1. LINEAR PROGRAMMING for Optimal Shipping
Mathematical Modeling Tool Used for Complex Problem Solving and Optimization of Limited Resources
We’ll be discussing problem solving using Linear Programming Models within Microsoft Excel spreadsheets through several examples and a hands-on exercise.
2/18/2019
OPERMGT-345

2. Agenda Introduction Creating LP Models Fertile-Earth Example
Spreadsheet Guidelines
Spud-Man Exercise
LP Model Summary
Reading/Reference List
Introduction: A brief review of several of the underlying reasons, tools, and techniques that support the development and use of LP Models.
Creating LP Models: A look at different approaches of LP Models and the steps involved in their creation.
Fertile-Earth Example: This is an optimization of a simple blending problem first shown Graphical and then developed in a MS-Excel Spreadsheet.
Spreadsheet Guidelines: Building LP Model spreadsheets is more art than science. We’ll discuss some effective design guidelines.
Spud-Man Exercise: This will be your chance to optimize a given shipping problem. Don’t worry, we’ll provide you with one the way of implementing a solution to the problem.
LP Model Summary: Wrap up of mathematical modeling using Linear Programming and how you can use it in your organization.
Reading/Reference List: Other sources of information and reading.
2/18/2019
OPERMGT-345

3. Introduction Problem Solving Mathematical Models Linear Programming
LP Model Applications
Problem Solving: Effective modeling requires understanding how modeling fits into a problem solving process.
Mathematical Models: What are these and how are they described.
Linear Programming: Sounds intimidating, but it’s just a type of mathematical model. We’ll define it and break it out into it’s components.
LP Model Applications: What can you use LP Models for?
2/18/2019
OPERMGT-345

4. Problem Solving 2/18/2019 OPERMGT-345
Define Problem: This is the most important step, as all the work that follows depends on the correct definition of the problem to be modeled. A well-defined statement of the problem is the end result.
Formulate Model: Create and “solve” the appropriate model using a spreadsheet. Spreadsheet models lend themselves to organizing the data in a way that is necessary for solving complex problems and the many changes involved in the iterative nature of the optimization.
Analyze Model: Generate and evaluate the alternatives that may lead to the best solution to the defined problem. Is the model a good representation of the environment of the problem?
Test Results: Test the feasibility and the quality of the results produced by the model verifying the solution against known results and simple common sense. If results are unsatisfactory, update, re-implement, and reanalyze the model.
Implement Solution: This is not as easy as it may seem because most people are resistant to change. Good interpersonal skills are needed now, and throughout the project in order to have a successful implementation. Early participation, continual buy in, and a shared sense of ownership during all phases of the project go a long way to ensure success.
2/18/2019
OPERMGT-345

5. Mathematical Models Definition Categories
A functional relationship, like the simple model of
PROFIT = ƒ(REVENUE – EXPENSES)
Categories
Prescriptive, Predictive, and Descriptive
Linear Programming is a prescriptive technique because it prescribes actions that should be taken
Definition: Usually describe functional relationships like:
PROFIT = ƒ(REVENUE – EXPENSES).
Categories: Mathematical models can be categorized as follows:
Prescriptive: Suggests to the decision maker to take certain actions. Linear Programming (LP) and Critical Path Method (CPM) are some techniques.
Predictive: Forecasting a dependent variable based on specific independent variable values. Regression and Time Series Analysis are some techniques.
Descriptive: Describing outcomes of systems. Project Evaluation and Review Technique (PERT) and Economic Order Quantity (EOQ) are some techniques.
2/18/2019
OPERMGT-345

6. Linear Programming Definition Elements Methods
A model of linear relationships representing an objective within specific resource constraints
Elements
Decision Variables, Constraints, Objective Function
Methods
Graphical or Spreadsheet
Definition: A common LP problem is to determine how much product to produce to maximize profit subject to resource constraints such as labor and materials. The components that make up the problem are expressed as mathematical linear relationships that together form a model.
Elements:
Decision Variables are mathematical symbols representing levels of activity.
Constraints are linear relationships representing decision making restrictions.
Objective Function is a linear relationship reflecting a specific objective.
Methods:
Graphical method of solving LP problems is valid but generally cumbersome and sometimes misleading.
Spreadsheets are the preferred method of solving LP problems for businesses. Today, LP solvers are built into spreadsheet packages in a way that makes them easy to construct and use.
2/18/2019
OPERMGT-345

7. LP Model Applications Optimization Applications The name of the game
Product Mix
Logistics
Manufacturing
Others
Optimization: Linear programming models find the most efficient ways of using limited resources to achieve an objective. In this way, the technique is referred to as “Optimization”.
Applications:
Product Mix is how much of each product to produce to maximize profit.
Logistics is determining the least costly method/route of shipping product.
Manufacturing is the optimizing of a manufacturing process or technique (i.e. minimizing the distance a drill must move in a repetitive drilling process consisting of thousands of holes that are to be drilled on a circuit board).
Others consists of a variety of ways to use LP Models that include: Financial, Forecasting, Marketing, Engineering, Scheduling, Capacity Utilization, etc..
2/18/2019
OPERMGT-345

