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IE 312 1
Solution Techniques Models
Thousands (or millions) of variables Thousands (or millions) of constraints
Complex models tend to be valid Is the model tractable?
Solution techniques
IE 312 2
Definitions Solution: a set of values for all variables n decision variables: solution n-vector Scalar is a single real number Vector is an array of scalar components
IE 312 3
Some Vector Operations Length (norm) of a vector x
Sum of vectors
Dot product of vectors
n
jjx
1
2x
nn yx
yx
11
yx
n
jjj yx
1
yx
IE 312 4
Neighborhood/Local Search
Find an initial solution
Loop: Look at “neighbors” of current solution
Select one of those neighbors
Decide if to move to selected solution
Check stopping criterion
IE 312 5
Example: DClub Location Location of a discount department
store Three population centers
Center 1 has 60,000 people Center 2 has 20,000 people Center 3 has 30,000 people
Maximize business Decision variables: coordinates of
location
IE 312 6
Constraints
222
21
222
21
222
21
)2/1())4(()0(
)2/1()3()1(
)2/1()3()1(
xx
xx
xx
Must avoid congested centers (within 1/2 mile):
IE 312 7
Objective Function
22
21
22
21
22
21
21
)3()(1
30
)3()1(1
20
)3()1(1
60),(max
xx
xx
xxxxp
IE 312 8
Objective Function
IE 312 9
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Searching
IE 312 10
Improving Search Begin at feasible solution Advance along a search path Ever-improving objective function
value
Neighborhood: points within small positive distance of current solution
Stopping criterion: no improvement
IE 312 11
Local Optima Improving search finds a local optimum
May not be a global optimum(heuristic solution)
Tractability: for some models there is only one local optimum (which hence is global)
IE 312 12
Selecting Next Solution Direction-step approach
Improving direction: objective function better for than for all value of that are sufficiently small
Feasible direction: not violate constraints
xxx )()1( tt
Search directionStep size
xx )(t )(tx
IE 312 13
Step Size How far do we move along the
selected direction? Usually:
Maintain feasibility Improve objective function value
Sometimes: Search for optimal value
IE 312 14
Detailed Algorithm
0 Initialize: choose initial solution x(0); t=01 If no improving direction x, stop.2 Find improving feasible direction x (t+1)
3 Choose largest step size t+1 that remains feasible and improves performance
4 Update
Let t=t+1 and return to Step 1
)1(1
)()1(
tt
tt xxx
IE 312 15
Stopping Search terminates at local optimum
If improving direction exists cannot be a local optimum
If no improving direction then local optimum
Potential problem with unboundedness Can improve performance forever Search does not terminate
IE 312 16
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Local Optimum
IE 312 17
Improving Search Still a bit abstract ‘Find improving feasible direction’
How?
IE 312 18
Smooth Functions Assume objective function is
smooth What does this mean? You can find the derivative
Smooth Not Smooth
IE 312 19
Gradients Function
The gradient is found from the partial derivatives
Note that the gradient is a vector
),...,,()( 21 nxxxff x
)/,...,/,/()( 21 nxfxfxff x
IE 312 20
Example
22
21
22
21
22
21
21
)3()(1
30
)3()1(1
20
)3()1(1
60),(max
xx
xx
xxxxp
IE 312 21
Partial Derivatives
...)3()1(1
)1(120
...)3()1(1
)3()1(160),(
222
21
1
222
21
22
21
1
1
21
xx
x
xx
xxx
x
xxp
IE 312 22
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Plotting Gradients
IE 312 23
Direction of Gradients Gradients are
perpendicular to contours of objective function
direction most rapid objective value increase
Using gradients as direction gives us the steepest descent/ascent (greedy)
IE 312 24
Effect of Moving When moving in direction x: The objective function is increased if
(Improving direction for maximization)
The objective function is decreased if
0)( xxf
0)( xxf
IE 312 25
1x
62 x
1 9x
Feasible Direction Make sure the direction is feasible Only have to worry about active
constraints (tight/binding constraints)
No active constrains
One active constraint
Active if equalitysign holds!
