Final answer:
To address the student's LP formulation problem, a budget constraint is established along with an objective to maximize the worst-case market value, and constraints for each scenario to ensure the market value is above a certain threshold. Non-negativity constraints are also employed to avoid short selling.
Step-by-step explanation:
To solve this investment problem using a Linear Programming (LP) formulation, we need to maximize the worst-case scenario of the market value among all the different future scenarios.
Budget Constraint
The company has a current budget constraint which can be expressed as the sum of the products of the quantity of each asset to be purchased times their current prices. This can be written as:
\(P_1 \times Q_1 + P_2 \times Q_2 + \ldots + P_r \times Q_r <= \text{Budget}\)
Objective Function
We want to maximize the minimum value of the portfolio in all scenarios, i.e., maximize \(W\) where \(W\) is the worst-case market value. Therefore, our objective function is:
Maximize \(W\)
Scenario Constraints
For each scenario \(i\), the market value of the investment in assets cannot be lower than \(W\). This introduces \(m\) constraints:
\(q_{i1} \times Q_1 + q_{i2} \times Q_2 + \ldots + q_{ir} \times Q_r >= W\), for all \(i = 1, \ldots, m\)
Non-Negativity Constraint
We must also include the non-negativity constraint, stating that no negative quantities of assets can be held (no short selling):
\(Q_j >= 0\), for all \(j = 1, \ldots, r\)
Combining these constraints and the objective function gives us the complete LP formulation to maximize the worst-case scenario market value, subject to the budget and non-negativity constraints.