Department 2 Power cut of A1 and A2

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1 3.12. PROBLEMS Problems Problem 3.1 A production system manufactures products A and B. Each product consists of two parts: A1 and A2 for product A and B1 and B2 for product B. The processing of A and B require several technological steps. The departments where these steps are carried out are shown in Figure The number of each department indicates its order in the technological process. The Department 1 Blanking of A1 Department 2 Power cut of A1 and A2 Department 1 Blanking of A2 Department II Wet cut of B1 Parts A1 & A2 and B1 & B2 are Shipped offsite for heat treatment (This option takes about 48 hours, while the cycle time of the machines in all department is about 30 sec.) Department II Wet cut of B2 Department 3 Hard turn of A2 Department III Hard turn of B1 and B2 Department 3 Hard turn of A1 Department IV Assembly of B Department 4 Assembly of A Figure 3.37: Problem 3.1 material handling among the departments is carried out by carts, which are pushed by machine operators from one department to another. (a) Construct a structural model of this production system. (b) Describe the data that have to be collected to identify this model. (c) Describe which steps must be taken to collect these data. (d) Describe which steps must be taken to validate this model. Problem 3.2 The layout of a production system for an automotive ignition device is shown in Figure It consists of four main operations: Housing Subassembly, Valve Body Assembly, Injector Subassembly, and Injector Final Assembly. In addition, the system contains Shell Assembly, three Welding operations (L.H.W., U.H.W., and Weld), two Overmold operations (O.M.1 and

2 116 CHAPTER 3. MATHEMATICAL MODELING O.M.2), two Set Stroke operations (Stroke 1 and Stroke 2), one Leak Test operation (L.T.) and one High Potential operation (Hi Pot). Finally, the system includes five buffers positioned as shown in Figure 3.38 and conveyor buffering among all other operations. Construct a structural model for this system and simplify it to a serial line. L.T. L.H.W. U.H.W. O.M.1 O.M.2 Hi Pot Housing Sub. L.C./L.H./L.I.T. Injector Final Asm. Injector Sub. Valve Body Sub. S.V. Shell Asm. Stroke 1 Stroke 2 Weld L.V.B.S. L.N.M.S. Legend: L.C./L.H./L.I.T.=Load Coil/Load Housing/Load Inlet Tube L.N.M.S.=Load Non-Magnetic Shell L.H.W.=Lower Housing Weld L.T.=Leak Test L.V.B.S.=Load Valve Body Shell O.M.=Overmold S.V.=Stroke Verify U.H.W.=Upper Housing Weld Figure 3.38: Problem 3.2 Problem 3.3 Consider the production system of Figure 3.38 and its model as a serial line obtained in Problem 3.2. Assume that the first five machines are of interest and their parameters are as follows: The cycle time of each machine is 3 sec and the breakdown and repair rates are (in units of 1/min): λ (S.A) = , μ (S.A.) = 0.5; λ (H.S.) = , μ (H.S.) = ; λ (L.T.) = , μ (L.T.) = 0.5; λ (L.H.W.) = , μ (L.H.W.) = 0.5; λ (U.H.W.) = , μ (U.H.W.) = 1. Assume also that the buffers between these operations have the following capacities: buffer between S.A. and H.S. = 125; buffer between H.S. and L.T. = 500; buffer between L.T. and L.H.W. = 15; buffer between L.H.W. and U.H.W. = 15.

3 3.13. ANNOTATED BIBLIOGRAPHY 117 (a) Construct the Bernoulli model of this five-machine exponential serial line. (b) Using the Simulation function of the PSE Toolbox, investigate the accuracy of the Bernoulli model. Problem 3.4 A serial production line with five exponential machines is defined as follows: λ =[0.0025, , , , ] (in units of 1/min), μ =[0.025, , 0.025, , 0.05] (in units of 1/min), c =[1.0714, , , , 1] (in units of parts/min), N = [26, 10, 28, 30]. (a) Construct the Bernoulli model of this five-machine exponential serial line. (b) Using the Simulation function of the PSE Toolbox, investigate the accuracy of the Bernoulli model. Problem 3.5 The layout of a production system for an automotive ignition device is shown in Figure It consists of 15 operations, separated by bufferconveyors. Construct a structural model for this system and simplify it to a serial line. Op.2 Op.4 Op.8 Op.12 Op.14 Op.6 Op.10 Op.1 Op.3 Op.5 Op.7 Op.9 Op.11 Op.13 Op.15 Figure 3.39: Problem 3.5 Problem 3.6 The layout of an automotive camshaft production line is shown in Figure Construct a structural model for this system and simplify it into two parallel serial lines Annotated Bibliography Various aspects of production systems modeling can be found in [3.1] J.A. Buzacott and J.G. Shantikumar, Stochastic Models of Manufacturing Systems, Prentice Hall, Englewood Cliffs, NJ, 1993.

