Markov

Markov gy duisbalc I ACk’a6pR 03, 2010 | 20 pagos Structure and Drive Paul A Jensen Copyright July 20, 2003 A system is made up of several operations with flow passing between them. The structure of the system describes the flow paths from inputs to outputs. In this section we recognize three structure alternatives, line, tree and network. The drive option speclfies the cause offlow through the system. Here we have two options, pull and push. For the pull option, flow is pulled from the outputs of operations. For the push option, flow is pushed into the inputs of the operations.

For each ofthe Six combinations of structure and drive, given the flows into or out of the structure, one can compute the flows through the system operations. The remainder of this section describes the analysis. Pull Line The line structure, illustrated in Fig. 1, is the simplest because flow enters at the first operation and leaves from the flows are determine y t 5. When there is noc nee i the flow that enters operations have the PACE 1 or20 ed ull line where the system at operation the operations, peration 5 and all he case where some mechanism allows the

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flow to be increased or decreased as it asses through an operation.

We assume the operations are numbered Swlpe to vlew next page numbered in order from 1 to m, with m the last operation. m is the flow withdrawn from operation m. In general we Will also allow flow to be withdrawn from the other operations as well, vvlth i defined as the flow pulled from the output of operation i. We call the flows pulled from or pushed into the system external flows to distinguish them from the flows within the system. Our goal is to find the flow through each operation as a function of the external flows. Define the following notation. Figure 1 .

The pull line xi = The flow passing into operation i Xi’ = The flow passing out of operation The flow ratio for an operation is the ratio between the flow leaving the operation and the flow entering. Then ri = xi ‘ . xi There are many situations where the ratio is other than 1 Perhaps the operation does inspection in a manufacturing system and faulty items are removed from the flow. Here the ratio would be less than 1 . In another situation, the operation may divide an item into two parts. Every entering item results in two leaving items, so the ratio is 2.

We assume the ratios are given. The alue of Xi’ is entirely dependent on the pull flow withdrawn at operation i and the amount required by the following operatlon, i xi’-x xi ‘ = ri Since the flow for an oper on the flow of its unique 2 OF followine ope Since the flow for an operation depends on the flow of its unique following operation, we can compute flows recursively, starting with the xm and continuing for each operation with sequentially decreasing operation index. An example based on Fig. 1 has operations 2 and 4 each causing a 10% loss in flow.

The resultant operation flows are in Table 1. In arder to pull 100 units from peration 5, more than 100 units must pass through operation 1 to accommodate for the losses. Table 1. Pull line parameters and operation flows with Index (i) 12345 -100. Pull Out ( i) 0000 100 Ratio (ri) 1 1 0. 91 123. 46 123. 46 111. 11 111. 11 100 Push Line The push line is illustrated in Fig. 2. The only difference between the push and the pull lines is the driver for flows. When all flow ratios are 1, there really is no difference between the two drive mechanisms because all operation flows are the same.

Again we assume the operations are numbered in order from 1 to m, with m the last operation. Flows enter operation 1 in the amount 1. In addition to node 1, products may also be pushed into the network at other operations. The flow entering at operation i is L. Note that push flow enters the process just before the operation, while pull fl process at the output of the operation. The notatio flow ratios remain the process at the output of the operation. The notation for flow and flow ratios remain the same as for the pull line.

The flow into an operation is equal to the external flow added at the operation plus the flow withdrawn from the previous operation. xi = xi = + x’ + r -1 xi —1 i Figure 2. The push line Starting with operation 1, Eq. 2 can be used sequentially to compute the flows on the line. The example for the pull line is repeated below with 100 units pushed into operation 1 instead of flow being pulled from operation 5. Comparison of Tables 1 and 2 Will show the difference between the two cases when non-unity flow ratios are present. Table 2. Push line parameters and operation flows with Index (i) Push in ( ü) 1000000 1 0. 1 0. 9 1 100 100 90 90 81 Pull Tree The generic pull ‘s illustrated in Fig. 3. For 4 this structure the flow thr ration goes to a unique example, in the amount 5. 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 identib,’ the amount pulled from the output of operation i as i, 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 r12 2 q12 q23 4 114 q3S q 45 05 Figure 3. The pull tree s OF For the pull tree we identi ion, qii, as the amount twodimensional table as illustrated below. Table 3. Pull tree parameters Index 1 2345 33 5 5 0 pull out 0 0 0 0 100 Ratio 0. 9 0. 9 0. 9 0. 9 0. 9 proportion 0. 5 0. 5 0. 5 0. 5 1 For this illustration, we are assuming that 10% of the units passing through each operation are scrapped. We alsa assume proportions of 0. 5 for operations 1 through 4. This means that operation 3 receives half of its input from each of operations 1 and 2.

