Multi-Objective Optimisation

Multi-Objective Optimisation for Health Resource Allocation

Multi-objective optimisation is a generalisation of mathematical programming that keeps several goals as separate objective functions, for example QALYs gained in a deprived population and QALYs gained elsewhere, instead of folding them into one score at the outset. Its answer is a set of efficient allocations, each of which can only be improved on one goal by giving something up on another, and the decision maker chooses among them. In health economics this matters because the usual single objective, total health or net monetary benefit, already contains a value judgement about how goals trade off, and planners are often unwilling to state that judgement as a number. The method shows which choices are worth debating and what each costs in forgone health.

Objectives, decisions and the efficient set

A multi-objective problem has the parts of any constrained optimisation problem, with decision variables, a feasible set fixed by budgets and other limits, and objectives, except that the objectives are several and conflict. The comparison between allocations rests on Pareto dominance, which compares two vectors of objective values without adding any preference information. The solutions that no feasible alternative dominates form the efficient set, and their objective values trace the Pareto frontier.

$$\max_{x \in X} \; F(x) = \big( f_1(x), f_2(x), \ldots, f_k(x) \big)$$

where $x$ is the vector of decisions, such as funding levels, $X$ is the feasible set allowed by the budget and any other constraints, $f_i(x)$ is the value of objective $i$ under decision $x$, oriented so that more is better, and $k$ is the number of objectives, at least two. A cost to be minimised enters as its negative.

Morton gives the health economic reading. Solving the problem means identifying every solution that is optimal for some aggregate objective within a family of possible functions, rather than optimising one fixed function, and when that family is all increasing functions the definition reduces to Pareto efficiency. The planner then sees a number of efficient, possibly very different, solutions and chooses between them directly.

Health gains to different groups as separate objectives

The most direct health economic use treats health gains to different population groups as the objectives. Morton starts from a health-related social welfare function and shows that, when each project has a small effect on individual health, the planner's problem becomes a weighted sum of subgroup health gains when all members of a group share one weight.

$$\max_{x \in X} \; \sum_{g=1}^{G} a_g \sum_{i=1}^{M} d_{ig} \, x_i$$

where $g$ indexes the $G$ population groups, $a_g$ is the social weight on a unit of health for group $g$, $i$ indexes the $M$ candidate programmes, $d_{ig}$ is the health gain, for instance in QALYs, that full funding of programme $i$ gives group $g$, and $x_i$ is the funding level of programme $i$, between 0 and 1 when partial funding is possible and either 0 or 1 when it is not. Equal weights give ordinary QALY maximisation.

The weights are the difficulty. Morton notes that planners are reluctant to state parameters valuing one section of the population above another, finding it hard to imagine a minister announcing that a QALY for a smoker counts as 80% of one for a non-smoker. Keeping each group's health gain as its own objective avoids fixing the weights.

Hynninen, Vilkkumaa and Salo modelled coronary heart disease prevention with Finnish register data under two alternative population objectives, maximising aggregate health (utilitarian) or maximising the health of those worst off (egalitarian). Rather than monetising health through a willingness-to-pay threshold, they generated, for each objective in turn, the whole frontier of Pareto optimal strategies trading expected health against cost.

Turning several objectives into solvable problems

Classical methods scalarise the problem by aggregating the objectives or recasting some of them as constraints, then solve the resulting single-objective problem repeatedly with different parameters, using ordinary methods of numerical optimisation. Emmerich and Deutz review several such forms and ask two questions of each, whether its solutions are efficient and whether every efficient solution can be reached.

$$\max_{x \in X} \; \sum_{i=1}^{k} w_i f_i(x), \qquad w_i > 0, \quad \sum_{i=1}^{k} w_i = 1$$

where $w_i$ is the weight on objective $i$ and the other symbols are as defined above. With strictly positive weights, the solution of this weighted-sum problem always lies on the Pareto front. When the front bulges outwards, the case Emmerich and Deutz call convex, every efficient point is the solution for some set of weights. When it is not, some efficient points are the solution for no weights at all, and on a front that dents inwards throughout, sweeping the weights tends to return only extreme solutions.

$$\max_{x \in X} \; f_1(x) \quad \text{subject to} \quad f_i(x) \ge \varepsilon_i, \quad i = 2, \ldots, k$$

where $\varepsilon_i$ is a required minimum level for objective $i$. This epsilon-constraint form optimises one objective while the others become floors, and varying the floors gives different points on the front, including points the weighted sum misses. Its drawbacks are that weakly dominated solutions can appear and that suitable floor values are hard to choose without some prior idea of where the front lies.