IE 312 26
Linear Constraints Direction x is feasible if and only
if (iff)
n
jjj
n
jjj
n
jjj
bxa
bxa
bxa
1
1
1
if,0
if,0
if,0
xa
xa
xa
IE 312 27
Comments The gradient gives us
A greedy improving direction Building block for other improving
directions (later) Conditions
Check if direction is improving (gradient)
Check if direction is feasible (linear)
IE 312 28
Validity versus Tractability Which models are tractable for
improving search? Stops when it encounters a local
optimum
This is guaranteed to be a global optimum only if it is unique
IE 312 29
Unimodal Functions Unimodal functions:
Straight line from a point to a better point is always an improving direction
IE 312 30
Typical Unimodal Function
IE 312 31
Linear Objective Linear objective functions
Unimodal for both maximization and minimization
xcx
n
jjj xcf
1
)(
IE 312 32
Check
)()(
)(
)()(
)1()2()1(
)1()2(
1
)1()2(
1
)1(
1
)2()2()1(
xxx
xxc
xx
f
xxc
xcxcff
n
jj
n
jj
n
jj
jj
jj
First calculate
Then apply our test to)1()2( xxx
IE 312 33
Optimality Assume our objective function is
unimodal
Then every unconstrained local optimum is an unconstrained global optimum
Note that none of the constraints can be active (tight)
IE 312 34
Convexity Now lets turn to the feasible region A feasible region is convex if any
line segment between two points in the region is contained in the region
IE 312 35
Line Segments Representing a line
To prove convexity, we have to show that for any points in the region, all points that can be represented as above are also in the region
]1,0[,)1()2()1( xxx
IE 312 36
Linear Constraints
If all constraints are linear then the feasible region is convex Suppose we have two feasible points:
How about a point on the line between:
bxabxan
jjj
n
jjj
1
)2(
1
)1( and
?)1( )2()1()1()2()1( xxxxxx
IE 312 37
Verify Constraints Hold
bbb
xaxa
xaxa
xaxaxxa
n
jjj
n
jjj
n
jjj
n
jjj
n
jjjjj
n
jjjj
)1(
)1(
)1(
)1()1(
1
)2(
1
)1(
1
)2(
1
)1(
1
)2()1(
1
)2()1(
IE 312 38
Example
1x
62 x
1 9x
2x
IE 312 39
Tractability
(This is the reason we are defining all these term!)
The most tractable models for improving search are models with unimodal objective function convex feasible region
Here every local optimum is global
IE 312 40
Improving Search Algorithm
0 Initialize: choose initial solution x(0); t=01 If no improving direction x, stop.2 Find improving feasible direction x (t+1)
3 Choose largest step size t+1 that remains feasible and improves performance
4 Update
Let t=t+1 and return to Step 1
)1(1
)()1(
tt
tt xxx
IE 312 41
Initial Feasible Solutions Not always trivial to find an initial
solution Thousands of constraints and variables
Initial analysis: Does a feasible solution exist? If yes, find a feasible solution
Two methods Two-phase method Big-M method
IE 312 42
Two-Phase Method Choose a solution to model and construct
a new model by adding artificial variables for each violated constraint
Assign values to artificial variables. Perform improving search to minimize
sum of artificial variables If terminate at zero then feasible model
and continue, otherwise stop Delete artificial components to get
feasible solution Start an improving search
Phase I
Phase II
IE 312 43
Crude Oil Refinery
0,
6
9
s)(lubricant 5.03.02.0
fuel)(jet 5.12.04.0
(gasoline) 0.24.03.0s.t.
21
2
1
21
21
21
xx
x
x
xx
xx
xx
21 1520min xx
IE 312 44
Artificial Variables Select a convenient solution
Add artificial variables for violated constraints
0,0 21 xx
0,,,,
6
9
5.03.02.0
5.12.04.0
0.24.03.0
54321
2
1
521
421
321
xxxxx
x
x
xxx
xxx
xxx
IE 312 45
Phase I Model
0,,,,
6
9
5.03.02.0
5.12.04.0
0.24.03.0
54321
2
1
521
421
321
xxxxx
x
x
xxx
xxx
xxxs.t.
