Jeremy Charlesworth could see the skepticism on his client’s face. She didn’t say it, but he knew what she was thinking: You’re wrong.
Charlesworth, a research analyst for The Nielsen Company, had been asked to recommend a name for the client’s new product. At first glance, one name stood out. Then Charlesworth started digging into the data and doing the mathand, according to his analysis, the initially appealing name wasn’t so appealing: it carried the risk of cannibalizing the company’s other brands. But his client was unconvinced.
“I felt so dejected,” Charlesworth says. “This was my first big project for this client, and I was sure my analysis showed what was best for the business.”
Fortunately, not long after their discussion, the client met with her team and reviewed Charlesworth’s report—and his math.
“When they saw the numbers, they were convinced that they should go with my recommendation,” Charlesworth says. “In the end, they appreciated my analysis.”
Much like Charlesworth, professionals at every level and in every industry are using math to help make better decisions. Most people agree that being able to do the math is an important, even essential, skill. Why then do so many—including many in the business world—recoil at the thought of using numbers to solve problems?
“People almost view it as a badge of honor that they can’t do math,” observes Laura Laing, a former high school math teacher and author of Math for Grownups. “It becomes a label they give themselves. But it’s funny: we don’t walk around saying, ‘I’m such a bad reader.’ If someone does, it’s likely that other people will want to help that person improve. The same thing needs to be true for math.”
Jeffrey Humpherys, associate professor of mathematics at BYU, points out that culture—particularly in America, where students’ math test scores lag far behind other industrialized nations—is a major factor in math’s unpopularity.
“Math is hard, no question,” he says. “But the problem is that many parents are excusing children when they struggle with math. The truth is that any subject worth pursuing is going to require effort.”
Making the effort to master math is certainly worthwhile in today’s increasingly complex world. If professionals want to make the most of the deluge of data, they’ll want to understand math. In addition, more and more employers are looking for people who can approach problems in a quantitative way.
“Even fields that have not traditionally been quantitative have become quantitative,” says Mark Herschberg, who uses math frequently in his work as a business turnaround executive. “Take marketing. The people I know working in marketing are looking at spreadsheets, trying to analyze how the market will respond if an ad is phrased a certain way. It’s all about the math.”
If you have less-than-fond memories of fumbling through second-year algebra, math may seem like an approaching storm: threatening, unpleasant, and best avoided if possible. But when you look at all the things math can help you do, it’s easier to see math not as a storm to suffer through but as a trusty compass.
The oft-overlooked beauty of math is that it provides a way to efficiently, even elegantly, make sense of the world around us. In the business world, managers are constantly confronted with problems involving constraints, uncertainty, complexity, and change—problems math is specifically designed to address.
Not sure you’re up to the task? Have no fear. Dust off your calculator, grab a pencil, and follow along as we explore the basics of four powerful math concepts—linear programming, probability, input-output analysis, and Markov chains—and how they help professionals make clear, actionable business decisions.
Linear Programming: Managing with Constraints
How can you feed an army on a budget?
For army organizers during World War II, it was important to keep food costs down. But they also needed to make sure soldiers’ food met standard dietary requirements. To find the right balance of nutritious yet cost-effective foods, they used linear programming.
The army’s meal planning was actually one of the first uses of linear programming, also called linear optimization, which is used to figure out the optimal outcome, such as maximum profit or lowest cost, given certain constraints.
Linear programming came to prominence thanks to mathematician George Bernard Dantzig, who worked as an advisor to the Pentagon. His contributions to the field evolved out of the military’s interest in optimizing resource allocation, deployment schedules, and, of course, soldiers’ meals.
Today linear programming is used in a variety of ways. Oil refineries use it to determine the optimal combination of gasoline grades; airlines use it to decide how many discounted seats they should sell; delivery companies use it to organize efficient shipping routes. Businesses are always trying to make the most of what they have, and linear programming is arguably the most widely used mathematical tool in business management.