8. Creating LP Models Understand the Problem
Identify the Decision Variables
State the Objective Function
State the Constraints
Identify Decision Variable Bounds
Understand the Problem: As stated earlier in problem solving, this also is the most important step in creating an LP Model. If you do not fully understand the problem, it is unlikely that your model of the problem will be correct.
Identify the Decision Variables: What are the fundamental decisions to be made in order to solve the problem? The answers to this questions will often help you identify appropriate decision variable for your model.
State the Objective Function: This function expresses the mathematical relationship between the decision variables to be maximized/minimized.
State the Constraints: There are usually some limitations on the values that can be assumed by the decision variables in an LP Model. These restrictions must be identified and stated in the form of constraints.
Identify Decision Variable Bounds: Often simple upper or lower bounds apply to decision variables. You can think of these as additional constraints.
2/18/2019
OPERMGT-345

9. Fertile-Earth Example Defining the Problem
Fertile-Earth wants to know how much of each of their fertilizer products it needs to produce and sell in order to make the most profit.
This is the introduction to the problem. Fertile-Earth is a fertilizer producer that wants to maximize its profits by mixing and selling it two products in an optimal way.
2/18/2019
OPERMGT-345

10. Fertile-Earth Example Defining the Problem
Fertilizer products
Giant-Grow (GG) and Super-Start (SS)
Net contribution
GG nets $18.50/ton and SS nets $20.00/ton
Ingredients
Nitrogen (N) 1100 Tons, Phosphorous (P) 1800 Tons
Potassium (K) 2000 Tons
Product recipe
GG: 5(N), 5(P), 10(K) and SS: 5(N), 10(P), 5(K)
Here are the elements needed to define the problem.
Fertilizer products: Decision Variables, products GG and SS.
Net contribution: Objective Function based on cost per ton of products.
Ingredients: Constraints, how much of each mineral.
Product recipe: Linear Relationship, percent of ingredient in each product.
2/18/2019
OPERMGT-345

11. Fertile-Earth Example Defining the Problem
Maximize:
(18.50)GG + (20.00)SS
Subject to:
N: (0.05)GG + (0.05)SS <= 1100
P: (0.05)GG + (0.10)SS <= 1800
K: (0.10)GG + (0.05)SS <= 2000
What are we trying to do, expressed algebraically.
2/18/2019
OPERMGT-345

12. Fertile-Earth Example Formulating the Model
Graphical Method
Here we have plotted each linear relationship to arrive at a “feasible” solution area on the graph.
2/18/2019
OPERMGT-345

13. Fertile-Earth Example Formulating the Model
Graphical Method
Of the “feasible” solution area, there are several reasonable points that could indicate the highest profit within the set constraints.
2/18/2019
OPERMGT-345

14. Fertile-Earth Example Formulating the Model
Graphical Method
After doing the math, we see that point “c” is the point where the highest profit is achieved within the existing constraints.
2/18/2019
OPERMGT-345

15. Fertile-Earth Example Formulating the Model
Spreadsheet Method
Now to build a Linear Programming Model in a spreadsheet for the same problem we just resolved graphically. We are using Microsoft’s Excel with the solver “add-in”.
Notice that we have organized the data in a way that is easy t-i understand in context to the problem. Decision Variables and Decision Bounds are together. Constraints and Linear Relationships are together.
Take some time and review the formulas in the various cells. If you they are not familiar with them, you will need to bone-up on them by referring to the HELP feature of MS-Excel. Be sure you understand the “SUMPRODUCT” function, as it is most critical to spreadsheet LP Modeling.
2/18/2019
OPERMGT-345

16. Fertile-Earth Example Formulating the Model
Spreadsheet Method
After the spreadsheet LP model has been set up, you need to invoke the “solver” in MS-Excel to get a solution to the problem.
2/18/2019
OPERMGT-345

17. Fertile-Earth Example Formulating the Model
Spreadsheet Method
There are several parameters that the solver needs to be supplied with. In addition, the solver needs to know if you are maximizing or minimizing to a solution
What cell is the Objective Function located in? .
2/18/2019
OPERMGT-345

18. Fertile-Earth Example Formulating the Model
Spreadsheet Method
What cell(s) are the Decision Variable located in?
2/18/2019
OPERMGT-345

19. Fertile-Earth Example Formulating the Model
Spreadsheet Method
What cell(s) are the Constraints located in?
2/18/2019
OPERMGT-345

