{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:APC6627SLJI4T6P522PCIA5MBN","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"35f694c63342e7de6f8365cb2fccf173655fbd0f1c318ee7e19f41c49ce468f6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-07-27T15:33:58Z","title_canon_sha256":"f89b3d8caed4d7d88acb4906587a11ade5c2f3c0a19bd8be731bfd2c6398da14"},"schema_version":"1.0","source":{"id":"1207.6572","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.6572","created_at":"2026-05-17T23:50:39Z"},{"alias_kind":"arxiv_version","alias_value":"1207.6572v1","created_at":"2026-05-17T23:50:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.6572","created_at":"2026-05-17T23:50:39Z"},{"alias_kind":"pith_short_12","alias_value":"APC6627SLJI4","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"APC6627SLJI4T6P5","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"APC6627S","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:5677fec22f9a57eaefb50c1aae7b37d989106fdb9bc02b479898a1eb121a4be8","target":"graph","created_at":"2026-05-17T23:50:39Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The Analytic Hierarchy Process (AHP) is widely used for decision making involving multiple criteria. Elsner and van den Driessche introduced a max-algebraic approach to the single criterion AHP. We extend this to the multi-criteria AHP, by considering multi-objective generalisations of the single objective optimisation problem solved in these earlier papers. We relate the existence of globally optimal solutions to the commutativity properties of the associated matrices; we relate min-max optimal solutions to the generalised spectral radius; and we prove that Pareto optimal solutions are guaran","authors_text":"Buket Benek Gursoy, Oliver Mason, Sergei Sergeev","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-07-27T15:33:58Z","title":"The Analytic Hierarchy Process, Max Algebra and Multi-objective Optimisation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.6572","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f97b1c46ddc49d88fc66fc0ca5c2f670904f24e088eefff744fa6af0ff511891","target":"record","created_at":"2026-05-17T23:50:39Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"35f694c63342e7de6f8365cb2fccf173655fbd0f1c318ee7e19f41c49ce468f6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-07-27T15:33:58Z","title_canon_sha256":"f89b3d8caed4d7d88acb4906587a11ade5c2f3c0a19bd8be731bfd2c6398da14"},"schema_version":"1.0","source":{"id":"1207.6572","kind":"arxiv","version":1}},"canonical_sha256":"03c5ef6bf25a51c9f9fdd69e2403ac0b506dfab87ea1459f6bf313344c449777","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"03c5ef6bf25a51c9f9fdd69e2403ac0b506dfab87ea1459f6bf313344c449777","first_computed_at":"2026-05-17T23:50:39.135790Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:50:39.135790Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4epMJidLgiGRKx2zEGfDET5gdXgrjkVNsEkMGMtFCs9GnFzCphPljZGObfC00LgEovqKBDfN1AgnTR1erOr3Dg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:50:39.136196Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.6572","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f97b1c46ddc49d88fc66fc0ca5c2f670904f24e088eefff744fa6af0ff511891","sha256:5677fec22f9a57eaefb50c1aae7b37d989106fdb9bc02b479898a1eb121a4be8"],"state_sha256":"bdf19f56612d62bc6cac98ba1b49ba84a9f78f418f0bf8a132c79746abc92f15"}