---
id: wiki-2026-0508-operations-research
title: Operations Research
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [OR, Management Science, Decision Science]
duplicate_of: none
source_trust_level: A
confidence_score: 0.9
verification_status: applied
tags: [optimization, decision-science, mathematical-programming]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: Python
  framework: 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
```python
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
```python
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
```python
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)
```python
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
```python
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
```python
# 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 |