Maximising investments of a startup¶
Problem definition¶
Your startup company has just raised 100M€ in a round of funding to reinvest in the company during the next period. Using regression and historical data, your team has found the following formulas for the Return of Investments (ROI) in different departments:
Research and Development:
\(10·\left(\sqrt[3]{2X-2}\right)\)
Advertisement:
\(10·\left(\ln(X)-2\right)\)
Technical Equipment:
\(10·\left(e^{X/50}-2\right)\)
Human Resources:
\(10·\dfrac{X}{50}\)
According to the collective agreement, the minimum investment in human resources is 20M€. Additionally, analysing the formulas obtained, it is clear that although significant investments in technical equipment yield a high ROI, for small investments the ROI of R&D or advertisement is actually larger. However, due to the pressure imposed by competitors, the company has decided that investments in technical equipment must be at least high enough to ensure that this concept does not have a negative impact in the ROI of the next period investments.
a. Write a Non-Linear Programming Model to maximise the ROI (1.5 points)
Decision variables \(x_n, n=[1,2,3,4]\): (non-negative) Investment in M€ in 1: R&D, 2: Advertisement, 3: Technical Equipment, 4: Human Resources
Maximise ROI.
\(\max z = 10·\left(\sqrt[3]{2x_1-2}\right) + 10·\left(\ln(x_2)-2\right) +10·\left(e^{x_3/50}-2\right) + 10·\dfrac{x_4}{50}\)
s.t.
\(x_1 + x_2 + x_3 + x_4 \leq 100\)
\(x_4 \geq 20\)
\(10·\left(e^{x_3/50}-2\right) \geq 0\)
\(x_n \geq 0 \quad \forall n\)
b. Write down the Kuhn-Tucker conditions (1.5 points)
First, we write down the problem in the canonical form:
\(\max z = 10·\left(\sqrt[3]{2x_1-2}\right) + 10·\left(\ln(x_2)-2\right) +10·\left(e^{x_3/50}-2\right) + 10·\dfrac{x_4}{50}\)
s.t.
\(x_1 + x_2 + x_3 + x_4 - 100 \leq 0\)
\(20 - x_4 \leq 0\)
\(-10·\left(e^{x_3/50}-2\right) \leq 0\)
\(x_n \geq 0 \quad \forall n\)
From here, we can write down the Lagrangian:
\(L = 10·\left(\sqrt[3]{2x_1-2}\right) + 10·\left(\ln(x_2)-2\right) +10·\left(e^{x_3/50}-2\right) + 10·\dfrac{x_4}{50}\)
\(\lambda_1·(x_1 + x_2 + x_3 + x_4 - 100)\)
\(\lambda_2·(20 - x_4)\)
\(\lambda_3·(-10·\left(e^{x_3/50}-2\right)\)
And, once we have computed the lagrangian, we can obtain the Kuhn-Tucker conditions:
Gradient condition
\(\nabla{L}=0\)
\(\dfrac{dL}{dx_1} = 20/3·(2·x_1-2)^{-2/3} + \lambda_1 = 0\)
\(\dfrac{dL}{dx_2} = 10·(1/x_2) + \lambda_1 = 0\)
\(\dfrac{dL}{dx_3} = 10/50·e^{x_3/50} + \lambda_1 -\lambda_3·10/50·e^{x_3/50} = 0\)
\(\dfrac{dL}{dx_4} = 10/50 + \lambda_1 -\lambda_2= 0\)
Orthogonality condition
\(\lambda_1·(x_1 + x_2 + x_3 + x_4 - 100) = 0\)
\(\lambda_2·(20 - x_4) = 0\)
\(\lambda_3·(-10·\left(e^{x_3/50}-2\right) = 0\)
Feasibility condition
\(x_1 + x_2 + x_3 + x_4 - 100 \leq 0\)
\(20 - x_4 \leq 0\)
\(-10·\left(e^{x_3/50}-2\right) \leq 0\)
non-negative condition:
\(\lambda_n \leq 0 \forall n\)
\(x_n \geq 0 \quad \forall n\)
c. Solving the problem yields a value of -0.29 for the lagrangian multiplier of the minimum investment in human resources constraint. Interpret this result. How should the company use this value in the negotiation of the next collective agreement? (1 point)
The lagrangian multiplier represents the change in the objective function per unitary change in the right-hand side of the corresponding constraint. Note that, in the canonical form, if we increase the right hand side 1 unit (from 0), then we need to increase the left hand side also one unit for the equation to remain the same, for instance making the investments in human resources raise to 21M€. This yields a decrease of the ROI of -0.29M€, so the impact of increasing the investments in human resources is negative in the return for the company. In the next collective agreement negotiation, the company could use this to argue against proposals or make a proposal to reduce the investment.