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Mathematical ProgrammingMathematical Programming
Linear Programming
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TThreehree Goals in this Chapter Goals in this Chapter
• Learn the principle of Simplex-Algorithm
• Learn how to formulate pbs as LP’s
– Like brainteasers: can be fun – Man problems that !o not look likestereotpical "pro!uct mix# problems can beformulate! as LP’s
– Spotting LP’s is an art
• Learn how to implement an! sol$e LP’sin %xcel
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Linear Linear ProgrammingProgramming
LP Models
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LP ModelLP Model
An optimi&ation mo!el is a linear
program 'or LP( if it has continuous$ariables) a single linear ob*ecti$efunction) an! all constraints are
linear e+ualities or ine+ualities,
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LP ModelLP Model
• Linearit
• i$isibilit '.ontinuous(
• Assumption of .ertaint
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LP ModelLP Model
21 2420: x x Maximize +
3224
6063..
21
21
≤+
≤+
x x
x xt s
;0;0: 21 ≥≥ x x NNC
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Standard FormStandard Form
4321 002420: x x x x Min ++−−−
3224
6063..
421
321
=++
=++
x x x
x x xt s
;0;0
;0;0:
43
21
≥≥
≥≥
x x
x x NNC
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Linear ProgrammingLinear Programming
For purposes of describing and analyzing algori!"s# !e
proble" is ofen saed in !e sandard for"
$0#:"in% ≥= xb Ax xcT
&!ere ' is !e (ecor of n un)no&ns# c is !e n di"ensional
cos (ecor# and * !e consrain "ari' +" ro&s and ncolu"ns,.
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Steps in Formulating a LinearSteps in Formulating a Linear
Programming ProblemProgramming Problem
• /n!erstan! the problem
• 0!entif the ecision Maker 'M(
• 0!entif the !ecision $ariables• State the ob*, function as a linear
combination of the !ecision $ariables
•State the constraints as a linearcombination of the !ecision $ariables
• 0!entif upper or lower boun!s on 1’s
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Linear Linear ProgrammingProgramming
Solving and Sensitivity
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Linear ProgrammingLinear Programming
!e feasible region described by !e consrains
is a polytope, or simplex# and a leas one
"e"ber of !e soluion se lies a a (ere' of !is polyope
/ac! consrain +euaion, defines a sraig! line in
!e space of !e un)no&ns '
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Ways of Solving an LPWays of Solving an LP
• 2raphical Metho!
• %numerating all extreme points
• Simplex metho! in$ente! b 2, ant&ig – Speciali&e! software such as L034 – 2eneral software such as %xcel sol$er
• 0nterior point metho!s of the sort propose! b
5armarkar • Speciali&e! algorithms for special tpes of
LP’s
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Sensitivity Analysis ofSensitivity Analysis of
LPsLPs
• Simplex Metho! is a stan!ar! wa to sol$e linearprograms
• 0ts solution iel!s a "simplex tableau# with "!ual$ariables# which sol$e a closel relate! "!ual# problem
6 which contain sensiti$it analsis information fororiginal problem concerning – 7ow much coul! ob*, fn, coefficients change without changing
optimal solution8
– 7ow woul! changing 97S $alues affect $alue of optimal
solution8 – how much coul! one change the 97S $alues without
changing the pattern '"nature#( of optimal solution8
– ;oul! it be optimal to pro!uce a new pro!uct if it werea$ailable8
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Simple!"#ased SensitivitySimple!"#ased Sensitivity
Analysis is $ery LimitedAnalysis is $ery Limited
• .an’t see "aroun! corners#
• 4nl rele$ant for linear programs
• ;as more rele$ant before .P/ cclesbecame cheap
• 3ow more con$enient to sol$e
iterati$el an! plot results) e,g,) with – Spi!er
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Linear Linear ProgrammingProgramming
Various types of LP Models
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%e!t Topic& Learning%e!t Topic& Learning
'o( to Formulate LPs'o( to Formulate LPs
• 2etting !ec $ars right is often the ke• on’t rein$ent the wheel= get familiar with
common tpes of LP’s
– Man problems are $ariants on thesecommon tpes
– Man other problems ha$e components whichare similar to one or more of the common
tpes•
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)evie( of Common)evie( of Common
Types of LP Pbs*Types of LP Pbs*
>( Pro!uct mix problems
?( Make $s, bu
@( 0n$estmentPortfolio allocation pbsB( Sche!uling
C(
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)evie( of Common)evie( of Common
Types of LP Pbs*Types of LP Pbs*
1) Product mix problems
?( Make $s, bu
@( 0n$estmentPortfolio allocation pbsB( Sche!uling
C(
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+,& Product Mi! -ecisions+,& Product Mi! -ecisions
.Li/e the 'o(ies Problem0.Li/e the 'o(ies Problem0
• ecision: 7ow man of each tpe of pro!uctshoul! be ma!e 'offere!(
• ecision $ariables – Gi H amount of pro!uct i to make 'offer(
•
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)evie( of Common)evie( of Common
Types of LP Pbs*Types of LP Pbs*
>( Pro!uct mix problems
2) Make vs. buy
@( 0n$estmentPortfolio allocation pbsB( Sche!uling
C(
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+1& Ma/e vs* #uy+1& Ma/e vs* #uy
-ecisions-ecisions
• Se$eral pro!ucts) each can be ma!e inhouse or purchase! from $en!ors
• ecision: 3ot *ust how much of each pro!uctto obtain but also how much to make an!how much to bu) so ,,,
• ecision $ariables
for each pro!uct i: – Mi H amount of pro!uct i to make in house
– i H amount of pro!uct i to purchase
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Ma/e vs* #uy -ecisionsMa/e vs* #uy -ecisions
• .onstraints – Meet !eman!
