Operations Research Guide

A skill for applying operations research (OR) methods to business, logistics, and resource allocation problems. Covers linear programming, integer programming, scheduling, network optimization, simulation, and decision analysis using Python optimization libraries.

Linear Programming

Problem Formulation and Solving

python
from scipy.optimize import linprog
import numpy as np

def solve_production_planning():
    """
    Example: A factory produces two products (A and B).
    Product A: profit $40, uses 2h labor + 1kg material
    Product B: profit $30, uses 1h labor + 2kg material
    Constraints: 100h labor available, 80kg material available
    Maximize total profit.
    """
    # linprog minimizes, so negate for maximization
    c = [-40, -30]  # objective coefficients (negated)

    # Inequality constraints: A_ub @ x <= b_ub
    A_ub = [
        [2, 1],   # labor constraint
        [1, 2],   # material constraint
    ]
    b_ub = [100, 80]

    # Non-negativity bounds
    bounds = [(0, None), (0, None)]

    result = linprog(c, A_ub=A_ub, b_ub=b_ub, bounds=bounds, method="highs")

    return {
        "product_A": result.x[0],
        "product_B": result.x[1],
        "max_profit": -result.fun,
        "status": "optimal" if result.success else "infeasible",
    }

Using PuLP for Readable Models

python
from pulp import LpProblem, LpMaximize, LpVariable, lpSum, value

def workforce_scheduling():
    """
    Workforce scheduling: minimize staffing cost while meeting
    demand for each day of the week. Workers work 5 consecutive days.
    """
    days = ["Mon", "Tue", "Wed", "Thu", "Fri", "Sat", "Sun"]
    demand = [17, 13, 15, 19, 14, 16, 11]
    cost_per_worker = 1  # uniform cost

    prob = LpProblem("workforce_scheduling", LpMaximize)

    # x[i] = number of workers starting on day i
    x = {i: LpVariable(f"start_{days[i]}", lowBound=0, cat="Integer")
         for i in range(7)}

    # Minimize total workers
    prob += -lpSum(x[i] for i in range(7))

    # Each day, workers starting on days [d-4, d-3, ..., d] are available
    for d in range(7):
        workers_available = lpSum(x[(d - j) % 7] for j in range(5))
        prob += workers_available >= demand[d], f"demand_{days[d]}"

    prob.solve()

    return {
        "status": prob.status,
        "schedule": {days[i]: int(value(x[i])) for i in range(7)},
        "total_workers": int(sum(value(x[i]) for i in range(7))),
    }

Integer and Mixed-Integer Programming

Vehicle Routing Problem

python
from itertools import combinations

def solve_tsp_mtz(distances: np.ndarray) -> dict:
    """
    Solve the Traveling Salesman Problem using Miller-Tucker-Zemlin formulation.
    distances: n x n distance matrix
    Returns optimal tour and total distance.
    """
    from pulp import LpProblem, LpMinimize, LpVariable, LpBinary, lpSum, value

    n = len(distances)
    prob = LpProblem("TSP", LpMinimize)

    # Binary variables: x[i][j] = 1 if edge (i,j) in tour
    x = {(i, j): LpVariable(f"x_{i}_{j}", cat=LpBinary)
         for i in range(n) for j in range(n) if i != j}

    # Subtour elimination variables
    u = {i: LpVariable(f"u_{i}", lowBound=1, upBound=n - 1)
         for i in range(1, n)}

    # Objective: minimize total distance
    prob += lpSum(distances[i][j] * x[i, j] for i, j in x)

    # Each city visited exactly once
    for i in range(n):
        prob += lpSum(x[i, j] for j in range(n) if j != i) == 1
        prob += lpSum(x[j, i] for j in range(n) if j != i) == 1

    # MTZ subtour elimination
    for i in range(1, n):
        for j in range(1, n):
            if i != j:
                prob += u[i] - u[j] + (n - 1) * x[i, j] <= n - 2

    prob.solve()

