How Much of Each Product Should I Make for the Most Profit?

Editor’s note

This problem is colloquially called ‘what should I produce?’, ‘which order has priority?’ or ‘product mix’. In academic terms it is the Product Mix Problem or Resource Allocation Linear Programming; it was the first wide industrial application of linear programming (LP — the math method that finds the best mix under constraints). After George B. Dantzig developed the simplex algorithm in 1947, the first industry to apply LP in the 1950s was oil refining (mixing gasoline / diesel / asphalt from refinery output for maximum profit); from there it spread to all manufacturing and food industry.

Not to be confused with problem 44 (Aggregate Production Planning — APP). APP sits one layer above: across a 3-18 month horizon it jointly decides production rate, inventory, workforce, hire-fire, overtime and subcontracting. Product Mix LP is smaller and single-period: this week or this month, with capacity and raw material treated as given, decide how much of each product to make. APP gives monthly product-family rates; Product Mix LP breaks those into weekly product-level quantities.

Not to be confused with problem 16 (Wagner-Whitin lot sizing) either. Wagner-Whitin is single-product, multi-period: “produce this month, how much?” — the trade-off between setup cost and holding cost. Product Mix LP is multi-product, single-period scarce-resource allocation. They combine: Product Mix LP picks how much of each product this period, then Wagner-Whitin sets each product’s lot size.

The most often missed point: The decision should be made by margin per bottleneck-machine hour, not by unit profit. Eliyahu Goldratt’s Theory of Constraints insists on this: use each bottleneck-machine hour for the product that brings the most margin, not the highest sale price. A product giving 1000 TRY profit per unit and using 4 hours of the bottleneck returns 250 TRY/hour on that machine. Another giving 600 TRY profit per unit and using 1.5 hours returns 400 TRY/hour. The second one wins. Gut feel almost always picks the first, because its unit profit is bigger — but per hour, the table flips.

Second often-missed point — reading the shadow price. The LP gives a marginal value for each bottleneck resource: “if I had one more hour of this resource, how much would my monthly margin grow?” Say it comes out as 380 TRY/hour. Compare with overtime premium: if the premium is 250 TRY/hour, opening overtime pays (net 130 TRY/hour). If the shadow price were 180 TRY/hour, overtime would lose money. The same number anchors the payback calculation for a new machine or a subcontractor deal.

Step-by-step path — for an SMB

Stage 1 — Measure first, plan second. For the next 4-12 weeks, collect per product: unit margin (sale price minus variable unit cost — read to the kuruş from accounting), bottleneck-machine hour/unit, kilograms raw material/unit, labour hour/unit, weekly customer demand ceiling. Capacity inventory: weekly net machine hours (minus planned maintenance, shift changes and stoppages), weekly raw-material stock arrivals, weekly net labour hours.

Stage 2 — Set up the model. In a spreadsheet, one row per product, one column per resource. With the Solver add-in, declare decision variables (quantity of each product), constraints (total consumption of each resource cannot exceed capacity, each product cannot exceed its demand ceiling, quantities cannot go below zero), and the objective (maximum total margin). As a pilot, solve last week’s real situation (real order pool, capacity, prices); compare the output to the call you made by hand. A first pilot typically shows 5-15% margin improvement.

Stage 3 — Automate. Data flow: unit margin from accounting, demand ceiling from sales, resource capacities from production. The weekly model runs on its own, the output drives the production schedule and the sales order-acceptance call. Shadow-price report appears on the monthly management dashboard.

Stage 4 — Roll out. Over 6-12 months: full product line, multi-bottleneck, automatic re-computation when raw-material prices change, shadow-price-based recommendations for extra-machine or extra-shift investment. If demand uncertainty is material, add a stochastic or robust extension as a separate module.

Risks — what can go wrong

1. Unit-cost error. Unit margin comes from accounting; if overhead is misallocated (e.g. shared machine cost split equally rather than by usage), the decision pushes the wrong product. Audit the cost system first.
2. Demand-ceiling error. Overstate the demand ceiling and the optimum turns into unsold inventory; understate it and the margin you could have earned slips away. Anchor the ceiling in sales actuals from the last 6-12 months, not a forecast.
3. Bottleneck drift. As the mix changes, the bottleneck machine can change too; the assumption that the same machine stays tight grows stale. Monthly re-solve plus a shadow-price report keeps an eye on the bottleneck.
4. Raw-material uncertainty. A single-point LP is fragile under price or supply shocks. For critical materials, run 2-3 scenarios (base / low / high) as multi-scenario or robust LP. Single-supplier dependence locks the product mix to that supplier’s state.

Solver landscape — a technical view

Objective choice:

• Maximize total margin: standard objective. Sum of (quantity × unit margin) across all products.
• Maximize customer fulfilment rate: if strategic-customer order completion is the priority, weight key products higher within the demand ceiling.
• Minimum bottleneck idle: maximum capacity utilization. Usually agrees with the total-margin objective; if it diverges, capacity is being kept full for strategic reasons.
• Risk-adjusted margin (margin under bad scenarios): protect margin under demand uncertainty — needs a stochastic or robust extension.

If multi-objective: either weighted sum (total margin + customer-priority weight), or hierarchical (strategic-customer fulfilment first, then total margin).

Academic references

Listed in the sources field at the top of the page.