321321 ),,(min xxxxxxf
IE 312 46
Phase I Initial Solutions As before
To assure feasibility
Thus, the initial solution is
0,0 )0(2
)0(1 xx
5.0,5.1,2 )0(4
)0(3
)0(3 xxx
)5.0,5.1,2,0,0()0( x
IE 312 47
Phase I Outcomes We cannot get negative numbers and
problem cannot be unbounded Three possibilities
Terminate with f =0 Start Phase II with final solution as initial
solution Terminate at global optimum with f > 0
Problem is infeasible Terminate at local optimum with f > 0
Cannot say anything
IE 312 48
Big-M Method Artificial variables as before Objective function
Combines Phase I and Phase II search
sconstraint ingfor violatpenalty
523
objective original
21 )(1520min xxxMxx
IE 312 49
Terminating Big-M If terminates at local optimum with all
artificial variables = 0, then also local optimum for original problem
If M is ‘big enough’ at terminates at global optimum with some artificial variables >0, then original problem infeasible
Cannot say anything in between
IE 312 50
Linear Programming We know what leads to high
tractability: Unimodal objective function Convex feasible region
We know how to guarantee this Linear objective function Linear constraints
When are linear programs valid?
Much stronger!
IE 312 51
Allocation Models Scarce resource
Land, capital, time, fuel, etc. Allocate resources to competing ‘jobs’ Decision variables:
How much to allocate to each job Constraints:
Limitation on resources Other dynamics
IE 312 52
Forest Service Allocation Manage 191 acres of national forest Jobs (Prescriptions):
Logging Grazing Wilderness preservation
7 analysis areas
This is a simplified version of an actual application (fewer variables/constraints)
IE 312 53
Analysis Goals
Maximize the net present value (NPV) of
the land, such that we produce at least 40 million board feet of
timber,
we have at least 5 thousand animal unit
months of grazing, and
we keep wilderness index at least 70.
IE 312 54
DataArea Acres Use NPV Timber Grazing Wild
s i j p i,j t i,j g i,j w i,j
1 75 1 503 310 0.01 402 140 50 0.04 803 203 0 0 95
2 90 1 675 198 0.03 552 100 46 0.06 603 45 0 0 65
3 140 1 630 210 0.04 452 105 57 0.07 553 40 0 0 60
4 60 1 330 112 0.01 302 40 30 0.02 353 295 0 0 90
5 212 1 105 40 0.05 602 460 32 0.08 603 120 0 0 70
6 98 1 490 105 0.02 352 55 25 0.03 503 180 0 0 75
7 113 1 705 213 0.02 402 60 40 0.04 453 400 0 0 95
IE 312 55
Objective: Maximize NPV
jix ji on prescriptiby managed areain acres of thousands,
3,72,71,7
3,1
7
1
3
12,11,1,,
40060705...
...203140503max
xxx
xxxxpi j
jiji
IE 312 56
Constraints All acres in an area are allocated
Performance requirements are met
isx jj
ji
,3
1,
s)(wildernes 70788
1
(grazing) 5
(timber) 000,40
7
1
3
1,,
7
1
3
1,,
7
1
3
1,,
i jjiji
i jjiji
i jjiji
xw
xg
xt
IE 312 57
Blending Models Best mix of ingredients Applications
chemicals animal feed metals diets
Example: Fagersta AB - Swedish Steel Mill
IE 312 58
Steel Mill Charging a furnish
1000 kilogram charge Combine
Scrap in inventory Pure blend
Blend with right % of chemical elements Iron Other: Carbon, Nickel, Chromium,
Molybdenum
IE 312 59
Formulation Decision variables
How much of each available ingredient to use
j=1,2,…,7
Four supplies of scrap (1,2,3,4) Three pure materials (5,6,7)
chargein included ingredient of kilograms jx j
IE 312 60
Formulation (cont.) Constraints
Correct amount
Composition constraints Properties of the resulting blend
10007654321 xxxxxxx
total
blend
blend in the
fraction allowedor
used ingredient
th ofamount
ingredientth
infraction j
jj
IE 312 61
Example Constraints
7
172
7
172
7
1621
7
1621
7
1521
7
1521
7
14321
7
14321
013.00.1001.0
011.00.1001.0
012.00.1011.0120.0
010.00.1011.0120.0
0035.00.1032.0180.0
0030.00.1032.0180.0
0075.00040.00085.00070.0008.0
0065.00040.00085.00070.0008.0
jj
jj
jj
jj
jj
jj
jj
jj
xxx
xxx
xxxx
xxxx
xxxx
xxxx
xxxxx
xxxxx
IE 312 62
Other Constraints Availability
Positive Amounts
250
75
2
1
x
x
7,...,2,1,0 jx j
IE 312 63
Objective Function Minimize Cost
Note that the pure additives are very expensive!
