A more general topology

For more general SC such as fig. 5.4, several supply topologies are incorporated in a single SC. For the ease of exposition, here we model an abstract version of fig. 1.1 in chapter 1. However, the same solution method applies to fig. 1.1 as well. We use several M/M/1 to represent each inbound logistics (Sj) of the supplier and we assume there is only

one M/M/1 outbound logistics (T1). The single distribution process (T2) is Hyper-exponential distributed. Notice, if we use a dedicated transportation route for each retailer, the analysis will be a slightly different. Under such configuration, we can use sequential refinement method of Lee and Zipkin (1995) for the analysis for distribution system. However, sequential refinement method is similar to L & Z. It differs by treating each dedicated transportation route as a M/M/1 queue and calculates it respectively. In the past, simulation seems to be the only choice for analyzing such complex structure.

However, with the assistance of other stochastic modeling procedure, we can still transform the original complex structure into a tractable tandem-form as explained in subsection 3.2.2 of chapter 3 and subsection 5.2.1 of chapter 5. Now the supplier may be from multiple sources. Suppose we have 5 functional stages: supply (S), assembly (A), manufacture (M), outbound logistics (T1) and distribution (T2). Note here T2 may include the processing needed for product differentiation and transporting tailored for non-identical customers’ needs as illustrated in inbound logistics of fig. 1.1 in chapter 1.

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Fig. 5.4 A more general SC model.

Suppose we want to solve BAP and CAP simultaneously with the restriction that the service level for each retailer is not larger than some threshold levels and the arithmetically averaged service level is not larger than some pre-specified level. We formulate the optimization process as follows.

.

All the parts from different supply sources will be collected whenever one from each supplier is available and immediately sent to the inbound warehouse, I1 of the producer.

We neglect the transferring process of inbound logistics and transit inside producer.

Suppose we have 2 suppliers and 4 retailers. The supply distributions are all exponential with identical rates. Thus the results of subsection 5.2.1 can be applied here. Under this setting, the mean response time of the FES is 3/2 times larger than that of the original M/M/1 queue (Bolch, 1998). Assume the parameters for retailer demand rates are

(λ, λ2, λ3, λ4) = (0.1, 0.2, 0.3, 0.4), distribution rates for T2 are u5=(1, 2, 3, 4). Notice here we assume non-identical retailers with different serving (distribution) rates. The u is set 1.01 to preserve the stability in a ququeing system. Remember the original formula of the

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number of order in the system of a M/PH/1 is L=(1−ρ)α_*R(I−R)⁻²1 (see (3.9) of chapter 3 for details), however we have to modify this performance measure to account for different distribution rate uj.Applying Little’s formula, we get the actual queue length for each retailer Lq= L−ρ , and so the actual sojourn time for each distribution process is

ui

Lq

Ws= /λ+1/ . The accuracy of the modified approximation model is verified through simulation study. We setup the experimental design in table 5.8 for this verification process. For the sake of simplicity, we neglect the cost incurred in capacity when making the comparison. Table 5.9 (related to TC and other measures) and 5.10 (related to SL) show the results. The deviations of the approximated model (App) and the simulation model (Sim) of all performance measures are very tight for most of the test cases. High deviations occur at some expected backorder levels and service level where both app and sim are extremely low. From the high accuracy of the tables, we are confident that the approximation model is also sufficient to act as the evaluation function for the optimization process of such more general SC topology.

From section 5.1, the roles played by buffer and capacity seem to be the same. The increase in one variable may decrease the other to maintain low cost while striving to preserve SL requirement. Observations derived from tables 5.9 and 5.10 justify our conjecture. The following lists the simple rules-of-thumb after observing the results of the tables. Note rules 5.1 and 5.2 relate to TC while rules 5.3 and 5.4 relate to SL.

Rule 5.1 If all resources are low, increase capacity only, not else to decrease cost. This measure decreases TC from 31.552 to 7.29 as seen from table 5.9. Any other measure results in more TC. This is because no matter what resource(s) is/are increased, E[B5] decreases. However, the “side effect” of increasing capacity is the mildest. It decreases WIP and increase a little E[I5] while all other measures increase either WIP and/or E[I5] significantly.

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Rule 5.2 If there is/are any resource(s), which is/are already high, it may be not necessary to increase any other resource(s) to prevent from increasing cost. When resource(s) is/are already high, it may be enough to fight against demand uncertainty and therefore adding extra resource(s) result(s) in marginal decrease in E[B5] but increase either WIP and/or E[I5] significantly. The overall result is not beneficial.

Rule 5.3 If all resources are low, increase any resource(s) as much as possible to increase service quality. Basically, SL is closely related to E[B5]. When E[B5] decreases service quality increases (i. e. SL decreases). Since any resource expansion plan lowers the risk of demand uncertainties, more resources increase service quality more.

Rule 5.4 If there is/are any resource(s), which is/are already high, the benefit of increasing any other resource(s) may be marginal. As can be seen from table 5.10, when resource(s) are high enough, adding more resource(s) is/are useless to decrease SL (adding intermediate buffer causes SL to remain 0.01).