Health allocation problems are often discrete, since a programme is funded or not, leading to integer programming and to fronts that are not convex. Morton calls the efficient solutions that no weighted sum can select unsupported, and notes that when the feasible set is convex the efficient and weight-supported solutions coincide, as they do for divisible programmes in linear programming. For small problems, enumerating every feasible portfolio is enough. For large combinatorial or non-convex problems, evolutionary methods such as NSGA-II, a multiobjective genetic algorithm, evolve a population of solutions that together approximate the front, handling constraints through penalties on infeasible solutions.

Net monetary benefit as a weighted sum of health and cost

Standard cost-effectiveness analysis already solves a two-objective problem, more health and less cost, by fixing the weight between them in advance. Writing $Q(x)$ for QALYs and $C(x)$ for cost, net monetary benefit is a weighted sum of the two objectives.

$$\mathrm{NMB}(x) = \lambda \, Q(x) - C(x)$$

where $\lambda$ is the value of one QALY, the cost-effectiveness threshold, so the weights on health and avoided cost stand in the ratio $\lambda$ to 1. Each value of $\lambda$ selects one supported point of the health and cost front. Morton recalls that ranking by cost per QALY has long been read as a heuristic solution to an implicit optimisation problem. Multi-objective optimisation goes one step further by leaving open the weights that the threshold, or any equity weighting, would otherwise fix.

Worked example with three portfolios and two population groups

All numbers here are illustrative. A commissioner has GBP 10 million and four candidate programmes, each funded fully or not at all. Group A lives in more deprived areas and group B in the rest of the population. The two objectives are QALYs gained in each group.

1. List the feasible portfolios. Within GBP 10 million the commissioner can fund nothing, P1, P2, P3, P4, or P3 with P4 (cost 5 + 5 = 10, group A 130 + 100 = 230, group B 100 + 130 = 230). P1 or P2 with any other programme costs at least 15 and is infeasible.

2. Remove dominated portfolios. P3 alone (130, 100) and P4 alone (100, 130) are dominated by P3 with P4 (230, 230), which is better on both objectives, and funding nothing is dominated by every other option. Three efficient portfolios remain, P1 (400, 100), P2 (90, 430) and P3 with P4 (230, 230).

3. Sweep the weights. With weight $w$ on group A and $(1 - w)$ on group B, P1 scores $S_1 = 400w + 100(1 - w) = 100 + 300w$, P2 scores $S_2 = 90w + 430(1 - w) = 430 - 340w$, and P3 with P4 scores 230 whatever the weight. At $w = 0.3$ the scores are 190, 328 and 230, so P2 wins. At $w = 0.5$ they are 250, 260 and 230, so P2 wins. At $w = 0.7$ they are 310, 192 and 230, so P1 wins.

4. Find the unsupported solution. P3 with P4 beats P1 only when $S_1$ falls below 230, that is $w < 130/300 = 0.433$, and beats P2 only when $S_2$ falls below 230, that is $w > 200/340 = 0.588$. No weight satisfies both, so the balanced portfolio is efficient but unsupported, and a weighted-sum analysis would never report it.

5. Use epsilon constraints. Maximising group A's QALYs with no floor for group B returns P1 (400). Adding the floor of at least 200 QALYs for group B leaves P2 (90) and P3 with P4 (230), so P3 with P4 is chosen. A floor of 300 leaves only P2. Three runs recover the whole efficient set.

6. Restrict the weights. Suppose the commissioner will only say that a QALY in group A counts at least as much as one in group B, so $w \ge 0.5$. P1 and P2 tie where $S_1 = S_2$, at $w = 330/640 = 0.516$, with both scoring 254.7. P2 is preferred when $w$ is at least 0.5 but below 0.516, and P1 above that, so the statement narrows the choice to two portfolios without fixing a weight.