Practice Problem #1: Linear Programming
The owners at snack company Happy Trails know their Classic trail mix has always been a strong seller, but they’re considering introducing a new mix, Fruitylicious. Suppose that each package of Classic mix uses 8 ounces of peanuts and 6 ounces of dried fruit and yields $1.50 of profit, and each package of Fruitylicious mix uses 4 ounces of peanuts and 10 ounces of dried fruit and yields $1.00 of profit. Suppose that Happy Trails has 20,000 ounces of peanuts and 30,000 ounces of dried fruit available each day. Should the company introduce a new line of trail mix? If so, how many packages of each variety should it produce daily to yield a maximum profit?
Setup
Here are the basic steps we’ll take to solve this problem: determine the variables, set up a system of inequalities, graph those inequalities, then plug the coordinates from the graph into the profit equation.
The variables we’re solving for are:
x = number of Classic packages
y = number of Fruitylicious packages
Each Classic package requires 8 ounces of peanuts and 6 ounces of dried fruit. Likewise, each Fruitylicious package requires 4 ounces of peanuts and 10 ounces of fruit. We’ll use this information to create a system of inequalities.
The requirements and constraints for the peanuts are represented
8x + 4y ≤ 20,000
The requirements and constraints for the fruit are represented
6x + 10y ≤ 30,000
We use the less-than-or-equal-to sign because the amount of peanuts and fruit needs to be less than or equal to the amount available.
Finally, we need to create an equation describing the profit per package. The profit for Happy Trails will be 1.5x for each x package of Classic mix and 1.0y for each y package of Fruitylicious mix. Thus:
P = 15x + 10y
Our goal is to find the x and y that make the profit as large as possible while satisfying the constraints of this system of inequalities (the last two are included because the answer cannot be negative):
8x + 4y ≤ 20,000
6x + 10y ≤ 30,000
x ≥ 0
y ≥ 0
Solve
Graph the system of inequalities. To find the coordinates for each inequality, you can set x or y to 0 and then solve for the remaining variable. For example, we can set x = 0 for 8x + 4y ≤ 20,000:
8(0) + 4y ≤ 20,000
4y ≤ 20,000
y ≤ 5,000
This will give us our first coordinates: (0, 5,000). If we set y = 0, we can solve for x and find the coordinates (2,500, 0). This means the line for 8x + 4y ≤ 20,000 will be from (0, 5,000) to (2,500, 0). Similarly, we can find that the line for 6x + 10y ≤ 30,000 is from (0, 3,000) to (5,000, 0).
The graph for these lines should look like this:
This can also be done on a graphing calculator. The lines intersect at (1,428.6, 2,142.8). We can plug these coordinates into the profit equation P = 1.5x + 1.0y.
P = 15(0) + 10(3,000) = 30,000
P = 15(1,428.6) + 10(2,142.8) = 42,857
P = 15(2,500) + 10(0) = 37,500
The highest profit comes from x = 1,428.6 and y = 2,142.8.
The bottom line: The Fruitylicious line has a promising futureassuming, of course, that there is market demand for it and that the company’s supplies remain consistent. If Happy Trails wants to maximize profits, it should produce 1,429 packages of Classic trail mix and 2,142 packages of Fruitylicious trail mix.
Probability: Managing with Uncertainty
How did your teen’s car insurance rates get so high? The insurance company would probably tell you not to blame them—instead, blame probability.
Insurance companies rely on probability to estimate the likelihood of certain events occurring, such as who will get into a traffic accident. They incorporate data regarding age, driving history, and other factors. They are in the business of taking on risk for a price. If they want to make a profit and have competitive prices, they must use probability to assess that risk.
