Search in book...
Toggle Font Controls
Create new playlist

Name your new playlist

Playlist description (optional)
Sign In

Email address

Password

Forgot Password?

or

Continue with Facebook

Continue with Google
Sign Up

Full Name

Email address

Confirm Email Address

Password

or

Continue with Facebook

Continue with Google

1.4 How to Develop a Quantitative Analysis Model

Developing a model is an important part of the quantitative analysis approach. Let’s see how we can use the following mathematical model, which represents profit:

Profit = Revenue - Expenses

$Profit = Revenue - Expenses$

In many cases, we can express revenue as the selling price per unit multiplied times the number of units sold. Expenses can often be determined by summing fixed cost and variable cost.

Modeling in The Real World Railroad Uses Optimization Models to Save Millions

1.4-2 Full Alternative Text

Defining the Problem

CSX Transportation, Inc., has 35,000 employees and annual revenue of $11 billion. It provides rail freight services to 23 states east of the Mississippi River, as well as parts of Canada. CSX receives orders for rail delivery service and must send empty railcars to customer locations. Moving these empty railcars results in hundreds of thousands of empty-car miles every day. If allocations of railcars to customers is not done properly, problems arise from excess costs, wear and tear on the system, and congestion on the tracks and at rail yards.

Developing a Model

In order to provide a more efficient scheduling system, CSX spent 2 years and $5 million developing its Dynamic Car-Planning (DCP) system. This model will minimize costs, including car travel distance, car handling costs at the rail yards, car travel time, and costs for being early or late. It does this while at the same time filling all orders, making sure the right type of car is assigned to the job, and getting the car to the destination in the allowable time.

Acquiring Input Data

In developing the model, the company used historical data for testing. In running the model, DCP uses three external sources to obtain information on the customer car orders, the available cars of the type needed, and the transit-time standards. In addition to these, two internal input sources provide information on customer priorities and preferences and on cost parameters.

Developing a Solution

This model takes about 1 minute to load but only 10 seconds to solve. Because supply and demand are constantly changing, the model is run about every 15 minutes. This allows final decisions to be delayed until absolutely necessary.

Testing the Solution

The model was validated and verified using existing data. The solutions found using DCP were determined to be very good compared to assignments made without DCP.

Analyzing the Results

Since the implementation of DCP in 1997, more than $51 million has been saved annually. Due to the improved efficiency, it is estimated that CSX avoided spending another $1.4 billion to purchase an additional 18,000 railcars that would have been needed without DCP. Other benefits include reduced congestion in the rail yards and reduced congestion on the tracks, which are major concerns. This greater efficiency means that more freight can ship by rail rather than by truck, resulting in significant public benefits. These benefits include reduced pollution and greenhouse gases, improved highway safety, and reduced road maintenance costs.

Implementing the Results

Both senior-level management who championed DCP and key car-distribution experts who supported the new approach were instrumental in gaining acceptance of the new system and overcoming problems during the implementation. The job description of the car distributors was changed from car allocators to cost technicians. They are responsible for seeing that accurate cost information is entered into DCP, and they also manage any exceptions that must be made. They were given extensive training on how DCP works so they could understand and better accept the new system. Due to the success of DCP, other railroads have implemented similar systems and achieved similar benefits. CSX continues to enhance DCP to make it even more customer friendly and to improve car-order forecasts.

Source: Based on M. F. Gorman et al., “CSX Railway Uses OR to Cash In on Optimized Equipment Distribution,” Interfaces 40, 1 (January–February 2010): 5–16, © Trevor S. Hale.

Variable cost is often expressed as the variable cost per unit multiplied times the number of units. Thus, we can also express profit in the following mathematical model:

Profit = Revenue - (Fixed cost + Variable cost)

$Profit = Revenue - (Fixed cost + Variable cost)$

Profit = (Selling price per unit) (Number of units sold)

$Profit = (Selling price per unit) (Number of units sold)$

- [Fixed cost + (Variable cost per unit) (Number of units sold)]

$- [Fixed cost + (Variable cost per unit) (Number of units sold)]$

Profit = s X - [f + ν X]

$Profit = s X - [f + ν X]$

Profit = s X - f - ν X

$Profit = s X - f - ν X$ (1-1)

where

s = selling price per unit

$s = selling price per unit$

f = fixed cost

$f = fixed cost$

ν = variable cost per unit

$ν = variable cost per unit$

X = number of units sold

$X = number of units sold$

The parameters in this model are f, $ν,$ $ν,$ and s, as these are inputs that are inherent in the model. The number of units sold (X) is the decision variable of interest.

