lp: example of formulations - dist.unige.it didattico 2017-18...transportation problem: formulation...

Operations Research – Massimo Paolucci – DIBRIS University of Genova

• Three classical decision problems in OR:

– Transportation problem– Product-mix problem– Production planning problem

LP: example of formulations


Transportation problem• The decision problem: considering a certain time period (e.g., an year),

determine the most convenient way to transport some available items (products) for satisfying a given demand in the same period– m suppliers producing s1,...,sm quantities of a product– n customers requiring r1,...,rn quantities of that product– Assume in general that the product can be transported from any supplier

to any customers– For each unit of product transported from supplier i to customer j the cost

is cij

– Problem: determine the quantities of product transported between each pair (i,j) supplier-customer such that the demand are satisfied, the availabilities are respected and the total transportation cost is minimized


• Graph representation

s1

si

sm

r1

r j

rn

c i j

Transportation problem


• Data feasibility condition

• Formulate TP as LP:– Define the decision variables– Define the objective as a function of decision variables– Define the set of constraints that impose that the variables must assume

only feasible values (satisfaction of operational conditions and compliancy with variables meaning)

≥==

n

jj

m

ii rs

11

Transportation problem


• Variables:– Product quantity transported on each arc (continuous variables)

• Objective function: – Total transportation cost

• Constraints: – Total quantity supplied by suppliers cannot exceed availability

– Total quantity received by customers must be equal to the demand

– Only positive quantities can be supplied

njmixij ,...,1;,...,1 ==∈ R

==

n

jijij

m

ixc

11

Transportation problem: formulation

)1(,...,11

misx i

n

jij =≤

=

)2(,...,11

njrx jm

iij ==

=

)3(,...,1;,..,10 njmixij ==≥


• TP formulated as LP problem

njmixnjmix

njrx

misx

ts

xc

ij

ij

j

m

iij

i

n

jij

n

jijij

m

i

,...,1;,...,1)3(,...,1;,...,10

)2(,...,1

)1(,...,1

..

min

1

1

11

==∈

==≥

==

=≤

=

=

==

R

Transportation problem: formulation


Transportation problem: variations

• Some possible variations of TP– Maximum transportation capacity for arcs in the considered period– Each customer can be served by a subset of suppliers– Transportation through intermediate distribution centers

s1

si

sm

r1

r j

rn

Distribution

centers

example


Product mix problem• The decision problem: determine the optimal level of production in a

reference time period for a set of products (activities) taking into account the limited availability of a set of resources in the same period– n different product types can be produced– m different production resources (i.e., materials) with maximum

availability b1,...,bm

– Each product need a certain quantity of a set of resources: specifically, each unit of product i need aij units of resource j

– For each product i the profit for unit produced is ci

– Problem: determine the quantities to produce (the mix of products) in order to maximize the total profit without exceeding the resources availability


Product mix problem: formulation• Variables:

– The quantities of products to produce (continuous variables)

• Objective function – Total profit from production

• Constraints: – For each resource, the total used quantity cannot exceed the maximum

availability

– Only positive quantities can be produced

nixi ,...,1=∈ R

=

n

iii xc

1

)1(,...,11

mjbxa j

n

iiij =≤

=

)2(,..,10 nixi =≥


• Product mix formulated as a LP problem

nixnix

mjbxa

ts

xc

i

i

j

n

iiij

n

iii

,...,1)2(,...,10

)1(,...,1

..

max

1

1

=∈=≥

=≤

=

=

R

Product mix problem: formulation


Production planning problem• The decision problem: plan the level of production for a single product

type over a time horizon(e.g., a year) determining the production for each time period (e.g., month) in which the horizon has been divided– The product demand is known for each period– Inventory is allowed at the end of a period to store product for next

periods– A horizon of N periods (months) is assumed– For each period are known:

• the production capacity m1,...,mN• the production costs c1,...,cN• the inventory costs r1,...,rN• the product demand d1,...,dN

– The initial inventory is given M0– Problem: determine the product quantity to be produced in the

considered periods to satisfy the demand, minimizing the overall production and inventory cost


• Variables: – The quantities of product planned to be produced in each period (cont. vars)

– The quantities of product stored at the end of each period (cont. vars)

• Objective function – The total cost for production and inventory in the N periods

