Slides simplexe

Arthur Charpentier, Master Statistique & Économétrie - Université Rennes 1 - 2017
Simplex Method
The beer problem: we want to produce beer, either blonde, or brown



barley : 14kg
corn : 2kg
price : 30e



barley : 10kg
corn : 5kg
price : 40e



barley : 280kg
corn : 100kg
Admissible sets :
10qbrown + 14qblond ≤ 280 (10x1 + 14x2 ≤ 280)
2qbrown +5qblond ≤ 100 (2x1 +5x2 ≤ 100)
What should we produce to maximize the proﬁt ?
max 40qbrown + 30qblond (max 40x1 + 30x2 )
@freakonometrics freakonometrics freakonometrics.hypotheses.org 1
0 5 10 15 20 25 30
01020304050
Brown Beer Barrel
BlondBeerBarrel

Simplex Method
First step: enlarge the space, 10x1 + 14x2 ≤ 280 becomes 10x1 + 14x2 − u1 = 280
(so called slack variables)
max 40x1 + 30x2
s.t. 10x1 + 14x2 + u1 = 280
s.t. 2x1 + 5x2 + u2 = 100
s.t. x1, x2, u1, u2 ≥ 0
summarized in the following table, see wikibook
x1 x2 u1 u2
(1) 10 14 1 0 280
(2) 2 5 0 1 100
max 40 30 0 0

Simplex Method
Consider a linear programming problem written in a standard form.
min cT
x (1a)
subject to
Ax = b , (1b)
x ≥ 0 . (1c)
Where x ∈ Rn
, A is am × n matrix, b ∈ Rm
and c ∈ Rn
.
Assume that rank(A) = m (rows of A are linearly independent)
Introduce slack variables to turn inequality constraints into equality constraints
with positive unknowns : any inequality a1 x1 + · · · + an xn ≤ c can be replaced
by a1 x1 + · · · + an xn + u = c with u ≥ 0.
Replace variables which are not sign-constrained by diﬀerences : any real number
x can be written as the diﬀerence of positive numbers x = u − v with u, v ≥ 0.

Simplex Method
Example :
maximize {x1 + 2 x2 + 3 x3}
subject to
x1 + x2 − x3 = 1 ,
−2 x1 + x2 + 2 x3 ≥ −5 ,
x1 − x2 ≤ 4 ,
x2 + x3 ≤ 5 ,
x1 ≥ 0 ,
x2 ≥ 0 .
minimize {−x1 − 2 x2 − 3 u + 3 v}
subject to
x1 + x2 − u + v = 1 ,
2 x1 − x2 − 2 u + 2 v + s1 = 5 ,
x1 − x2 + s2 = 4 ,
x2 + u − v + s3 = 5 ,
x1, x2, u, v, s1, s2, s3 ≥ 0 .

Simplex Method
Write the coeﬃcients of the problem into a tableau
x1 x2 u v s1 s2 s3
1 1 −1 1 0 0 0 1
2 −1 −2 2 1 0 0 5
1 −1 0 0 0 1 0 4
0 1 1 −1 0 0 1 5
−1 −2 −3 3 0 0 0 0
with constraints on top and coeﬃcients of the objective function are written in a
separate bottom row (with a 0 in the right hand column)
we need to choose an initial set of basic variables which corresponds to a point in
the feasible region of the linear program-ming problem.
E.g. x1 and s1, s2, s3

Simplex Method
Use Gaussian elimination to (1) reduce the selected columns to a permutation of
the identity matrix (2) eliminate the coeﬃcients of the objective function
x1 x2 u v s1 s2 s3
1 1 −1 1 0 0 0 1
0 −3 0 0 1 0 0 3
0 −2 1 −1 0 1 0 3
0 1 1 −1 0 0 1 5
0 −1 −4 4 0 0 0 1
the objective function row has at least one negative entry

Simplex Method
x1 x2 u v s1 s2 s3
1 1 −1 1 0 0 0 1
0 −3 0 0 1 0 0 3
0 −2 1 −1 0 1 0 3
0 1 1 −1 0 0 1 5
0 −1 −4 4 0 0 0 1
This new basic variable is called the entering variable. Correspondingly, one
formerly basic variable has then to become nonbasic, this variable is called the
leaving variable.

