Linear%20Programming%20Formulation

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Linear Programming Formulation

• I. PRODUCT MIX
• II. DIET
• III. FINANCE
• IV. MARKETING
• V. TRANSPORTATION
• VII. Data Envelopment Analysis

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PRODUCT MIX EXAMPLE
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4 TYPES OF SHIRTS

• X1 NUMBER BOXES OF SWEATSHIRTS WITH PRINTING ON
  FRONT
• X2 NUMBER BOXES SWEATSHIRTS PRINTING ON BOTH
  SIDES
• X3 NUMBER BOXES T SHIRTS FRONT
• X4 NUMBER BOXES T SHIRTS BOTH

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SHIRT PRODUCTION

• ENOUGH CAPACITY FOR 1200 STANDARD SIZED BOXES
• BUT STANDARD SIZE HOLDS DOZEN T SHIRTS, WHILE A
  BOX OF A DOZEN SWEAT SHIRTS IS 3 TIMES THE SIZE
  OF A STANDARD SIZE BOX
• 3X13X2X3X4 lt 1200

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PROCESSING TIME
TYPE HOUR PER BOX
X1 .1
X2 .25
X3 .08
X4 .21
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ALL PRODUCTION WITHIN 72 HOURS

• .1X1.25X2.08X3.21X4 lt72

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BUDGET25000
TYPE COST PER BOX
X1 36
X2 48
X3 25
X4 35
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36X148X225X335X4lt25000
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CAPACITY500 FOR SWEATSHIRTS AND FOR TSHIRTS

• X1X2 lt 500
• X3X4 lt 500

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PROFIT PER BOX
TYPE PROFIT
X1 90
X2 125
X3 45
X4 65
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OBJECTIVE FUNCTION

• MAX 90X1125X245X365X4

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DIET PROBLEM
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DECISION VARIABLES

• X1NUMBER CUPS OATMEAL
• X2NUMBER EGGS
• X3NUMBER CUPS MILK
• X4NUMBER SLICES WHEAT TOAST

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MINIMIZE COST
FOOD COST
OATMEAL .10
EGG .10
MILK .16
TOAST .07
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OBJECTIVE FUNCTION

• MIN .10X1.10X2.16X3.07X4

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AT LEAST 420 CALORIES
FOOD CALORIES
OATMEAL 100
EGG 75
MILK 100
TOAST 65
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100X175X2100X365X4gt420
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NO MORE THAN 30 CHOLESTEROL
FOOD CHOL
OATMEAL 0
EGG 270
MILK 12
TOAST 0
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270X212X3lt30
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FINANCE EXAMPLE
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DECISION VARIABLES

• X1 IN MUNI BONDS
• X2 IN CERTIF OF DEPOSIT (CD)
• X3 IN T BILLS
• X4 IN STOCK

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RETURN ON INVESTMENT
INVESTMENT ANNUAL RETURN
MUNI .085
CD .05
T .065
STOCK .13
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OBJECTIVE FUNCTION

• MAX .085X1.05X2.065X3.13X4

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NO MORE THAN 20 OF TOTAL INVESTMENT IN MUNI

• GIVEN TOTAL INVESTMENT 70,000
• YOU CALCULATE (.20)(70000)14000
• X1 lt 14000

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CD CONSTRAINT

• AMOUNT INVESTED IN CD SHOULD NOT EXCEED AMOUNT
  INVESTED IN OTHER 3 ALTERNATIVES
• X2ltX1X3X4

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AT LEAST 30 OF INVESTMENT IN T BILL AND CD

• GIVEN 70,000 INVESTMENT
• YOU CALCULATE (.30)(70000)21000
• X2X3 gt 21000

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RATIO CONSTRAINT

• MORE SHOULD BE INVESTED IN
  CDs AND T BILLS THAN IN MUNI AND STOCKS BY A
  RATIO OF AT LEAST 1.2 TO 1
• X2 X3 gt 1.2
  X1X4
• NOTE EXCEL REQUIRES LINEAR CONSTRAINT
• X2X3 gt 1.2x11.2x4
  
  

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INVEST ENTIRE 70,000

• X1X2X3X470000

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MARKETING EXAMPLE
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DECISION VARIABLES