Nixi ,...,1=∈ R

Nisi ,...,1=∈R

Production planning problem: formulation

( ) +=

N

iiiii srxc

1


• Constraints: – Product flow conservation for each period

– Production in each period cannot exceed production capacity

– Starting inventory level

– Production and inventory levels cannot be negative


)1(,...,11 Nisdsx iiii =+=+ −

)2(,...,1 Nimx ii =≤

)3(00 Ms =

)4(,..,100 Nisx ii =≥≥


• Planning problem (Single Item capacitated lot-sizing problem) formulated as LP

( )

NisxNisx

MsNimx

Nidssxts

srxc

ii

ii

ii

iiii

N

iiiii

,...,1)4(,...,100)3()2(,...,1)1(,...,1

..

min

00

1

1

=∈∈=≥≥

==≤

==−+

+

−

=

RR



• Possible variations:– n different products (multi-item production)– Backlogging is allowed – Inventory capacities – Multi-stage production (semi-finished products and raw

material – Bill of material - Material Requirement Planning -MRP)



LP – theory and solution methods• Main assumptions for formulating a problem as LP

– Proportionality (variables multiplied by constants)– Additivity (sum of variables by constants)– Divisibility (variables assume real values)– Certainty (all coefficients are assumed deterministic)

• Next concepts:– Fundamental definitions– Geometrical aspects– Simplex method (George Dantzig, 1947)


LP: standard and canonical forms

• Two canonical forms for LP problems– Canonical form of maximization

– Canonical form of minimization

n

T

Rx

xbxA

xcx

∈

≥≤=

0

max 0

n

T

Rx

xbxA

xcx

∈

≥≥=

0

min 0


Any LP problem can be expressed in standard form

Where:– x n×1 decision variables vector– c n×1 objective function coefficients vector– b m×1 right hand side (rhs) constraint coefficients vector– A m×n matrix of constraints coefficients A=[aij], i=1,...,n j=1,...,mAssumptions:1. b ≥ 0 bj ≥0 ∀j=1,...,m2. m<n3. m=rank(A)

n

T

Rx

xbxA

xcx

∈

≥==

)2(0)1(

max 0

LP: standard and canonical forms


Definitions:– Constraints (1) define x as solution of the LP problem– Constraints (1) and (2) define x as feasible solution of LP problem– A LP problem can be:

– Feasible with bounded optimal solutions• The feasibility region is a non empty polyhedron

• A sufficient condition: if X is closed (Polytope)

LP: types of solutions

{ } ∅≠≤∈= bxAxX n :R

X


Definitions:– A LP problem can be:

– Feasible without bounded optimal solutions (unbounded problem)• The feasibility region is a non closed non empty polyhedron (not

sufficient condition)

– Not feasible• The feasibility region is an empty polyhedron

LP: types of solutions

∅≠X Xand open

XxxX n ∈∈∃∅= such thatR

X1

X2

∅=∩= 21 XXXexample


LP: polyhedra

Definitions:• A halfspace in Rn the set

where

• A polyhedron is defined by the intersection of a set of halfspaces

• A polytope is a bounded polyhedron (for M>0, )• A set X is a convex set ⇔ given two points

any point y generated as

is (is a convex combination of )

}:{ bxax Tn ≤∈RRR ∈∈ ba n

}:{ bxAxP n ≤∈= R

Xxx ba ∈,

10)1( ≤≤−+= λλλ ba xxy

Xy∈ Xxx ba ∈,

XxMx ∈∀<||||


Graphic representation of a convex combination of two points

Definitions:• A halfspace is a convex set • The intersection of convex set is a convex set• A polyhedron is a convex set

10)1( ≤≤−+= λλλ ba xxy

Xax

bxy

LP: polyhedra


Definition:• A point of a convex set (polyhedron) X is an extreme point

(vertex) ⇔ do not exist such that for some

Vertices of X

X

LP: polyhedra

baba xxXxx ≠∈ ,,Xy∈

ba xxy )1( λλ −+= 10 << λ


Definitions:• Given a set of points the convex hull

is the smallest convex set including such points

Theorem (property of extreme points of closed polyhedra)

Any point where X is a closed polyhedron (polytope) with

extreme points (vertices) can be expressed as

convex combination of such points

LP: polyhedra

nkxx R∈,....,1

}0,1,:{),....,(11

1 ixxxxxconv i

k

ii

k

i

ii

nk ∀≥==∈= ==

λλλR

Xx∈

Eixie ,...,1, =

ixx i

E

ii

E

iei i

∀≥== ==

01 11

λλλ


Example: convex combination of 1ex2ex

3ex

X

xe1xe2

xe3

X’X

LP: polyhedra

lp: example of formulations - dist.unige.it didattico 2017-18...transportation problem: formulation...

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