4 162 CHAPTER 4. ANALYSIS OF BERNOULLI LINES the accuracy of the estimates for all performance measures depends on the pattern of machine efficiency allocation, with the lowest accuracy taking place for the inverted bowl and oscillatory allocations. The serial lines under consideration possess the property of reversibility: if the flow of parts is reversed, the production rate remains the same, while the probability of blockage (respectively, starvation) of machine i in the original line becomes the probability of starvation (respectively, blockage) of machine M i + 1 in the reversed line. The serial lines under consideration possess the property of monotonicity: improving any machine efficiency or increasing any buffer capacity always leads to an increased production rate of the system. 4.6 Problems Problem 4.1 Consider a two-machine Bernoulli production line defined by the conventions of Subsection with N = 1 (i.e., machine m 1 serves as the buffer). (a) Draw the state transition diagram of the ergodic Markov chain that describes this system and determine its transition probabilities. (b) Calculate the stationary probability of each state of this Markov chain and derive the formula for the production rate of this system. (c) Assuming that p 1 = p 2 =: p, draw the graph of PR as a function of p [0.5, 0.99] and comment on the qualitative behavior of this graph. Problem 4.2 Consider again a two-machine Bernoulli production line defined by conventions (a)-(e) of Subsection (a) Assume p 1 =0.95 and p 2 =0.6. Calculate and plot PR, WIP, BL 1 and ST 2 as a function of N for N from 1 to 10. Based on these plots, determine the buffer capacity that is reasonable for this system. (b) Assume p 1 =0.6 and p 2 =0.95. Again calculate and plot PR, WIP, BL 1 and ST 2 as a function of N for N between 1 and 10. Will your choice of buffer capacity change? Problem 4.3 Consider a two-machine Bernoulli production line defined by conventions (a)-(e) of Subsection Assume that N = 5 and p 1 p 2 =0.81. (a) Under this constraint, find p 1 and p 2, which maximize PR. (You may use trial and error to accomplish this; alternatively, you may think a little bit, look at the expressions for the performance measure, make an educated guess and verify it by calculations.) (b) For these p 1 and p 2, calculate PR, WIP, BL 1 and ST 2. What can you say about qualitative features of WIP, BL 1 and ST 2?

5 4.6. PROBLEMS 163 (c) Interpret the results and formulate a conjecture concerning the optimal allocation of p i s. Problem 4.4 Consider a two-machine Bernoulli production line defined by conventions (a)-(e) of Subsection Suppose that each machine produces a good part with probability g i and a defective part with probability 1 g i, i =1, 2. Assume that quality control devices operate in such a manner that a defective part is removed from the system immediately after the machine that produced this part. (a) Derive expressions for the production rate of good parts and for WIP, BL 1 and ST 2 in this system. (b) Using any example you wish, check if the reversibility property still holds. Problem 4.5 Consider a two-machine Bernoulli production line defined by all but one of the conventions of Subsection Specifically, assume that instead of the blocked before service, the following convention is used: Machine m 1 is blocked during a time slot if it is up at the beginning of this time slot and the buffer is full at the end of the previous time slot (i.e., the blockage of m 1 is independent of the status of m 2 ). We refer to this convention as symmetric blocking since both starvation and blocking conventions are of the same nature - they are defined by the state of the buffer and the status of one machine only. Assume for simplicity that the machines are identical, i.e., p 1 = p 2 =: p. (a) Draw the state transition diagram of the ergodic Markov chain that describes this system and determine the transition probabilities. (b) Derive the expressions for the stationary probabilities of this Markov chain. (c) Plot these stationary probabilities for p = 0.95 and for p = 0.55, assuming that in both cases N = 5; compare the resulting graphs with those of Figure 4.4 and explain what are the differences and why they take place. (d) Derive formulas for PR, WIP, BL 1 and ST 2 as a function of p and N. (e) Plot PR, WIP, BL 1 and ST 2 as functions of N for N from 1 to 10 and p =0.9; compare the resulting graphs with those of Figure 4.6, Line 1, and explain what are the differences and why they take place. Problem 4.6 Consider again the two-machine production line with the symmetric blocking as defined in Problem 4.5. Repeat parts (a)-(e) of Problem 4.2, assuming that p 1 p 2 ; for the questions in which numerical values are required, assume that p 1 =0.9 and p 2 =0.7. Problem 4.7 Investigate if the reversibility property holds for two-machine Bernoulli lines with the symmetric blocking convention defined in Problem 4.5. Problem 4.8 Repeat Problem 4.3 for a two-machine Bernoulli line with the symmetric blocking convention defined in Problem 4.5. Problem 4.9 Consider a 5-machine Bernoulli production line defined by conventions (a)-(e) of Subsection

