IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of...
-
Upload
rosamond-washington -
Category
Documents
-
view
215 -
download
1
Transcript of IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of...
![Page 1: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/1.jpg)
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
![Page 2: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/2.jpg)
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
![Page 3: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/3.jpg)
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
![Page 4: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/4.jpg)
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
![Page 5: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/5.jpg)
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
![Page 6: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/6.jpg)
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):
![Page 7: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/7.jpg)
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
![Page 8: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/8.jpg)
IE 312 8
Objective Function
![Page 9: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/9.jpg)
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
![Page 10: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/10.jpg)
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
![Page 11: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/11.jpg)
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)
![Page 12: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/12.jpg)
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
![Page 13: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/13.jpg)
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
![Page 14: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/14.jpg)
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
![Page 15: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/15.jpg)
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
![Page 16: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/16.jpg)
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
![Page 17: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/17.jpg)
IE 312 17
Improving Search Still a bit abstract ‘Find improving feasible direction’
How?
![Page 18: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/18.jpg)
IE 312 18
Smooth Functions Assume objective function is
smooth What does this mean? You can find the derivative
Smooth Not Smooth
![Page 19: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/19.jpg)
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
![Page 20: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/20.jpg)
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
![Page 21: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/21.jpg)
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
![Page 22: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/22.jpg)
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
![Page 23: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/23.jpg)
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)
![Page 24: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/24.jpg)
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
![Page 25: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/25.jpg)
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!
![Page 26: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/26.jpg)
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
![Page 27: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/27.jpg)
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)
![Page 28: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/28.jpg)
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
![Page 29: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/29.jpg)
IE 312 29
Unimodal Functions Unimodal functions:
Straight line from a point to a better point is always an improving direction
![Page 30: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/30.jpg)
IE 312 30
Typical Unimodal Function
![Page 31: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/31.jpg)
IE 312 31
Linear Objective Linear objective functions
Unimodal for both maximization and minimization
xcx
n
jjj xcf
1
)(
![Page 32: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/32.jpg)
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
![Page 33: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/33.jpg)
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)
![Page 34: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/34.jpg)
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
![Page 35: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/35.jpg)
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
![Page 36: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/36.jpg)
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
![Page 37: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/37.jpg)
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(
![Page 38: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/38.jpg)
IE 312 38
Example
1x
62 x
1 9x
2x
![Page 39: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/39.jpg)
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
![Page 40: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/40.jpg)
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
![Page 41: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/41.jpg)
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
![Page 42: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/42.jpg)
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
![Page 43: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/43.jpg)
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
![Page 44: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/44.jpg)
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
![Page 45: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/45.jpg)
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
![Page 46: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/46.jpg)
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
![Page 47: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/47.jpg)
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
![Page 48: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/48.jpg)
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
![Page 49: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/49.jpg)
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
![Page 50: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/50.jpg)
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!
![Page 51: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/51.jpg)
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
![Page 52: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/52.jpg)
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)
![Page 53: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/53.jpg)
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.
![Page 54: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/54.jpg)
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
![Page 55: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/55.jpg)
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
![Page 56: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/56.jpg)
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
![Page 57: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/57.jpg)
IE 312 57
Blending Models Best mix of ingredients Applications
chemicals animal feed metals diets
Example: Fagersta AB - Swedish Steel Mill
![Page 58: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/58.jpg)
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
![Page 59: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/59.jpg)
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
![Page 60: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/60.jpg)
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
![Page 61: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/61.jpg)
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
![Page 62: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/62.jpg)
IE 312 62
Other Constraints Availability
Positive Amounts
250
75
2
1
x
x
7,...,2,1,0 jx j
![Page 63: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/63.jpg)
IE 312 63
Objective Function Minimize Cost
Note that the pure additives are very expensive!
7654321 536048981016min xxxxxxx
![Page 64: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/64.jpg)
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
![Page 65: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/65.jpg)
IE 312 65
Linear Programming Models
Allocation
Blending
Operations Planning
Shift Scheduling
![Page 66: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/66.jpg)
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?
