The correct answer is: C. Both of the above
The stepping stone method and the modified distribution method are both methods that can be used to test a solution for optimality in the transportation model of operations research.
The stepping stone method is a graphical method that can be used to trace out a path of flow between two origins and two destinations. If the path of flow has a negative value, then the solution is not optimal and can be improved by increasing the flow along that path.
The modified distribution method is an algebraic method that can be used to test a solution for optimality. The method involves calculating the reduced cost for each arc in the network. If the reduced cost for an arc is negative, then the solution is not optimal and can be improved by increasing the flow along that arc.
The slack variable method is not a method that can be used to test a solution for optimality in the transportation model of operations research. The slack variable method is a method that can be used to solve the transportation model.
Here is a more detailed explanation of each method:
Stepping Stone Method
The stepping stone method is a graphical method that can be used to trace out a path of flow between two origins and two destinations. If the path of flow has a negative value, then the solution is not optimal and can be improved by increasing the flow along that path.
To use the stepping stone method, first draw a network with the origins and destinations as nodes and the arcs representing the possible flows of goods. Then, calculate the flow along each arc. The flow along an arc is equal to the difference between the supply at the origin and the demand at the destination.
Next, identify the two origins and two destinations that you want to test. Draw a line between these four nodes. Then, for each arc that is connected to one of the four nodes, calculate the reduced cost. The reduced cost is equal to the difference between the actual flow along the arc and the minimum possible flow along the arc.
If the reduced cost for an arc is negative, then the solution is not optimal and can be improved by increasing the flow along that arc. To do this, simply increase the flow along the arc by one unit and recalculate the reduced costs. If any of the reduced costs are now negative, then you can continue to improve the solution by increasing the flows along those arcs.
Modified Distribution Method
The modified distribution method is an algebraic method that can be used to test a solution for optimality. The method involves calculating the reduced cost for each arc in the network. If the reduced cost for an arc is negative, then the solution is not optimal and can be improved by increasing the flow along that arc.
To use the modified distribution method, first calculate the reduced cost for each arc in the network. The reduced cost is equal to the difference between the actual flow along the arc and the minimum possible flow along the arc.
Next, identify the arc with the lowest reduced cost. If the reduced cost for this arc is negative, then the solution is not optimal and can be improved by increasing the flow along that arc. To do this, simply increase the flow along the arc by one unit and recalculate the reduced costs. If any of the reduced costs are now negative, then you can continue to improve the solution by increasing the flows along those arcs.
Slack Variable Method
The slack variable method is a method that can be used to solve the transportation model. The method involves introducing slack variables into the model to represent the unused capacity at the origins and destinations. The slack variables are then used to solve the model for the optimal flow of goods.
The slack variable method is not a method that can be used to test a solution for optimality in the transportation model of operations research.