Advanced Mixed-Integer Programming Modeling Techniques

Assignment Task

1. We have the constraints ????1 ≤ x1 ≤ b1, a2 ≤ x2 ≤ b2, where a1, a2, b1, and 563 b2 are all positive parameters, x3 ∈ R, and z ∈ {0, 1}. It is required that if z = 1 564 then x3 = x1 and otherwise x3 = x2. We can formulate the requirements as MIP 565 constraints:

2. We have the constraints x ∈ {0, 1}, y ∈ {0, 1}, z ∈ {0, 1}, w ∈ {0, 1}, and 568 w = xyz. Formulate the nonlinear constraint w = xyz as MIP constraints.

3. We have the constraints x ∈ {0, 1}, y ∈ {0, 1}, z ∈ {0, 1}, w ∈ {0, 1}, and w = x + y 2 + z 3 570 . Formulate the last constraint as MIP constraints

4. Let Zt∈ Z+ be the number of ships a company plans to deploy in year 572 t = 1, ..., T, in which T is the number of years in the planning horizon. The company 573 would like to deploy a steady fleet size and thus it is required that the difference 574 between the numbers of ships deployed in two consecutive years (e.g., year 1 and 575 year 2, year 2 and year 3) cannot exceed 3. Formulate the requirement as MIP/IP/LP 576 constraints. 577

5. Suppose that you are Assistant Director of Academic Secretariat of University 578 P. Its Department of Logistics and Maritime Studies has N new undergraduate 579 students, and you are to divide them into K classes, K ≥ 2, N ≥ k + 1, in a way that 580 fulfills the following requirements:

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