Environmental Risk Analysis: Methods and Applications ' 20082009 Lecturer Prof' Eugene Levner Lectur

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Environmental Risk Analysis Methods and
Applications ????? ??????? ???????? ?????? ????
??? ????? ?' 2008/2009 Lecturer- Prof.
Eugene Levner Lecture 6. Preparation to the
Test LP Models in Environmental Projects.
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• Many Methods of ERA
• Quantitative Modeling
• L1. Risks in supply chains.Knapsack problem
• L2. Classification by using histograms.
• Decision tables and decision trees.
  Sensitivity
• analysis
• L3. Borda ranking method. Linear programming.
• L.4. PERT and CPM in large-scale projects. Risks
  in projects
• Advanced mathematical programming models
• Transportation problem. Diet problem.
  Knapsack problem.
• TEST (Class-work) 14/5/2009.

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Program for today

• PART 1.
• 1. The diet problem
• 2. The knapsack problem
• 3. The kibbutzim problem (The regional planning)
• PART 2.
• Feedback to Bohan Imunim

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Diet Problem

• Bob wants to plan a nutritious diet, but he is on
  a limited budget, so he wants to spend as little
  money as possible. His nutritional requirements
  are as follows

• 2000 kcal
• 55 g protein
• 800 mg calcium

From Linear Programming, by Vasek Chvátal
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Diet Problem
Nutritional values
Bob is considering the following foods
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Diet Problem
Variables
We can represent the number of servings of each
type of food in the diet by the variables
x1 servings of oatmeal x2 servings of chicken x3
servings of eggs x4 servings of milk x5 servings
of cherry pie x6 servings of pork and beans
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Diet Problem
Nutritional values
Bob is considering the following foods
x1 x2 x3 x4 x5 x6
KCAL constraint 110x1 205x2 160x3
160x4 420x5 260x6 ? 2000 (110x1 kcals
in oatmeal)
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Diet Problem
LP formulation
Minimize
Cost
0.3x1 2.40x2 1.30x3 0.90x4 2.0x5 1.9x6

subject to
Nutritional requirements
110x1 205x2 160x3 160x4 420x5
260x6 ? 2000 4x1 32x2 13x3
8x4 4x5 14x6 ? 55 2x1
12x2 54x3 285x4 22x5 80x6 ? 800
Bounds
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Diet Problem
Solution
When we solve the preceding LP (using CPLEX, of
course) we get a solution value of 6.71, which
is achieved with the following menu
14.24 servings of oatmeal 0 servings of
chicken 0 servings of eggs 2.71
servings of milk 0 servings of cherry pie
0 servings of pork and beans
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The Knapsack Problem

• The classic Knapsack problem is typically put
  forth as
• A thief breaks into a store and wants to fill his
  knapsack with as much value in goods as possible
  before making his escape. Given the following
  list of items available, what should he take?
• Item A, weighing wA pounds and valued at vA
• Item B, weighing wB pounds and valued at vB
• Item C, weighing wC pounds and valued at vC
• ? ? ?

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The Knapsack Problem

• Input
• Capacity K
• n items with weights wi and values vi
• Goal
• Output a set of items S such that
• the sum of weights of items in S is at most K
• and the sum of values of items in S is maximized

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The Simplest Versions
Can items be divided up such that only a portion
is taken? The thief can hold 5 pounds and has to
choose from 3 pounds of gold dust at
379.22/pound 6 pounds of silver dust at
188.89/pound 1/9 pound of platinum dust at
433.25/pound Are all of the weights or total
values identical? The thief breaks into a ring
shop where all of the rings weight 1oz. He can
hold 12 ounces which should he take?
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An Easier Version...
What if all of the sizes we are working with are
relatively small integers? For example, if we
could fit 10 pounds and Object A is 2
pounds and worth 40 Object B is 3 pounds
and worth 50 Object C is 1 pound and worth
100 Object D is 5 pounds and worth 95
Object E is 3 pounds and worth 30
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Knapsack Problem

• You have a knapsack that has capacity (weight) C.
• You have several items I1,,In.
• Each item Ij has a weight wj and a benefit bj.
• You want to place a certain number of copies of
  each item Ij in the knapsack so that
• The knapsack weight capacity is not exceeded and
• The total benefit is maximal.