Simplex Method
The entering variable shall correspond to the column which has the most
negative entry in the cost function row
the most negative cost function coefficient in column 3, thus u shall be the
entering variable
The leaving variable shall be chosen as follows : Compute for each row the ratio
of its right hand coefficient to the corresponding coefficient in the entering
variable column. Select the row with the smallest finite positive ratio. The
leaving variable is then determined by the column which currently owns the pivot
in this row.
The smallest positive ratio of right hand column to entering variable column is in
row 3, as
3
1
<
5
1
. The pivot in this row points to s2 as the leaving variable.

Simplex Method
x1 x2 u v s1 s2 s3
1 1 −1 1 0 0 0 1
0 −3 0 0 1 0 0 3
0 −2 1 −1 0 1 0 3
0 1 1 −1 0 0 1 5
0 −1 −4 4 0 0 0 1

Simplex Method
After going through the Gaussian elimination once more, we arrive at
x1 x2 u v s1 s2 s3
1 −1 0 0 0 1 0 4
0 −3 0 0 1 0 0 3
0 −2 1 −1 0 1 0 3
0 3 0 0 0 −1 1 2
0 −9 0 0 0 4 0 13
Here x2 will enter and s3 will leave

Simplex Method
After Gaussian elimination, we ﬁnd
x1 x2 u v s1 s2 s3
1 0 0 0 0 2
3
1
3
14
3
0 0 0 0 1 −1 1 5
0 0 1 −1 0 1
3
2
3
13
3
0 1 0 0 0 −1
3
1
3
2
3
0 0 0 0 0 1 3 19
There is no more negative entry in the last row, the cost cannot be lowered

Simplex Method
The algorithm is over, we now have to read oﬀ the solution (in the last column)
x1 =
14
3
, x2 =
2
3
, x3 = u =
13
3
, s1 = 5, v = s2 = s3 = 0
and the minimal value is −19

Duality
Consider a transportation problem.
Some good is available at location A (at no cost) and may be transported to
locations B, C, and D according to the following directed graph
B
4
!!
3

A
2
**
1
44
D
C
5
==
On each of the edges, the unit cost of transportation is cj for j = 1, . . . , 5.
At each of the vertices, bi units of the good are sold, where i = B, C, D.
How can the transport be done most eﬃciently?

Arthur Charpentier, Master Statistique Économétrie - Université Rennes 1 - 2017
Duality
Let xj denotes the amount of good transported through edge j
We have to solve
minimize {c1 x1 + · · · + c5 x5} (2)
subject to
x1 − x3 − x4 = bB , (3)
x2 + x3 − x5 = bC , (4)
x4 + x5 = bD . (5)
Constraints mean here that nothing gets lost at nodes B, C, and D, except what
is sold.

Duality
Alternatively, instead of looking at minimizing the cost of transportation, we seek
to maximize the income from selling the good.
maximize {yB bB + yC bC + yD bD} (6)
subject to
yB − yA ≤ c1 , (7)
yC − yA ≤ c2 , (8)
yC − yB ≤ c3 , (9)
yD − yB ≤ c4 , (10)
yD − yC ≤ c5 . (11)
Constraints mean here that the price diﬀerence cannot not exceed the cost of
transportation.

Duality
Set
x =





x1
...
x5





, y =





yB
yC
yD





, and A =





1 0 −1 −1 0
0 1 1 0 −1
0 0 0 1 1





,
The ﬁrst problem - primal problem - is here
minimize {cT
x}
subject to Ax = b, x ≥ 0 .
and the second problem - dual problem - is here
maximize {yT
b}
subject to yT
A ≤ cT
.

Duality
The minimal cost and the maximal income coincide, i.e., the two problems are
equivalent. More precisely, there is a strong duality theorem
Theorem The primal problem has a nondegenerate solution x if and only if the
dual problem has a nondegenerate solution y. And in this case yT
b = cT
x.
See Dantzig Thapa (1997) Linear Programming

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