Example: Pritchett’s Precious Time Pieces

We will use the Bill Pritchett clock repair shop example to demonstrate the use of mathematical models. Bill’s company, Pritchett’s Precious Time Pieces, buys, sells, and repairs old clocks and clock parts. Bill sells rebuilt springs for a price per unit of $8. The fixed cost of the equipment to build the springs is $1,000. The variable cost per unit is $3 for spring material. In this example,

s = 8

$s = 8$

f = 1, 000

$f = 1, 000$

ν = 3

$ν = 3$

The number of springs sold is X, and our profit model becomes

Profit = $ 8 X - $ 1, 000 - $ 3 X

$Profit = $ 8 X - $ 1, 000 - $ 3 X$

If sales are 0, Bill will realize a $1,000 loss. If sales are 1,000 units, he will realize a profit of $$ 4, 000 [$ 4, 000 = ($ 8) (1, 000) - $ 1, 000 - ($ 3) (1, 000)] .$ $$ 4, 000 [$ 4, 000 = ($ 8) (1, 000) - $ 1, 000 - ($ 3) (1, 000)] .$ See if you can determine the profit for other values of units sold.

In addition to the profit model shown here, decision makers are often interested in the break-even point (BEP). The BEP is the number of units sold that will result in $0 profits. We set profits equal to $0 and solve for X, the number of units at the BEP:

0 = s X - f - ν X

$0 = s X - f - ν X$

This can be written as

0 = (s - ν) X - f

$0 = (s - ν) X - f$

Solving for X, we have

f = (s - ν) X

$f = (s - ν) X$

X = \frac{f}{s - ν}

$X = \frac{f}{s - ν}$

This quantity (X) that results in a profit of zero is the BEP, and we now have this model for the BEP:

BEP = \frac{Fixed cost}{(Selling price per unit) - (Variable cost per unit)}

$BEP = \frac{Fixed cost}{(Selling price per unit) - (Variable cost per unit)}$

BEP = \frac{f}{s - ν}

$BEP = \frac{f}{s - ν}$ (1-2)

For the Pritchett’s Precious Time Pieces example, the BEP can be computed as follows:

BEP = $ 1, 000 / ($ 8 - $ 3) = 200 units, or springs,

$BEP = $ 1, 000 / ($ 8 - $ 3) = 200 units, or springs,$

The Advantages of Mathematical Modeling

There are a number of advantages of using mathematical models:

Models can accurately represent reality. If properly formulated, a model can be extremely accurate. A valid model is one that is accurate and correctly represents the problem or system under investigation. The profit model in the example is accurate and valid for many business problems.
Models can help a decision maker formulate problems. In the profit model, for example, a decision maker can determine the important factors or contributors to revenues and expenses, such as sales, returns, selling expenses, production costs, and transportation costs.
Models can give us insight and information. For example, using the profit model, we can see what impact changes in revenue and expenses will have on profits. As discussed in the previous section, studying the impact of changes in a model, such as a profit model, is called sensitivity analysis.
Models can save time and money in decision making and problem solving. It usually takes less time, effort, and expense to analyze a model. We can use a profit model to analyze the impact of a new marketing campaign on profits, revenues, and expenses. In most cases, using models is faster and less expensive than actually trying a new marketing campaign in a real business setting and observing the results.
A model may be the only way to solve some large or complex problems in a timely fashion. A large company, for example, may produce literally thousands of sizes of nuts, bolts, and fasteners. The company may want to make the highest profits possible given its manufacturing constraints. A mathematical model may be the only way to determine the highest profits the company can achieve under these circumstances.
A model can be used to communicate problems and solutions to others. A decision analyst can share his or her work with other decision analysts. Solutions to a mathematical model can be given to managers and executives to help them make final decisions.

Mathematical Models Categorized by Risk

Some mathematical models, like the profit and break-even models previously discussed, do not involve risk or chance. We assume that we know all values used in the model with complete certainty. These are called deterministic models. A company, for example, might want to minimize manufacturing costs while maintaining a certain quality level. If we know all these values with certainty, the model is deterministic.

Other models involve risk or chance. For example, the market for a new product might be “good” with a chance of 60% (a probability of 0.6) or “not good” with a chance of 40% (a probability of 0.4). Models that involve chance or risk, often measured as a probability value, are called probabilistic models. In this book, we will investigate both deterministic and probabilistic models.

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.

Table of Contents for 1.4 How to Develop a Quantitative Analysis Model

Create new playlist

Sign In

Sign Up

Table of Contents for
1.4 How to Develop a Quantitative Analysis Model