Structure and Drive Paul A. Jensen Copyright July 20, 2003

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1 Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs. In ths secton we recognze three structure alternatves, lne, tree and network. The drve opton specfes the cause of flow through the system. Here we have two optons, pull and push. For the pull opton, flow s pulled from the outputs of operatons. For the push opton, flow s pushed nto the nputs of the operatons. For each of the sx combnatons of structure and drve, gven the flows nto or out of the structure, one can compute the flows through the system operatons. The remander of ths secton descrbes the analyss. Pull Lne The lne structure, llustrated n Fg., s the smplest because flow enters at the frst operaton and leaves from the last. Here we show the pull lne where the flows are determned by the flow leavng the system at operaton 5. When there s no change n flow caused by the operatons, the flow that enters operaton also leaves operaton 5 and all operatons have the same flow. We analyze the case where some mechansm allows the flow to be ncreased or decreased as t passes through an operaton. We assume the operatons are numbered n order from to m, wth m the last operaton. m s the flow wthdrawn from operaton m. In general we wll also allow flow to be wthdrawn from the other operatons as well, wth defned as the flow pulled from the output of operaton. We call the flows pulled from or pushed nto the system external flows to dstngush them from the flows wthn the system. Our goal s to fnd the flow through each operaton as a functon of the external flows. Defne the followng notaton. Fgure. The pull lne The flow passng nto operaton The flow passng out of operaton

2 The flow rato for an operaton s the rato between the flow leavng the operaton and the flow enterng. Then r. There are many stuatons where the rato s other than. Perhaps the operaton does nspecton n a manufacturng system and faulty tems are removed from the flow. Here the rato would be less than. In another stuaton, the operaton may dvde an tem nto two parts. Every enterng tem results n two leavng tems, so the rato s 2. We assume the ratos are gven. The value of s entrely dependent on the pull flow wthdrawn at operaton and the amount reured by the followng operaton, r + + r () Snce the flow for an operaton depends on the flow of ts unue followng operaton, we can compute flows recursvely, startng wth the x m and contnung for each operaton wth seuentally decreasng operaton ndex. An example based on Fg. has operatons 2 and 4 each causng a 0% loss n flow. The resultant operaton flows are n Table. In order to pull 00 unts from operaton 5, more than 00 unts must pass through operaton to accommodate for the losses. Table. Pull lne parameters and operaton flows wth Index () Pull Out ( ) Rato (r ) Flow ( )

3 Push Lne The push lne s llustrated n Fg. 2. The only dfference between the push and the pull lnes s the drver for flows. When all flow ratos are, there really s no dfference between the two drve mechansms because all operaton flows are the same. Agan we assume the operatons are numbered n order from to m, wth m the last operaton. Flows enter operaton n the amount. In addton to node, products may also be pushed nto the network at other operatons. The flow enterng at operaton s ι. Note that push flow enters the process just before the operaton, whle pull flow leaves the process at the output of the operaton. The notaton for flow and flow ratos reman the same as for the pull lne. The flow nto an operaton s eual to the external flow added at the operaton plus the flow wthdrawn from the prevous operaton. Fgure 2. The push lne + x (2) + r Startng wth operaton, E. 2 can be used seuentally to compute the flows on the lne. The example for the pull lne s repeated below wth 00 unts pushed nto operaton nstead of flow beng pulled from operaton 5. Comparson of Tables and 2 wll show the dfference between the two cases when non-unty flow ratos are present. Table 2. Push lne parameters and operaton flows wth Index () Push n ( ι ) Rato (r ) Flow ( )