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Knapsack Problem

• Truck 10t capacity
• Optimum cargo combination
• Item 1 5 (3t)
• Item 2 7 (4t)
• Item 3 8 (5t)

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Integer Programming - Example
Classic Knapsack Problem You want to maximize
the value of items you can pack into a single
suitcase (or knapsack). However, you are limited
to a weight of 50 lbs. xi 0 if not chosen
1 if put in knapsack
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Integer Programming - Example
Classic Knapsack Problem If integer constraint
relaxed -
LP OPTIMUM FOUND AT STEP 4
OBJECTIVE FUNCTION VALUE 1)
60.42857 VARIABLE VALUE
REDUCED COST XA 0.000000
0.428571 XB
1.000000 0.000000 XC
0.857143 0.000000 XD
1.000000 0.000000 XE
1.000000 0.000000
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Integer Programming - Example
Classic Knapsack Problem With integer constraint
-
OBJECTIVE FUNCTION VALUE 1)
60.00000 VARIABLE VALUE
REDUCED COST XA 1.000000
-15.000000 XB
1.000000 -20.000000 XC
0.000000 -18.000000 XD
1.000000 -13.000000 XE
1.000000 -12.000000
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Integer Programming - Extensions
Classic Knapsack Problem How would you
modify this formulation if you realized that you
could not select item A without selecting item
D? Add constraint xA lt xD How would you
modify if you could select at most 3 items? Add
constraint xA xB xC xD xE lt 3
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Kibbutzim Problem ( REGIONAL PLANNING)

• The output of each kibbutz is limited by the
  amount of available irrigable land and by the
  quantity of water allocated for irrigation by the
  Water Commissioner. The crops are sugar beets,
  cotton, and sorghum. These crops differ in their
  expected net return per acre and their
  consumption of water. The Ministry of Ag. has
  set a max quota for the total acreage that can be
  devoted to each of these crops. The three
  kibbutzim have agreed that every kibbutz will
  plant the same proportion of its available
  irrigable land. For example, if kibbutz 1 plants
  200 of its available 400 acres, then kibbutz 2
  must plant 300 of its 600 acres, while kibbutz 3
  plants 150 acres of its 300 acres. How many
  acres to devote to each crop at the respective
  kibbutzim while satisfying the given
  restrictions. The objective is to maximize the
  total net return to all the kibbutzim as a whole.

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Tables

• Table 1 Resource Data

Usable Land (Acre)
Water Alloc. (Acre Feet)
Kibbutz
1
400
600
2
600
800
3
300
375
Table 2 Crop Data
Max Qta (Ac)
Profit (to Acre Feet)
Max Qta (Ac)
Crop
Sg Beets
600
3
1000
Cotton
500
2
750
Sorghum
325
1
250
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Kibbutzim Problem ( REGIONAL PLANNING)
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TOPICS of Home Tasks

• 1. Risk analysis management in a supply chain.
• 2. Evaluation of the environmental risk in a
  region and creation of the ecological map.
• 3. Evaluation of the Jordan River and the Dead
  Sea at risk.
• 4. Evaluation of the water quality in Israel and
  creation of the ecological map of water resources
  at risk.
• 5. Estimation of the integrated quality (ranking)
  of the airport (a company, university)
• 6. Quality-based location of warehouses (or water
  reservoirs)
• 7. Quality of education (classes, courses,
  universities)
• 7. Topic suggested by a student
• Students who present 25 of work on April 23,
  wull get a BONUS

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MORE OF PROJECTS

• 1. Classification of airports and estimate of
  risks.
• 2. Classification and identification of the spam
• 3. Analysis of water quality and estimate of
  water quality in Jerusalem (in Tel-Aviv) (in
  BeerSheva)
• 4. Analyis of risky wastewater systems in
  Tel-Aviv (in Haifa) (in Eilat)
• 5. Analysis of household water usage in Israel
• 6. Analysis of water utilization technologies in
  the Dead Sea
• 7. Risks in water utilization technologies in the
  Jordan River basin.
• IMPORTANT DATES
• TO SUBMIT 25 of the work - APRIL 23.
• (including Problem Description, 25 of real
  input data found or collected, 1 or 2 related
  papers (in journals or Internet)
• To present and defend your project (homework)
  not later than June 25, 2009.

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• THATS ALL FOR TODAY!