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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