In short: Choosing which localisation projects to fund from a long candidate list is a constrained selection problem with a knapsack structure, and solving it as one produces better portfolios than trimming the list in a workshop. The hard part is agreeing the objective, since maximising score contribution, domestic value added or jobs created will each return a different portfolio from the same inputs. The investment estimate, the benefit estimate and the domestic capability assessment decide the result, and an optimiser amplifies errors in any of them rather than correcting for them. A knapsack ignores sequencing, so projects that depend on capability another project creates need precedence constraints or a staged budget.
A localisation team ends up with a long list. Two hundred candidate categories, each with an estimated investment, an estimated score contribution, a domestic capability assessment and a sponsor who believes theirs should go first.
The budget funds perhaps forty of them. The selection method, in most organisations, is a workshop where the loudest sponsors win and the list gets trimmed until the total fits.
This is a constrained selection problem with a known structure, and treating it as one produces materially better portfolios than a workshop does. The mathematics is not difficult. The difficulty is agreeing what to maximise.
The shape of the problem
Each candidate has a cost and a value, you have a budget, and each candidate is either funded or not. That is a zero-one knapsack, one of the most studied problems in combinatorial optimisation, and it has an exact dynamic programming solution that runs in time proportional to the number of items times the budget.
For two hundred candidates and a budget expressed in units of a hundred thousand, that is a table of a few million cells. It solves in under a second, and it gives the exact optimum rather than an approximation.
The instinctive alternative, ranking by value and funding down the list until the money runs out, is the fractional greedy solution. It is optimal when items can be partially funded and it is not optimal when they cannot, which is the actual situation since half a smelter is worth nothing. The gap between greedy and exact is usually a few percent of total value, which on a large capital programme is a real number.
Six candidates and a budget of a hundred million make the difference visible. Costs and values both in millions, with value being whatever your objective happens to measure:
- A: cost 45, value 58.5
- B: cost 42, value 52.5
- C: cost 40, value 48.0
- D: cost 30, value 34.5
- E: cost 25, value 27.5
- F: cost 20, value 21.0
Value densities run from 1.30 down to 1.05 in even steps, so the ranking is unambiguous and everyone agrees on it. Greedy takes A, leaving 55. It takes B, leaving 13. Nothing else fits inside 13. The ranked-list version stops there with two funded projects, 111 of value, and 13 million sitting unspent because there was nothing small enough to buy with it.
The exact answer funds A, D and E. Cost is 45 plus 30 plus 25, exactly 100, and value is 58.5 plus 34.5 plus 27.5, or 120.5. Nothing else clears it: four items cannot fit, since the four cheapest already cost 115, and every other feasible triple comes in below, with B, D and E the closest at 114.5.
The instructive part is which project got rejected. B has a better density than either D or E and a sponsor who can point at the ranking to prove it. B loses because funding it strands 13 million that nothing on the list can absorb, and no argument about B's own merits touches that. Density ranks candidates. The budget line ranks combinations, and those are different orderings.
A six item instance overstates the gap. With two hundred candidates and a long tail of small ones, greedy usually finds something to fill the remainder and the shortfall falls back to a few percent. It does not fall to zero, and it is widest exactly where the money is, on portfolios dominated by a handful of large projects. Martello and Toth's Knapsack Problems: Algorithms and Computer Implementations, 1990, has the dynamic programme written out along with the bounding arguments that make the large instances tractable.
Two refinements make it match reality better.
Value density rather than raw value. Ranking by value per unit cost is the right ordering for the greedy version and it is also the right way to present the exact solution, because it makes the marginal decision legible. The projects that fall just outside the budget line are the interesting ones, and the density ordering shows how close they were.
Multiple constraints. Real portfolios are constrained by more than money. A minimum score contribution, a maximum concentration in any one sector, a floor on projects in a particular region, a limit on how many can run simultaneously given engineering capacity. Each of these adds a dimension to the knapsack and the exact solution gets expensive quickly. In practice a Lagrangian relaxation or a straightforward heuristic with the constraints checked as filters gets you close enough, and the honest presentation says which it was.
What to maximise, and why this is the actual argument
The mathematics is settled. The objective is not, and this is where the real disagreement lives.
Maximise score contribution and you will fund the cheapest points, which tend to be categories where the score moves easily and the economic substance is thin. Assembly operations and repackaging score well per dollar.
Maximise economic contribution using sector multipliers and you will fund deeper supply chains, which cost more per point of score and generate more activity. This puts you in tension with the headline commitment, and somebody senior has to be willing to accept a slower score trajectory in exchange for a better outcome.