– Pro!uction capacit constraints
– 3onnegati$it
• 4b*ecti$e: Minimi&e cost 'or max profit(
• %xamples
– .7AMP/S '.i$ilian 7ealth an! Me!ical Programof the /niforme! Ser$ices(
– 4utsourcingpri$ati&ation
– Staffing courses with a!*uncts
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Ma/e vs* #uy -ecisionsMa/e vs* #uy -ecisions
More General PerspectiveMore General Perspective
• Se$eral pro!ucts) each can be obtaine!through one or more sources
• ecision $ariables – Gi* H amount of pro!uct i obtaine! from source *
• .onstraints – Suppl constraints on each source *
– eman! constraints on each pro!uct i – Pro!uction capacit constraints
– 3onnegati$it
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)evie( of Common)evie( of Common
Types of LP Pbs*Types of LP Pbs*
>( Pro!uct mix problems
?( Make $s, bu
) !nvestment"Portfolio allocation pbsB( Sche!uling
C(
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+2& 3nvestment4Portfolio+2& 3nvestment4Portfolio
AllocationAllocation
• Pool of resources 'e,g,) mone orworkers( nee!s to be allocate! across anumber of a$ailable "instruments#
• ecision: how much to put 'e,g,) in$est(in each instrument) so ,,,
• ecision $ariables
for each pro!uct i: – Gi H amount of in$este! in instrument i
3 t t4P tf li3 t t4P tf li
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3nvestment4Portfolio3nvestment4Portfolio
Allocation -ecisionsAllocation -ecisions
• .onstraints – All resources allocate!
– i$ersit constraints on amount in$este! in an
one instrument or tpe of instrument – 3onnegati$it
• 4b*ecti$e: Maximi&e returnbenefit
• %xamples – ollars to financial in$estments
– ollars to !e$elopment pro*ects
– Staffpersonnel to work pro*ects
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)evie( of Common)evie( of Common
Types of LP Pbs*Types of LP Pbs*
>( Pro!uct mix problems
?( Make $s, bu
@( 0n$estmentPortfolio allocation pbs#) Sc$eduling
C(
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+5& Scheduling Personnel .or+5& Scheduling Personnel .or
6ther )esources0 to Shifts6ther )esources0 to Shifts
• 4utline of problem: eman! for ser$ices $arieso$er time 'time of !a or !a of week(, Iou canha$e emploees start at an time) but ou ha$e
less control o$er when the stop,
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Personnel Scheduling %aturalPersonnel Scheduling %atural
Language -escriptionLanguage -escription
• ecisions: 7ow man people shoul! beassigne! to each shift 'or shift tpe( – 'Must explicitl i!entif shiftsJ(
• 4b*ecti$e: Minimi&e the cost of theassignment) which is the sum o$er all shifts ofthe number of people working that shift timesthe costperson assigne! to that shift
• .onstraints: 4ne for e$er time perio!, Meet!eman! in that perio! '6 nonneg(
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Scheduling 7!ample&Scheduling 7!ample&
Police Shift AssignmentPolice Shift Assignment
• eman! for police is !efine! for four hourblocks throughout a ?B hour !a – 'DF hour week an!
pa attention to K of consecuti$e !as officersworke! too,(
• 4fficers can work F or >? hour shifts
• Police are pai! !ouble time for workingbeon! F hours) so >? hour shift costs twiceas much as an F hour shift
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-emand -ata for Police-emand -ata for Police
Shift Assignment 7!ampleShift Assignment 7!ample
Perio! >> pm - @ am @
? @ am - E am >?