To conclude, adding more capacity seems to be the most beneficial measure to reduce TC if buffer is low and the capacity incurred cost is neglected. However, when SL is also

concerned, the decision may be different. Decision maker has to make adequate measure to trade-off between cost reduction and service requirement. In this example our rules-of-thumb cope up with the reasoning of system dynamics. However, it may not be true under different assumptions. For example, if different cost structures and other factors not concerned herein are adopted it may result in different conclusions. For the ease of decision, adequate optimizer such as introduced in this chapter seems to fit in. Suppose the required service level for retailers is β = (0.1, 0.2, 0.3, 0.4) and β = 0.15. Note β is the averaged service requirements. Table 5.11 is such exercise. Interestingly, the positioning of the workload sequence in distribution stage and the positioning of the buffer in other stages are consistent with the previous results.

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Table 5.8 DOE setting for approximation validation of more general topology

Decision Factors Levels

Low: (2.25, 1.5, 1.5, 1.5) Capacity (u1, u2, u3, u4)

High: (6, 4, 4, 4) Low: ((1, 1, 1, 1) Intermediate buffer (S1, S2, S3, S4)

High: (4, 4, 4, 4) Low: ((1, 1, 1, 1) Ending buffer (S5)

High: (4, 4, 4, 4)

Note: (u1, u2, u3, u4) are service rates for supply, assembly, manufacture, and outbound logistics respectively;

(S1, S2, S3, S4) are base stock levels for I1, I2, I3, and DC respectively. S5 is a vector, representing base stock levels for different retailers.

Table 5.9 Approximation validation (1).

TC WIP E[I5] E[B5]

Capacity Intermediate buffer

Ending

buffer Sim App %Err Sim App %Err Sim App %Err Sim App %Err Low Low Low 31.552 31.602 0.16 7.471 7.679 2.79 1.822 1.725 -5.35 2.443 2.604 6.58 Low Low High 19.369 19.053 -1.63 7.471 7.679 2.79 11.729 11.493 -2.01 0.350 0.372 6.31 Low High Low 15.792 15.882 0.57 16.233 16.236 0.02 3.080 3.036 -1.42 0.471 0.473 0.36 Low High High 23.054 23.056 0.01 16.233 16.236 0.02 14.645 14.598 -0.32 0.035 0.034 -2.86 High Low Low 7.290 7.165 -1.72 4.552 4.594 0.93 3.318 3.266 -1.57 0.164 0.160 -2.32 High Low High 17.450 17.413 -0.21 4.552 4.594 0.93 15.156 15.107 -0.32 0.003 0.001 -70.00 High High Low 13.028 12.813 -1.65 16.448 16.449 0.00 3.392 3.373 -0.56 0.136 0.122 -10.66 High High High 23.487 23.482 -0.02 16.448 16.449 0.00 15.257 15.252 -0.03 0.002 0.001 -70.00

Note: %Err = (App – Sim) / Sim ×·100 %

Table 5.10 Approximation validation (2).

(SL1, SL2, SL3, SL4, SL) Capacity Intermediate

buffer

Ending

buffer Sim App %Err

Low Low Low (0.384, 0.544, 0.642, 0.706, 0.569) (0.360, 0.524, 0.619, 0.672, 0.543) -4.57 Low Low High (0.008, 0.047, 0.109, 0.18, 0.086) (0.007, 0.043, 0.100 , 0.169, 0.080) -6.98 Low High Low (0.167, 0.224, 0.268, 0.304, 0.241) (0.167, 0.212, 0.259, 0.280, 0.230) -4.77 Low High High (0.001, 0.005, 0.013, 0.023, 0.011) (0.001, 0.005, 0.014, 0.024, 0.011) 0.00 High Low Low (0.131, 0.168, 0.202, 0.233, 0.184) (0.133, 0.157, 0.183, 0.208, 0.170) -7.61 High Low High (0, 0.001, 0.001, 0.002, 0.001) (0.000, 0.001, 0.002, 0.004, 0.002) 100.00 High High Low (0.119, 0.145, 0.17, 0.193, 0.157) (0.120, 0.145, 0.164, 0.177, 0.152) -3.18 High High High (0, 0, 0.001, 0.001, 0.001) (0.000, 0.000, 0.001, 0.003, 0.001) 0.00

Note: SL = (SL1 + SL2 + SL3 + SL4) / 4.

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Table 5.11 Optimal buffer and workload allocations for more general SC.

J (S1, S2, S3, S4, S5)* (ρ1, ρ 2, ρ 3, ρ 4, ρ5)* TC* SL*

5 (0, 0, 2, 3, 2, 2, 2, 2) (0.46, 0.48, 0.47, 0.46, 0.84, 0.63, 0.52, 0.45) 26.946 (0.03, 0.08, 0.14, 0.18, 0.11) 6 (0, 0, 1, 2, 3, 2, 2, 2, 2) (0.46, 0.48, 0.47, 0.46, 0.45, 0.84, 0.63, 0.52, 0.45) 30.875 (0.03, 0.09, 0.14, 0.19, 0.11) 7 (0, 0, 1, 1, 2, 3, 2, 2, 2, 2) (0.47, 0.48, 0.47, 0.47, 0.46, 0.44, 0.83, 0.62, 0.52, 0.45) 34.946 (0.04, 0.09, 0.14, 0.19, 0.12)

Note: CSA achieved convergence after 71 200 iterations for J = 5, 81 000 iterations for J = 6 and 90 000 iterations for J = 7.