linear programming in investment allocation
Linear programming in investment allocation is the use of mathematical optimization techniques to determine the optimal allocation of investment resources among different assets or projects, subject to certain constraints and objectives.
A linear programming problem typically consists of 3 parts:
1. The objective function: a linear function to be maximized (or minimized. For example, f(x1, x2) = c1 x1 + c2 x2
2. Problem constraints: the values of the decision variables must satisfy a set of constraints, which must be a linear equation or inequality. For example,
a11 x1 + a12 x2 <= b1
a21 x1 + a22 x2 <= b2
a31 x1 + a32 x2 <= b3
3. Sign restriction for each variable. For example, for any variable xi, the sign restriction must be either non-negative or unrestricted in signs
In many LP problems, variables are typically defined as non-negative because they represent quantities such as quantities of goods, amounts of resources, or durations of activities, which cannot be negative in real-world contexts.
However, there are cases where variables should be left unrestricted in sign. For example, in certain financial or economic models, variables representing investments, cash flows, or returns may have positive or negative values, reflecting gains or losses. Similarly, if decision variables represent differences or deviations from a reference point, variables can be unrestricted to represent positive or negative deviations (from such a reference/anchor point).
Pros of Linear Programming:
Mathematical Optimization: LP is a scientific approach to problem-solving, using a mathematical framework for optimizing resource allocation based on quantitative analysis and objective functions
Flexibility: Linear programming can accommodate various types of constraints, such as budget constraints, production constraints, and resource constraints, making it applicable to a wide range of industries and problem domains
Sensitivity Analysis: Linear programming allows for sensitivity analysis, which examines how changes in the input parameters affect the optimal solution. This analysis helps assess the robustness and reliability of the solution.
CONs of Linear programming:
Optimal solution: depending on the problem, there could be none, a unique, or indefinite number of optimal solutions
Linearity Assumption: LP assumes that the objective function and constraints are linear but, many real-world problems involve non-linear relationships. LP also ignores factors such as uncertainty and randomness, which may limit the accuracy and realism of the solutions
Constant value of objective and constraint equations: LP technique can only be applied to a given problem once the coefficients of the objective function and the constraint equations are all known with certainty. In real life scenarios, these variables may lie on a probability distribution curve and their probability can only be approximated, not one single constant. LP also assumes that these values do not change while in real life, these values might change due to both external and internal factors
Single goal: organizations’ long-term goals are often not limited to a single goal. A firm wants profit maximization, but it also wants market share expansion, stakeholder relationship improvements, etc. But in LP problem, only 1 objective function is allowed.
Linear Programming’s disadvantages in finding optimal cashflow stream of long term horizon
With regards to the problem of finding optimal cashflow stream when those streams have long-term horizons, LP has some disadvantages below:
Constant value of objective and constraint equations: the cashflow amounts in each period might not be deterministic (known with absolute certainty). The NPV itself is also highly sensitive to interest rates, which can be change substantially within a short period of time (like now).
Integer vs fractional variables: the investment variables might be constrained to integers only (such as those in our quiz) but if fractional variables are allowed (doing a similar project of half such scale), a better optimal solution could be found.
Linearity Assumption: if the cashflow stream involves complex nonlinear relationships, the LP model might not capture all aspects of the problem. For example, multiple projects might share fixed-cost resources, thus are not completely independent from one another.
Incorporating Long-Term Considerations: If the long-term horizons involve considerations such as uncertainty, risk management, or changing conditions, these factors may not be easily identified or explicitly incorporated into the model’s objective function
Updating and Adaptability: Long-term horizons often require periodic updates and adjustments when new information becomes available or circumstances change. LP models can be adapted to incorporate these updates, but it may involve recalibrating the model, solving it again, and verifying the optimality of the solution, which can be different from the original solution suggested at time 0.