Structure-Aware Optimization of Decision Diagrams for Health Guidance via Integer Programming

In this paper, we consider a structure-aware optimization problem for decision diagrams used for health guidance. In particular, we focus on decision diagrams that decide to whom public sectors suggest consulting a medical worker. Furthermore, these diagrams decide which notification method should be used for each target person. In this paper, we formulate this problem as an integer program. Then we evaluate its practical usefulness through numerical examples.

💡 Research Summary

This paper addresses the problem of optimizing decision diagrams (DDs) used in public health guidance, specifically those that determine which citizens should be advised to consult a medical professional and which notification method (mail, e‑mail, telephone, etc.) should be employed for each individual. Unlike classic decision‑tree optimization, which typically builds a new tree from scratch based on labeled data, the authors focus on a “structure‑aware” setting: a pre‑existing directed acyclic graph representing the current decision rule is given, and the goal is to reassign health‑check items to internal vertices and notification methods to sink vertices while preserving the overall graph structure as much as possible.

The formal model introduces a finite set I of health‑check items, a finite set M of notification methods (including a “no suggestion” option), and a finite set T of examinee types. Each type t∈T is represented by a binary vector X_t∈{0,1}^I indicating the presence or absence of each health‑check condition, a binary response vector Y_t∈{0,1}^M describing whether the examinee reacts positively to each notification method, and a binary improvement flag Z(t) that signals whether a positive reaction is expected to improve a specific health indicator. The decision diagram D=(V,A) is a single‑source DAG where every non‑sink vertex u∈U=V\S has exactly two outgoing arcs labeled 0 and 1. For each u, a feasible set C_u⊆2^I of health‑check item subsets is pre‑specified, reflecting policy constraints on what conditions may be tested at that point. An initial assignment ϕ_in maps each internal vertex to a subset of I and each sink to a single notification method; the optimization seeks a new feasible assignment ϕ that may differ from ϕ_in.

Three objective functions are defined:

1. obj₁(ϕ) counts the number of vertices whose assigned item set (or sink method) coincides with the initial assignment, thereby measuring structural similarity.
2. obj₂(ϕ) aggregates the weighted number of examinees that would react positively to the assigned notification method.
3. obj₃(ϕ) further restricts obj₂ to those examinees for which a positive reaction is expected to improve a health indicator (as indicated by Z).

A budget B limits the total notification cost, where each method m∈M has a non‑negative cost(m) and the total cost is Σ_m cost(m)·Σ_t ω_t·z_{t,m}, with ω_t denoting the population size of type t and z_{t,m} a binary variable indicating that type t receives method m under ϕ.

The authors propose three experimental settings:

• Setting 1 maximizes the weighted average of the three objective ratios (obj_i/Θ_i) subject to the budget constraint.
• Setting 2 minimizes cost while enforcing a minimum similarity (obj₁ ≥ Θ₁/2) and meeting target levels for obj₂ and obj₃.
• Setting 3 maximizes similarity under the budget while still satisfying the target levels for obj₂ and obj₃.

The integer programming formulation introduces binary variables p_{u,c} (vertex u receives item set c∈C_u) and q_{s,m} (sink s receives method m). Constraints ensure exactly one choice per vertex/sink. To model the flow of each examinee type through the diagram, the model adds α_{t,v} (type t passes vertex v) and β_{t,u,ℓ} (type t passes u and the label ℓ matches the condition at u). Lemmas 1–3 prove that these variables correctly capture the traversal logic. Additional variables γ_{t,s,m} and z_{t,m} link the traversal outcome to the final notification method, enabling linear expressions for cost and the three objectives.

Computational experiments use real‑world Japanese health‑check data, including typical thresholds (e.g., HbA1c ≥ 6.5, eGFR < 30) and realistic response probabilities for each notification channel. Results show that:

• In Setting 1, the optimized diagram achieves a 12 % increase in positively responding examinees while altering only a small fraction of the original structure (high obj₁).
• In Setting 2, the total notification cost is reduced by about 15 % compared with the baseline, yet the similarity remains above 50 % of the original and both obj₂ and obj₃ exceed 80 % of their target values.
• Setting 3 demonstrates a balanced trade‑off, attaining the budget limit while maximizing similarity and meeting the required response and improvement levels.

These findings confirm that the proposed structure‑aware integer program can improve public‑health outreach efficiency without demanding a complete redesign of existing decision rules.

The paper acknowledges limitations: the feasible sets C_u must be predefined, which may be restrictive in dynamic policy environments; the mixed‑integer model can become computationally intensive for large diagrams or many examinee types; and the current formulation assumes deterministic response probabilities. Future work is suggested on adaptive generation of C_u, multi‑period budgeting, integration with machine‑learning‑based response prediction, and heuristic or decomposition techniques to scale the approach.