4 Pull Tree Fgure 3. The pull tree 4 45 The generc pull tree process s llustrated n Fg. 3. For ths structure the flow through each operaton goes to a unue followng operaton, whle each operaton may have several nput flows from other operatons. Ths structure s approprate for modelng manufacturng processes where raw materals are combned or mxed to produce a sngle product. Product s wthdrawn or pulled from the operaton wth the greatest ndex, operaton 5 for the example, n the amount 5. In addton to the fnal operaton of the process, our models also allow flow to be pulled from the other operatons. These flows represent ntermedate products. In general, we dentfy the amount pulled from the output of operaton as, the pull flow at operaton. For the tree structures we reure that the operatons be ndexed so that when flow passes from operaton to operaton j, < j. The greatest ndes m. For the pull tree we dentfy the proporton, j, as the amount of the output of operaton reured for each unt of product passng through operaton j. The value of j may be any postve amount to represent a varety of manufacturng stuatons. An assembly operaton that reures one unt of each nput to be combned to produce one unt of a subassembly would have the proportons eual to for each nput. A mxng operaton that combnes nputs nto a mxture would have nput proportons that sum to. An operaton that reures more than one unt of some nput would be modeled wth a proporton greater than on the assocated nput. Tabular Representaton We can represent the data for a pull tree of Fg. 3 wth a twodmensonal table as llustrated below.

5 Table 3. Pull tree parameters Index Next Pull Out Rato Proporton For ths llustraton, we are assumng that 0% of the unts passng through each operaton are scrapped. We also assume proportons of 0.5 for operatons through 4. Ths means that operaton 3 receves half of ts nput from each of operatons and 2. Further, operaton 5 receves half of ts nput from each of operatons 3 and 4. For the example, 00 unts are pulled from operaton 5. For the pull tree, an operaton can send ts output to no more than one other operaton, so the column labeled Next s suffcent to descrbe the tree structure. The column labeled Proporton gves the proporton of the nput of the next-operaton that s obtaned from the operaton. The number (0.5) n the row for operaton, holds the value of 3. For the pull tree structure we defne the followng notaton. We use for the general operaton ndex. a the ndex of the operaton followng (or after) operaton. Ths s the number n the Next column. the flow pulled from the output of operaton. r the rato between the output and nput flows for operaton. a the proporton of the nput of operaton a that s obtaned from operaton. We use the symbol a as the second subscrpt on a to ndcate that t s the proporton of the nput of the followng operaton that must come from operaton. When an operaton has no followng operaton we assgn the value 0 to a, and the value of a s wll have no effect. Flow Gven pull flows for the operatons, we want to compute the flow through each operaton. We use the notaton, and r as prevously defned.

6 To llustrate the computaton of the unt flows we use an example wth three operatons as n Fg. 4. k u u j k k u k Fgure 4. Porton of a pull tree j j jk The value of the flow out of operaton,, s entrely dependent on the pull flow wthdrawn at operaton and the amount reured by the followng operaton k. Snce the amount reured s k x k, we have the relaton between the flows at and k. + k x k r + k x k r Notce that the flow for an operaton depends on the flow of ts unue followng operaton. For the pull tree process, the unt flows can be computed recursvely, startng wth x m and contnung for each operaton wth seuentally decreasng operaton ndex.

7 2 / /2 / /2 Fgure 5. Example pull tree To llustrate a specfc case, consder the pull tree process shown n Fg. 5. The numbers on the arcs are proportons. One unt s pulled from node 5. For the llustraton we use the data of Table 3. We start wth the operaton wth the greatest ndex. By defnton, ths operaton wll have no followers so: x 5 5 r Consderng the operaton wth the next lower ndex, we compute x 4 ( ) x 5 r 4 (0.5)(.) Contnung n order of decreasng ndex, x 3 x 2 ( ) x 5 ( ) x 3 r 3 (0.5)(.) r 2 (0.5)(6.73) ( x + x 3 3) r (0.5)(6.73) The results are shown n Table 4 Table 4. Pull tree parameters and operaton flows wth a a r

