🤖 machine learning

Optimizing ARDL Models for Retail Sales Forecasting and Fair Pricing

This paper proposes a fairness-aware retail pricing framework that utilizes log-log ARDL models with CPI-anchored constraints and Simulated Annealing optimization to prevent consumer exploitation by addressing inflation-driven positive price elasticities while balancing sales targets with consumer welfare.

Original authors: Sujay Uday Rittikar

Published 2026-07-14✓ Author reviewed ⓘ

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Sujay Uday Rittikar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are the captain of a giant grocery ship sailing through the foggy waters of the Canadian market. Your job is to sell food—apples, eggs, milk, bread, and coffee—to millions of people. You want to make sure your ship is full (maximizing sales), but you also want to make sure you aren't robbing your passengers blind.

For a long time, some captains have used "dynamic pricing," which is like a surfer riding the waves of demand: if everyone wants a surfboard, you hike the price up to the moon. But this paper asks a tough question: Is that fair? If you hike prices too high, you might be exploiting people, especially when the cost of living is already rising.

The Magic Crystal Ball (The ARDL Model)

To solve this, the author, Sujay, built a special crystal ball called an ARDL model. Think of this model as a time-traveling calculator that looks at the past to guess the future. It doesn't just look at how many apples were sold last month; it looks at how the price of apples then affected sales now, and how the price before that affected sales then.

The author fed this crystal ball data from January 2017 to August 2024, covering the whole of Canada. The goal was simple: Predict how to set prices to sell the most food, but with a strict rule: You cannot charge more than what the "Consumer Price Index" (CPI) says is fair. The CPI is like a government thermometer that measures how much prices are rising across the country due to inflation.

The Surprise: The "Upward Slope" Glitch

Here is where the story gets weird. When the author ran the numbers on the "nominal" prices (the actual dollar amounts on the shelf, not adjusted for inflation), the crystal ball told a strange story.

It suggested that higher prices actually lead to higher sales.

Yes, you read that right. In this specific simulation, the model thought that if you raised the price of milk or eggs, people would buy more of them. This is like a video game glitch where buying a sword makes your character stronger, even though in real life, expensive swords usually make you buy fewer of them.

The paper explains that this isn't because people love paying more; it's because everything is going up at the same time. Prices are rising, and total sales are rising, all because of general inflation. The model got confused and thought the rising prices were causing the rising sales.

The Two Navigators: The Robot vs. The Human

To figure out the best price, the author used two different "navigators" to steer the ship:

The Robot (Linear Programming): This navigator is a strict logic machine. It sees the glitchy "higher price = more sales" rule and thinks, "Okay, to sell the most, I should charge the absolute maximum allowed price!" So, it pushes every price up to the very top of the CPI limit. It's like a robot that says, "If the speed limit is 100, I will drive at 100.0001." It finds the mathematical extreme, but it's not very friendly to the passengers.
The Human (Simulated Annealing): This navigator is a bit more cautious. It uses a method called "Simulated Annealing," which is like a metal cooling down. It starts with a random price and slowly "cools" its choices, testing different prices to see what works. It doesn't just rush to the maximum limit. Instead, it finds a "sweet spot" in the middle. It suggests prices that are lower than the maximum limit, saving money for the shoppers, while still hitting the sales targets.

The Results: What Actually Worked?

The author tested these methods on five specific foods: Apples, Eggs, Milk, White Bread, and Roasted/Ground Coffee.

The Robot's Plan: It wanted to push prices up to the ceiling. For example, for apples, it suggested a price jump of +25%.
The Human's Plan: The Simulated Annealing method suggested much more conservative prices. For apples, it suggested a price change of -6.57% (meaning cheaper than the maximum allowed).

When the author checked how well these methods predicted the future sales, the "Human" method (Simulated Annealing) did a better job of balancing the books. It found prices that were fairer to consumers but still met the sales goals.

The Reality Check: The "Inflation Artifact"

The paper is very honest about a major limitation. The author admits that the "higher price = more sales" finding is likely a glitch caused by inflation.

To prove this, the author ran the experiment again, but this time they "deflated" the numbers. Imagine taking a photo of a rising tide and then subtracting the water level to see the actual rocks underneath. When they adjusted for inflation (looking at "real" prices instead of just dollar amounts), the weird glitch disappeared.

For White Bread and Coffee, the relationship flipped to normal: higher real prices meant lower sales.
For the other items, the effect became tiny or zero.

This means the "Robot" was reacting to a fake signal (inflation), while the "Human" method, by staying in the middle, accidentally avoided the worst of that trap.

The Bottom Line

This paper doesn't claim to have solved the mystery of pricing forever. In fact, the author admits that if you just want to predict how many items will be sold next month, a simple guess (like "it will be the same as last month") is actually more accurate than this fancy model.

However, the paper's real victory is fairness.
It shows that if you use a standard computer program to maximize sales, it will try to charge you the highest possible price allowed by law. But by using a smarter, "human-like" search method (Simulated Annealing) and anchoring prices to the CPI, retailers can find a middle ground. They can still sell their goods without pushing prices to the absolute limit, keeping the "exploitation" out of the equation.

The paper suggests that while the math can get tricky and sometimes confused by inflation, there is a way to use these tools to keep grocery prices fair, transparent, and anchored to the real cost of living, rather than just chasing the highest possible profit.

