Consider a Nigerian woven-packaging manufacturer producing four types of bags:

• Standard 50kg rice sacks

• Printed 50kg rice sacks

• Fertilizer bags

• Animal-feed bags

Each product earns a different contribution per bag. Each also consumes different amounts of the factory's scarce resources.

The management problem is therefore not simply:

“Which product makes the most money?”

It is:

“Given the resources we have available, what combination of products produces the greatest total contribution?”

And the answer turns out to be less obvious than it looks.

The factory

For this illustrative case, assume the manufacturer has three important production resources:

1. Polypropylene — the primary material used in the bags

2. Loom capacity — the time available for weaving

3. Printing capacity — the time available for printing bags

The factory also faces a maximum monthly demand for each product.

Here are the assumptions:

The objective is straightforward:

Maximize total contribution from the products manufactured.

Subject to the factory's resource and demand constraints.

At first glance, the answer seems easy.

Printed 50kg sacks have the highest contribution:

₦263 per bag.

So perhaps the factory should prioritize printed sacks.

But this is where production optimization becomes interesting.

A product doesn't consume money directly.

It consumes scarce resources.

And those resources have opportunity costs.

Now the answer looks different.

If loom capacity were the only constraint, feed bags would look like the best choice.

But loom capacity isn't the only constraint, printing capacity matters too.

Now fertilizer bags look like the obvious winner.

So we have a problem; the product with the highest contribution per bag is printed 50kg.

The product with the highest contribution per loom-hour is feed.

The product with the highest contribution per printing-hour is fertilizer.

Which one should the factory prioritize?

None of those rankings, by themselves, gives us the answer.

Solve the whole system

This is where linear programming becomes useful.

Rather than ranking products according to a single metric, the optimization model considers all products and all constraints simultaneously.

The model asks:

If I produce one more unit of this product, what resources does it consume, what resources remain for the other products, and what combination ultimately produces the highest total contribution?

Total contribution:

₦19,844,242 per month

Notice the result.

The model recommends producing the entire demand ceiling for printed sacks, even though printed sacks are not the most productive use of either bottleneck resource individually, and it recommends producing zero standard sacks, despite the fact that standard sacks are profitable, which is interesting.

Why is the standard sack eliminated?

The standard 50kg sack contributes:

₦132 per bag.

So it is clearly profitable.

But profitability per bag isn't the relevant question when capacity is scarce.

The optimization engine calculates the marginal value of the factory's constrained resources:

In other words, at the optimal solution, another hour of loom capacity could generate approximately ₦107,500 in additional contribution, while another hour of printing capacity could generate approximately ₦60,000.

Polypropylene, meanwhile, has no marginal value at the current solution because the factory isn't running out of it.

This is an important distinction.

The factory's problem isn't lack of material.

It is scarce machine capacity.

The hidden cost of making one standard sack

A standard sack consumes:

• 0.0018 loom hours

• 0 printing hours

At the marginal value of loom capacity, those resources have an opportunity cost of:

0.0018 × ₦107,500 = ₦193.50

But the sack contributes only:

₦132.

So producing that additional standard sack would effectively sacrifice more contribution elsewhere than it creates.

That's why the optimization recommends:

0 standard sacks.

This does not mean standard sacks are a bad product.

It means that, under these particular capacity constraints, their available contribution isn't high enough to justify the scarce resource they consume.

Why printed sacks still make the cut

This is where simple heuristics become especially dangerous.

Printed sacks don't have the highest contribution per loom-hour.

They don't have the highest contribution per printing-hour either.

Yet the model recommends:

41,248 printed sacks; the full demand ceiling.

Why?

Because the two bottlenecks interact.

Every product consumes a particular combination of loom and printing capacity. Increasing one product can change the amount of capacity available to the others.

There is therefore no single “best product” that can be identified by looking at one ratio.

The optimal solution is a combination.

The factory needs the entire portfolio of production decisions to work together.

What the factory should actually ask

This is the fundamental lesson of the case.

A manager might ask:

“Which product has the highest margin?”

A slightly more sophisticated manager might ask:

“Which product makes the most money per machine-hour?”

Both are reasonable questions.

But when several resources are simultaneously constrained, neither question is sufficient.

The better question is:

“What combination of products gives us the highest return from all of our scarce resources simultaneously?”

That is what the optimization model answers.

What happens if capacity changes?

The shadow prices also give management another useful piece of information.

Suppose someone offers the factory additional loom capacity.

The model estimates that an additional hour of loom capacity is worth approximately:

₦107,500

in additional contribution around the current solution.

That gives management a basis for evaluating capacity decisions.

If an additional hour of capacity costs ₦20,000, it could be economically attractive.

If it costs ₦150,000, it probably isn't.

The same logic applies to printing capacity, whose marginal value is approximately ₦60,000 per additional hour.

This turns optimization from a production scheduling exercise into a capital and capacity planning tool.

The bigger lesson

The factory doesn't need to ask:

“What is our most profitable product?”

It needs to ask:

“What is the most profitable use of the resources we currently have?”

Those are different questions.

A product can have a healthy margin and still be a poor use of scarce capacity.

Another product can have a lower margin per unit and still belong in the optimal production plan.

And when multiple resources constrain production simultaneously, even profit-per-bottleneck-hour can fail as a decision rule.

The answer lives in the system, not in any individual ratio.

That's the value of mathematical optimization.

A note on the numbers: This is an illustrative case study.

The production quantities, resource consumption rates, capacities, contribution figures and demand limits above are constructed for demonstration. They are not data from an audited Nigerian manufacturing company.

In a real engagement, the model would use the manufacturer's actual:

• material consumption

• machine speeds and capacities

• production times

• printing requirements

• contribution margins

• demand limits

• operating constraints

• available production hours

The mathematics is the easy part.

The quality of the decision depends on the quality of the operational data going into the model.

Conclusion

A factory operating at full capacity doesn't necessarily need another machine.

It may first need to answer a more fundamental question:

Are we using the capacity we already have on the right products?

In this illustrative case, optimizing the production mix increased monthly contribution from roughly ₦15.82 million to ₦19.84 million under the same stated production constraints; a difference of approximately ₦4.02 million per month.

No new machine was added.

No additional raw-material capacity was purchased.

The improvement came from changing the allocation of existing scarce resources.

That's the promise of production optimization: making better use of what you already have.