Probability has come a long way since its humble beginnings around 1654, when, fittingly, the first comprehensive theory of probability emerged from a pair of mathematicians—Pierre Fermat and
Blaise Pascal—and their friendly dispute over a gambling game.1 But what happens with probability doesn’t just stay in Vegas; it has countless applications throughout the business world, such as the valuation of capital expenditures, option pricing in finance, and quality control in manufacturing. If you’re dealing with uncertainty, your best bet is to use probability.
Practice Problem #2: Probability
Tomorrow the FDA is going to announce whether or not it has approved a new drug owned by the privately held XYZ drug company. If the drug is approved, a major shareholder will agree to sell his stock in XYZ for $50 per share. If the drug is rejected, he will accept $20 per share. A hedge fund has offered him $30 today, before the FDA announcement. At what probability is the expected payoff equal to the offer price?
Setup
To solve this probability problem, we’ll use a simple algebraic equation. P will represent the probability that the drug will be approved and the share price will be $50, while (1 - P) will represent the probability that the drug will be rejected and the share price will be $20. All of this will equal the hedge fund’s offer of $30. Thus
P(50) + (1 - P)(20) = 30
Solve
Use algebra to solve for P. First, we’ll simplify the (1 - P)(20) using order of operations, which gives us 20 - P(20). If we subtract the P(20) from the P(50) that we already have, the equation reads
P(30) + 20 = 30
We can then subtract 20 on both sides and divide both sides by 30, which gives us
P = 1/3, or 33.3%.
The bottom line: The hedge fund is estimating that the probability of the drug being approved is at least 33.3 percent.
Input-Output Analysis: Managing with Complexity
Does a Grateful Dead concert benefit the local economy?
According to a study in Las Vegas, it does—to the tune of at least $17 million.2 To determine the economic impact of the concerts, researchers used input-output analysis, a method designed to analyze the interactions between the various industries within an overall economy. The idea is that outputs from one industry often become inputs for another industry and vice versa. If we’re aware of a change in one sector of the economy—such as a concert—we can use input-output analysis to predict how that change will affect the overall economy.
The theory of input-output models was proposed by economist Wassily Leontief, who received a Nobel Prize in economics for his work. Leontief developed a matrix that detailed more than forty sectors of the economy and the web of relationships between their inputs and outputs.3 The resulting model captured the complexity and connectedness of the market and provided a way to view how changes in technology and demand affect production within a regional economy or business sector.
Practice Problem #3: Input-Output Analysis
Next year the city of Numbersville will celebrate its one hundredth anniversary. The director of the visitors bureau is deciding how the city should allocate the funds for its centennial. The projected amount of money coming into the city per day is $100,000 in space (lodging, attractions), $100,000 in consumables (food, gas, retail), and $40,000 in services. It has also been determined that a dollar of space requires 0.2 dollars of space (administration), 0.1 dollars of consumables (cleaning supplies, linens), and 0.2 dollars of services (cleanup, hotel staff). A dollar of consumables requires 0.3 dollars of space (back office, break rooms, storage), 0.2 dollars of consumables (boxes, dishes), and 0.2 dollars of services (wait staff). A dollar of services uses 0.3 dollars of space (worker housing), 0.3 dollars of consumables (workers’ meals), and 0.2 dollars of services (management, administration). How many total dollars in space, consumables, and services should be supplied for the celebration?
Setup
To solve this problem, we’ll need to solve a system of equations. An input-output analysis is represented by the equation X = AX + D, where X is a list of amounts of goods to be produced, A is a consumption matrix, and D is the external demand. Another way of saying this is that total consumption = internal consumption + external demand.
What we’re trying to find here is X. Specifically, we’re solving for x1, the dollars to be supplied for space; x2, for consumables; and x3, for services.
