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Transcript of 0 Product Mix Example Static Workforce Planning models Blending Problem Aggregate Production...
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Product Mix Example
Static Workforce Planning models
Blending Problem
Aggregate Production Planning
Reference:
Winston and Albright, Chapters 2 & 3
Modeling and Solving LPs by Excel Solver
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Product Mix Problem
Monet company makes four types of frames.
The following table gives the manufacturing data:
How many frames of each type should be made per week to maximize the profit?
Labor Metal Glass Profit Upperbound
Frame 1 2 4 6 $6 1000
Frame 2 1 2 2 $2 2000
Frame 3 3 1 1 $4 500
Frame 4 2 2 2 $3 1000
Resourcelimits perweek
4000 6000 10000
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Product Mix Problem (contd.)
Linear Programming Formulation:
Max z = 6x1 + 2x2 + 4x3 + 3x4
subject to
6x1 + 2x2 + 4x3 + 3x4 4000 (labor)
4x1 + 2x2 + x3 + 2x4 6000 (metal)
6x1 + 2x2 + x3 + 2x4 10000 (glass)
x1 1000x2 2000x3 500x4 1000
x1, x2, x3 , x4 0
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Product Mix Problem (contd.)
Product Mix Problem with Optimal Solution
Input dataFrame Type
1 2 3 4 Total used Total availableLabor hours per frame 2 1 3 2 4000 <= 4000Metal (oz.) per frame 4 2 1 2 6000 <= 6000Glass (oz.) per frame 6 2 1 2 8000 <= 10000
Total profit
Profit per frame $6.00 $2.00 $4.00 $3.00 $9,200.00
Production planFrame Type
1 2 3 4
Frames produced 1000 800 400 0
<= <= <= <=Maximum sales 1000 2000 500 1000
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Answer & Sensitivity ReportsTarget Cell (Max)
Cell Name Original Value Final Value$F$10 Total profit $9,200.00 $9,200.00
Adjustable CellsCell Name Original Value Final Value
$B$15 Type 1 frames produced 1000 1000$C$15 Type 2 frames produced 800 800$D$15 Type 3 frames produced 400 400$E$15 Type 4 frames produced 0 0
ConstraintsCell Name Cell Value Formula Status Slack
$F$6 Labor hours 4000 $F$6<=$H$6 Binding 0$F$7 Metal (oz.) 6000 $F$7<=$H$7 Binding 0$F$8 Glass (oz.) 8000 $F$8<=$H$8 Not Binding 2000$B$15 Type 1 frames nonnegativity 1000 $B$15>=0 Not Binding 1000$C$15 Type 2 frames nonnegativity 800 $C$15>=0 Not Binding 800$D$15 Type 3 frames nonnegativity 400 $D$15>=0 Not Binding 400$E$15 Type 4 frames nonnegativity 0 $E$15>=0 Binding 0$B$15 Maximum type 1 frames 1000 $B$15<=$B$17 Binding 0$C$15 Maximum type 2 frames 800 $C$15<=$C$17 Not Binding 1200$D$15 Maximum type 3 frames 400 $D$15<=$D$17 Not Binding 100$E$15 Maximum type 4 frames 0 $E$15<=$E$17 Not Binding 1000
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Answer & Sensitivity Reports
Changing CellsFinal Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease$B$15 Type 1 frames produced1000 2 6 1E+30 2$C$15 Type 2 frames produced800 0 2 6 0.25$D$15 Type 3 frames produced400 0 4 2 0.5$E$15 Type 4 frames produced 0 -0.2 3 0.2 1E+30
ConstraintsFinal Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease$F$6 Labor hours 4000 1.2 4000 250 1000$F$7 Metal (oz.) 6000 0.4 6000 2000 500
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Static Work Scheduling Problem
Number of full time employees on different days of the week
Each employee must work five consecutive days and then receive two days off
Meet the requirements by minimizing the total number of full time employees
Day 1 = Monday 17
Day 2 = Tues. 13
Day 3 = Wedn. 15
Day 4 = Thurs. 19
Day 5 = Friday 14
Day 6 = Satur. 16
Day 7 = Sunday 11
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Static Workforce Scheduling
LP Formulation:
Min. z = x1+ x2 + x3 + x4 + x5 + x6 + x7
subject to
x1 + x4 + x5 + x6 + x7 17x1+ x2 + x5 + x6 + x7 13x1+ x2 + x3 + x6 + x7 15x1+ x2 + x3 + x4 + x7 19x1+ x2 + x3 + x4 + x5 14 x2 + x3 + x4 + x5 + x6 16 x3 + x4 + x5 + x6 + x7 11
x1, x2, x3, x4, x5, x6, x7 0
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Static Workforce Scheduling (contd.)