7654321 536048981016min xxxxxxx
IE 312 64
Ration Constraints Bound quotients of linear functions
by a constant
Not linear! Can often be converted to linear
constraints
0065.00040.00085.00070.0008.0
0065.00040.00085.00070.0008.0
7654321
4321
7
14321
xxxxxxx
xxxx
xxxxxj
j
IE 312 65
Linear Programming Models
Allocation
Blending
Operations Planning
Shift Scheduling
IE 312 66
Operations Planning
What to do and when to do it?
Tubular Products Division of Babcock and Wilcox
New mill
Three existing mills
Optimal distribution of work?
IE 312 67
Problem DataWeekly
Cost Hours Cost Hours Cost Hours Cost Hours Demand
1 .5" thick 90 0.8 75 0.7 70 0.5 63 0.6 1002 .5" thin 80 0.8 70 0.7 65 0.5 60 0.6 6303 1" thick 104 0.8 85 0.7 83 0.5 77 0.6 5004 1" thin 98 0.8 79 0.7 80 0.5 74 0.6 9805 2" thick 123 0.8 101 0.7 110 0.5 99 0.6 7206 2" thin 113 0.8 94 0.7 100 0.5 84 0.6 2407 8" thick - - 160 0.9 156 0.5 140 0.6 758 8" thin - - 142 0.9 150 0.5 130 0.6 22
1 .5" thick 140 1.5 110 0.9 - - 122 1.2 502 .5" thin 124 1.5 96 0.9 - - 101 1.2 223 1" thick 160 1.5 133 0.9 - - 138 1.2 3534 1" thin 143 1.5 127 0.9 - - 133 1.2 555 2" thick 202 1.5 150 0.9 - - 160 1.2 1256 2" thin 190 1.5 141 0.9 - - 140 1.2 357 8" thick - - 190 1.0 - - 220 1.5 1008 8" thin - - 175 1.0 - - 200 1.5 10
Mill 2 Mill 3 Mill 4
Pressure
StandardProduct
Mill 1
IE 312 68
Formulation Decision Variables
Objective
ek)(pounds/we
millat produced product ofamount , mpx mp
16
1
product ofcost
4
1,,min
p
p
mmpmp xc
IE 312 69
Constraints Demand is met
Capacity is ok
Non-negativity
16,...,2,1,4
1,
pdx pm
mp
4,3,2,1,16
1,
mbx mp
mp
mpx mp ,,0,
IE 312 70
Canadian Forest Products
purchaselogs
peellogs
purchaseveneer
dry andprocess
presssheets
sand andtrim
IE 312 71
Study Objectives Stages of Production - More Complex Objective of OR analysis
How to operate production facilities Maximize contributed margin
Sales - Wood cost
Fixed costs: labor, maintenance, etc. Constraints: availability of wood, market for
products, capacity of plants
IE 312 72
Production Yield Data
Good Fair Good Fair
Available per month 200 300 100 1000Cost per log ($ Canadian) 340 190 490 140
A 1/16" green veneer (ft2) 400 200 400 200
B 1/16" green veneer (ft2) 700 500 700 500
C 1/16" green veneer (ft2) 900 1300 900 1300
A 1/8" green veneer (ft2) 200 100 200 100
B 1/8" green veneer (ft2) 350 250 350 250
C 1/8" green veneer (ft2) 450 650 450 650
Vendor 1 Vendor 2Veneer Yield (ft2)
IE 312 73
Availability and Costs
A B C A B C
Available (ft2/month) 5000 25000 40000 10000 40000 50000
Cost ($/ft2) 1.00 0.30 0.10 2.20 0.60 0.20
1/8" Green Veneer1/16" Green Veneer
IE 312 74
Composition of Products
AB AC BC AB AC BC
Front veneer 1/16 A 1/16 B 1/16 B 1/16 A 1/16 B 1/16 B
1/8 C 1/8 C 1/8 C1/8 C 1/8 C 1/8 C 1/8 B 1/8 B 1/8 B
1/8 C 1/8 C 1/8 C
Back veneer 1/16 B 1/16 C 1/16 C 1/16 B 1/16 C 1/16 C
Market/month 1000 4000 8000 1000 5000 8000
Price 45.00$ 40.00$ 33.00$ 75.00$ 65.00$ 50.00$
Pressing time 0.25 0.25 0.25 0.40 0.40 0.40
1/4" Plywood Sheets 1/2" Plywood Sheets
Core veneer
IE 312 75
Decision Variables
sold and pressed , gradeback , gradefront , thicknessof sheets
grade from processingafter grade as used thicknessof sheets
directly purchased veneer grade , thicknessof ft
thicknessinto peeled and vendor frombought quality of logs
',,
',,
2,
,,
g'gtz
gg'ty
gtx
tvqw
ggt
ggt
gt
twq
IE 312 76
Continuous Variables Treat all variables as continuous Continuous models more tractable Is it equally valid?