8 Push Tree Push Flows γ 2 γ p 2 p 3 γ p p γ 4 γ The generc push tree s llustrated n Fg. 6. For ths structure the flow nto an operaton comes from a unue precedng operaton, whle the operaton may have several output flows gong to other operatons. Ths structure s approprate for modelng servce systems where customers arrve at a source node, node n the example n the amount. In addton to node, flow may also be pushed nto the network at other operatons. The push flow enterng at operaton s ι. Note that push flow enters the process just before the operaton, whle pull flow leaves the process after passng through the operaton. Fgure 6. The push tree The flow that passes through an operaton may be splt to go to other operatons to receve dfferent types of processng. Unts pass through the tree untl fnally they are wthdrawn to the nodes that have no successors, nodes 2, 4 and 5 n the fgure. For the tree structures we reure that the operatons be numbered so that when flow passes from operaton to operaton j, < j. The greatest ndes m. For the push tree we dentfy the proporton, p j as the proporton of the output of operaton that s passed to operaton j. The value of p j may be any postve amount to represent a varety of manufacturng stuatons. For a splttng operaton that separates the total flow passng through operaton nto several paths, the sum of the proportons leavng would eual. Tabular Representaton We can represent the data for a push tree wth a two-dmensonal table as llustrated for the example below.

9 Table 5. Push tree parameters Index Prevous Push Flow Rato Proporton For ths llustraton, we are assumng that 0% of the unts passng through each operaton are scrapped. We also assume proportons of 0.5 for operatons 2 through 5. Ths means that the output of operaton s splt wth half gong to operaton 2 and half gong to operaton 3. Further, the operaton of operaton 3 s splt wth half gong to operaton 4 and half to operaton 5. For the example, 00 unts are pushed nto operaton. For the push tree, an operaton receves ts nput from no more than one other operaton, so the column labeled Prevous s suffcent to descrbe the tree structure. For row, the column labeled Proporton gves the proporton of the output of the prevous operaton that goes to operaton. The number (0.5) n the row for operaton 2, holds the value of p 3. For the pull tree structure we defne the followng notaton. We use for the general operaton ndex. b the ndex of the operaton precedng (or before) operaton. Ths s ndes appears n the Prevous column of the table. the flow pushed nto the nput of operaton. r the rato between the output and nput flows for operaton. p b the proporton of the output of operaton b that goes to operaton. For p b we use the b to represent the prevous operaton. When an operaton has no precedng operaton we assgn the value 0 to b, and the value of p b has no effect. Flow To llustrate the computaton of the unt flows we use an example wth three operatons as n Fg. 7.

10 γ k γj p jk u k k u j j p j γ u For operaton n Fg. 7, the value of s entrely dependent on the push flow added at operaton and the amount passed from operaton j. Snce the amount from operaton j s p j x j we have the relaton between the flows at and j. + p j x j (2) + p j r j x j For the push tree, the flows can be computed recursvely, startng wth the x and contnung for each operaton wth seuentally ncreasng ndexes. Fgure 7. Porton of a push tree γ 00 /2 /2 2 3 /2 /2 4 5 Fgure 8. Example push tree To llustrate a specfc case, consder the push tree shown n Fg. 8. The numbers on the arcs are proportons. The flow pushed nto node s 00. For the llustraton we use the data of Table 5. We start wth the operaton wth the smallest ndex. By defnton, ths operaton wll have no predecessor operatons so: x 00 Consderng the operaton wth the next greater ndex we compute x p 2 r x (0.5)(0.9)(00) 45 Contnung n order of ncreasng ndces, x p 3 r x (0.5)(0.9)(00) 45 x p 4 r 3 x 3 (0.5)(0.9)(45) x p 5 r 3 x 3 (0.5)(0.9)(45) The results are shown n Table 6.