Maximise capability created and you will fund things that score badly for years, because capability building has a long lag and no clean measurement.
The productive move is to stop pretending there is one objective and to build a frontier instead. Solve the portfolio repeatedly across a range of weightings between score and economic contribution, and plot the results. What you get is a set of portfolios where each one is efficient given a particular preference, and no portfolio inside the frontier should ever be chosen because something dominates it.
That frontier turns an argument about values into a choice between concrete options. The executive conversation stops being about whether economics or score matters more in the abstract, and becomes about which of these six portfolios to fund, with the trade-off between them quantified in units both sides recognise.
The inputs that decide whether it works
Three estimates carry the whole exercise and they have very different reliability.
Investment cost. Reasonably estimable, usually with an engineering basis, though early-stage estimates in this domain are routinely low by a wide margin. Carry a contingency and show the portfolio's sensitivity to it, because a portfolio that falls apart when costs rise twenty percent is not stable enough to commit to.
Score contribution. Mechanically computable if the scoring formula is known, and most of these formulas are published in supplier guidance. Where the formula weights certain levers more heavily, supplier development and research and development frequently carry a multiple with a cap, the arithmetic rewards specific behaviours strongly and the portfolio should exploit that deliberately rather than by accident.
Economic contribution. This comes from sector multipliers, and it inherits every assumption in the input-output model. Upper bound rather than expectation, and the sensible treatment is a range rather than a point.
Probability of success. The one most often omitted. Not every funded project delivers, and the failure rate differs sharply by category. A project requiring capability that does not exist domestically and cannot be acquired through a partner has a materially lower success probability than one extending an existing supplier into an adjacent product. Multiplying expected value by a success probability changes the ranking noticeably, and refusing to estimate the probability is itself an estimate of one.
A first pass at that probability is available from your own record. Take every localisation project approved in the last five years, mark each as delivering to specification, delivering more than two years late, or quietly abandoned, and split the population by whether the domestic capability existed at the point of approval. The two groups usually separate far enough to be worth carrying as a prior, and the exercise takes an afternoon with the capital approval register.
Sequencing, which the knapsack ignores
A knapsack chooses a set. It says nothing about order, and order matters here for two reasons.
Some projects unlock others. Localising a component makes localising the assembly that uses it more attractive, because the assembly's own domestic content rises without further investment. Those dependencies form a graph, and funding a prerequisite early raises the value of everything downstream of it. A portfolio optimiser blind to dependencies will fund the assembly and the component in the wrong order, or fund the assembly and never fund the component.
Some projects have windows. A capital programme with a defined construction period creates demand for a period and then stops. A domestic supplier established after the demand peak has missed it. Time-boxing candidates against the demand profile they are meant to serve is part of the selection, not a scheduling detail to sort out later.
The practical treatment is to solve the portfolio in annual tranches with the dependency graph enforced as precedence constraints, rolling forward each year as costs and capabilities update. That is closer to how the money is actually released and it handles both problems.
What the output should look like
A defensible portfolio recommendation has four parts, and the last one is the one that gets skipped.
The funded set, with cost and expected contribution for each.
The frontier, showing what a different weighting would have chosen, so the reader can see the alternatives that were rejected and why.
The near-miss list, the projects that fell just outside the budget line, ranked by density. When additional funds appear mid-year, and they usually do, this list is the answer rather than a fresh workshop.
The sensitivity, showing how the selection changes when costs rise, when multipliers are at the low end of their range, and when success probabilities are cut. A portfolio whose composition is stable across those variations is a strong recommendation. One that reshuffles completely is telling you the differences between candidates are inside the noise, and that is worth knowing before committing capital.
The limit
Optimisation allocates a budget across a candidate list. It cannot tell you whether the candidate list is any good, and the quality of the list dominates the quality of the selection.
If the two hundred candidates were generated by asking existing suppliers what they would like to make, the portfolio will be a good selection from a list shaped by incumbent interests. Generating candidates from the spend base itself, looking at what is imported in volume and asking what could plausibly be made domestically, produces a different and usually better list. That generation step is unglamorous, and it matters more than the optimisation that follows it.
The other limit is political and there is no analytical answer to it. Some projects are funded for reasons that are not on the spreadsheet: regional development commitments, diplomatic relationships, a promise made in a speech. Attempting to hide those inside the objective function corrupts the model. Better to fund them explicitly, deduct their cost from the budget before optimising, and let the remainder be allocated on the stated criteria. Then the trade-off is visible instead of laundered.
Start by building the frontier on last year's portfolio, using the decisions that were actually made. Showing where the chosen portfolio sat relative to the efficient set is the most persuasive argument available for doing it differently this year.