Does one unit back or forth really matter?
If no, then ok to use continuous approximation
Example: Does $0.01 ever matter?
IE 312 77
Objective
income) (sales cost) veneer (purchased - cost) (log- max
IE 312 78
Constraints Log availability
Purchased veneer availability
Pressing capacity
,...2008/1,1,16/1,1, GG ww
,...5000,16/1 Ax
,...1000,,4/1 BAz
IE 312 79
Balance Constraints Must link log and veneer
purchasing Balance constraint
assures in-flows equal or exceed out-flows for each stage of production
BAAA
AFGFG
yy
xwwww
,,16/1,,16/1
,16/116/1,2,16/1,2,16/1,1,16/1,1,
3535
200400200400
Etc.
IE 312 80
Assembly for Two Products
Assembly2
Part3
Part3
Part4
Assembly1
Part3
Part3
Part4
Assembly2
IE 312 81
Balance Equations
produced ofnumber jx j Decision Variable
Assembly 2 must be at least assembly 1
Must have enough parts
12 xx
214
213
21
12
xxx
xxx
IE 312 82
Shift Scheduling
Work fixed
Plan resources to do work
Shift scheduling/staff planning
models
How many types of workers/shifts
IE 312 83
Ohio National Bank
Hour Arrivals Hour Arrivals11:00 AM 10 5:00 PM 3212:00 PM 11 6:00 PM 501:00 PM 15 7:00 PM 302:00 PM 20 8:00 PM 203:00 PM 25 9:00 PM 84:00 PM 28 - -
Check processing center All checks must clear by 10 PM Check arrivals:
IE 312 84
Workers Two types:
Full time (8 hour shift, 1 hour lunch) $11 hour + $1 night 150% overtime 1000 checks/hour
Part time (4 hour shift) $7 hour 800 check/hour
IE 312 85
All Possible Shifts
11 12 13 11 12 13 14 15 16 17 18
11:00 AM R - - R - - - - - - -
12:00 PM R R - R R - - - - - -
1:00 PM R R R R R R - - - - -
2:00 PM R R R R R R R - - - -
3:00 PM - R R - R R R R - - -4:00 PM R - R - - R R R R - -5:00 PM R R - - - - R R R R -6:00 PM RN RN RN - - - - - RN RN RN7:00 PM RN RN RN - - - - RN RN RN8:00 PM OT RN RN - - - - - - RN RN9:00 PM - OT RN - - - - - - - RN
Start Part Time ShiftsFull Time Shifts
IE 312 86
Decision Variables
18...11at starting workerspart time
1211at starting e w/overtim workers timefull
131211at starting workers timefull
,,hz
,hy
,,hx
h
h
h
IE 312 87
Objective Function
181716151413
12111211131211
323130292828
28281818929190min
zzzzzz
zzyyxxx
IE 312 88
Constraints No more than 35 operators on duty
Overtime limits
machines) AM (11 351111 zx
202
12
1
1211
1212
1111
yy
xy
xy
IE 312 89
Covering Constraints Shift provides enough workers
131212111211
121111
shifts
118.08.011
108.01
trequiremenduty)on er rker)(numb(output/wo
wwzzxx
wzx
IE 312 90
Example Government agency Clerical workers can work 4x 10-hour
days/week
Need: Monday - 10 workers Friday 9 workers, and Tuesday - Thursday 7 workers
j = 1 Monday-Wednesday-Thursday-Fridayj = 2 Monday-Tuesday-Thursday-Fridayj = 3 Monday-Tuesday-Wednesday-Friday
IE 312 91
Formulation Decision variables
LP Model
jx j pattern workingemployees ofnumber
0,,
9
7
7
7
10s.t.min
321
321
21
31
32
321
321
xxx
xxx
xx
xx
xx
xxxxxx
Friday) (cover
Thursday) (cover
Wednesday) (cover
Tuesday) (cover
Monday) (cover
IE 312 92
Time-Phased Models
Static Models One point in time
Planning or design stage decisions
Dynamic Models Account for decision over time
Operational decisions
IE 312 93
Institutional Food Services (IFS) Cash Flow Supplies food to restaurants,
schools, etc. Cash flow decisions
