Computation Section
Subunit Inventory Analysis
 Systems - Push Tree




A worksheet with the push tree structure shown is below. The example has the same structure as the figure at the left. The example represents a product distribution system with a depot (1), two distribution centers (2, 3) and two local distributors (4, 5). The distribution center at 2 is also a local distributor. The flows into the system are described in row 23 where only the depot has external flow. The branching structure of the tree is in row 24. Each component has no more than one branch entering, and the cell in row 24 gives the index of the component where the branch starts. Only the depot has a 0 in this row, indicating that this is the root of the tree. Row 25 holds the transfer proportions. For the example, 50% of the output of the depot goes to each of the distribution centers. 75% of the output of center 3 goes to local distributor 4 with the remaining 25% going to local distributor 5. The resultant flows through the various components are computed and displayed in row 26. The contents of row 26 are transferred to row 4 for analysis of the WIP components.

We model the depot and distribution centers as processes and the local distributors as queues. Queues are the same as processors except we assume some variability of the arrival and service processes for queues. Rows 5 through 11 provide the data for the components. Rows 12 through 14 give the lot sizes that link the components. For the push tree the input lot size of a component is set equal to the output lot size of the previous tree component. An exception is the depot where the input lot size is set by the modeler. Rows 15 through 19 compute the relevant results for each component.

In the following we describe the individual components with reference to the worksheet.


The depot is described in column J on the worksheet. We model the depot as a process because items enter as a lot from the supplier and must be processed before being sent to the distributors. The weekly demand of 1500 items is pushed into the system at the depot in cell J23. The items have the value of $100/unit at the depot. We choose to replenish the inventory every two weeks with a lot of 3000 units. The setup cost for an order is $1000. The setup cost as well as the holding cost for WIP is included in the WIP cost reported in J17. The purchase cost of the items and the revenues from their sale are not included because these do not affect the inventory policy.

Each lot entering the depot must be processed. The processing time includes a fixed time of 0.5 weeks. The variable time of 0.0001 weeks is the time required to check each incoming item for quality. Items are shipped from the depot to the distribution centers in lots of 600 units.

The distribution centers are described in columns K and L. They are similar except that the center in column K interacts directly with local customers and the center in column L distributes to another set of local distributors. The unit processing time for Dist. 2 assumes that individual customers require additional processing time. The local distributors receive deliveries in lots of 200. After processing, the lots are delivered to individual customers.

Row 18 shows the number of processors necessary to provide the service generated by the flow rate. Row 19 shows the integer number of servers given the maximum utilization rate of each server. For queuing components this utilization determines the number of servers used in the WIP model, and the utilization must be strictly less than 100%.

The system results are in column O. The system WIP is the sum of the component WIP's. The system cycle time is the system WIP divided by the total flow entering the system (1500). This is the residence time in the system averaged over all units. Notice that the cycle time for the distribution center at 2 is almost 7.5 weeks, which is greater than the number in cell O16.

The WIP cost in O17 is a result of interest. The modeler may want to search over the controllable variables, such as the lot sizes, to lower this cost. The add-in evaluates a particular plan, but it does not directly determine the optimum plan. The cost of WIP is a very nonlinear function of the lot sizes, and the existence of local minimum solutions makes the optimization problem very difficult.


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tree roots

Operations Management / Industrial Engineering
by Paul A. Jensen
Copyright 2004 - All rights reserved