@ E am - >> am >D
B >> am - @ pm ?
C @ pm - E pm @D
D E pm - >> pm @B
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Algebraic Formulation ofAlgebraic Formulation of
Police Scheduling 7!amplePolice Scheduling 7!ample
• ecision 1ariables
– Gi H officers starting F hour shift in perio! i
– Ii H officers starting >? hour shift in perio! i• 4b*ecti$e: Minimi&e Labor .ost
– Assume base pa e+ual for all officers) an!
measure in terms of multiples of base pafor one shift
– Min H G> N ? I> N G? N ? I? N ,,,
Al b i F l ti
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Algebraic FormulationAlgebraic Formulation
.cont*0.cont*0
• .onstraintsGD N G> N IC N ID N I> OH @ 'Perio! >(
G> N G? N ID N I> N I? OH >? 'Perio! ?(G? N G@ N I> N I? N I@ OH >D 'Perio! @(
G@ N GB N I? N I@ N IB OH ? 'Perio! B(
GB N GC N I@ N IB N IC OH @D 'Perio! C(GC N GD N IB N IC N ID OH @B 'Perio! D(
Gi) Ii OH for all i
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)evie( of Common)evie( of Common
Types of LP Pbs*Types of LP Pbs*
>( Pro!uct mix problems
?( Make $s, bu
@( 0n$estmentPortfolio allocation pbsB( Sche!uling
%) &ransportation"'ssignment problems
D( len!ing
E( Multi-perio! planning
F( .utting stock problems
+ 4+8 T t ti 4
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+8& Transportation4+8& Transportation4
Assignment ProblemsAssignment Problems
• 7a$e $arious +uantities of a commo!it atmultiple sources, 3ee! to meet !eman! forthat commo!it at "sinks# '!estinations(, 7ow
much shoul! ou mo$e that from each sourceto each sink in or!er to minimi&e the cost ofmeeting !eman! at each !estination,
• ecision $ariables
– Gi* H amount sent from source i to sink *• .ost parameters
– ci* H cost per unit of shipping from source i to sink *
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)evie( of Common)evie( of Common
Types of LP Pbs*Types of LP Pbs*
>( Pro!uct mix problems
?( Make $s, bu
@( 0n$estmentPortfolio allocation pbsB( Sche!uling
C(
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+9& #lending Problems+9& #lending Problems
• Se$eral ingre!ients 'fee!stocks( are mixe! tocreate !ifferent final pro!ucts, 7ow much ofeach ingre!ient shoul! go into each pro!uct
in or!er to minimi&e pro!uction costs whilesatisfing +ualit constraints on the pro!ucts8%,g) – oil refining
– pro!ucing animal fee! mixes – pro!uction of !air pro!ucts – allocation of coal to power plants
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#lending Problems#lending Problems
• ecision 1ariables
– Gi* Hunits of ingre!ient i use! in pro!uct *
– units coul! be poun!s) tons) gallons) etc,• 4b*ecti$e
– Minimi&e cost of pro!ucing re+uire!
amounts of each pro!uct 'tpicall !ri$enb cost of ingre!ients(
#l di P bl#l di P bl
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#lending Problemslending Problems&
Constraint FormulationConstraint Formulation
• eman! constraint example – 3ee! at least F) poun!s of pro!uct >
– G>> N G?> OH F)• ualit constraint example
– 0ngre!ient > is ?Q corn, 0ngre!ient ? isCQ corn,
– Pro!uct > must be at least @Q corn,
– ,? G>> N ,C G?> OH ,@ 'G>> N G?>(
f C) i f C
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)evie( of Common)evie( of Common
Types of LP Pbs*Types of LP Pbs*
>( Pro!uct mix problems
?( Make $s, bu
@( 0n$estmentPortfolio allocation pbsB( Sche!uling
C(
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+:& Multi"period Planning+:& Multi"period Planning
ProblemsProblems
•
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+:a& Multi"period Financial+:a& Multi"period Financial
Planning ProblemsPlanning Problems
• 0n$est mone to maximi&e return)manage cash flow !uring construction
pro*ect) etc,• ecision $ariables – Gi*Hamount in$este! in instrument i at time *
•
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Multi"period PlanningMulti"period Planning
ConstraintsConstraints
• "Mass balance# constraint on mone for eachtime perio!
