supplemental reading: chapter 5.3 and 5.4 overheads 4 different types of lp formulations part 1: the...
DESCRIPTION
Transportation Model Basic Concept Involves the shipment of homogeneous products from a number of supply locations to a number of demand locations Problem: given needs at the demand locations how should I take limited supply at supply locations and move the goods to meet needs. Supply LocationsDemand Locations … m n 3TRANSCRIPT
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SUPPLEMENTAL READING: CHAPTER 5 .3 AND 5 .4
Overheads 4Different types of LP Formulations Part 1:
The Transportation ModelThe Feed Mix Model
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Basic LP Formulations
LP formulations are typically composed of a number of standard problem types We will cover 4 basic models
Transportation Feed mix Joint products Disassembly problems
Examining: Basic structure Formulation Example applications Answer interpretation
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Transportation Model
Basic Concept Involves the shipment of homogeneous products from a
number of supply locations to a number of demand locations Problem: given needs at the demand locations how should I
take limited supply at supply locations and move the goods to meet needs.
Supply Locations Demand Locations 1 1 2 2
… …m n
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Transportation Model
Features Objective: minimize costs Variables: quantity of goods to move from each
supply point to each demand point Restrictions:
Non-negative shipments Supply availability at each supply point Demand needed at each demand point
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Formulating the Transportation Problem
Decision Variables Fundamental decision variable
The set of individual shipment quantities from each supply location to each demand location
Basic notation for the decision variables Let “i” represent a supply location Let “j” represent a demand location
Denote the decision variables as: Movesupplyi,demandj
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Formulating the Transportation Model
Objective Function- Minimize transportation costs We need an expression for shipping costs
Define Costsupplyi, demandj as the per unit cost of shipments
from each supply location to each demand location Objective is to minimize shipping costs over all
possible supply and demand locations
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Formulating the Transportation Model
Constraints- 3 main types Supply availability: limiting shipments from each
supply point so that the sum of outgoing shipments from the supplyith supply point to all possible destinations (demandj) to not exceed supplysupplyi
Minimum demand: requiring shipments at the demandjth location to be greater than or equal to the demand at that location
Non-negative Shipments:
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Formulating the Transportation Model
Putting all the pieces together
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Example Transportation Model
Problem details Three plants:
New York, Chicago, Los Angeles Four demand locations:
Miami, Houston, Minneapolis, Portland Quantities:
Supply Available Demand RequiredNew York 100 Miami 30Chicago 75 Houston 75Los Angeles 90 Minneapolis 90
Portland 50
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Example Transportation Model
Problem detail, cont. Distances between locations
Transportation Costs = 5 + 5*Distance
Miami Houston Minneapolis PortlandNew York 3 7 6 23Chicago 9 11 3 13Los Angeles 17 6 13 7
Miami Houston Minneapolis PortlandNew York 20 40 35 120Chicago 50 60 20 70Los Angeles 90 35 70 40
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Example Transportation Model
Fitting the information for this problem into a transportation model structure
i’s = New York (NY=1), Chicago (C=2), Los Angeles (LA=3) j’s = Miami (MF=1), Houston (HT=2), Minneapolis (MM=3), Portland (PO=4)
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Example Transportation Model
Move11 Move 12 Move 13 Move14 Move21 Move22 Move23 Move24 Move31 Move32 Move33 Move3420 40 35 120 50 60 20 70 90 35 70 40 Minimize
100759030759050
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Example Transportation Model
Move11 Move 12 Move 13 Move14 Move21 Move22 Move23 Move24 Move31 Move32 Move33 Move3420 40 35 120 50 60 20 70 90 35 70 40 Minimize
1 1 1 1 ≤ 1001 1 1 1 ≤ 75
1 1 1 1 ≤ 901 1 1 ≥ 30
1 1 1 ≥ 751 1 1 ≥ 90
1 1 1 ≥ 501, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ≥ 0
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Example Transportation Model
Solution objective function value of $7,425 Shadow price represents marginal values of the
resources (i.e. marginal value of additional units in Chicago = $15)
Reduced cost represents marginal costs of forcing non-basic variable into the solution (i.e. shipments from New York to Portland costs $75)
Twenty units are left in New York
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Example Transportation Model
Variable Value Reduced CostMoveNY,MF 30 0MoveNY,HT 35 0MoveNY,MM 15 0MoveNY,PO 0 75MoveC,MF 0 45MoveC,HT 0 35MoveC,MM 75 0MoveC,PO 0 40MoveLA,MF 0 75MoveLA,HT 40 0MoveLA,MM 0 40MoveLA,PO 50 0
Equation Slack Shadow Price1 20 02 0 -153 0 -54 0 205 0 406 0 357 0 45
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Feeding Problem
Basic concept Involves composing a minimum cost diet from a set of