20. Fertile-Earth Example Formulating the Model
Spreadsheet Method
Generally, you should keep to the default settings when just starting. Once you’ve gotten the hang of LP Modeling in spreadsheets, you may want to experiment with the various options.
2/18/2019
OPERMGT-345

21. Fertile-Earth Example Formulating the Model
Spreadsheet Method
Here is the “solved” LP Model. Notice that the Objective Function is the same as the highest profit point in the graphical solution (it should be).
2/18/2019
OPERMGT-345

22. Spreadsheet Guidelines
Organize the Data
Don’t Imbed Constants
Group Related Items
Be Able to Copy Formulas
Now we need to take some time and discuss guidelines for effective design of spreadsheet LP Models.
Organize the Data: First organize the data, then build the model around the data. Once the data is arranged, logical locations for the Decision Variables, Constraints, and Objective Function tend to naturally suggest themselves.
Don’t Imbed Constants: Numeric constants should be placed in separate cells and labeled appropriately.
Group Related Items: Things that are logically related should be arranged in close proximity to one another. This enhances functionality and readability.
Be Able to Copy Formulas: Design the model so that formulas can be copied. Once you understand one cell’s formula, you would then understand a series of similar cells and would be able to then just copy the formula to them.
2/18/2019
OPERMGT-345

23. Spreadsheet Guidelines
Put Totals Close to Sum Fields
Organize Left-Right, Top-Bottom
Use Distinguishable Formatting
Document Model Elements
Put Totals Close to Sum Fields: Column or row totals should be in close proximity to the column or rows being totaled.
Organize Left-Right, Top-Bottom: English reading people scan from left to right, top to bottom. This fact should be reflected in your spreadsheet design.
Use Distinguishable Formatting: Use color, shading, borders, and protection to distinguish changeable parameters from other elements of the model.
Document Model Elements: Use text boxes and cell comments to document various elements of the model.
2/18/2019
OPERMGT-345

24. Spud-Man Exercise Defining the Problem
Spud-Man is a local trucking company specializing in hauling potatoes from local fields to collection points in Buhl, Gooding, and Rupert Idaho. Currently, they have three clients: Mr. Doe, Mr. Jones, and Mr. Smith.
Spud-Man wants to know how many pounds of potatoes to truck from each client’s field to each city’s collection point in order to minimize to total number of miles the spuds must be shipped.
The mileage saved translates into fuel savings, labor time savings, savings on vehicle wear and tear, etc..
Now, we have an exercise we would like you to try. Please read the problem carefully, define the problem, formulate a spreadsheet LP Model, and solve for the Objective Function.
We have provided you with our model formulation and solution if you get stuck and need some guidance or want to compare your work against ours.
2/18/2019
OPERMGT-345

25. Spud-Man Exercise Defining the Problem
Mr. Doe’s field has 275,000 lbs of potatoes Mr. Jones’ field has 400,000 lbs of potatoes Mr. Smith’s field has 300,000 lbs of potatoes.
Buhl collection point can handle 200,000 lbs Gooding collection point can handle 600,000 lbs Rupert collection point can handle 225,000 lbs.
Distance between fields and collection points:
Remember, there are Decision Variables, Constraints, and an Objective Function to defined and formulate into a model.
Generally, if you try and think intuitively about what Spud-Man wants to do, you’ll be able to define and formulate the problem easier.
2/18/2019
OPERMGT-345

26. Spud-Man Exercise Formulating the Model
Spreadsheet Method
Here’s how we formulated the model.
2/18/2019
OPERMGT-345

27. Spud-Man Exercise Formulating the Model
Spreadsheet Method
Here’s the formulas that we placed in the model.
2/18/2019
OPERMGT-345

28. Spud-Man Exercise Formulating the Model
Spreadsheet Method
Here’s the solver parameters that we used.
2/18/2019
OPERMGT-345

29. Spud-Man Exercise Formulating the Model
Spreadsheet Method
Here’s the solution we came up with to Spud-Man’s problem.
2/18/2019
OPERMGT-345

30. LP Model Summary
LP Models are one of the best ways to solve complex problems.
Building LP Models is dependent on a well defined and understood problem.
A Spreadsheet LP Model represents the problem in a way that tends to clearly communicate it’s purpose.
Spreadsheet LP Models are easy to build with today’s packaged tools.
2/18/2019
OPERMGT-345

31. Reference List Software Resources 2/18/2019 OPERMGT-345
Software: In addition to Microsoft Excel, there are many companies that specialize in Linear Programming.
LINDO Systems is one of the oldest, and in my opinion, one of the best. Their “What’s Best” product is relatively inexpensive, and can fit almost any level of model development. Phone: (312)
Resources: Some that were used in this presentation are:
What’s Best (user manual), LINDO Systems
Operations Management (textbook), Russell, Taylor
Boise Cascade Corporation, Zalucha
2/18/2019
OPERMGT-345