2nd/10_Wiki/Topics/Computer_Science_and_Theory/Operations-Research.md

id, title, category, status, canonical_id, aliases, duplicate_of, source_trust_level, confidence_score, verification_status, tags, raw_sources, last_reinforced, github_commit, tech_stack

id	title	category	status	canonical_id	aliases	duplicate_of	source_trust_level	confidence_score	verification_status	tags	raw_sources	last_reinforced	github_commit	tech_stack
wiki-2026-0508-operations-research	Operations Research	10_Wiki/Topics	verified	self	OR Management Science Decision Science	none	A	0.9	applied	optimization decision-science mathematical-programming		2026-05-10	pending	language framework Python PuLP/Gurobi/OR-Tools

Operations Research

매 한 줄

"매 decision-making을 mathematical model로". 매 WWII 군사 logistics에서 시작된 OR은 매 LP/MIP/network/queueing/simulation으로 확장, 매 2026 현재 supply chain·routing·scheduling·revenue management의 backbone이며 Gurobi 12 / OR-Tools 9.10 / Mosek가 매 industrial workhorse.

매 핵심

매 4 sub-disciplines

• Mathematical Programming: LP, MIP, NLP, SOCP, SDP.
• Network Models: shortest path, max flow, min-cost flow, assignment.
• Stochastic Models: queueing (M/M/c), Markov chains, MDP.
• Simulation: discrete event, Monte Carlo, agent-based.

매 LP duality

• Primal min cᵀx s.t. Ax ≥ b, x ≥ 0.
• Dual max bᵀy s.t. Aᵀy ≤ c, y ≥ 0.
• Strong duality: optimal values equal (Slater's condition).
• Dual = shadow prices of constraints.

매 응용

1. Supply chain: facility location, inventory, transportation.
2. Airline: crew scheduling, fleet assignment, revenue management.
3. Energy: unit commitment, economic dispatch.
4. Healthcare: OR scheduling, ambulance routing.
5. ML 교차: structured prediction, MIP-based interpretability.

💻 패턴

Linear Programming with PuLP

import pulp
prob = pulp.LpProblem("Diet", pulp.LpMinimize)
x1 = pulp.LpVariable("bread", lowBound=0)
x2 = pulp.LpVariable("milk", lowBound=0)
prob += 2*x1 + 3*x2  # cost
prob += 4*x1 + 3*x2 >= 100  # protein
prob += 2*x1 + 5*x2 >= 80   # vitamin
prob.solve(pulp.GUROBI_CMD(msg=0))
print(x1.varValue, x2.varValue, pulp.value(prob.objective))

MIP — Facility Location

from gurobipy import Model, GRB, quicksum
m = Model()
y = m.addVars(facilities, vtype=GRB.BINARY, name="open")
x = m.addVars(facilities, customers, lb=0, name="ship")
m.setObjective(
    quicksum(f[i]*y[i] for i in facilities) +
    quicksum(c[i,j]*x[i,j] for i in facilities for j in customers),
    GRB.MINIMIZE)
m.addConstrs(quicksum(x[i,j] for i in facilities) == d[j] for j in customers)
m.addConstrs(x[i,j] <= d[j]*y[i] for i in facilities for j in customers)
m.optimize()

VRP with OR-Tools

from ortools.constraint_solver import pywrapcp, routing_enums_pb2
manager = pywrapcp.RoutingIndexManager(len(dist), num_vehicles, depot)
routing = pywrapcp.RoutingModel(manager)
def cb(i, j): return dist[manager.IndexToNode(i)][manager.IndexToNode(j)]
idx = routing.RegisterTransitCallback(cb)
routing.SetArcCostEvaluatorOfAllVehicles(idx)
params = pywrapcp.DefaultRoutingSearchParameters()
params.first_solution_strategy = routing_enums_pb2.FirstSolutionStrategy.PATH_CHEAPEST_ARC
solution = routing.SolveWithParameters(params)

Min-Cost Flow (NetworkX)

import networkx as nx
G = nx.DiGraph()
G.add_edge('s', 'a', capacity=4, weight=2)
G.add_edge('s', 'b', capacity=2, weight=4)
G.add_edge('a', 't', capacity=3, weight=1)
G.add_edge('b', 't', capacity=5, weight=3)
flow_cost, flow_dict = nx.network_simplex(G)

M/M/c queue analysis

import math
def mmc(lam, mu, c):
    rho = lam/(c*mu)
    p0_inv = sum((c*rho)**n / math.factorial(n) for n in range(c))
    p0_inv += (c*rho)**c / (math.factorial(c)*(1-rho))
    p0 = 1/p0_inv
    Lq = p0 * (c*rho)**c * rho / (math.factorial(c)*(1-rho)**2)
    Wq = Lq/lam
    return {'utilization': rho, 'Lq': Lq, 'Wq': Wq}

Stochastic 2-stage LP

# x: first-stage, y_s: scenario s recourse
# min cᵀx + Σ pₛ qᵀyₛ  s.t. Tx + Wyₛ = hₛ
# Benders decomposition으로 solve 매 large-scale.

매 결정 기준

상황	Approach
Continuous, linear	LP (simplex/interior point)
Discrete decisions	MIP (branch & cut)
Nonconvex	Global solver (BARON, Gurobi 12) or heuristic
Stochastic	2-stage SP / robust optimization
Sequential decision	MDP / RL
Combinatorial/routing	OR-Tools CP-SAT

기본값: Gurobi 12 (commercial) or HiGHS (open-source) for LP/MIP; OR-Tools CP-SAT for combinatorial.

🔗 Graph

• 부모: Optimization · Mathematical-Programming
• 변형: Linear-Programming · Integer-Programming
• 응용: Supply-Chain
• Adjacent: Reinforcement-Learning · Combinatorial-Optimization

🤖 LLM 활용

언제: 매 problem formulation translation (NL→model), constraint extraction, MIP warm-start heuristics, post-hoc 해석. 언제 X: 매 numerical solving 자체 (LLM은 solver 호출 매 wrapping role).

❌ 안티패턴

• Pure LLM solving: 매 LLM은 OR solver 아님. 매 Gurobi/OR-Tools 매 사용.
• Ignoring duality: 매 shadow price 매 sensitivity analysis 의 핵심.
• Over-tight constraints: 매 infeasibility → IIS (irreducible inconsistent subsystem) 매 분석.
• Symmetry-blind MIP: 매 symmetry breaking constraint 매 추가, branch tree 매 collapse.

🧪 검증 / 중복

• Verified (Hillier & Lieberman 11e, Bertsimas & Tsitsiklis, Gurobi docs).
• 신뢰도 A.

🕓 Changelog

날짜	변경
2026-05-08	Phase 1
2026-05-10	Manual cleanup — full OR overview with LP/MIP/VRP/queueing patterns