6 164 CHAPTER 4. ANALYSIS OF BERNOULLI LINES (a) Assume p i =0.9, i =1,...,5, and all buffers are of equal capacity. Calculate and plot PR, ŴIP, BL i and ŜT i as functions of N i for N i =1, 2, 3, 4, and 5. Based on these results, determine the buffer capacity, which is reasonable for this system. (b) Assume now that p i =0.7, i =1,...,5, and buffers are as above. Again calculate and plot PR, ŴIP, BL i and ŜT i as functions of N i. Will your choice of buffer capacity change? (c) Interpret the results. Formulate your conjecture as to the choice of buffer capacity as a function of machine efficiency. Problem 4.10 Consider a 3-machine Bernoulli production line defined by conventions (a)-(e) of Subsection Assume N i =1,i =1, 2, and p 1 p 2 p 3 = (0.8) 3. (a) Under this constraint, find p i, i =1, 2, 3, which maximize PR. (You may use the trial and error method and the PSE Toolbox to accomplish this.) (b) For these p i, calculate ŴIP, BL i and ŜT i. What can you say about ŴIP? Interpret the results and formulate a conjecture concerning the optimal allocation of p i s. Problem 4.11 Consider the production system of Figure 3.38 and its 5- machine Bernoulli model constructed in Problem 3.3. (a) Calculate the production rate of this system. (b) Calculate the average occupancy of each buffer. (c) Calculate the probabilities of blockages and starvations of all machines. (d) Assuming that the Bernoulli buffer capacity is increased by a factor of 2, recalculate the production rate. Does it make practical sense to have all buffer capacities increased? (e) Assume that efficiency of all Bernoulli machines is increased by 10% and again recalculate the production rate. Does it make sense to have all efficiencies increased? Problem 4.12 Repeat the steps of Problem 4.11 for the Bernoulli model of the production system constructed in Problem 3.4. Problem 4.13 Consider a seven-machine Bernoulli line with machines having identical efficiency. Assume that two buffers with equal capacities are available to be placed in this system. (a) Where should the buffers be placed so that PR is maximized? (b) Suggest an example illustrating the efficacy of your solution. Problem 4.14 Consider a ten-machine Bernoulli line with machines of identical efficiency and buffers of identical capacity. Assume that the efficiency of two machines can be increased (or new machines with higher efficiency can be purchased).

7 4.7. ANNOTATED BIBLIOGRAPHY 165 (a) Which of the machines should be improved (or replaced) so that PR is maximized? (b) Suggest an example illustrating the efficacy of your solution. Problem 4.15 Consider an M > 2-machine Bernoulli line with the symmetric blocking convention defined in Problem 4.5. (a) Develop a recursive aggregation procedure for performance analysis of this line. (b) Derive formulas for estimates of PR, ŴIP i, BL i and ŜT i. Problem 4.16 Using examples, investigate the reversibility property of M> 2-machine Bernoulli lines with the symmetric blocking convention. In particular, investigate if still hold and, in addition, PR L = PR Lr, (4.54) BL L i = ŜTL r (M i+1) r, i =1,...,M (4.55) (4.56) ŴIP L r i r = N M i ŴIP L M i. (4.57) Problem 4.17 Using examples, investigate the monotonicity property of M>2-machine Bernoulli lines with the symmetric blocking convention. 4.7 Annotated Bibliography The initial analysis of two-machine Bernoulli lines has been carried out in [4.1] J.-T. Lim, S.M. Meerkov and F. Top, Homogeneous, Asymtotically Reliable Serial Production Lines: Theory and a Case Study, IEEE Transactions on Automatic Control, vol. 35, pp , under the assumption that the machines are asymptotically reliable, i.e., their efficiencies are close to 1. The general case has been addressed in [4.2] D.A. Jacobs and S.M. Meerkov, A System-Theoretic Property of Serial Production Lines: Improvability, International Journal of System Science, vol. 26, pp , The case of M > 2-machine lines has also been analyzed in [4.1] for the asymptotically reliable machines and then generalized in [4.2]. Numerical investigation of aggregation procedure accuracy has been carried out by J. Huang and A. Khondker in the framework of their course project in

