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Setup a network Flow for this Problem using nodes and links but do not solve for optimal solution

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A manufacturer must produce a certain product in sufficient quantity to meet contracted sales in the next four months. The production facilities available for this product are limited, but by different amounts in the respective months. The unit cost of production also varies according to the facilities and personnel available. The product can be produced in one month and then held for sale in a later month, but at an estimated storage cost of $1 per unit per month. No storage cost is incurred for goods sold in the same month in which they are produced. There is presently no inventory of this product, and none is desired at the end of the four months. Pertinent data are given in the table below.

Month.....Contracted Sales.....Max Production.....Unit Cost of Production.....Unit Storage Cost Per Month
1.................20................................40.............................14............................................1
2.................30................................50.............................16............................................1
3.................50................................30.............................15............................................1
4.................40................................50.............................17............................................1

The approach is to create a node for each month, plus a supply node from which the production comes, and a demand node which receives the contracted sales. Lower, upper bounds and cost should be given to each link. An artificial back link from the demand node to the supply node is needed for a fully circulating network. Links connecting the nodes for each month represent carryover inventory.

Setup a network Flow for this Problem using nodes and links but do not solve for optimal solution.

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Solution Summary

The solution assists with setting up a network flow for the problem using nodes and links but not solving for the optimal solution.

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Formulate the following network problem

(See the attached file for the diagram)

- In the above network, a given flow Q is transferred through the network to a demand node N. The goal is to route the flows in such a way that total channel loss is minimized, where channel loss coefficients clij per unit length are given for each link or arc of length Lij. Ignore the maximum capacities of the links for this. Specify how the arc parameters [lij, uij, cij] for any link (i,j) in this network should be defined, as well as the demand node parameters (i.e., priority and demand), that will result in minimization of total channel loss.

- Instead of Q being specified in the above network, suppose that now the objective is to find the maximum flow that can be delivered to demand node N, assuming that upper bounds uij are now specified on the flow in each link or arc (i,j) and channel losses are ignored. Specify the arc parameters and demand node parameters that would be required to solve this problem [Hint: you may need to add a supply node such as a dummy reservoir].

- How would you define the arc parameters and demand node parameters in order to find the shortest path through this network, assuming each link is of length Lij? We are not trying to route flows in this network, but simply identify the shortest connected path through the network.

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