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The derivations above do not include the impact of variation in demand, setup times, processing times per unit and reject rates These phenomena can be approximated by formulas from Queuing Theory, but due to their limited range of application, we prefer to calculate the effects of variation using computer Discrete Event Simulations
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Steps beyond the Simplified Complexity Equation
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In 8, we discussed examples of how the Complexity Equation can be used to estimate the impact of process or product improvements, such as the reduction in the number of different offering or reduction in average setup time, all other things being held constant But what if you want to know the impact of reducing only the number of low-volume products via commonization or outsourcing, etc The derivation becomes a bit more complicated, but the math itself remains in the realm of the second year of High School algebra We start by divide the offering into two groups (high-volume vs low-volume) using Pareto analysis The high-volume group will constitute about 20% of the total number of offerings, and have an average demand of DH per offering per unit of time The lowvolume offerings (which constitute about 20% of total demand, and 80% of the number of offerings) will have an average of DL demand We will use the same nomenclature as above: NH = number of different offering in the high-volume group, etc (It doesn t matter if the split is 70/30, we will first derive a completely general expression for process cycle efficiency We will then apply it to the case of breaking the product line into two groups, then as many as you like Remember though, or goal is to increase Process Cycle Efficiency, not build models)
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To keep the math simple, we will return to the original derivation earlier in this Appendix The addition of quality and process complexity complicates the formulas2 We start with the definition of Process Cycle Efficiency (numbers correspond to equation numbers from the patent applications):
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Process Cycle Efficiency = Value-add Time Total Lead Time
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(15)
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Now the average value-add time is
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1 = N
(16)
The total lead time is, from Little s law:
Total lead time
Total Number of Things in Process Completion Rate
(17)
The components of this equation can be denoted as:
Total Number of Things in Process
TIP
Completion rate
(18, 19)
From the Patent3 using the usual rule for matrix multiplication:
(TIP1 TIP2 TIPN )= 2
i =1
1 ( X1 - D1P1 ) D 2 P1 (1 - X2 - D 2 P2 ) D1P2 S (2A + 1)(D1 D 2 D N ) i D P - D 2 PN 1 N
D N P2 (1 XN DN PN )
D N P1
(20)
(20)
This assumes that each of the N products has unique demand, processing time, etc So now lets apply this to the problem we posed, breaking the product line into two segments, with NH being the number of offerings with high volume demand DH high volume and NL the number of low volume products, etc:
(1 - XH - NHDHPH ) - NLDLPH (TIPH TIPL) = 2(SH + SL )(2A + 1)(DH DL) - NHDHPL (1 - XL - NLDLPL
(21)
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Now any 2x2 matrix can be inverted according to the rule
(22)
Thus we have:
a b 1 d - b c d = ad bc - c a
(23)
1 (1 - XL - NLDLPL) NHDHPL ( - NHDHPH ) - NLDLPH 1 ( - NHDHPL (1 - NLDLPL) = ( - NHDHPH )(1 - NLDLPL) - (NHDHPL)(NLDLPH) NLDLPH (1 - XH - NHDHPH ) 1
We can now substitute (23) into (21) and compute TIPH and TIPL, use these values in (19), (17) and finally compute PCE in (15) A 2x2 analysis is often an easy first cut at the problem, because it captures a lot that is currently hidden from management It allows you to vary the number of low-volume parts NL and compute the impact on Process Cycle Efficiency From the complexity matrix you can decide whether an initiative related to commonization or pruning (reduction of NL) or a lean initiative in reducing setup time (SH and/or SL) will be most effective in reducing PCE Given the cost data from the Complexity Value Stream Map, you can estimate just what chunks of non-value-add cost could be removed Equation (9) in the patent allows us to have a Processing Time per unit PH for the high volume and a separate PL for the low volume products, and in fact we would expect PH<PL, because of the volume and repetition differences, and similarly SH<SL Now if you believe that you need to break the demands into more than two groups to more closely mirror your offering, the math isn t any more difficult its just bigger, and gets beyond the realm of pencil and paper The problem of inverting an NxN matrix in a reasonable amount of time used to require mainframe computers However, ExcelTM now has solvers with add on packages that make virtually all practical problems within the reach of PentiumTM PC
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