Sales Discount Optimization

Purpose

This calculator recommends a discount that maximizes expected monthly profit for a sales offer.

It captures a realistic trade-off:

• Higher discount can increase Conversion Rate.
• Higher discount also reduces Unit Contribution Margin.
• Capacity (Maximum Fulfillable Units) and policy limits (Minimum Unit Margin) can constrain the best decision.

Inputs

• Number of Prospects: Potential customers exposed to the offer in the analysis period.
• List Price: Undiscounted selling price per unit.
• Unit Cost: Variable cost per unit.
• Base Conversion Rate: Conversion rate at zero discount.
• Maximum Conversion Rate: Saturation ceiling for conversion as discount increases.
• Sensitivity Factor (k): Controls how quickly conversion rises with discount. This is dimensionless (not a percent input). Suggested practical range: 3 to 20.
• Sales Cost per Prospect: Acquisition or outreach cost per prospect.
• Current Discount: Existing discount policy used for comparison outputs.
• Minimum Discount: Lower bound of the discount search range.
• Maximum Discount: Upper bound of the discount search range.
• Maximum Fulfillable Units: Capacity cap on sellable units in the period.
• Minimum Unit Margin: Minimum allowed unit margin for discount feasibility.
• Optimization Points: Number of trial discount points used in the optimization search.
• Scenario Points: Number of points shown in the scenario table and chart curves.

Results

• Recommended Discount: Best discount found by optimization.
• Current Discount: Reference discount used for comparison.
• Profit at Current Discount: Expected profit at the current discount.
• Profit at Optimal Discount: Expected profit at the recommended discount.
• Profit Lift vs Current Discount: Absolute monthly gain from moving to the recommended discount.
• Profit Lift % vs Current Discount: Percent profit improvement versus current discount.
• Expected Conversion Rate: Expected conversion at recommended discount.
• Expected Units Sold (Demand): Demand-side expected units before capacity cap.
• Expected Units Sold (Capped): Units sold after applying capacity cap.
• Net Selling Price: Effective selling price after recommended discount.
• Unit Contribution Margin: Margin per sold unit at recommended discount.
• Revenue: Expected revenue at recommended discount.
• Gross Contribution: Units sold times unit contribution margin.
• Sales Cost Total: Prospect-level selling/acquisition cost for the period.
• Profit per Prospect: Profit normalized by number of prospects.
• Profit at Zero Discount: Baseline profit at zero discount.
• Profit Lift vs Zero Discount: Improvement versus zero-discount baseline.
• Capacity-Limited at Optimum: Indicates demand exceeds fulfillable units at optimum.
• Margin-Floor Active at Optimum: Indicates optimum is on the minimum-margin boundary.
• Boundary Warning: Indicates optimum occurs at search boundary (Minimum Discount or Maximum Discount).
• Discount Scenarios: Table across discount points with conversion, units, pricing, and profit metrics.

Charts

The calculator prepares chart outputs for visual analysis:

• Optimization Chart: Profit/cost behavior versus discount.
• Conversion vs Margin Chart: Conversion response and margin compression versus discount.
• Demand vs Capacity Chart: Demand-side units versus capacity-capped sold units.

Understanding the Optimization

• The model computes expected results for trial discounts from Minimum Discount to Maximum Discount.
• Only points satisfying Minimum Unit Margin are considered feasible.
• The feasible trial point with highest Total Profit becomes Recommended Discount.
• Optimization Points controls search resolution; larger values can slightly shift the reported optimum.
• Scenario Points controls reporting granularity (table/chart density), not optimization logic.

Input Clarification

• Sensitivity Factor (k) is not a % field. Enter plain numeric values such as 5, 10, or 15.
• Larger k means faster conversion response at low discount levels; smaller k means a flatter response curve.
• Start with k in the 3 to 20 range and calibrate from historical data.

Example (Current Defaults)

With current defaults, the calculator returns approximately:

• Recommended Discount: 10.85 pct
• Current Discount: 15.0 pct
• Profit at Current Discount: 5816.31 USD/mo
• Profit at Optimal Discount: 6762.62 USD/mo
• Profit Lift vs Current Discount: 946.31 USD/mo (16.27 pct)
• Expected Conversion Rate: 14.93 pct
• Expected Units Sold (Demand): 1493.15 unit/mo
• Expected Units Sold (Capped): 1493.15 unit/mo
• Capacity-Limited at Optimum: False
• Boundary Warning: False

Interpretation:

• In this configuration, reducing discount from current policy improves profit by recovering margin faster than conversion drops.
• Capacity is not binding at the optimum (Expected Units Sold (Demand) equals Expected Units Sold (Capped)).

Important Assumptions and Interpretation

• Conversion follows a saturating response curve with discount; it does not increase indefinitely.
• Sales Cost per Prospect is modeled as fixed per prospect and independent of discount.
• Capacity is represented as a hard cap (Maximum Fulfillable Units).
• This is a single-period offer optimization; seasonality, competitor response, and inventory dynamics are not modeled.
• Results are decision support estimates and should be validated with market/segment context.

Formula References

Objective:

$$\max_d\ Profit(d)$$

Conversion model:

$$CR(d)=CR_0 + (CR_{max}-CR_0)\left(1-e^{-k d}\right)$$

Demand and sold units:

$$Q_{demand}(d)=Prospects \times CR(d)$$

$$Q_{sold}(d)=\min\left(Q_{demand}(d),\ Q_{cap}\right)$$

Price and margin:

$$P_{net}(d)=P_{list}(1-d)$$

$$M(d)=P_{net}(d)-C_{unit}$$

Profit:

$$Profit(d)=Q_{sold}(d)\times M(d)-Prospects\times C_{sales}$$

Feasibility rule:

$$M(d) \ge M_{min}$$