WFM 6207 : Water Resources System Analysis
Homework
Homework 1
The Water Management Association (WMA) of a cooperative irrigation project wants to optimize the
use of its groundwater and surface water supplies. Assume that the cost to pump and deliver
groundwater to the irrigation region is Tk. 30 per acre-foot, and that the cost of surface water
importation and delivery from outside the region is Tk. 60 per acre-foot. The WMA has agreed to pay
for at least 50 acre-feet of water, but will use (pay for) more water if the local government allocates
it.
Assume also that the local government can provide no more than 40 acre-feet of groundwater and 70
acre-feet of surface water to the region. In addition, the local government cannot deliver a ratio of
surface water to groundwater greater than 7/8 at any one time due to physical constraints of the
system.
a. Determine the least-cost combination of surface water and groundwater supplies that the WMA
would like to avail. (Write down the decision variables, the objective function, and the constraints;
then solve graphically. Verify your results by LINDO)
b. Write down the ‘Primal’ and ‘Dual’, and determine the shadow prices on all constraints including
the non-negativity constraints. Discuss the economic implications of the shadow prices. What is the
actual ‘physical’ meaning of a zero shadow price in this case?
Homework 2
Write down the formulation of the ‘dual’ for the following problem:
minimize z = 30 x1 + 60 x2
subject to
x1 + x2  50
7 x1 – 8 x2  0
x1  40
x2  70
x1  0
x2  0
Be sure to write the constraints of the primal in standard form for a minimization problem in order to
obtain the correct signs of the coefficients for the dual.
a. Solve the dual on LINDO for shadow prices of the constraints of the original problem.
b. Check the ‘Sensitivity’ of:
(i) Changing the coefficients of the objective function
(ii) Changing the RHS of the constraints
Attach all supporting printouts of LINDO for full credit.
Homework 3
Problem 1:
As the leader of a team surveying the feasibility of rubber-dam projects, you must determine the
least-cost selection of five out of 10 possible sites. Label the sites S1, S2, ..., S10, and the
development costs associated with each as C1, C2, ..., C10. Regional development restrictions are
such that:
(a) Selecting sites S1 and S7 will prevent you from selecting site S8.
(b) Selecting site S3 or S4 prevents you from selecting site S5.
(c) Of the group S5, S6, S7, S8, at most two sites may be selected.
(d) If site S9 is selected, site 10 also must be selected (the reverse is not mandatory).
Formulate an integer program (IP) to determine the minimum-cost scheme that satisfies these
restrictions.
Problem 2:
Solve the following problem (by hand) using the branch-and-bound technique. You can solve the
associated linear program required at each step by using LINDO in the LP mode. Just remember to
use the appropriate set of constraints as you subdivide the domain.
minimize z = 2x1 -3x2 - 4x3
subject to
-x1 + x2+ 3x3 
3x1 + 2x2 - x3 
x1, x2, x3 
Problem 3:
Explore the integer-programming capabilities of LINDO by solving the branch-and-bound example
presented in class on LINDO. Invoke the integer program-solving mode per the instructions in the
LINDO manual (using the ‘GIN’ command). Interpret the output and compare the results to those
found in class.
Homework 4
A total of 425 acres of land is available for the production of three crops. The total seasonal irrigation
water allotment is 1000 acre-ft. A farmer has traditionally allocated 275 acres to crop A, 100 acres to
crop B, and 50 acres to crop C. Using a dynamic programming approach, find how much water should
be distributed to each crop in order to maximize returns to the farmer, using the data given by the
benefit functions in the figure below. In order to simplify the problem, you may want to assume that
the water can be delivered only in increments of 200 acre-ft.