7. Price the equity choice. Total QALYs are 520 for P2, 500 for P1 and 460 for P3 with P4. Choosing P1 over P2 gives group A 310 more QALYs at a cost of 20 QALYs in total, and the balanced portfolio gives up 60 QALYs in total relative to P2. These are the health losses accepted for each distribution, which a single pre-weighted objective hides.

Equity constraints, weight bounds and the cost of equity

A common alternative keeps one objective and writes equity in as a constraint, for example a minimum share of resources or health for a group. Stinnett and Paltiel showed how such ethical constraints can be built into the programme with an explicit cost of equity, and Epstein, Chalabi, Claxton and Sculpher represented equity concerns as constraints and estimated the opportunity cost of each. Hynninen and colleagues define the cost of equity as the difference in health outcomes at a given resource level, or the difference in resources needed for a given health outcome. In their heart disease example, against the utilitarian strategy at an ICER of about 50,000 EUR per QALY, it was roughly 160 QALYs at a budget of 168 million EUR, or about 6 million EUR to keep the expected outcome of 763,420 QALYs.

Morton warns that constraints and objectives are not interchangeable. In a discrete problem a floor on one group's health can select a solution that no set of positive linear weights would choose, as the floor of 200 QALYs for group B selects the unsupported balanced portfolio in step 5. He suggests instead restricting the weights to a set the decision maker can accept, for example that one group's weight is at most three quarters of another's, and reporting the efficient solutions under that set, as in step 6.

Where it sits beside MCDA and cost-effectiveness frontiers

Morton places multi-objective optimisation within the broad field of multi-criteria decision analysis, and notes that multicriteria prioritisation in healthcare has generally used additive scoring and weighting, sometimes within multi-attribute utility theory. Such scoring ranks a list of named options, whereas multi-objective optimisation searches a feasible set defined by a budget and decision variables, often with too many portfolios to list. The cost-effectiveness frontier is the special case with two objectives, health and cost, for a set of mutually exclusive options. Distributional cost-effectiveness analysis tackles the same equity-efficiency trade-off by estimating how health effects fall across groups, which can supply the group-level gains used as objectives here.

Limitations and common misreadings

The efficient set depends on the objectives chosen. An outcome left out, such as access, plays no part in the dominance test, so a portfolio good on it can be discarded as dominated.

1. Treating the front as the decision. Efficiency rules out dominated options but does not choose among the rest, so a value judgement is still needed, even if only a range of weights.
2. Sweeping weights on a discrete problem. A weighted-sum sweep finds only supported solutions and can miss balanced portfolios, as in step 4.
3. Reading an equity floor as a neutral device. The floor value is itself a value judgement and, in discrete problems, can pick solutions that no linear weighting supports.
4. Ignoring uncertainty. Objective values are model estimates, so a portfolio efficient at point estimates may not be efficient across plausible parameter values, and the front should be recomputed in scenarios or probabilistic draws.
5. Assuming an evolutionary solver has found the true front. Its output is an approximation whose coverage depends on the algorithm and its settings.

Sources

• Emmerich MTM, Deutz AH. A tutorial on multiobjective optimization: fundamentals and evolutionary methods. Natural Computing. 2018;17(3):585-609. https://doi.org/10.1007/s11047-018-9685-y
• Epstein DM, Chalabi Z, Claxton K, Sculpher M. Efficiency, equity, and budgetary policies: informing decisions using mathematical programming. Medical Decision Making. 2007;27(2):128-137. https://doi.org/10.1177/0272989X06297396
• Hynninen Y, Vilkkumaa E, Salo A. Operationalization of utilitarian and egalitarian objectives for optimal allocation of health care resources. Decision Sciences. 2021;52(5):1169-1208. https://doi.org/10.1111/deci.12448
• Morton A. Aversion to health inequalities in healthcare prioritisation: a multicriteria optimisation perspective. Journal of Health Economics. 2014;36:164-173. https://doi.org/10.1016/j.jhealeco.2014.04.005
• Stinnett AA, Paltiel AD. Mathematical programming for the efficient allocation of health care resources. Journal of Health Economics. 1996;15(5):641-653. https://doi.org/10.1016/S0167-6296(96)00493-6