![Page 67: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/67.jpg)
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
![Page 68: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/68.jpg)
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
![Page 69: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/69.jpg)
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,
![Page 70: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/70.jpg)
IE 312 70
Canadian Forest Products
purchaselogs
peellogs
purchaseveneer
dry andprocess
presssheets
sand andtrim
![Page 71: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/71.jpg)
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
![Page 72: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/72.jpg)
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)
![Page 73: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/73.jpg)
IE 312 73
Availability and Costs
A B C A B C
Available (ft2/month) 5000 25000 40000 10000 40000 50000
Cost ($/ft2) 1.00 0.30 0.10 2.20 0.60 0.20
1/8" Green Veneer1/16" Green Veneer
![Page 74: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/74.jpg)
IE 312 74
Composition of Products
AB AC BC AB AC BC
Front veneer 1/16 A 1/16 B 1/16 B 1/16 A 1/16 B 1/16 B
1/8 C 1/8 C 1/8 C1/8 C 1/8 C 1/8 C 1/8 B 1/8 B 1/8 B
1/8 C 1/8 C 1/8 C
Back veneer 1/16 B 1/16 C 1/16 C 1/16 B 1/16 C 1/16 C
Market/month 1000 4000 8000 1000 5000 8000
Price 45.00$ 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
![Page 75: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/75.jpg)
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
![Page 76: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/76.jpg)
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?
![Page 77: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/77.jpg)
IE 312 77
Objective
income) (sales cost) veneer (purchased - cost) (log- max
![Page 78: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/78.jpg)
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
![Page 79: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/79.jpg)
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.
![Page 80: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/80.jpg)
IE 312 80
Assembly for Two Products
Assembly2
Part3
Part3
Part4
Assembly1
Part3
Part3
Part4
Assembly2
![Page 81: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/81.jpg)
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
![Page 82: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/82.jpg)
IE 312 82
Shift Scheduling
Work fixed
Plan resources to do work
Shift scheduling/staff planning
models
How many types of workers/shifts
![Page 83: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/83.jpg)
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:
![Page 84: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/84.jpg)
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
![Page 85: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/85.jpg)
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
![Page 86: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/86.jpg)
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
![Page 87: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/87.jpg)
IE 312 87
Objective Function
181716151413
12111211131211
323130292828
28281818929190min
zzzzzz
zzyyxxx
![Page 88: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/88.jpg)
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
![Page 89: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/89.jpg)
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
![Page 90: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/90.jpg)
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
![Page 91: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/91.jpg)
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
![Page 92: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/92.jpg)
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
![Page 93: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/93.jpg)
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
![Page 94: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/94.jpg)
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
![Page 95: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/95.jpg)
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
![Page 96: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/96.jpg)
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
![Page 97: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/97.jpg)
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
![Page 98: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/98.jpg)
IE 312 98
Balance Constraints Cash balance
Debt balance
3
1111
)(98.0
002.0001.0
tttttt
tttttttt
wwpres
yxxxhgzz
tttt hgyy 1
![Page 99: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/99.jpg)
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
![Page 100: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/100.jpg)
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
![Page 101: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/101.jpg)
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)
![Page 102: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/102.jpg)
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
![Page 103: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/103.jpg)
IE 312 103
“Typical” Initial Inventory
443
332
221
11
15
64
48
119
inventory
endingdemandproduction
inventory
beginning
ixi
ixi
ixi
ix
![Page 104: IE 3121 Solution Techniques Models Thousands (or millions) of variables Thousands (or millions) of constraints Complex models tend to be valid Is the model.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d6e5503460f94a4f7b5/html5/thumbnails/104.jpg)
IE 312 104
Wrapper (Infinite Horizon)
443
332
221
114
15
64
48
11
inventory
endingdemandproduction
inventory
beginning
ixi
ixi
ixi
ixi