D is the projected total consumption mentioned earlier. If we use d1 for the demand for space, d2 for consumables, and d3 for services, we could say:
d1 = 100,000
d2 = 100,000
d3 = 40,000
A will be represented by an input-output matrix, which is basically a table expressing the relationship between each category.
x1 x2 x3
spac cons serv
spac 0.2 0.1 0.2
A = cons 0.3 0.2 0.2
serv 0.3 0.3 0.2
Now we’ll plug our figures into the equation X = AX + D to get this system of equations:
x1 = 0.2x1 + 0.3x2 + 0.3x3 + 100,000
x2 = 0.1x1 + 0.2x2 + 0.3x3 + 100,000
x3 = 0.2x1 + 0.2x2 + 0.2x3 + 40,000
Solve
We can use a method called Gaussian Elimination (or row reduction) to solve this system of equations. First we’ll isolate the variables on one side of the equation and multiply the first two equations by 10 and the third equation by -5. Thus we have equation #1 in scan
Adding the third row to the second and -8 times the third row to the first gives equation #2 in scan
Multiplying the second equation by 11 and adding that to 9 times equation one gives equation #3 in scan
If we divide both sides of the first equation by 184, we see that x3 = 175,000. Now we can plug that value into the other two equations and simplify to get that x2 = 225,000. Further simplification yields the solution with x1 = 275,000, x2 = 225,000, and x3 = 175,000.
The bottom line: The city can prepare for a celebration that will require $275,000 in space, $225,000 in consumables, and $175,000 in services.
MARKOV CHAINS: Managing with Change
What’s the secret behind the PageRank algorithm that launched Google to search engine stardom?
Although many of the details are under wraps, what isn’t a secret is that the concept of Markov chains influences PageRank. Markov chains provide an efficient framework for understanding the uncertain evolution of a system. They’re used to describe a process in which there is one possible “state” at any given time and to analyze the transitions between those states. In the case of Google, each web page would be considered a state. Thus the PageRank rating of a particular page would reflect the probability that a random web user would transition to that state, or web page.
The concept of Markov chains, which was first developed in 1906 by a mathematician named Andrey Markov, is used in a variety of ways. Its practical applications include market research analysis, stock and bond pricing models, and speech recognition programs.
Practice Problem #4: Markov Chains
The marketing team at Sparkle, a toothpaste company, wants to launch a new ad campaign: “Seize the Sparkle!” Before they spend their advertising dollars though, they want to know how much of an effect the campaign is likely to have. According to their focus-group research, someone who has seen the ad and has previously purchased Sparkle toothpaste has a 70 percent chance of continuing to purchase it, while someone who has seen the ad and has most recently purchased a different brand has a 48 percent chance of switching to Sparkle. Assuming that consumers buy toothpaste once a month, how much can Sparkle, which currently holds 32 percent of the market share, expect to see its market share increase after a month-long ad campaign?
Setup
To solve this problem we’ll determine our transition probabilities, put them into a transition matrix, then multiply the matrices we create.
Markov chains are all about calculating the probability that the system will move from one state to another. Here the state in question is which toothpaste brand a consumer uses. We’ll represent the options like this:
S = uses Sparkle brand
A = uses another brand
The probability of the system moving from one state to another is called the transition probability. We can illustrate these probabilities in two ways: a transition diagram and a transition matrix. For the diagram, we’ll start with two circles, one with S and one with A.
First, we’ll use an arrow to indicate the probability of moving from A to S (48 percent will become .48).
The complement would be that there is a 52 percent chance of staying at A.
We’ll also include arrows to indicate the probability of moving from S to A (30 percent) and of staying at S (70 percent).
Now we’ll illustrate these probabilities in a different way, as a transition matrix,
(to)
S A
(from) S .70 .30
A .48 .52
The S and A to the left of the bracketed section represent the current state (from); the S and A above the bracketed section represent the next state (to).
We also need to create a transition matrix to reflect the information about Sparkle’s market share of 32 percent. In probability terms, we could say there is a 32 percent chance that a random consumer will be a user of Sparkle toothpaste. We can express it
S A
S0 = [ .32 .68 ]
The solution we’re seeking will reveal Sparkle’s likely market share after one month of the campaign.
