables

Variables which will decide output that represent the ultimate solution. First need to identify the decision variables to solve any problem