available ingredients while maintaining nutritional characteristics within certain bounds
Determine how much of each feedstuff (ingredient) is used in a diet to minimize costs while satisfying nutritional requirements
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Feeding Problem
Features Objective: minimize total diet costs Variables: how much of each feedstuff is used in the
diet Constraints:
Non-negative feedstuff Minimum requirements by nutrient Maximum requirements by nutrient Total volume of diet
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Feeding Problem
Model requires 2 types of indices Type of feed ingredients available from which the diet
can be composed Ingredient j = {corn, soybeans, salt, etc.}
Type of nutritional characteristics which must fall within certain limits Nutrient i= {protein, calories, etc.}
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Feeding Problem
Decision Variable Amount of each of the “j” feedstuffs to use in the diet
Denote as: Feedingredientj
Objective Minimize cost across all the different feedstuffs Data item for cost per unit denoted as: Costingredientj Objective is then to:
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Feeding Problem
Exogenous Parameters Needed Parameters representing how much of each nutrient is
present in each feedstuff Dietary minimum and maximum requirements for
each nutrient
Let: anutrienti,ingredientj be the amount of the ith nutrient present
in one unit of the jth feed ingredient ULnutrienti and LLnutrienti be the maximum and minimum
amount of the ith nutrient in the diet
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Feeding Problem
Constraints Formed by summing the nutrients generated from
each feedstuff (anutrienti,ingredientj* Feedingredientj) and requiring these to exceed the dietary minimum and/or be less than the maximum
Four general types Min nutrient requirements Max nutrient requirements Total volume of feedstuffs Non-negative feedstuffs
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Feeding Problem - Constraints
Minimum nutrient requirements Sum of the nutrients generated from each feedstuff
(anutrienti,ingredientjFingredientj) to meet the dietary minimum
Maximum nutrient requirements Sum of the nutrients generated from each feedstuff
(anutrienti,ingredientjFingredientj) to not exceed the dietary max
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Feeding Problem - Constraints
Total volume of the diet Requires the ingredients in the diet equal the required
weight of the diet If we suppose the weight of the formulated diet and
the feedstuffs are the same, then:
Non-negative feedstuffs
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Example – Cattle Feeding Problem
Seven nutritional characteristics energy, digestible protein, fat, vitamin A, calcium, salt,
dical, phosphorus Seven feed ingredients available + one new
ingredient corn, hay, soybeans, urea, dical phosphate, salt,
vitamin A New ingredient: potato slurry Ingredient costs per kilogram (Cingredientj)
Corn $0.13Dical $0.50
Alfalfa Hay $0.08Salt $0.11
Soybeans $0.30Vitamin A $0.29
Urea $0.33
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Example – Cattle Feeding Problem
Nutrient Unit Minimum MaximumNet Energy Mega Calories 1.34 na
Digestible Protein Kilograms 0.071 0.13Fat Kilograms na 0.05
Vitamin A International Units 2200 naSalt Kilograms 0.015 0.02
Calcium Kilograms 0.0025 0.01Phosphorus Kilograms 0.0035 0.012
Weight Kilograms 1 1
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Example – Cattle Feeding Program
Nutrient compositions of 1 kg of each feed
Nutrient Corn Hay Soybean Urea Dical Salt Vitamin A PotatoCharacteristic Phosphate Concentrate Slurry
Net Energy 1.48 0.49 1.29 1.39Protein 0.075 0.127 0.438 2.62 0.032
Fat 0.0357 0.022 0.013 0.009Vitamin A 600 50880 80 2204600
Salt 1Calcium 0.0002 0.0125 0.0036 0.2313 0.002
Phosphorus 0.0035 0.0023 0.0075 0.68 0.1865 0.0024
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Example – Cattle Feeding Problem
Set up in ExcelDecision Variables Corn Hay Soybeans Urea Dical Salt Vit. A Slurry SumChangeObjective to Min 0.133 0.077 0.3 0.332 0.498 0.11 0.286 0 0ConstraintsMax Nutrient
Protein 0.075 0.127 0.438 2.62 0.032 0 LE 0.13Fat 0.0357 0.022 0.013 0.009 0 LE 0.05Salt 1 0 LE 0.02
Calcium 0.0002 0.0125 0.0036 0.2313 0.002 0 LE 0.01Phosphorus 0.0035 0.0023 0.0075 0.68 0.1865 0.0024 0 LE 0.012
Min NutrientEnergy 1.48 0.49 1.29 1.39 0 GE 1.34Protein 0.075 0.127 0.438 2.62 0.032 0 GE 0.071Vit. A 600 50880 80 2204600 0 GE 2200Salt 1 0 GE 0.015
Calcium 0.0002 0.0125 0.0036 0.2313 0.002 0 GE 0.0025Phosphorus 0.0035 0.0023 0.0075 0.68 0.1863 0.0024 0 GE 0.0035
Volume 1 1 1 1 1 1 1 1 0 E 1Non-Negativity 1, 1, 1, 1, 1, 1, 1, 1, GE 0
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Example – Cattle Feeding Problem
Variable Value Reduced CostXC 0 0.095XH 0.001 0XSB 0.011 0XU 0.014 0XD 0.002 0XSLT 0.015 0XVA 0.001 0XSL 0.956 0
Equation Shadow PriceProtein L Max 0Fat Max 0Salt Max 0Calcium Max 0Phosphorus -2.207Net Engy Min 0.065Protein Min 0.741Vita Lim Min 0Salt Lim Min 0.218Calcium Min 4.4Phosphorus 0Weight -0.108
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Example – Cattle Feeding Problem
Reduced costs of feeding corn is 0.095 cents
Shadow prices: nonzero values indicate the binding constraint
How much would it cost us to increase the min requirement on Energy? It would cost us $0.065
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Next Time…
Joint Products
Disassembly Problem