Solution
To determine the new market share, we’ll want to multiply the two matrices.
S0 = [ .32 .68 ] .70 .30
.48 .52
If you’re familiar with matrix multiplication, either by hand or on a calculator, you can go ahead and do that. Otherwise, one way to solve this problem is using a tree diagram. First, we’ll create two branches indicating Sparkle’s current market share.
Next, we’ll include branches to indicate the probability of moving from each state to another.
Viewing the probabilities this way can help us write it out in equation form:
P = (.32)(.70) + (.68)(.48)
= .22 + .33
= .55
The first part of the market-share matrix is
We can simply subtract .55 from 1.00 to determine the remainder of the matrix to create
S A
S1 = [ .55 .45 ]
The bottom line: After a month-long campaign, Sparkle is likely to see a significant increase in market share—from 32 percent to 55 percent. Clearly something to smile about!
Adding it all up
Tough decisions are a fact of life in the business world, but you don’t have to rely solely on intuition to make the right choice. When faced with challenges, an understanding of math concepts can help you make more informed decisions and, like Charlesworth, make a convincing case for the decisions you make.
When you pay attention, you’ll see that numbers really do tell a story. You may even find that—heaven forbid—you enjoy math. After struggling through a lengthy and difficult equation, it’s hard not to feel an exhilarating sense of accomplishment.
In his job as an analyst, Charlesworth uses math every day. But in his eyes, that’s a good thing. “Math is your ally,” he says. “You just have to know how to use it.”
New Class Adds Math to Management
When students walk into a math class, they’re usually less than thrilled to be there. This was often the case with a calculus course that was, until recently, required for students applying to the Marriott School. Now the school is requiring a new course called Finite Math.
“I think students will see this class differently,” says associate professor Jeffrey Humpherys of the BYU Math Department. “They’ll find that it will be reinforced in their business courses.”
Although the calculus taught in the previous course provided intellectual rigor, faculty from the Marriott School and the math department saw a need for curriculum focused on practical skills. Humpherys, who taught a similar course at another university, worked with Brent Wilson, professor of finance and Marriott School director of undergraduate programs, and other faculty to fine-tune Finite Math, which premiered last semester.
The class gives students an arsenal of practical math skills that will help them solve real-world business problemsand tackle more advanced concepts in later courses. Some of the topics covered include permutations, probability, linear programming, and Markov chains.
In its first semester, the class received positive feedback from students. “I think this is a more realistic approach,” says Kevin Hatch, a sophomore premanagement major who plans to study strategy. “The class emphasizes making businesses more efficient, and I think that will help me be a better consultant for companies in the future.”
Wilson says he hopes students will not only gain knowledge of specific principles but also a new perspective of math.
“I’ve taught for thirty years,” he explains, “and I’ve noticed a lot of students get scared when I put an equation on the board. We want students to be comfortable with math. Most important, we want them to be able to think quantitatively as they approach management decisions.”
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Article written by Holly Munson
Illustrated by Serge Bloch
About the Author
Holly Munson is a writer trying to love solving equations as much as she loves diagramming sentences. She graduated from BYU in 2010 with a degree in communications and lives in Philadelphia with her husband, Dave.
Notes
- Lorraine Daston, Classical Probability in the Enlightenment. Princeton, New Jersey. Princeton University Press (1988): 15–18.
- Ricardo C. Gazel and R. Keith Schwer, “Beyond Rock and Roll: The Economic Impact of the Grateful Dead on a Local Economy,” Journal of Cultural Economics, 21.5 (1997): 41–55.
- J. Steven Landefeld and Stephanie H. McCulla, “Wassily Leontief and His Contributions to Economic Accounting,” Survey of Current Business, 79.3 (March 1999): 9–11.
Marriott Alumni Magazine expresses appreciation to professor Jeffrey Humpherys for lending his technical expertise to this article
