Homework #4 - Optimization
CE 385D - McKinney
1. Consider the allocation problem illustrated below.
User 2
Gage site
User 1
User 3
The allocation priority in each simulation period t is:
First 10 units of streamflow at the gage remain in the stream.
Next 20 units go to User 3.
Next 60 units are equally shared by Users 1 and 2.
Next 10 units go to User 2.
Remainder goes downstream.
Assume no incremental flow along the stream and no return flow from users. Define the
allocation policy at each site. This will be a graph of allocation as a function of the flow at the
allocation site.
2. A vertical cylindrical steel tank of height h and inside diameter D is to be constructed. The
tank is open at the top, and it is known that the bottom must be twice as thick as the side
thickness, t. Determine the dimensions of the minimum cost tank of given volume V, where the
weight of the material represents the total cost of the tank.
3. Consider the river shown in the figure below. There are j = 1, …, J water users along the river. Each
user receives an actual allocation of water xj j = 1, …, J and has a desired or known target allocations Tj j
= 1, …, J. If a user receives an excess of water xj > Tj or an deficit of water xj < Tj, then the user incurs a
penalty (reduced benefit). The water users all want to minimize these penalties; that is their goal
(objective). Given a supply of water Q, you are to write an optimization model for this problem. Assume
that the goal (i.e., objective) is to minimize the sum of squared deviations (i.e., the excesses or deficits) of
the actual allocations xj from the target allocations Tj. for all users j = 1, …, J. Write the corresponding
Lagrangian function for this problem and solve for the optimal values of xj, j=1,…,J.
Homework #4 - Optimization
CE 385D - McKinney
4. Assume water can be allocated to three users. The allocation, xj, to each use j provides the
following returns: R(x1) = (12 x1 – x12), R(x2) = (8 x2 – x22) and R(x3) = (18 x3 – 3 x32). Assume
that the objective is to maximize the total return, F(X), from all three allocations and that the
sum of all allocations cannot exceed 10. a) How much would each use like to have? b) Show
that at the maximum total return solution the marginal values, ∂(R(xj))/ ∂xj, are each equal to the
shadow price or Lagrange multiplier (dual variable) λ associated with the constraint on the
amount of water available. c) Finally, without resolving a Lagrange multiplier problem, what
would the solution be if 15 units of water were available to allocate to the three users and what
would be the value of the Lagrange multiplier?