Modeling Resource Allocation Problems
In analytics, a typical resource allocation problem involves distributing a limited set of resources—such as budget, personnel, or materials—across a range of competing projects or activities to maximize a specific business objective. For example, a company may want to allocate its annual budget among several marketing campaigns to maximize overall expected return, while ensuring that each campaign receives at least a minimum investment and the total spend does not exceed the available funds. Translating such business requirements into a linear programming (LP) model involves identifying the decision variables (e.g., how much to allocate to each campaign), formulating the objective function (e.g., maximize total expected return), and expressing business rules as constraints (e.g., budget limits, minimum allocation per campaign).
1234567891011121314151617181920212223242526272829303132333435import pulp # List of projects and their expected returns per unit allocation projects = ["A", "B", "C"] returns = {"A": 5, "B": 3, "C": 4} # Minimum allocation required for each project min_alloc = {"A": 10, "B": 5, "C": 8} # Total available budget total_budget = 30 # Define decision variables: allocation for each project alloc = pulp.LpVariable.dicts("alloc", projects, lowBound=0, cat="Continuous") # Create the LP problem: maximize total expected return model = pulp.LpProblem("Resource_Allocation", pulp.LpMaximize) # Objective function: sum of (allocation * return) for all projects model += pulp.lpSum([returns[p] * alloc[p] for p in projects]), "Total_Expected_Return" # Budget constraint: total allocation cannot exceed budget model += pulp.lpSum([alloc[p] for p in projects]) <= total_budget, "Budget_Limit" # Minimum allocation constraints for each project for p in projects: model += alloc[p] >= min_alloc[p], f"Min_Alloc_{p}" # Solve the model model.solve() # Output the allocations for p in projects: print(f"Allocate {alloc[p].varValue:.2f} units to project {p}") print(f"Total expected return: {pulp.value(model.objective):.2f}")
Examining the solution, you see how the model allocates resources to each project in a way that maximizes total expected return while strictly respecting the business constraints: the overall budget is not exceeded, and each project receives at least its minimum required allocation. This approach ensures that business goals—such as achieving a certain level of investment in every campaign—are met. However, trade-offs are often involved: increasing the minimum allocation for a lower-return project may reduce the overall return, while relaxing the budget constraint could allow for greater returns but may not be feasible in practice. The LP model helps you quantify these trade-offs and make informed decisions.
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In analytics, a typical resource allocation problem involves distributing a limited set of resources—such as budget, personnel, or materials—across a range of competing projects or activities to maximize a specific business objective. For example, a company may want to allocate its annual budget among several marketing campaigns to maximize overall expected return, while ensuring that each campaign receives at least a minimum investment and the total spend does not exceed the available funds. Translating such business requirements into a linear programming (LP) model involves identifying the decision variables (e.g., how much to allocate to each campaign), formulating the objective function (e.g., maximize total expected return), and expressing business rules as constraints (e.g., budget limits, minimum allocation per campaign).
1234567891011121314151617181920212223242526272829303132333435import pulp # List of projects and their expected returns per unit allocation projects = ["A", "B", "C"] returns = {"A": 5, "B": 3, "C": 4} # Minimum allocation required for each project min_alloc = {"A": 10, "B": 5, "C": 8} # Total available budget total_budget = 30 # Define decision variables: allocation for each project alloc = pulp.LpVariable.dicts("alloc", projects, lowBound=0, cat="Continuous") # Create the LP problem: maximize total expected return model = pulp.LpProblem("Resource_Allocation", pulp.LpMaximize) # Objective function: sum of (allocation * return) for all projects model += pulp.lpSum([returns[p] * alloc[p] for p in projects]), "Total_Expected_Return" # Budget constraint: total allocation cannot exceed budget model += pulp.lpSum([alloc[p] for p in projects]) <= total_budget, "Budget_Limit" # Minimum allocation constraints for each project for p in projects: model += alloc[p] >= min_alloc[p], f"Min_Alloc_{p}" # Solve the model model.solve() # Output the allocations for p in projects: print(f"Allocate {alloc[p].varValue:.2f} units to project {p}") print(f"Total expected return: {pulp.value(model.objective):.2f}")
Examining the solution, you see how the model allocates resources to each project in a way that maximizes total expected return while strictly respecting the business constraints: the overall budget is not exceeded, and each project receives at least its minimum required allocation. This approach ensures that business goals—such as achieving a certain level of investment in every campaign—are met. However, trade-offs are often involved: increasing the minimum allocation for a lower-return project may reduce the overall return, while relaxing the budget constraint could allow for greater returns but may not be feasible in practice. The LP model helps you quantify these trade-offs and make informed decisions.
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