Further, operation 5 receives half of its input from each of operations 3 and 4. For the example, 100 units are pulled from operation 5. For the pull tree, an operation can send its output to no more than one other operation, so the column labeled Next is sufficient to describe the tree structure. The column labeled Proportion gives the proportion of the input of the next- operation that is obtained from the operation. The number (0. 5) in the row for operation 1, holds the value of q13. For the pull tree structure we define the following notation.

We use i for the general operation index. ai = the index of the operation following (or after) operation i. This is the number in the Next columna = the flow pulled from the output of operation i. ri = the ratio between the output and input flows for operation i. qia = the proportion of he input of operation ai that is obtained from operation i. We use the symbol a as the second subscript on 6 OF obtained from operation i. We use the symbol a as the second subscript on qia to indicate that it is the proportion of the input of the following operation that must come from operation i.

When an operation has no following operation we assign the value O to ai, and the value of qia is Will have no effect. Flow Given pull flows for the operations, we want to compute the flow through each operation. We use the notation xi, Xi’ and ri as previously defined. To illustrate the computation of the unit flows we use an xample with three operations as in Fig. 4. ui ni i qik uk nk k qjj qjk uj The value of the flow out of operation i, xi’, is entirely dependent on the pull flow withdrawn at operation i and the amount required by the following operation k.

Since the amount required is qik xk , we have the relation between the flows at and k. xi ‘ – + qikxk +qikxkri Figure 4. Portion of a pull tree Notice that the flow for an operation depends on the flow of its unique following operation. For the pull tree process, the unit flows can be computed recursively, starting with xm and continuing for each operat- entially decreasing operation index. umbers on the arcs are proportions. One unit is pulled from node 5. 12 For the illustration we use the data of Table 3. We start with the operation with the greatest index.

By definition, this operation Will have no followers so: x5 = 0. 9=111. 1 Considering the operation with the next Iower index, we compute x4 — + q45 ) 0. 9 = 61. 73 Continuing in order of decreasing index, customers arrive at a source node, nade 1 in the example in the amount 1. In addition to node 1, flow may also be pushed into the network at other operations. The push flow entering at operation i is Note that push flow enters the process just before he operation, while pull flow leaves the process after passing through the operation. Figure 6.

The push tree The flow that passes through an operation may be Split to go to other operations to receive diferent types of processing. Units pass through the tree until finally they are withdrawn to the nodes that have no successors, nodes 2, 4 and 5 in the figure. 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 the proportion, pij as the proportion of he output of operation i that is passed to operation j. e value of pij 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 would equal 1. Tabular Representation We can represent the data for a push tree with a two-dmensional table as illustrated for the example below. Table 5. push tree parameters Index 1 2345 previousol 1 33 push Flow 1000 O Ratio 0. 9 0. 9 0. 9 0 tree parameters Index 1 2345 previousol 1 33 push Flow 100000 0 Ratio 0. 9 0. 9 0. 9 0. 0. Proportion 1 0. 5 0. 5 0. 5 0. 5 passing through each operation are scrapped. We also assume proportions of 0. 5 for operations 2 through 5. This means that the output of operation 1 is Split with half going to operation 2 and half going to operation 3. Further, the operation of operation 3 is Split with half going to operation 4 and half to operation 5. For the example, 100 units are pushed into operation 1. For the push tree, an operation receives its input from no more than one other operation, so the column labeled previous is suficient to describe the tree structure.

For row i, the column labeled Proportion ives the proportion ofthe output of the previous operation that goes to operation i. The number (0. 5) in the row for operation 2, holds the value ofp13. For the pull tree structure we define the following notation. We use for the general operation index. bi = the index of the operation preceding (or befare) operation i. This is index is appears in the Previous column of the table. = the flow pushed into the input of operation i. ri = the ratio between the output and input flows for operation i. Pbi the proportion of the output of operation bi that goes to operation i. For Pbi we use the b to represen