Tutorial 1 Exercises

Part A: Supply Chain Concepts and Roles

Question 1

Outline the main responsibilities of manufacturers, distributors, and retailers in evaluating and enhancing their operations, decision-making procedures, and management of the supply chain.

Question 2

Define the objectives pursued by manufacturers, distributors, and retailers within the supply chains of the following products.

  1. Smartphone
  2. Computer
  3. Kitchenware

Part B: Optimisation Concepts

Question 1

Consider the function

\[ f(x,y)=\frac{x+y}{x^2+y^2+1}. \]

  1. Find the gradient of the function.
  2. Determine the critical (stationary) points.
  3. Find the Hessian of the function.
  4. Classify the stationary points as saddle points, strict or non-strict local/global minima, or strict or non-strict local/global maxima.

Question 2

A Cobb-Douglas production function for a new company is given by

\[ f(x,y)=50x^{2/5}y^{3/5}, \]

where \(x\) represents units of labour and \(y\) represents units of capital. Suppose each unit of labour costs \(\$100\) and each unit of capital costs \(\$200\). If the budget constraint is \(\$30{,}000\), find the maximum production level for this manufacturer.

Question 3

Consider the optimisation problem

\[ \begin{aligned} \text{Minimise}\quad & f(x_1,x_2)=x_1^2+x_2^2+x_1x_2-5x_2\\ \text{subject to}\quad & x_1+2x_2=5. \end{aligned} \]

  1. Write down the Lagrangian formula \(L(x_1,x_2,\lambda)\).
  2. Determine all constraints of \(L(x_1,x_2,\lambda)\).
  3. Find the values of \(x_1\) and \(x_2\).

Part C: Critical Points of Functions

For each function below:

  1. Find all critical points by setting the partial derivatives to zero.
  2. Form the Hessian matrix at each critical point.
  3. Classify each critical point as a local minimum, local maximum, saddle point, or inconclusive using the second derivative test.

Functions:

  1. \(f(x,y)=(x-2)^2+2(y-1)^2\)
  2. \(f(x,y)=x^3+y^3-3xy\)
  3. \(f(x,y)=2xy-x^4-x^2-y^2\)
  4. \(f(x,y)=e^{-(x^2+y^2)}\)
  5. \(f(x,y)=\sin(x)\sin(y)\)
  6. \(f(x,y)=\ln(1+x^2+y^2)\)
  7. \(f(x,y,z)=x^2+y^2-z^2\)

Part D: Lagrange Multipliers

Use the method of Lagrange multipliers to find and classify the extrema, local or global, for the following problems.

  1. \(f(x,y)=x^2+y^2\), subject to \(x+y=1\).
  2. \(f(x,y)=xy\), subject to \(x^2+y^2=1\).
  3. \(f(x,y)=x^2+4xy+y^2\), subject to \(x+y=3\).
  4. \(f(x,y)=e^{xy}\), subject to \(x^2+y^2=1\).
  5. \(f(x,y)=x^3y\), subject to \(x^2+y^2=1\).
  6. \(f(x,y)=\sin(x)+\cos(y)\), subject to \(x+y=\pi/2\).
  7. \(f(x,y,z)=x+y+z\), subject to \(x^2+y^2+z^2=1\).

Part E: KKT Conditions

Use the KKT conditions to find and classify the extrema for the following problems. For inequalities, rewrite them in the form \(g(x)\le 0\) if necessary.

Question 1

\[ \begin{aligned} \text{Minimise}\quad & f(x,y)=(x-1)^2+(y-2)^2\\ \text{subject to}\quad & -x-y+3\le 0,\\ & -x\le 0. \end{aligned} \]

Question 2

\[ \begin{aligned} \text{Maximise}\quad & f(x,y)=\ln(x)+\ln(y)\\ \text{subject to}\quad & x+y-1=0,\\ & -x\le 0,\\ & -y\le 0. \end{aligned} \]

Question 3

\[ \begin{aligned} \text{Minimise}\quad & f(x,y)=e^x+e^y\\ \text{subject to}\quad & x+y=0,\\ & x-1\le 0. \end{aligned} \]

Question 4

\[ \begin{aligned} \text{Minimise}\quad & f(x,y,z)=x^2+y^2+z^2\\ \text{subject to}\quad & x+y+z-1=0,\\ & -x\le 0. \end{aligned} \]