To build the algebraic and excel model, we made several assumptions to simplify the problem.
First, there is little information given about the number of zones, depots and service centers. To get a specific number of our optimization problem, we set the number of zones to be 7, the number of depots to be 3, and the number of service centers to be 3 as well.
The original paper is optimizing the transportation from zones all the way to airports, and we are using the same direction in our model as well.
Each one unit of package in our problem represents one equivalent shipment , and the unit cost is calculated on this basis. Unit cost includes all the cost related to one package, such as handling cost, transit cost and human resource cost.
Under these assumptions, we built the algebraic model as follows.
The network model can be represented by this graph. Here, the decision variables are which depots are responsible for which zones, which service centers are responsible for which depots, and their corresponding flow number.
The objective is to minimize the variable cost and fixed cost. Variable cost is the sumproduct of flows on a arc and the unit cost of transporting one package on that arc, and the fixed cost is the total cost of building facilities. If a facility has no flow, we assume that it’s not necessary, and the fixed cost of it would not be considered.
All the demands must be satisfied, and for depots and service centers, their inflow must equal to outflow. We assumed that each zone and depot should have be served by at least one facilities, and each depot and service center has a maximum coverage. In addition, depots and service centers have a max capacity, and the flow should not exceed that capacity.
Next, we’ll use Excel to implement this model.
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