Sustainable optimization

Most optimization models chase one goal: lowest cost. Sustainable optimization adds emissions alongside cost and service, and accepts that you can’t usually minimize all three at once. Cheaper often means fuller trucks and fewer sites (good for carbon too), but faster often means air freight and more stock near customers (bad for carbon).

There are three standard ways to handle several goals:

• Weighted objective: put a price on carbon, say ₹X per tonne of CO₂, and minimize cost + carbon price × emissions.
• Constraint: minimize cost subject to emissions staying below a cap, such as 20% below last year (the “epsilon-constraint” method).
• Frontier: solve the model many times with different caps or prices and plot the results.

The third gives the most useful output, a Pareto frontier: each point is a network plan where you can’t cut emissions further without raising cost, or the reverse. A typical shape:

Plan	Cost	Emissions
Lowest cost	₹100 crore	50,000 t CO₂
−15% emissions	₹101 crore	42,500 t
−30% emissions	₹104 crore	35,000 t
−45% emissions	₹112 crore	27,500 t

The first cuts are cheap (better loads, rail on long lanes); deeper cuts get expensive. Showing leadership the whole curve turns “go green” from a slogan into a choice with a known price.

Two ways to add carbon

As a price (weighted objective):

$$\min \; \text{Cost} + \lambda \times \text{Emissions}$$

where $\lambda$ is a carbon price in rupees per tonne. Raising $\lambda$ moves the solution towards lower emissions.

As a cap (epsilon-constraint):

$$\min \; \text{Cost} \quad \text{subject to} \quad \text{Emissions} \le \varepsilon$$

Solving for a range of caps $\varepsilon$ traces the frontier point by point.

Reading the frontier

Step	Extra cost	Emissions cut	Cost per tonne avoided
Lowest cost → −15%	₹1 crore	7,500 t	about ₹1,333
−15% → −30%	₹3 crore	7,500 t	₹4,000
−30% → −45%	₹8 crore	7,500 t	about ₹10,667

Each step costs more per tonne than the last: the marginal abatement cost rises. If the company uses an internal carbon price of ₹5,000 a tonne, the −30% plan is the right choice: the step to −30% costs less than ₹5,000 a tonne, while the step to −45% costs more than twice that.

What the model needs

• Emission factors for every lane, mode and vehicle type, and energy use at every site.
• The levers the model can choose: modes, consolidation, site locations, energy sources, packaging.
• Cost and service data consistent with the emission data, so trade-offs are real.

Pareto optimality

A plan is Pareto optimal if no other plan is better on one objective without being worse on another. Plans that are worse on every objective than some other plan are dominated and can be dropped. Decision makers should only ever choose among Pareto-optimal plans; the frontier shows them all.

Why weighted sums can miss plans

Minimizing cost plus a weighted carbon term finds only the frontier points on its convex hull. In mixed-integer network models, the frontier often has dents: good compromise plans that no single weight will select. The epsilon-constraint method, solving with a cap on emissions, finds those too, which is why it is preferred for discrete network decisions.

The shadow price of a cap

When emissions are capped, the solver also reports the cap’s shadow price: how much total cost would fall if the cap were loosened by one tonne. That is the marginal abatement cost at that point, an implied carbon price. Comparing it with the company’s internal carbon price, or with expected regulatory prices, shows whether a target is cheap or expensive at the margin.

Beyond carbon

• Service is often the third objective: faster delivery usually means more emissions and cost. A three-way frontier, or a service constraint, keeps the trade-off visible.
• Social goals (local sourcing, jobs in a region, supplier diversity) are harder to quantify but can enter as constraints or weighted scores.
• Uncertainty: emission factors carry uncertainty too; test whether the chosen plan stays good if factors are 20% higher or lower.

Intensity versus absolute emissions

A plan can cut emissions per unit while total emissions rise with growth. Targets aligned with climate science are usually absolute, so models should report both, and leadership should know which one a target refers to.