Queuing Analysis

#### Optimization Models

Drive/Structure

 Pull/Tree Push/Tree Pull/Network Push/Network

Models

Subassemblies

 Process Flow Analysis - Drive/Structure In this section we recognize two structure alternatives, tree and network, and two drive alternatives, pull and push. To find the flows in the operations of a process, the structure of the process and the driver of the flows must be specified. We review the four possibilities on this page. More detailed pages are reached through the links on the titles.
 The structure and drive options are set with buttons on the Add Process dialog. The structure determines the arrangement of the flow paths in the operations chart. A tree either starts with a single raw material and the flow diverges to produce multiple products, or starts with several raw materials and the flow converges to produce a single product. A network allows an arbitrary interconnection between operations. The drive option specifies the cause of flow through the process. For the pull option, products are pulled from the outputs of operations. For the push option, items are pushed into the inputs of the operations. The generic pull/tree process is illustrated in Figure 1. For this structure the flow through each operation goes to a unique following operation, while each operation may have several input flows from other operations. This structure is appropriate for modeling many manufacturing processes where raw materials are combined or mixed to produce a single product. Product is withdrawn or pulled from the operation with the greatest index, operation 5 for the example, in the amount . For analysis purposes we will usually assume that this amount is 1, so we can compute the flow through each operation per unit of finished good. In addition to the final operation of the process, our models also allow flow to be pulled from the other operations. These flows represent intermediate products. In general, we identify the amount pulled from the output of operation i as , the pull flow at operation i. For the tree structures we require that the operations be indexed so that when flow passes from operation i to operation j, i < j. The greatest index is m. For the example m is 5. For the pull tree, we identify as the amount of the output of operation i required for each unit of product passing through operation j. The value of may be any positive amount to represent a variety of manufacturing situations. An assembly operation that requires one unit of each input to be combined to produce one unit of a subassembly would have the proportions equal to 1 for each input. A mixing operation that combines inputs into a mixture would have input proportions that sum to 1. An operation that requires more than one unit of some input would be modeled with a proportion greater than 1 on the associated input. Figure 2 The generic push/tree process is illustrated in Fig. 2. For this structure the flow into an operation comes from a unique preceding operation, while the operation may have several output flows going to other operations. This structure is appropriate for modeling service systems where customers arrive at a source node, node 1 in the example in the amount . We call the items entering the system, units of product. In addition to node 1, products may also be pushed into the network at other operations. The flow entering at operation i is . Note that push flow enters the process just before an operation. The flow that passes through an operation may be split to go to other operations to receive different types of processing. Units pass through the tree until finally they are withdrawn at the nodes that have no successors, nodes 2, 4 and 5 in the figure. For analysis purposes, we will usually assume that = 1 and that all other are zero, so that we can compute the flows passing through the operations per unit of flow entering node 1. We call these amounts the unit flows. For the tree structures we require that the operations be numbered so that when flow passes from operation i to operation j, i < j. The greatest index is m. For the push tree, we identify as the proportion of the output of operation i that is passed to operation j. The value of may be any positive amount to represent a variety of manufacturing situations. For a splitting operation that separates the total flow passing through operation i into several paths, the sum of the proportions leaving i would equal 1. Pull/Network Process Figure 3 The pull/network process is illustrated in Fig. 3. For this structure the flow through each operation may go to more than one operation, and each operation may have several input flows from other operations. This is a more general structure than the pull/tree structure. Product is withdrawn or pulled from any of the operations. Again we use as the amount pulled from operation i. With m operations we assign the value of 1 to . The pull flows from the other operations are given as relative to . Indices are assigned to the operations arbitrarily, however, it is often convenient to assign the indices to be increasing in the direction of primary product flow. For the pull network, we identify as the amount of the output of operation i required for each unit of product passing through operation j. The value of may be any nonnegative amount to represent a variety of manufacturing situations. An assembly operation that requires one unit of each input to be combined to produce one unit of a subassembly would have the proportions equal to 1 for each input. A mixing operation that combines inputs into a mixture would have input proportions that sum to 1. An operation that requires more than one unit of some input would be modeled with a proportion greater than 1 on the associated input. The example shows an arc passing from operation 5 back to operation 4. In a practical instance, this might represent the reworking of some part. Although we might be tempted to identify as the proportion of the output of operation 5 returned to operation 4, this is not correct for a pull network. is the proportion of the flow through operation 4 that comes from operation 5. Similarly, is the proportion of the flow through operation 4 that comes from operation 2. Push/Network Process Figure 4 The push/network process is illustrated in Fig. 4. For this structure the flow through each operation may go to more than one operation, and each operation may have several input flows from other operations. This is a more general structure than the push/tree process. Product is inserted or pushed into any of the operations. We use as the amount pushed into operation i. We assign the value of 1 to . The push flows to the other operations are given as relative to . Indices are assigned to the operations arbitrarily, however, it is often convenient to assign the indices to be increasing in the direction of primary product flow. For the push network we identify as the amount of the output of operation i that is passed to operation j for each unit of product passing through operation i. The value of may be any nonnegative amount. Typically for a service system, the sum of the proportions leaving an operation is equal to 1. This means that the flow is split among the several following operations. It may be necessary to use other combinations of proportions to represent different systems. The example shows an arc passing from operation 5 back to operation 3. The value of is the proportion of the output of operation 5 returned to operation 3. In a practical instance, this might represent the reworking of some part. It is not necessary to define a proportion for the flow leaving the system at operation 5.

Operations Management / Industrial Engineering
Internet
by Paul A. Jensen