11 Table 6. Push tree parameters and flows wth 00. b r Pull Network The pull network s llustrated n Fg. 9. For ths structure the flow through each operaton may go to more than one operaton, and each operaton may have several nput flows from other operatons. Ths s a more general structure than the pull tree structure. Product s wthdrawn or pulled from any of the operatons. Agan we use as the amount pulled from operaton. Indces are assgned to the operatons arbtrarly, however, t s often convenent to assgn the ndces to be ncreasng n the drecton of prmary product flow Fgure 9. The pull network For the pull network we dentfy the proporton, j, as the amount of the output of operaton reured for each unt of product passng through operaton j. The value of j may be any nonnegatve amount to represent a varety of manufacturng stuatons. An assembly operaton that reures one unt of each nput to be combned to produce one unt of a subassembly would have the proportons eual to for each nput. A mxng operaton that combnes nputs nto a mxture would have nput proportons that sum to. An operaton that reures more than one unt of some nput would be modeled wth a proporton greater than on the assocated nput. The example shows an arc passng from operaton 5 back to operaton 4. In a practcal nstance, ths mght represent the reworkng

12 of some part. Although we mght be tempted to dentfy 54 as the proporton of the output of operaton 5 returned to operaton 4, ths s not correct for a pull network. 54 s the proporton of the flow through operaton 4 that comes from operaton 5. Smlarly, 24 s the proporton of the flow through operaton 4 that comes from operaton 2. We use the term proporton here n a slghtly dfferent sense than normal. When we dentfy the flow enterng operaton 4 as x 4, the flow on the arc from 5 to 4 s 54 x 4. The flow on the arc from 2 to 4 s 24 x 4. There s no reurement that the proportons add to as s usually the case. Matrx Representaton Although the external flows and ratos can be represented n a table, t s necessary to represent the proportons on a suare matrx. We call ths proporton matrx Q. The matrx Q descrbes both followng nodes and proportons. 2 L m 2 22 L 2 m Q M M M m m 2 L mm For the pull network structure we defne the followng notaton. We use for the general operaton ndex. the flow pulled from the output of operaton. r the rato between the output and nput flows for operaton. j the proporton of the nput of operaton j that s obtaned from operaton.

13 Flow j u k To compute the unt flows for a general network, we wll construct and solve a system of lnear euatons. To llustrate, consder the example n Fg. 0. The value of, the output flow from operaton s dependent on the pull flow wthdrawn at operaton and the amounts reured by the followng operatons, j and k. j u j j k k u k + j x j + k x k We wrte the euatons entrely n terms of the nput flows by usng the flow rato. Fgure 0. Porton of a pull network r r j x j k x k + j x j + k x k Ths generalzes to the expresson that must hold for each operaton. m r j x j for Km j We defne the augmented proporton matrx as Q a r 2 3 L m 2 r L 2m 3 32 r 3 33 L 3m M M M M m m2 m3 L r m mm Also defne the column vector x of flows. Then the unt flows are the soluton to the lnear set of euatons Q a x or x Q a

14 To llustrate a specfc case, consder the pull network shown n Fg.. The numbers on the arcs are proportons. 00 unts are pulled from node 5. The flow ratos are all. The proportons ndcate that for each unt passng through operaton 3, 0.5 unts must pass through each of operatons and 2. The proportons enterng operaton 4 ndcate that for each unt passng through operaton 4, 0.9 unts must pass through operaton 2 and 0. must pass through operaton 5. The flow transfers for the network of Fg. are represented by the Q matrx. The pull flows and operaton flows are the column matrces and x Fgure. Example pull network Q x 0 00 x x 2 x 3 x 4 x 5 The augmented matrs: Q a

15 The set of lnear euatons determnng the unt flows s: Q a x Solvng we fnd: x x 55.56, x , x 3., x 4., x 5. x 2 x 3 x 4 x 5 Push Network γ 2 γ p 2 p 3 γ p 53 The push network s llustrated n Fg. 2. For ths structure the flow through each operaton may go to more than one operaton, and each operaton may have several nput flows from other operatons. Ths s a more general structure than the push tree. Flow s nserted or pushed nto any of the operatons. We use as the amount pushed nto operaton. Indces are assgned to the operatons arbtrarly, however, t s often convenent to assgn the ndces to be ncreasng n the drecton of prmary product flow. p 24 p 34 p 35 γ 4 γ Fgure 2. The Push Network For the push network we dentfy the proporton, p j, as the amount of the output of operaton that s passed to operaton j for each unt passng through operaton. The value of p j may be any nonnegatve amount. Typcally for a servce system, the sum of the proportons leavng an operaton s eual to. Ths means that the flow s splt among the several followng operatons. It may be necessary to use other combnatons of proportons to represent dfferent systems.

