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linear Programming Problems - Water Resources Systems Planning and Management - Lecture Notes, Study notes of Business Management and Analysis

Some concept of Water Resources Systems Planning and Management are Common Water Management, Compromise Programming Method, Flood Protection Planning. Main points of this lecture are: Linear Programming Problems, Feasible Region, Graphical Method, Optimal Point, Inactive, Constraints, Optimal Solution, Simplex Method, Manufacturing Factory, Treatment Plant

Typology: Study notes

2012/2013

Uploaded on 04/27/2013

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Module 3:
1. Solving the following linear programming problems by Graphical Method and identify in-
active constraints, if any. Also show the feasible region and optimal point on the plot.
(a) (b)
(c) (d)
(e) In problem 1(d), if the objective function is changed to Z=x+8y, will there be a
different optimal solution ?
2. Solve problems 1(a), 1(b) and 1(c) by Simplex method.
3. Consider a system composed of a manufacturing factory and a waste treatment plant owned
by the manufacturer. The manufacturing plant produces finished goods that sell for a unit
price of Rs 10,000. However, the finished goods cost Rs 3,000 per unit to produce. In the
manufacturing process two units of waste are generated for each unit of finished goods
produced. In addition to deciding how many units of goods to produce, the plant manager
must also decide how much waste will be discharged into a river without treatment so that
the total net benefit to the company can be maximised and the water quality requirement of
the water course is met. The treatment plant has a maximum capacity of treating ten units of
waste with 80% waste removal efficiency at a treatment cost of Rs 600 per unit of waste.
There is also an effluent tax imposed on the waste discharged to the receiving water body (Rs
2,000 for each unit of waste discharged). The water pollution control authority has set an
upper limit of four units on the amount of waste the company may discharge. Formulate an
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Partial preview of the text

Download linear Programming Problems - Water Resources Systems Planning and Management - Lecture Notes and more Study notes Business Management and Analysis in PDF only on Docsity!

Module 3:

  1. Solving the following linear programming problems by Graphical Method and identify in- active constraints, if any. Also show the feasible region and optimal point on the plot. (a) (b)

(c) (d)

(e) In problem 1(d), if the objective function is changed to Z = x +8 y , will there be a different optimal solution?

  1. Solve problems 1(a), 1(b) and 1(c) by Simplex method.
  2. Consider a system composed of a manufacturing factory and a waste treatment plant owned by the manufacturer. The manufacturing plant produces finished goods that sell for a unit price of Rs 10,000. However, the finished goods cost Rs 3,000 per unit to produce. In the manufacturing process two units of waste are generated for each unit of finished goods produced. In addition to deciding how many units of goods to produce, the plant manager must also decide how much waste will be discharged into a river without treatment so that the total net benefit to the company can be maximised and the water quality requirement of the water course is met. The treatment plant has a maximum capacity of treating ten units of waste with 80% waste removal efficiency at a treatment cost of Rs 600 per unit of waste. There is also an effluent tax imposed on the waste discharged to the receiving water body (Rs 2,000 for each unit of waste discharged). The water pollution control authority has set an upper limit of four units on the amount of waste the company may discharge. Formulate an

a b

a b

a b

a b

subjectto

MaximizeZ a b

x y

x y

x y

x y

subjectto

MinimizeZ x y

1 2

1 2

1 2

1 2

x x

x x

x x

subjectto

MaximizeZ x x

x y

x y

x y

subjectto

MinimizeZ x y

LP model clearly specifying the decision variables, Objective function and constraints and solve it using both graphical method as well as simplex method.

  1. A reservoir is to be constructed to supply water at a maximum constant rate per season for a city. The inflows in the six seasons of the year are 3, 12, 7, 3, 2 and 3 respectively. Determine the minimum required reservoir capacity using (i) Mass diagram method and (ii) Sequent Peak Method neglecting all losses.
  2. Solve the following problem by using (i) Mass diagram method (ii) Sequent Peak Method and (iii) Linear Programming to estimate the reservoir capacity. (Neglect evaporation losses). Period, t 1 2 3 4 5 6 Inflow, Qt 4 8 7 3 2 0 Demand, Dt (=Rt) 5 0 5 6 2 6
  3. Solve the following problem by using Linear Programming to estimate the reservoir capacity. Monthly Inflows and demands are in Mm^3 and et in mm. Area at dead storage level, A 0 = 38.52 Mm^2. Slope, a = 0.127. June July Aug Sept Oct Nov Qt 86.52 425.75 360.60 159.39 122.85 56. Dt 55.69 139.68 138.76 71.26 39.59 220. et 230.53 151.60 150.52 154.45 120.56 119. Dec Jan Feb Mar Apr May Qt 22.65 17.38 12.99 9.58 10.81 21. Dt 220.15 191.30 90.19 0 0 0 et 96.69 95.45 100.59 150.89 225.42 245.