Question

UK climate projections suggest a change in rainfall patterns. This will increase risk of flooding of cities from rivers. As a mitigation, a local lead flood authority is considering replacing the current temporary flood barriers with three different permanent gates to control the flood in different flood situations. The time required to complete each, and the number of workers required to be on the job at all times are shown in Table A below.

Table A. Specification ofthe flood barrier projects

Flood barrier gate - Duration of work [months] - Number of workers required

1 - 2 - 30

2 - 2 - 20

3 - 3 - 30

Once a flood gate is completed, it is estimated that it can control the following amount of water measured in cubic metres per month (cmm): gate I, 50cmm; gate 2, 30cmm; gate 3, 40cmm. For building these three gates, 60 workers are available during each month.

(a) Formulate a mathematical model in order to maximise, in the case of flood, the total water control capacity at the end of month 4.

(b) Formulate the following additional constraints forced by the tlood management committee:
(i) At most, construction on only one gate can be started during anyone month.
(ii) Gate 2 must be completed by the end of month 4.

Answer 1

(a)

Let Xij be the binary integer such that Xij=1 when gate-i is built in month-j and Xij=0 otherwise; i=1,2,3; j=1,2,3,4

Also, let Yi be the binary integer such that Yi=1 when gate-i is built at all and Yi=0 otherwise.

Objective function: Maximize Z = total cmm
Z = 50Y1 + 30Y2 + 40Y3

Subject to,

Xi - Mi*Yi = 0
where Mi are the required months for the gate-i i.e. Mi: [2; 2; 3] and Xi is the sum of Xij's for gate-i

30*X1j + 20*X2j + 30*X3j <= 60 for j=1,2,3,4

Xij, Yi = {0,1}

------------------------------

Implementation

Result

			Month
		Yi	1	2	3	4	Xi	Mi	Xi - Mi*Yi	cmm
Gate	1	1	1	0	1	0	2	2	0	50
	2	1	1	1	0	0	2	2	0	30
	3	1	0	1	1	1	3	3	0	40
										120
		Workers
Gate	1	30	30	0	30	0
	2	20	20	20	0	0
	3	30	0	30	30	30
			50	50	60	30

Conclusion: Make gate-1 in Month1, and 3; make gate-2 in Month1, and 2; make gate-3 in Month 2, 3, and 4. in this way, all gates are made with maximum cmm of 120 units.

(b)

Additional constraints

(i) X1j + X2j + X3j <= 1 for j=1,2,3,4

(ii) Y2 = 1

			Month
		Yi	1	2	3	4	Xi	Mi	Xi - Mi*Yi	cmm
Gate	1	1	0	0	1	1	2	2	0	50
	2	1	1	1	0	0	2	2	0	30
	3	0	0	0	0	0	0	3	0	40
		Total	1	1	1	1				80
		Workers
Gate	1	30	0	0	30	30
	2	20	20	20	0	0
	3	30	0	0	0	0
			20	20	30	30