Post Office Scheduling Problem
Starting day of 5-day shift
Monday Tuesday Wednesday Thursday Friday Saturday Sunday
Total employee
sNumber starting 6.33 3.33 2.00 7.33 0.00 3.33 0.00 22.33
Number working on each
day 17.00 13.00 15.00 19.00 19.00 16.00 12.67>= >= >= >= >= >= >=
Minimal number required
each day 17 13 15 19 14 16 11
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Static Workforce Scheduling (contd.)
Post Office Scheduling Problem with Integer Constraints
Starting day of 5-day shift
Monday Tuesday Wednesday Thursday Friday Saturday Sunday
Total employee
sNumber starting 7 4 2 8 0 2 0 23
Number working on each
day 17 13 15 21 21 16 12>= >= >= >= >= >= >=
Minimal number required
each day 17 13 15 19 14 16 11
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Static Workforce Scheduling (contd.)
Generally, such problems have multiple optimal solution. We may be interested in an optimal solution which maximizes the number of days with weekends off. How to obtain such a solution?
How to create a fair schedule for employees so that all employees get weekends off?
We assumed demands are static with time. What if demands are functions of time? Such problems are called dynamic workforce scheduling problems.
How to allocate overtimes, and allow part-time employees?
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Blending Problems
Situations where various inputs must be blended in some desirable proportion to produce goods for sale are called blending problems.
Examples of blending problems:
Blending various crude oils to produce different types of gasoline
Blending various types of metal alloys to various types of steels
Blending various livestock feeds to produce minimum-cost feed mixture for cattle
Mixing various types of paper to produce recycled paper of varying quality
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Blending Problems (contd.)
Chandler Oil manufactures gasoline and heating oil
These products are produced by blending two types of crude oil (crude 1, and crude 2)
The following table gives the data for quality points and sales and purchase prices:
Determine the production quantities of gasolines and heating oils to maximize the profit
Sellingprice perbarrel
Minimumqualitypoints
Advertizingcost perbarrel
Gasoline $25.00 8 $0.20
Heating Oil $20.00 6 $0.10
Cost perbarrel
Qualitypoints
Availabilityin barrels
Crude 1 $25.00 10 5000
Crude 2 $20.00 5 10000
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Blending Problem (contd.)
Decision Variables:
x11 : Number of barrels of crude 1 used in the manufacturing of gasoline
x12 : Number of barrels of crude 1 used in the manufacturing of heating oil
x21 : Number of barrels of crude 2 used in the manufacturing of gasoline
x22 : Number of barrels of crude 2 used in the manufacturing of heating oil crude 2
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Blending Problem (contd.)
Max. (25 - .02) (x11 + x21) + (20 - .01) (x12 + x22)
s. t.x11 + x12 5000x21 + x22 10000
10x11 + 5x21 8(x11 + x21) 10x12 + 5x22 6(x12 + x22)
x11, x12 , x21, x22 0
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Blending Problem (contd.)Chandler Blending Problem (nonoptimal solution)
Monetary inputs Gasoline Heating oilSelling price/barrel $25.00 $20.00
Advertising cost/barrel $0.20 $0.10
Quality level per barrel of crudes
Crude oil 1 10Crude oil 2 5
Required quality level per barrel of productGasoline Heating oil
8 6
Blending plan (barrels of crudes in each product)Gasoline Heating oil Barrels used Barrels available
Crude oil 1 0 3000 3000 <= 5000Crude oil 2 5000 0 5000 <= 10000
Barrels sold 5000 3000
Constraints on qualityGasoline Heating oil
Quality "points" obtained 25000 30000>= >=
Quality "points" required 40000 18000
Profit summaryGasoline Heating oil
Profit/barrel $24.80 $19.90
Total profit $183,700
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Blending Problem (contd.)Cell Name Value Cost Coefficient Increase Decrease$B$17 Crude oil 1 Gasoline 3000 0 24.8 58.1666667 8.166666667$C$17 Crude oil 1 Heating oil 2000 0 19.9 8.16666667 58.16666667$B$18 Crude oil 2 Gasoline 2000 0 24.8 87.25 6.125$C$18 Crude oil 2 Heating oil 8000 0 19.9 6.125 14.54166667
ConstraintsFinal Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease$D$17 Crude oil 1 Barrels used5000 29.7 5000 10000 2500$D$18 Crude oil 2 Barrels used10000 17.45 10000 10000 6666.666667$B$23 Quality points obtained Gasoline40000 -2.45 0 5000 20000$C$23 Quality points obtained Heating oil60000 -2.45 0 10000 6666.666667
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Blending Problem (contd.)
MODELING ISSUES:
Blending problems in practice have many more inputs and outputs
Quality level of gasoline and heating oil may not a linear function of the fractions of the inputs used
Blending problems are periodically solved on the basis of current inventories and demand forecasts