Question

1. Canning Transport is to move goods from three factories to three distribution centers. Information about the move is given below. Give the network model and the linear programming model for this problem.

Source	Supply	Destination	Demand
A	250	X	350
B	130	Y	125
C	170	Z	175

Shipping costs are:

		Destination
Source	X	Y	Z
A	3	2	6
B	8	--	9
C	5	6	8
	(Source B cannot ship to destination Y)

Answer 1

ANSWER::

Below is the network model based on details given:

Formulate the linear programming model

Decision variables:

The decision variable are number of units shipped from Source i to Destination j

where i= A , B, C

j= X, Y, Z.

UAX= Number of units shipped from Source A to Destination X

UAY= Number of units shipped from Source A to Destination Y

UAZ= Number of units shipped from Source A to Destination Z

UBX= Number of units shipped from Source A to Destination X

UBZ= Number of units shipped from Source A to Destination Z

UCX= Number of units shipped from Source C to Destination X

UCY= Number of units shipped from Source C to Destination Y

UCZ= Number of units shipped from Source C to Destination Z

Objective function:

The objective is to minimize total shipping cost. Total shipping cost is the sum of the product of number of units and shipping cost per unit.

MIN 3UAX +2UAY + 6UAZ + 8UBX +9UBZ + 5UCX + 6UCY + 8UCZ

Constraints:

Supply constraints are

UAX+UAY+ UAZ <= 250

UBX+ UBZ <= 130

UCX+UCY+ UCZ <= 170

Demand constraints are

UAX+UBX+ UCX >= 350

UAY+UCY >= 125

UCX+UCX+ UCZ >= 175

Non negativity constraints are

UAX>=0, UAY>=0, UAZ>=0, UBX>=0, UBZ>=0, UCX>=0, UCY>=0, UCZ>=0

(OR) TRY THIS ANSWER FOR BETER UNDERSTANDING

there are 3*3 = 9 variables {although from B to Y is 0}

xij represent source i to Destination j

i can be A,B and C

j can be X,Y and Z

Minimize Z = 3 xAX + 2 xAY + 6 xAZ + 8 xBX + 9 xBZ + 5 xCX + 6 xCY + 8 xCZ

constraints

xAX + xAY+ xAZ <= 250

xBX + xBZ <= 130

xCX + xCY + xCZ <= 170

xAX + xBX + xCX >= 350

xAY + xBY + xCY >= 125

xAZ + xBZ + xCZ >= 175

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