9e$enue at time t N R maturing at time t H
amount in$este! at time t N paments
!ue at time t 'for e$er perio! t(
• .onstraints on mix of in$estments
– o not excee! maximum risk threshol! – Maintain minimum le$el of li+ui!it
– %tc,
+:b M lti P i d Pl i
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+:b& Multi"Period Planning&+:b& Multi"Period Planning&
Trading 6ff 6T and 3nventoryTrading 6ff 6T and 3nventory
• Suppose that in some perio!s !eman!excee!s normal pro!uction capacit,
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%otation%otation
• ecision $ariables
– Gi H regular pro!uction in perio! i
– Ii H 4< pro!uction in perio! i
– i H shortage in perio! i
– 0i H in$entor carrie! into perio! i
• Parameters
– i H !eman! in perio! i – 0> H initial in$entor
– .ost parameters
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FormulationFormulation
• Minimi&e weighte! sum of G’s) I’s) ’s)an! 0’s
• Sub*ect to – Gi TH regular pro!uction capacit
– Ii TH 4< pro!uction capacit
– 0i TH storage capacit
– 0i N Gi N Ii N i H i N 0iN>
mass balance constraint
+:c& Multi Period Planning&+:c& Multi Period Planning&
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+:c& Multi"Period Planning&+:c& Multi"Period Planning&
Wor/ Force SchedulingWor/ Force Scheduling
• Si&e of 'traine!( labor force re+uire! $arieso$er time 'e,g,) 09S staff(
• .an
– 7ire new emploees) but the nee! to be traine!b experience! workers 'which takes time awafrom primar task( an! ma not sta withorgani&ation
– 'Sometimes( can la-off emploees – 'Sometimes( can outsource or hire temps) buttpicall at a high cost
– 'or proacti$el balance the workloa!(
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%otation%otation
• ecision $ariables
– ;i H K of contractors hire! in perio! i
– Gi H K emploees hire! 6 traine! in perio! i
– Ii H K of experience! workers in perio! i
– i H K lae! off at beginning of perio! i
• Parameters
– i H !eman! for exp, workers in perio! i
– I H initial number of experience! workers
– .ost parameters
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FormulationFormulation
• Min weighte! sum of ;’s) G’s) I’s) 6’s
• Suppose – 4ne exp worker can train four new hires
– UCQ retention of experience! workers
– CQ retention of trainees
• N ,C Gi-> mass balance
– nonnegati$it
) i f C)evie( of Common
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)evie( of Common)evie( of Common
Types of LP Pbs*Types of LP Pbs*
>( Pro!uct mix problems
?( Make $s, bu
@( 0n$estmentPortfolio allocation pbsB( Sche!uling
C(
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+;& Cutting Stoc/ Problem+;& Cutting Stoc/ Problem
• Suppose ou pro!uce a wi!e)continuous sheet of material 'steel) film)
paper) fabric) etc,(, .ustomers !eman!$arious +uantities 'lengths( of thinnerstrips, 7ow shoul! ou cut the wi!esheet into strips to meet !eman! whileminimi&ing either amount of rawmaterial cut or amount waste!8
Cutting Stoc/ Problem&Cutting Stoc/ Problem&
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Cutting Stoc/ Problem&Cutting Stoc/ Problem&
-ecision $ariables-ecision $ariables
•
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7!ample of Patterns for 297!ample of Patterns for 29
Foot Wide )a( MaterialFoot Wide )a( Material
Pattern FV pieces >?V pieces >DV pieces ;aste
K> B L L B
K? @ > L LK@ ? L > B
KB > ? L B
KC > > > L
KD L @ L LKE L L ? B
Cutting Stoc/Cutting Stoc/
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Cutting Stoc/Cutting Stoc/
Problem ConstraintsProblem Constraints
• Meet !eman! for F foot stripsB G> N @ G? N ? G@ N GB N GC OH b>
• Meet !eman! for >? foot stripsG? N ? GB N GC N @ GD OH b?
• Meet !eman! for >D foot strips
G@ N GC N ? GE OH b@• 3onnegati$itGi OH for all i
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Linear Linear ProgrammingProgramming
Solving LPs in MS /xcel
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Solving LPs in 7!celSolving LPs in 7!cel
• 4rgani&e !ata for mo!el in sprea!sheet
• 9eser$e cells for !ecision $ariables
• .reate formula for the ob*, function $al,• or each constraint) create
– algebraic expression for L7S $alue
– 97S constraint $alue
• Label e$erthingJ
) ll C Mi 7 l
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)ecall Course Mi! 7!amples)ecall Course Mi! 7!amples
Algebraic FormulationAlgebraic Formulation
• ecision $ariables – G> H K of seminars offere! – G? H K of lectures offere!
