MIME 6740/8740 Homework 1 9/27/2014

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Due 10/12/2014

  1. A factory has two production lines available for a product. The first line can produce one unit in t1 hours at cost c1, while the second can produce one unit of the same product in t2 hours at cost c2. The company that owns the factory wants to produce b units at the lowest possible cost in T hours. Assume that both lines can operate simultaneously.
  1. Formulate the optimization problem for finding the number of units to be produced from lines 1 and 2, x1 and x2, respectively. These variables assume integer values only. Specifically, determine the design variables, the objective function to be minimized and the constraints.
  2. Consider the case where t1=10, t2=20, c1=500, c2=100, b=3 and T=40. Plot the constraints, the feasible region and lines corresponding to constant values of the objective function in the space of the design variables.
  3. Find the optimum number of units to be produced from each line and show the optimum in the plot you made in question b.
  4. Calculate the gradients of the objective function and of the constraints at the optimum. Check if the Kuhn-Tucker conditions are satisfied at the optimum.
  5. Consider the point x1=2, x2=1 in the space of the design variables. Check if the Kuhn-Tucker conditions are satisfied at that point.
  1. Answer to the following true-false questions. You do not need to justify your answers, but you can write something if you think that a question is vague.
  1. An optimization problem can have multiple optimum solutions. (T-F)
  2. The local optimum solution of a minimization problem is denoted X*. Then, we cannot improve the design by perturbing the solution by an infinitesimal vector , without violating at least one of the constraints. In other words, design X*+ is infeasible (violates at least one of the constraints) or has objective function greater than or equal to the objective function of X*. (T-F)
  3. The local optimum solution of a minimization problem is denoted X*. Then any other design is infeasible or has objective function greater than or equal to the objective function of X*. (T-F)
  4. The global optimum solution of a minimization problem is denoted X*. Then any other design is infeasible or has objective function greater than or equal to the objective function of X*. (T-F)
  5. Consider an unconstrained minimization problem. If the gradient at X* is zero, then X* is a global minimum. (T-F)
  6. Consider an unconstrained minimization problem. If the gradient at X* is zero, and the Hessian matrix of the objective function is positive definite at X*, then X* is a global minimum. (T-F)
  7. Consider an unconstrained minimization problem. If the gradient at X* is zero, and the Hessian matrix of the objective function is positive definite everywhere in the space of the design variables, then X* is a global minimum. (T-F)

3. Problem 1-5, page 36 of the text.