Question

Write mathematical form of transportation problem

Answer 1

Let there be m origins and n destinations.

Let the amount of supply at th i th origin is a_i.

Let the demand at j th destination is b_j.

The cost of transporting one unit of an item from origin i to destination j is C_ij and is known for all combination (i,j).

Quantity transported from origin i to destination j be x_ij.

The objective is to determine the quantity x_ij to be transported overall routes (i,j) so as to minimize the total transportation cost.

The supply limits at the origins and the demand requirements at the destinations must be satisfied.

The above transportation problem can be written in the following tabular form:

		Destinations
		1	2	3	…	n	Supply
	1	`""^((x_11))"C"_11`	`""^((x_12))"C"_12`	`""^((x_13))"C"_13`	…	`""^((x_(1n)))("C"_(1n))`	a1
	2	`""^((x_21))"C"_21`	`""^((x_22))"C"_22`	`""^((x_23))"C"_23`	…	`""^((x_(2n)))("C"_(2n))`	a2
Origins	:		:	:	:	:	:
	m	`""^((x_(m1)))"C"_("m"1)`	`""^((x_(m2)))"C"_("m"2)`	`""^((x_(m3)))"C"_("m"3)`	…	`""^((x_(mn)))("C"_"mn")`	am
Demand		b1	b2	b3	…	bn

Now the linear programming model representing the transportation problem is given by

The objective function is Minimize Z =  `sum_("i" = 1)^"m", sum_("J" = 1)^"n" "c"_"ij" "X"_"ij"`

Subject to the constraints

`sum_("j" = 1)^"n"` = x_ij = a_i, i = 1, 2 …….. m (Supply constraints)

`sum_("i" = 1)^"m"` = x_ij = b_j, i = 1, 2 …….. n (Demand constraints)

x_ij ≥ 0 for all i, j (non- negative restrictions)