By LinLin Yang, Principal Data Scientist, Aimpoint Digital, A Gurobi Alliance Premier Service Partner

Effective communication is crucial in business interactions, especially when explaining complex technical ideas to non-technical decision-makers. However, there is often a disconnect between the languages spoken, with technical jargon incomprehensible to those without the same background and training. This results in unproductive meetings and hinders business decision-making.

One particularly challenging communication area is modeling decision-making under uncertainty in operations research. Ironically, most business decisions involve varying degrees of uncertainty in demand, cost, and disruptions. Stochastic programming (SP) is a modeling approach that optimizes decision-making based on quantified probability distributions of uncertainties.

However, explaining stochastic programming to a non-technical audience, who may prefer more deterministic approaches, can take time and effort. Additionally, solving stochastic programming models mathematically is a complex task. After investing significant effort in data collection, model development, and solution analysis to align with business goals, effective communication with stakeholders is crucial to gain support.


The Key Is to Consider the Audience

Figure 1 – Typical Academic Journal Paper Outline

Consider the audience when preparing communication, focusing on what will resonate and motivate them. In academia, most people follow a specific sequence for presenting research work and writing journal papers (as depicted in Figure 1). This may involve starting with a literature review, followed by providing the model background, introducing a novel algorithm for solving the model, showcasing its performance improvement, and concluding.

However, in business communications (oral or written), the sequence of narration should be revised and instead focus on the recommendation and the why (Figure 2). Let’s break it down. First, we would begin with the recommendation. Many factors could distract the audience on any given day, and placing key recommendations in the body of communication risks burying them.

Then we need to build trust with our audience. We should consider other models or heuristics that the audience might believe to be promising approaches and explain why our approach is better. The goal here is to help the audience build intuition for the recommendation without having to explain the mathematics behind the model. 

Figure 2 – Recommended Presentation Sequence in Business Settings

Once the recommendation is deemed sensible, the last step is understanding the factors driving the solution. By pinpointing parameters with the most significant impact on decision-making, we can better focus the discussion on these parameters.

Depending on the audience’s background and meeting cadence, we expect some variation in the delivery sequence, but in general, it’s better to start with the business recipe outline.


Illustrative Example: Delivery Contract Under Uncertainty

Let’s explore how we can utilize the two sequences to explain a stochastic programming solution to a non-technical audience.

We have an online chocolatier who relies on 3rd party carriers to deliver their online orders. The vendor expects a large order quantity in February and wants to secure a more favorable pre-order delivery contract with the carrier in January. The major decision for the vendor to make is the pre-order delivery quantity.

Table 1 – Tiered Delivery Rate for Pre-Order Contract

If the vendor waits until demand is known in February, the retail delivery cost will be $8 per box. On the other hand, the pre-order delivery contract follows a tiered pricing scheme, as shown in Table 1. As the pre-order quantity increases, the marginal quantity gets a further delivery discount per item. If the order quantity is <5 M boxes, the price is the same as the retail cost, and there’s no incentive to pre-order. The downside to the pre-order contract is that it’s use-it-or-lose-it, meaning if the actual demand is lower than the pre-order quantity, there is no refund for the chocolate vendor.

The vendor’s demand forecast for online sales in February follows a normal distribution with expected demand of 10 M boxes and a standard deviation of 2 M boxes (Figure 3).

Figure 3 – Demand Distribution


You can solve the problem with a two-stage stochastic programming model; we generated 100 scenarios based on demand distribution. The result indicates that a pre-order quantity of 7.7 M boxes is optimal.


Explaining the Solution Following the Academic Recipe

How would an optimization modeler typically present the solution?

A two-stage stochastic programming model is a mathematical optimization model used to solve decision-making problems under uncertainty. The model consists of two stages: the first stage involves making a near-term decision before knowing the realization of the uncertain parameter, while the second stage involves adjusting the decision based on the actual realization of the uncertain parameter.

Stochastic programming has the benefit that it results in a single solution for the first stage decision instead of a different decision with each scenario (as one would if scenarios with varying quantities of demand are solved independently).


