i j = = i 1 j 1 Shipments from source i to destination j - - PDF document

i j i 1 j 1 shipments from source i to destination j
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i j = = i 1 j 1 Shipments from source i to destination j - - PDF document

Introduction to the transportation problem s Supply at source i : i d Contractual commitment to deliver to destination j : j For feasibility we require Total supply total demand m n s d i j = = i 1 j 1 Shipments


slide-1
SLIDE 1

1

Introduction to the transportation problem Supply at source i:

i

s

Contractual commitment to deliver to destination j:

j

d

For feasibility we require Total supply ≥ total demand

1 1 m n i j i j

s d

= =

∑ ∑

Shipments from source i to destination j Shipments from

ij

x

  • dest. 1
  • dest. 2
  • dest. 3

(source) supply to HK to Madrid to London source 1 from LA

11

x

12

x

13

x

LA 210.00 source 2 from NY

21

x

22

x

23

x

NY 100.00 HK Madrid London Shipments to (destination) Total shipments demand 60.00 60.00 180.00 Transportation costs from source i to destination j ij

t

  • dest. 1
  • dest. 2
  • dest. 3

to HK to Madrid to London source 1 from LA 4.00 7.00 8.00 Total Cost source 2 from NY 6.00 3.00 5.00

11 11 12 12 13 13 21 21 22 22 23 23

{ }

x

Min t x t x t x t x t x t x + + + + +

Supply constraints

1 2 3 i i i i

x x x s + + ≤ 1,2 i =

Demand constraints

1 2 j j j

x x d + ≥ 1,2,3 j =

slide-2
SLIDE 2

2

The dual problem Economic approach: Shadow prices are “as if” prices If route ij is used

ij ij

MR MC =

, if not,

ij ij

MR MC ≤

Let

i

θ be the price the shipper pays at source i.

Let

j

µ be the price the shipper receives at destination j.

ij j

MR µ =

,

ij i ij

MC t θ = +

Thus if

* ij

x >

, then

j i ij

t µ θ = +

And if

* ij

x =

, then

j i ij

t µ θ ≤ +

Summary: Shadow price at source + cost of shipping is at least equal to the shadow price at the destination.

11 12 13 1

x x x s + + ≤

(LA) shadow price

1

θ

21 22 23 2

x x x s + + ≤

(NY) shadow price

2

θ

11 21 1

x x d + ≥

(HK) shadow price

1

µ

12 22 2

x x d + ≥

(Madrid) shadow price

2

µ

13 23 3

x x d + ≥

(London) shadow price

3

µ

slide-3
SLIDE 3

3

Using solver

Shipments from source i to destination j x_ij destinations (source) supply d1 (HK) d2 (Madrid) d3 (London) sources s1 (LA) x_11 x_12 x_13 row sum 200.00 s2 (NY) X_21 x_22 x_23 row sum 100.00 (destination) col sum col sum col sum demand 60.00 60.00 180.00 Transportation costs from source i to destination j t_ij

  • dest. 1
  • dest. 2
  • dest. 3

to HK to Madrid to London source 1 from LA 4.00 7.00 8.00 Total Cost source 2 from NY 6.00 3.00 5.00 T

Target cell: =sumproduct(green block, yellow block) CONSTRAINTS Supply: red col <= yellow col Demand: red row >= yellow row Non-negativity: green block >= 0 Solvers printout of imputed prices. Increasing a supply lowers transportation cost so each source shadow price will be shown as negative. Increasing a demand raises transportation cost so each destination shadow price will be shown as positive. In our approach we define shadow prices as the effect on value when a constraint is relaxed so all shadow prices are non-negative

slide-4
SLIDE 4

4

Selecting the initial feasible shipments: Start in the top left corner and put in the smallest number which satisfies one

  • f the constraints with equality. Move to the next row or column and repeat.

The first such step is shown below.

x_ij destinations (source) supply d1 (HK) d2 (Madrid) d3 (London) sources s1 (LA) 60 * 60 210.00 s2 (NY) 100.00 (destination) 60 demand 60.00 60.00 180.00