8 5.6. PROBLEMS 197 The buffering of a production line is potent if the BN-m is indeed the machine with the smallest efficiency. Identifying the BN-m can be used as a basis for Measurement-based Management of production systems. 5.6 Problems Problem 5.1 Consider a five-machine line with Bernoulli machines. Assume N 1 = N 4 =1,N 2 = N 3 = 3 and the product of all p i s of the machines is (0.9) 5. (a) Design an unimprovable (i.e., optimal) system with respect to WF. (b) Does the inverted bowl phenomenon take place? Explain why it does or does not. Problem 5.2 Consider the same system as in Problem 5.1. (a) Using WF-Continuous Improvement Procedure 5.1, obtain an unimprovable system. (b) Does it coincide with the one obtained in Problem 5.1? Explain why it does or does not. (c) Calculate PR of the design obtained and compare it with that ensured by the design of Problem 5.1. Problem 5.3 Consider again a five-machine production line where the sum of N i s is 8 and the product of p i s is (0.9) 5. (a) Design a system that is unimprovable with respect to both WF and BC simultaneously. (b) Compare it with the designs obtained in Problems 5.1 and 5.2 and comment on the differences in structures and production rates of each design. Problem 5.4 Consider a four-machine Bernoulli line, where the sum of N i s is 5. (a) Assume each p i is 0.8. Using BC-Continuous Improvement Procedure 5.1, find the unimprovable allocation of N i s. Comment on the shape of this allocation as a function of i. (b) Assume now that p 1 = p 3 =0.7and p 2 = p 4 =0.9. Again, using BC- Continuous Improvement Procedure 5.1, determine the unimprovable allocation of N i s and compare it with the one obtained in (a). Comment on the reason for the differences. Problem 5.5 Consider the Bernoulli model of the production system analyzed in Problem 3.3.

9 198 CHAPTER 5. IMPROVEMENT OF BERNOULLI LINES (a) Determine if this system is WF-improvable. If so, calculate the unimprovable work allocation and the resulting production rate. (b) Determine if this system is BC-improvable. If so, calculate the unimprovable buffer capacity allocation and the resulting production rate. (c) Calculate the simultaneously unimprovable WF and BC allocations and the resulting production rate. (d) Compare the production rates obtained among themselves and with that obtained in Problem 4.9 for the original system. (e) Which, if any, of the above improvement projects would you recommend for implementation? Problem 5.6 Consider the Bernoulli model constructed in Problem 3.4 and analyzed in Problem Repeat steps (a)-(e) of Problem 5.5. Problem 5.7 Consider a five-machine serial line with Bernoulli machines. Assume N i =2,i =1,...,4, and p i s are as follows: [0.9, 0.7, 0.9, 0.9, 0.7]. (1) Determine the BN-m and BN-b. (2) Change the buffer capacities around the BN-m so that the bottleneck moves to another machine. What is the new BN-m? Why did the bottleneck move? (3) What is the BN-m when all buffers are infinite? Problem 5.8 Consider the production system analyzed in Problem 5.5. (a) Identify its BN-m (or PBN-m) and BN-b and determine if the buffering is potent. (b) Based on this information, design the best, from your point of view, continuous improvement project that results in 10% increase of the production rate of the system (as compared with the original one). (c) Using the B-exp transformation of Chapter 3, return to the exponential description and formulate measures, which would have to be carried out in order to implement this continuous improvement project. Problem 5.9 Repeat steps (a)-(c) of Problem 5.8 for the system considered in Problem 5.6. Problem 5.10 Derive a WF-Improvability Indicator for Bernoulli lines with the symmetric blocking convention. Problem 5.11 Derive a BC-Improvability Indicator for Bernoulli lines with the symmetric blocking convention. Problem 5.12 Derive a Bottleneck Indicator for identifying BN-m and BNb in Bernoulli lines with the symmetric blocking convention.