Figure 4 – Two-Stage Stochastic Programming


Equation 1 – General Form of SP

In this case, the first-stage decision is the pre-order contract quantity, and the uncertain parameter is the actual demand. The model takes a general form, as shown in Equation 1.

Where the objective is to minimize the summation of [Equation] – the first stage cost (contract cost) and [Equation]– the expected cost of all 2nd stage scenarios.

The recommendation is to pre-order a delivery contract of 7.7 M boxes.

When comparing the solution against pre-purchasing 10M boxes for delivery, several demand scenarios yield a significant saving of > \$1 per item (as shown in Figure 5), with an expected total saving across all scenarios being \$2.6 M.

Figure 5 – SP Savings Compared to Heuristic


There are two main issues with this version of result delivery.  First, by starting the communication with an explanation of how stochastic programming works, the modeler risks losing the audience.  The assumption of introduction stochastic programming concepts includes:

  • One needs to explain the mathematical principle to be considered technical and to be trusted
  • The audience can learn how stochastic programming works with a brief and abstract explanation
  • The audience is interested in the explanation and mechanics of the model


These assumptions don’t typically hold in business settings.

The second issue is the chart presented to demonstrate stochastic programming benefits is severely underselling the result. While the expected saving is a large positive number, the chart visually tells a different story. The large bar in the [-0.3-0] $/box interval is so much taller than the other bars that the audience would immediately fixate on the significant “negative cost savings.”

This sequence of narration only works well with a technical audience.


The Better Approach: Explaining the Same Solution Using the Business Recipe

Now let’s go through an explanation of the solution the way we would in a business setting.

We recommend purchasing a pre-order contract with 7.7 M boxes. It’s the best solution, as can be seen in Figure 6. Our solution offers a significant cost saving compared to using the average demand as the pre-order quantity. We would not recommend pre-ordering more than 9 M boxes because that would result in a higher cost than waiting for February to pay the retail delivery price after demand is known.

Figure 6 – Expected Cost by Pre-Order Quantity


We can take it a step further and examine the sensitivity of the solution. We assume a demand forecast with a standard deviation of 2 M boxes. Figure 7 shows how the confidence level in the forecast (i.e., change in the standard deviation) influences the decision.

Figure 7 – Cost Sensitivity to Forecast Confidence


In the current scenario, the expected cost with 2 M boxes of standard deviation is not very sensitive to pre-purchase quantity (areas around the dotted line). A slightly higher pre-order quantity will yield a similar expected cost if the forecast has more confidence.

Let’s also get a sense of the extremities. The graph’s left side represents absolute confidence in the demand forecast. In this case, we recommend pre-ordering 10 M boxes (since we’re confident that the demand will be 10 M) to reap the maximum benefit of the pre-order contract discount.

On the other hand, if there’s no confidence in the forecast, we recommend not pre-ordering the delivery contract. The reason for that is the risk is asymmetric – the cost of over-purchasing pre-order quantity (\$8/box) is significantly higher than any discount (<\$2.5/box) from under-purchasing pre-order quantity.

Note that we don’t mention stochastic programming at all in this version of the explanation. Skipping the explanation of how SP works allows the audience to be better focused on the main priority – the pre-order quantity decision. In addition, the explanation of sensitivity gives the audience a better understanding of the forecast assumption’s impact on the decision.


Enhancing Business Decision-Making with Aimpoint Digital

The goal that we aim for is to improve business decision-making with technology. Letting the audience dictate the storytelling and rethinking the communication sequence could simplify and expedite technology adoption.

As we’ve discussed, mathematical optimization is a powerful technology for a wide range of practical use cases. While it is over 70 years old, many organizations are still learning to make the most of its capabilities. That is why Gurobi established the Value Accelerator Program. We want you to be successful with optimization, so start with 12 free hours of guidance with Aimpoint Digital at no cost.

Please check out the Value Accelerator Program to learn more and visit Aimpoint Digital if you want to learn more about how they can support your business. Aimpoint Digital is a Gurobi Alliance Premier Service Partner.


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