Technical Summary: Optimizing ARDL Models for Retail Sales Forecasting and Fair Pricing

Problem Statement
Retailers face a dual challenge: maximizing revenue through dynamic pricing while avoiding consumer exploitation and perceived unfairness. While dynamic pricing is effective for profit maximization, it often relies on behavioral biases or opaque strategies that can lead to unethical outcomes, particularly when prices are decoupled from genuine consumer valuations. The paper addresses the specific risk of consumer exploitation in the Canadian food retail sector. The core problem is to develop a forecasting and pricing framework that maximizes retail sales but strictly adheres to fairness constraints, ensuring prices do not exceed levels justified by inflation trends.

Methodology
The proposed methodology integrates econometric modeling with constrained optimization to create a fairness-aware pricing system.

Data and Variables: The study utilizes monthly data from Statistics Canada (January 2017 – August 2024), including the Consumer Price Index (CPI) for food, retail prices for five specific food items (apples, eggs, milk, white bread, roasted/ground coffee), and total retail trade sales.
Model Specification: The authors employ a log–log Autoregressive Distributed Lag (ARDL) model. This specification is chosen because:
- It accommodates non-stationary time series (confirmed via Augmented Dickey–Fuller tests showing I(1) integration).
- It allows for the inclusion of exogenous variables (prices) and lagged dependent variables (sales).
- The coefficient on the log-price ( $\gamma$ ) directly represents the sales price elasticity.
- The model is tested in two configurations: Single-product (one price as an exogenous variable) and Multi-product (all five prices as exogenous variables).
Optimization Framework: The pricing problem is formulated as maximizing forecasted sales ( $\ln \hat{y}$ $ln \overset{y}{^}$ ) subject to two sets of constraints:
- Fairness Constraints: Prices are bounded by a CPI-adjusted ceiling ( $p_{max}$ ) and a lower bound ( $p_{min} = 0.65 \times p_{max}$ ). This anchors prices to inflation, preventing unbounded increases.
- Plausibility Constraints: Forecasted sales are bounded within a realistic range derived from historical volatility.
Solution Algorithms:
- Linear Programming (LP): Used to solve the optimization problem exactly. Given the linearity of the objective function and the box constraints, the LP solution yields "corner points," driving prices to their upper or lower bounds depending on the sign of the elasticity.
- Simulated Annealing (SA): A metaheuristic approach used to explore the solution space stochastically. Unlike LP, SA does not commit to a vertex solution. It is employed to find "interior" prices that are lower than the CPI ceiling while still meeting sales targets, thereby trading a marginal amount of predicted sales for significant consumer cost reductions.

Key Results

Positive Nominal Elasticities: A critical diagnostic finding is that the fitted nominal price elasticities ( $\gamma$ ) in the single-product models are positive. This implies that, within the raw data, higher prices correlate with higher aggregate sales. Consequently, an unconstrained sales-maximizer (the LP solution) would drive every price to its maximum allowable limit (the CPI ceiling).
Inflation Artifact: The paper demonstrates that these positive elasticities are largely an artifact of general inflation. When sales and prices are deflated by the CPI (real-terms specification), the elasticities for bread and coffee become negative (economically sensible), and others shrink toward zero. This confirms that the nominal model captures inflationary trends rather than a causal demand response.
Algorithmic Performance:
- LP vs. SA: The LP models produce extreme price recommendations (often +10% to +25% above baseline) to maximize sales. In contrast, the SA models settle on conservative, interior prices, resulting in lower consumer costs (negative percentage changes relative to the CPI ceiling) while still achieving the sales targets.
- Forecast Accuracy: The ARDL models do not outperform simple univariate baselines (Naive, Seasonal Naive, ARIMA, SARIMA) in terms of raw forecast accuracy (RMSE/MAPE). In fact, the simple baselines achieved lower error rates.
- Multi-product vs. Single-product: The Multi-product ARDL-SA model generally showed better percentage change and MAPE scores for specific products compared to the single-product configurations.

Significance and Claims
The paper explicitly frames its contribution as interpretability and fairness rather than raw predictive power.

Transparency: The primary value of the ARDL model is not its ability to predict sales more accurately than a naive forecast, but its ability to provide an interpretable, elasticity-based relationship between price and sales. This transparency is essential for embedding fairness constraints into the optimization process.
Fairness Mechanism: The study proves that without external constraints (like the CPI ceiling), a data-driven sales maximizer would systematically exploit consumers by raising prices to the maximum allowable limit due to the inflation-driven correlation in the data.
Practical Utility: The Simulated Annealing approach offers a practical mechanism for retailers to meet sales objectives while voluntarily adopting more consumer-friendly, interior pricing strategies, effectively balancing business goals with consumer welfare.

Limitations
The authors modestly acknowledge several limitations:

The analysis relies on aggregated nationwide data with a limited test set (18 months), restricting statistical power.
The model is descriptive, not causal; it does not perform formal cointegration testing or address price endogeneity.
The positive nominal elasticities are recognized as an inflation-driven confounding factor rather than a genuine demand phenomenon.
The optimization objective is linear, making the LP solution a degenerate corner point; thus, the metaheuristic (SA) is used not to escape local optima, but to generate feasible interior solutions.