• X1 NUMBER OF TV COMMERCIALS
• X2 NUMBER OF RADIO COMMERCIALS
• X3 NUMBER OF NEWSPAPER AD

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AUDIENCE
MEDIUM NUMBER OF PEOPLE (EXPOSURE)
TV 20000
RADIO 12000
PAPER 9000
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OBJECTIVE FUNCTION

• MAX 20000X112000X29000X3

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BUDGET CONSTRAINT 100,000
MEDIUM COST
TV 15,000
RADIO 6,000
PAPER 4,000
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15000X16000X24000X3 lt 100000
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TV CONSTRAINT

• TV STATION HAS TIME AVAILABLE FOR 4 COMMERCIALS
• X1 lt 4

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RADIO CONSTRAINT

• RADIO STATION HAS TIME AVAILABLE FOR 10
  COMMERCIALS
• X2 lt 10

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AD AGENCY CONSTRAINT

• AGENCY HAS STAFF AVAILABLE FOR PRODUCING NO MORE
  THAN A TOTAL OF 15 ADS
• X1X2X3 lt 15

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TRANSPORTATION EXAMPLE

• LOGISTICS, SUPPLY CHAIN MANAGEMENT

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DECISION VARIABLES

• X1A NUMBER OF UNITS SHIPPED FROM WAREHOUSE 1 TO
  STORE A
• X1B NUMBER OF UNITS SHIPPED FROM WAREHOUSE 1 TO
  STORE B
• X1C TO STORE C
• X2A FROM WAREHOUSE 2 TO STORE A
• X3C FROM W 3 TO STORE C

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COST TO SHIP FROM WAREHOUSE TO STORE
WARE-HOUSE TO STORE SUPPLY
A B C
1 16 18 11 300
2 14 12 13 200
3 13 15 17 200
DEMAND 150 250 200
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OBJECTIVE FUNCTION

• MIMINIZE COST
• MIN 16X1A 14X2A 13X3A
• 18X1B 11X1C17X3C

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CONSTRAINTS

• (1) SUPPLY
• (2) DEMAND

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2 CASES

• TOTAL SUPPLYgt TOTAL DEMAND
• SHIPMENTS lt SUPPLY
• SHIPMENTS DEMAND
• THIS EXAMPLE

• TOTAL SUPPLY lt TOTAL DEMAND
• SHIPMENTS SUPPLY
• SHIPMENTS lt DEMAND

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SUPPLY CONSTRAINTS

• (1) WAREHOUSE 1 X1AX1BX1C lt 300
• (2) W. 2 X2AX2BX2C lt 200
• (3) W. 3 X3AX3BX3C lt 200

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DEMAND CONSTRAINTS

• (A)STORE A X1AX2AX3A 150
• (B) STORE B X1BX2BX3B 250
• (C) STORE C X1CX2CX3C 200

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DEA Data Envelopment Analysis

• Compares service units for efficiency
• Inputs teacher/student ratio funds/student
• Outputs test scores
• LP will compare efficiency of 1 school with
  other school.

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Inputs
Teacher/student ratio Funds/student
Carey school .08 340
Delancey school .06 460
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Outputs
School Reading test score Math test score
C 81 79
D 81 73
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Decision Variables

• X1 opportunity cost of output 1 (reading score)
• X2 o.c. of output 2 ( math score)
• Y1 o.c. of input 1 (teacher/student ratio)
• Y2 o.c. of input 2 (funds/student)

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Objective Function

• MAXIMIZE Value of Delancey outputs
• We will measure Delancey efficiency
• Data from previous slide output
• MAX 81x1 73x2

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Constraint 1

• Scale Delancey inputs to one, so answer is an
  efficiency ratio
• Data from previous slide input
• .06y1 460y2 1

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Constraint 2

• Efficiency value of outputs/value of inputs
• Max efficiency 100, so
• value of outputs lt value of inputs
• Carey constraint Carey outputs lt Carey inputs
  81x1 79x2 lt .08y1
  340y2
• Delancey constraint Del outputs lt Del inputs
  81x1 73x2 lt .06y1 460y2

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output

• Case 1 Maximum objective function value 1, so
  Delancey is efficient
• Case 2 Maximum objective function value lt 1, so
  Delancey is not efficient

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EXCEL USE SOLVER FOR LP

• MEMO asnt