Item 1 2 3 4 5 6 7 8
Cash sales s t 600$ 750$ 1,200$ 2,100$ 2,250$ 180$ 330$ 540$
Accounts receivable r t 770$ 1,260$ 1,400$ 1,750$ 2,800$ 4,900$ 5,250$ 420$
Accounts payable p t 3,200$ 5,600$ 6,000$ 480$ 880$ 1,440$ 1,600$ 2,000$
Expenses e t 350$ 400$ 550$ 940$ 990$ 350$ 350$ 410$
Projected Weekly Amount/Week
IE 312 94
System Dynamics Cash sales and accounts receivable result in
immediate income to checking account Accounts payable are due in 3 weeks but
get 2% discount if paid immediately Bank
$4 million line of credit 0.2% interest/week Must have 20% of borrowed amount in checking
Short-term money market with 0.1%/week $20,000 safety balance in checking
IE 312 95
Decision Variables
tz
ty
tx
tw
th
tg
t
t
t
t
t
t
week during handon cash
in week debt credit of line cumulative
in week market money in investedamount
3 week until delayed payable accounts
in week paiddebt ofamount
in week borrowedamount
Decision Variables
Define for convenience
IE 312 96
Balance Constraints
1 period
incash
period
ingain net
period
incash
1 period
in level
starting
decisions
period
ofimpact
period
in level
starting
ttt
t
t
t
General time-phased models
Cash flow models
IE 312 97
Net Gain in Each Period Note that we carry both cash and debt
Cost Increments Cash DecrementsFunds borrowed in week t Borrowing paid off in week tInvestment principal from week t-1 Investment in week tInterest on investment in week t-1 Interest on debt in week t-1Cash sales from week t Expenses paid in week tAccounts receivable for week t Accounts payable paid with discount for week t
Accounts payable paid without discount for week t-3
IE 312 98
Balance Constraints Cash balance
Debt balance
3
1111
)(98.0
002.0001.0
tttttt
tttttttt
wwpres
yxxxhgzz
tttt hgyy 1
IE 312 99
Linear Programming Model
tttttt
t
tt
tt
t
ttt
tttttt
tttttttt
tt
tt
tt
zyxwhg
z
pw
yz
y
gyy
wwpres
yxxxhgzz
x.w.y.
,,,,,0
20
20.0
4000
)(98.0
002.0001.0s.t.
00100200020min
1
3
1111
8
1
8
1
8
1
IE 312 100
Optimal Solution
Decision Variable 1 2 3 4 5 6 7 8Borrowing 100.0$ 505.7$ 3,394.3$ -$ 442.6$ -$ -$ -$ Debt payment -$ -$ -$ 442.5$ -$ 2,715.3$ 1,284.7$ -$ Payables delayed 2,077.6$ 3,544.5$ 1,138.5$ -$ -$ -$ -$ -$ Short-term investments -$ -$ -$ -$ -$ -$ 2,611.7$ 1,204.3$ Cumulative debt 100.0$ 605.7$ 4,000.0$ 3,557.4$ 4,000.0$ 1,284.7$ -$ -$ Cumulative cash 20.0$ 121.1$ 800.0$ 711.5$ 800.0$ 256.9$ 20.0$ 20.0$
Optimal Weekly Amount for Weeks
IE 312 101
Time Horizons Fixed time horizon
usual case problems with boundaries start with ‘typical’ numbers condition to assure ends ‘reasonably’
Infinite time horizon wrap around (output of last stage input to the first
stage)
IE 312 102
Example: Balance Constraints
Decision variables
Demand: 11000, 48000, 64000, 15000
inventoryin held shovels snow of thousands
quarter in produced shovels snow of thousands
q
q
i
qx
443
332
221
11
15
64
48
110
inventory
endingdemandproduction
inventory
beginning
ixi
ixi
ixi
ix
IE 312 103
“Typical” Initial Inventory
443
332
221
11
15
64
48
119
inventory
endingdemandproduction
inventory
beginning
ixi
ixi
ixi
ix
IE 312 104
Wrapper (Infinite Horizon)
443
332
221
114
15
64
48
11
inventory
endingdemandproduction
inventory
beginning
ixi
ixi
ixi
ixi