    # Extract tour
    tour = [0]
    current = 0
    for _ in range(n - 1):
        for j in range(n):
            if j != current and (current, j) in x and value(x[current, j]) > 0.5:
                tour.append(j)
                current = j
                break

    return {
        "tour": tour,
        "total_distance": value(prob.objective),
    }

Queuing Theory

M/M/c Queue Analysis

python
from math import factorial, exp

def mmc_queue(arrival_rate: float, service_rate: float,
              n_servers: int) -> dict:
    """
    Analyze an M/M/c queue (Poisson arrivals, exponential service, c servers).
    arrival_rate: lambda (customers per unit time)
    service_rate: mu (customers served per unit time per server)
    n_servers: c (number of parallel servers)
    """
    rho = arrival_rate / (n_servers * service_rate)

    if rho >= 1:
        return {"stable": False, "utilization": rho}

    # Erlang C formula: probability of waiting
    a = arrival_rate / service_rate
    sum_terms = sum(a ** k / factorial(k) for k in range(n_servers))
    erlang_c = (a ** n_servers / factorial(n_servers)) / (
        (a ** n_servers / factorial(n_servers)) + (1 - rho) * sum_terms
    )

    # Performance metrics
    Lq = erlang_c * rho / (1 - rho)         # avg queue length
    Wq = Lq / arrival_rate                    # avg wait time
    W = Wq + 1 / service_rate                 # avg time in system
    L = arrival_rate * W                      # avg number in system

    return {
        "stable": True,
        "utilization": round(rho, 4),
        "prob_wait": round(erlang_c, 4),
        "avg_queue_length": round(Lq, 4),
        "avg_wait_time": round(Wq, 4),
        "avg_system_time": round(W, 4),
        "avg_in_system": round(L, 4),
    }

Simulation Methods

Discrete-Event Simulation

python
import simpy
import random

def simulate_service_center(n_servers: int, arrival_rate: float,
                             service_rate: float, sim_time: float = 480):
    """
    Discrete-event simulation of a service center using SimPy.
    sim_time: simulation duration in minutes (default 8-hour day).
    """
    wait_times = []

    def customer(env, server):
        arrival_time = env.now
        with server.request() as req:
            yield req
            wait = env.now - arrival_time
            wait_times.append(wait)
            yield env.timeout(random.expovariate(service_rate))

    def customer_generator(env, server):
        customer_id = 0
        while True:
            yield env.timeout(random.expovariate(arrival_rate))
            customer_id += 1
            env.process(customer(env, server))

    env = simpy.Environment()
    server = simpy.Resource(env, capacity=n_servers)
    env.process(customer_generator(env, server))
    env.run(until=sim_time)

    return {
        "customers_served": len(wait_times),
        "avg_wait": np.mean(wait_times) if wait_times else 0,
        "max_wait": max(wait_times) if wait_times else 0,
        "pct_waited": sum(1 for w in wait_times if w > 0) / len(wait_times) * 100,
    }

Decision Analysis

Multi-Criteria Decision Making

| Method | Description | Best For | |--------|-------------|----------| | AHP (Analytic Hierarchy Process) | Pairwise comparison matrix | Structured group decisions | | TOPSIS | Distance to ideal/anti-ideal solution | Ranking alternatives | | Weighted scoring | Simple weighted sum | Quick comparisons | | Decision trees | Sequential decision under uncertainty | Multi-stage problems |

Tools and Libraries

• PuLP: Python LP/MIP modeling with multiple solver backends
• OR-Tools (Google): Constraint programming, routing, scheduling
• Gurobi / CPLEX: Commercial high-performance MIP solvers (free academic licenses)
• SimPy: Python discrete-event simulation framework
• SciPy optimize: Linear programming, nonlinear optimization
• Pyomo: Algebraic modeling language for optimization in Python
• AMPL: Commercial algebraic modeling language