16 The example shows an arc passng from operaton 5 back to operaton 3. In a practcal nstance, ths mght represent the reworkng of some part. p 53 as the proporton of the output of operaton 5 returned to operaton 3. It s not necessary to defne a proporton for the flow leavng the system at operaton 5. Matrx Representaton Although we can represent much of the data for a push network wth a two-dmensonal table, t s necessary to represent the proportons on a suare matrx. We call ths proporton matrx P. Agan, proportons have dfferent meanngs for push and pull networks. p p 2 L p m p 2 p 22 L p 2m P M M M p m p m2 L p mm For the push network structure we defne the followng notaton. We use for the general operaton ndex. the flow pushed to the nput of operaton. r the rato between the output and nput flows for operaton. p j the proporton of the flow through of operaton that s passed to operaton j.

17 Flow γ j j u j p j γ γ k u k p k u k Fgure 3. Porton of a push network To compute the unt flows for a general network, we wll construct and solve a system of lnear euatons. To llustrate, consder the example n Fg. 3. The value of, the nput flow from operaton depends on the push flow at operaton and the amounts provded by the precedng operatons, j and k. + p j x j + pk x k We wrte the euatons entrely n terms of the operaton flows by usng the flow ratos. + p j r j x j + p k r k x k p j r j x j p k r k x k Ths generalzes to the expresson that must hold for each operaton. m p j r j x j for Km j We defne the augmented proporton matrx as r p r p 2 r p 3 L r p m r 2 p 2 r 2 p 22 r 2 p 23 L r 2 p 2m P a r 3 p 3 r 3 p 32 r 3 p 33 L r 3 p 3m M M M M r m p m r m p m2 r m p m3 L r m p mm Also defne x, the column vector of operaton flows, and, the column vector of push flows. Then the unt flows are the soluton to the set of lnear set of euatons x T P a T or x T T P a The T superscrpt ndcates the matrx transpose operaton.

18 γ To llustrate a specfc case, consder the push network shown n Fg. 4. The numbers on the arcs are proportons. 00 unts are pushed nto node. All flow ratos are all. The proportons p 3 p ndcate that for each unt passng through operaton, 0.5 unts must be sent to each of operatons 2 and 3. The proportons leavng operaton 3 ndcate that for each unt passng through operaton 3, 0.5 unts wll go to operaton 4 and 0.5 wll go to operaton 5. The value of p means that 0% of the output of operaton 5 wll return to 3. The remander of the flow out of operaton 5 leaves the network. It s not necessary to defne specfc flows to leave the network. The network of Fg.4 s represented by the P matrx below. Fgure 4. Example push network P x x x 2 x 3 x 4 x P a x T P a T

19 Solvng we fnd: x x 2 x 3 x 4 x 5 T T x 00, x 2 50, x , x , x Use n the Add-ns Three add-ns use the structure and drve optons descrbed n ths paper. The Inventory add-n uses all three structures and the two drve optons as they are descrbed above. The values of the flow ratos are provded drectly as data. The Process Flow Analyss add-n uses the tree and network structures and both drve optons. The lne structure can be handled as a specal case of the tree structure. The flow rato s not nput drectly, but t s computed from other operaton parameters. Ths add-n has the parameters: w the proporton of flow removed at operaton. g the number of tems grouped at operaton. The flow rato s computed wth the euaton r u u ( w ) g Ths add-n computes unt flow, the flow n each operaton per unt of the drvng flow. For the pull drve, we set m. The other pull flows are relatve to the pull flow at operaton m. For the push drve we set. The other push flows are relatve to the push flow at operaton. Otherwse all formulas descrbed above are used for ths add-n. The Queue add-n uses only the lne and the push network structures. All ratos are set to for ths add-n.

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