Lecture 1 Introduction to Lo-models
1.Standard form of Lo-model
·
decision variables :
( ,
kn
&n is
these variables are real-valued a
positive integer
·
objective
function :2 e Cnen ,
+ +
a linear function ofa decision variables ; C ..... In are objective constraints
& are real umbers
~
if maximised : max ( , x + Chin +
or minimised min G Chin : + +
·
constraints are :
technology : di, + + dinin Di
all technology constraints are linear combos of the decision variables
nonnegativity Ti
: - 0 Fies1, . . . . n3
>
example : Model Dovetail
Max 311 , + 2 the S t .
.
7 + 72 = 9 (1 .
1)
374 + 1118 (1 2) .
x [ 7 (1 3) .
> = 6(7 4) .
X, 12 8
S
the model is in standard ,
since there are only 2 variables : solve graphically
·
(1 3) is redundant
.
·
the purple area is the feasible region :
X X
the area within we will find our optimal solution
X
·
the points marked X are the vertices .
the intersection of 2 binding constraints
X X
the optimal point is a vertex of the feasible region
* Remarks :
constraint is binding if it holds with equality
if a constraint doesn't hold for c then it is violated ,
.
2 slack variables
slack variables are added to the constraints to transform inequalities
to equalities
~
example Model Dovetail :
x T = 9 +
= ( + +z +
xy = 9
here the slack z is added so that the sumx +
2 +
Ty =
9
· if the inequality was ,
we would add -
>s
, .
3 Lo-models
non-standard
·
5 steps to standardise :
1. min
> max : minz =
-max -
z
.
2 :
multiply both sides by -1
3 .
3
= = :
replace with ,
then apply ·
rule 2
nopositivity nonnega Subw
-Y =
&
<
-
with 1% 1, 0
,
> make sure that (( . = 0 i e
.
. either es = 0 or
< = 0
·
6 equivalent formulations
03 Standard (primal)
3
7. max[cix Ax = b
, x >
max(c Ax = b3
*
.
2 all variables free more away from origin when
.
3 max [C * Ax b, = > 03 Standard ea .
looking for optimal vertex
03 (dual)
3
min Ec Ax b , Standard
↳ * b3
·
· Min [c * Ax more toward origin when
6. minEct Ax = b ,
x 03 looking for optimal vertex
4.
linearising nonlinear functions
·
ratio constraint : < 2% - (, 12 % (th +
Sha)
24 + 72
·
abs .
value in objective :
Subxi =
-: with , 0 & ci =
0
convex piecewise f(x)
<3D :3
· : =
5 1 , 212
redefine variables ,
see pg
.
19-21
1.Standard form of Lo-model
·
decision variables :
( ,
kn
&n is
these variables are real-valued a
positive integer
·
objective
function :2 e Cnen ,
+ +
a linear function ofa decision variables ; C ..... In are objective constraints
& are real umbers
~
if maximised : max ( , x + Chin +
or minimised min G Chin : + +
·
constraints are :
technology : di, + + dinin Di
all technology constraints are linear combos of the decision variables
nonnegativity Ti
: - 0 Fies1, . . . . n3
>
example : Model Dovetail
Max 311 , + 2 the S t .
.
7 + 72 = 9 (1 .
1)
374 + 1118 (1 2) .
x [ 7 (1 3) .
> = 6(7 4) .
X, 12 8
S
the model is in standard ,
since there are only 2 variables : solve graphically
·
(1 3) is redundant
.
·
the purple area is the feasible region :
X X
the area within we will find our optimal solution
X
·
the points marked X are the vertices .
the intersection of 2 binding constraints
X X
the optimal point is a vertex of the feasible region
* Remarks :
constraint is binding if it holds with equality
if a constraint doesn't hold for c then it is violated ,
.
2 slack variables
slack variables are added to the constraints to transform inequalities
to equalities
~
example Model Dovetail :
x T = 9 +
= ( + +z +
xy = 9
here the slack z is added so that the sumx +
2 +
Ty =
9
· if the inequality was ,
we would add -
>s
, .
3 Lo-models
non-standard
·
5 steps to standardise :
1. min
> max : minz =
-max -
z
.
2 :
multiply both sides by -1
3 .
3
= = :
replace with ,
then apply ·
rule 2
nopositivity nonnega Subw
-Y =
&
<
-
with 1% 1, 0
,
> make sure that (( . = 0 i e
.
. either es = 0 or
< = 0
·
6 equivalent formulations
03 Standard (primal)
3
7. max[cix Ax = b
, x >
max(c Ax = b3
*
.
2 all variables free more away from origin when
.
3 max [C * Ax b, = > 03 Standard ea .
looking for optimal vertex
03 (dual)
3
min Ec Ax b , Standard
↳ * b3
·
· Min [c * Ax more toward origin when
6. minEct Ax = b ,
x 03 looking for optimal vertex
4.
linearising nonlinear functions
·
ratio constraint : < 2% - (, 12 % (th +
Sha)
24 + 72
·
abs .
value in objective :
Subxi =
-: with , 0 & ci =
0
convex piecewise f(x)
<3D :3
· : =
5 1 , 212
redefine variables ,
see pg
.
19-21