• Maximi&e H @ G> N ? G?s,t,
G> N ? G? TH @B 'facult time(
G? TH > 'lecture rooms(G> TH ? 'seminar rooms(
?C G> N > G? OH F 'enrollment(
G>) G? OH 'non-negati$it(
Course Mi! 7!ample&Course Mi! 7!ample&
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Course Mi! 7!ample&Course Mi! 7!ample&
Spreadsheet FormulationSpreadsheet Formulation
G> H K of G? H K of
Seminars Lectures .ourse
K offere! 1ariet
@ ? /nit 1ariet Score 0n!ex
1ariet ? @B acult > Lecture 7alls
> ? Seminar 9ooms
OH constraints A$ailable 3ee!e!
?C > F %nrollment
Course Mi! 7!ample&Course Mi! 7!ample&
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Course Mi! 7!ample&Course Mi! 7!ample&
Spreadsheet FormulasSpreadsheet Formulas
G> H K of G? H K of
Seminars Lectures .ourse
K offere! 1ariet
@ ? /nit 1ariet Score 0n!ex
HA@WAB H:@W:B 1ariet ? HAR@WAFN:R@W:F @B acult HAR@WAUN:R@W:U > Lecture 7alls
> HAR@WA>N:R@W:> ? Seminar 9ooms
OH constraints A$ailable 3ee!e!
?C > HAR@WA>@N:R@W:>@ F %nrollment
Spreadsheet Solution&Spreadsheet Solution&
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Spreadsheet Solution&Spreadsheet Solution&
Cells Specified to Solver Cells Specified to Solver
G> H K of G? H K of Seminars Lectures .ourse
K offere! 1ariet@ ? /nit 1ariet Score 0n!ex 1ariet ? @B acult > Lecture 7alls
> ? Seminar 9ooms
OH constraints A$ailable 3ee!e!?C > F %nrollment
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The Solver -ialog #o!The Solver -ialog #o!
• %nter target cell) changing cells) an!constraint cells information – can *ust point to cells= no nee! to tpe
– can highlight sets of constraints at once
– nonnegati$it constraints are *ust like otherconstraints= for 97S *ust tpe in
• /n!er options – .heck "Assume linear mo!el#
– All other !efaults shoul! be fine
Spreadsheet Solution&Spreadsheet Solution&
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Spreadsheet Solution&Spreadsheet Solution&
6btained by Solver 6btained by Solver
G> H K of G? H K of Seminars Lectures .ourse
? E K offere! 1ariet@ ? /nit 1ariet Score 0n!ex
D >B 1ariet ? @B @B acult E > Lecture 7alls
> ? ? Seminar 9ooms
OH constraints A$ailable 3ee!e!?C > >? F %nrollment
Suppose 'ad 6ne MoreSuppose 'ad 6ne More
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Suppose 'ad 6ne MoreSuppose 'ad 6ne More
Faculty& 7nter B 1ariet ? @B % acult E > Lecture 7alls
> ? ? Seminar 9ooms
OH constraints A$ailable 3ee!e!
?C > >? F %nrollment
Pull -o(n Solver andPull -o(n Solver and
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Pull -o(n Solver andPull -o(n Solver and
Clic/ on C 1ariet ? @C @C acult E,C > Lecture 7alls> ? ? Seminar 9ooms
OH constraints A$ailable 3ee!e!?C > >?C F %nrollment
Spreadsheet -esignSpreadsheet -esign
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Spreadsheet -esignSpreadsheet -esign
GuidelinesGuidelines
• uil! mo!el aroun! !ispla of !ata
• on’t bur constants in formulas
• Logicall close +uantities shoul! bephsicall close
• esign so formulas can be copie!
• /se color) sha!ing) bor!ers) an!protection
• ocument with text boxes 6 cell notes
Some 7!cel Functions ThatSome 7!cel Functions That
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Some 7!cel Functions ThatSome 7!cel Functions That
Are =seful in LP FormulationAre =seful in LP Formulation
• S/MP94/.
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Summary of Solving LP sSummary of Solving LP s
(ith 7!cels Solver (ith 7!cels Solver
• %xcel facilitates entering 6 sol$ing LP’s
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Mathematical ProgrammingMathematical Programming
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Linear Linear ProgrammingProgramming
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