#### Summary

An earlier Sudoku blog recognizing Puzzle Day, provided an overview of solving Sudoku using MILP optimization and mentions these methods are helpful to find solutions in supply or central planning. This blog elaborates on “binary variables” which is the connecting technology between Sudoku and Supply Chain Management (SCM). A binary variable is only allowed the values 0 or 1 and provides a computational method to handle qualitative information. For example, is the factory producing red shirts today or not producing red shirts today. RS_B would have the value 1 when “it is producing red shirts” and 0 when it is not. This can be used to support the business policy that if red shirts are being produced today no other color shirts can be produced today. To get the most out of Binary technology requires an experienced team.

#### Introduction

An earlier Sudoku blog recognized National Puzzle Day by providing information on solving the Sudoku Puzzle using MILP (mixed-integer linear programming) optimization methods in Arkieva and mentioned that the methods used to solve Sudoku are helpful with supply chain management (SCM) challenges. The purpose of this blog is to elaborate on this connection which is the “binary variable”: basic explanation, the importance this technology offers for smarter solutions in supply or central planning, and the importance of an experienced team to bring “smart” to your demand-supply network (DSN).

#### What is a binary variable?

It is a variable that can have one of two values: zero (0) or one (1) and has a long analytics history as a numeric way to represent the status of a certain situation, sometimes called an indicator or logic value. Most variables represent quantities – how many red shirts am I planning to produce today, how much milk do I have in stock. If the price for a red shirt is \$5 per shirt, then my revenue is calculated as: revenue = \$5 x RS_Q, where RS_Q is the quantity of red shirts.

What if I only wanted to know if the factory was producing red shirts today or not? We could call this variable RS_B, where it had one of two values: “red shirts are being produced” or “red shirts are not being produced”. A variable can have a character string as a value that has meaningful information. However, we cannot do any numeric or logical computations on character strings and they require lots of extra writing. The concept of a binary variable was developed where 0 and 1 are assigned arbitrarily to represent each of the two possible states. For this example:

1. 0 – red shirts are not being produced.
2. 1 – red shirts are being produced.

The “B” in RS_B stands for binary (sometimes the term Boolean is used). RS_B must have the value 0 or 1. Binaries have a long and distinguished history in statistics (binomial and dummy variables) in addition to MILP.

#### How can a binary variable help in SCM?

Assume my factory can make red shirts, yellow shirts, or pink shirts, but on any given day it can only make one of the three colors. It is too expensive and risky to change over the machines to handle a different color during the day. How can the binaries help? I create three binary variables: RS_B, YS_B, and PS_B.

1. RS_B
1. RS_B = 0, red shirts are not being produced.
2. RS_B= 1, red shirts are being produced.
2. YS_B
1. YS_B = 0, yellow shirts are not being produced.
2. YS_B= 1, yellow shirts are being produced.
3. PS_B
1. PS_B = 0, pink shirts are not being produced.
2. PS_B= 1, pink shirts are being produced.

I use equation (a) to ensure no more than one color is produced on a given day.

(a) RS_B + YS_B + PS_B =< 1, each variable can only have the values 0 or 1.

What if I wanted to ensure the factory was making one and only one color each day, that is no idle capacity. Then I would slightly modify equation (a) to be =1, shown in equation (b)

(a) RS_B + YS_B + PS_B =1, each variable can only have the values 0 or 1.

The binary variable provides a computational method to capture an important business policy, essentially a method to represent knowledge often working in conjunction with a quantitative variable.

#### The quantitative partner for the binary variable.

For the factory, a critical decision is determining how much of each color shirt to make each day. These are RS_Q, YS_Q, and PS_Q – the quantity of each color to make on a given day. Obviously, the quantitative partner can only be greater than 0 if its binary partner is 1. Equation (c) captures this relationship for red shirts.

(c)  RS_Q = (RS_B x RS_Q)

When RS_B is 0, then RS_Q must be 0 for this to be true. When RS_B is 1, then this statement is true for any value of RS_Q. The hiccup with equation (c) is it is not a linear equation – this is a topic for another time.

#### Life Before Binary Variables

Although the binary variables have been part of the theory of MILP models since the early days, the ability of solvers (finding optimal production quantities to meet demand and business requirements without exceeding material and capacity constraints) to effectively handle the binary variables is reasonably recent. Before binaries, the optimization used only the quantitative variables to find production quantities for large time buckets (7 to 30 days). Then various rules of thumb and manual processes were employed to find a solution that met the business requirement of one product per day without concern for the “smarts” of a solution. Given the amount of effort to find a “feasible solution”, there was no time to search for a “smarter” solution.

#### Binaries are valuable and tricky.

Binaries are clearly a technology that can help a firm find “smart or intelligent” solutions. At Arkieva some of our customers use this technology regularly to represent and solve very complex situations. However, as with many advanced technologies, to be successful requires partnering with a team that has years of applied experience mixed with the computational theory to ensure this and other technologies provide an answer of value – capturing the complexity needed but solving in a time that works within the rest of the business process. If not, the firm will find itself in one of the two traps:

1. A firm that does not dominate this technology will recommend ignoring the complexity and the firm will later need workarounds in Excel or simply not use the model.
2. A firm that loves the technology, but lacks applied experience, will deliver models that outstrip the organizational cognitive mindset and take too long to solve.

#### Related Technologies

An alternative technology to binaries has emerged to handle the numerous “Sudoku” type challenges (assignment, in Sudoku we assign one number or each cell) that populate the demand-supply landscape. Instead of binaries, this technology uses sets. Arkieva, as your trusted advisor on analytics technology without borders, is up to speed on this technology. Look for a forthcoming blog on sets and Sudoku. Interestingly the origin of this technology occurs simultaneously as binaries and often both can be applied to similar computational challenges.

#### Conclusion

Although we often do not associate the ongoing challenge of improved performance of a supply chain or demand-supply network with puzzles, in fact, these challenges are “supersized” puzzles where success requires the application of “smart technology” without concern with the origins of this technology. Successful use of this technology requires working with an experienced team.