CHAPTER 4 ECONOMIC ORDER QUANTITY MODEL FOR ITEMS WITH IMPERFECT
4.5 Numerical Analysis
102
103 The sensitivity of the total forgetting time will also be analyzed later. The rest of the data for this numerical analysis is taken from Salameh and Jaber (2000). The probability density function for the percentage of defectives is taken to be
π1(πΎ) =οΏ½1/(0.04β0),
0, 0β€ πΎ β€0.04 ππ‘βπππ€ππ π
Microsoft Excel Solver is used to obtain the optimal annual profit for a certain learning exponent and the percentage of defectives. It is observed that the annual profit plateaus to some extent after ten cycles, as a result of learning and forgetting. Therefore, in case of partial or total transfer of learning, an average of the results from the first ten cycles is used to compare them with those from the other scenarios of learning. To illustrate this, Table 4.1 shows the changes in the cycle length, the order size, the forgetting exponent and the annual profit from cycle to cycle, for the case of partial transfer of learning.
Table 4.1 Results of partial transfer of learning from cycle to cycle
Lost Sales Backorders
Ti Qi Ξ²i E[TPU(Qi)] Ti Qi Ξ²i E[TPU( Qi)] 0.0711 3625 0.4277 1216192 0.0573 2923 0.4025 1218890 0.0493 2517 0.3914 1220513 0.0488 2491 0.3909 1220626 0.0450 2297 0.3886 1221502 0.0447 2279 0.3885 1221589 0.0419 2137 0.3889 1222305 0.0416 2123 0.3890 1222379 0.0394 2012 0.3913 1223009 0.0392 2001 0.3917 1223076 0.0375 1912 0.3956 1223650 0.0373 1903 0.3961 1223711 0.0359 1831 0.4013 1224244 0.0357 1824 0.4020 1224301 0.0346 1765 0.4084 1224802 0.0345 1759 0.4092 1224856 0.0335 1710 0.4168 1225332 0.0334 1705 0.4177 1225383 0.0431 2201 0.4011 1222394 0.0414 2112 0.3986 1222757
(the bold values are the averages of the columns)
The input data and the results of the numerical example are shown in Table 4.2. It should be noted that the total transfer of learning is a better approach for both lost sales and backorders.
The rationale is that while retaining the previous knowledge in screening, the buyer orders less.
Total forgetting in both lost sales and the backorders does not result as a profitable option which is an understandable finding as the buyer has to screen with no previous experience.
104 The difference between lost sales and the backorders is not huge in case of total or partial transfer of learning. The reason is that the reported results are an average of the values in ten cycles. This minimizes the difference in the two approaches to a great extent. On the other hand, this difference in case of no transfer of learning is a noticeable one.
Table 4.2 Input data and results of the numerical examples
D x1 K π1 π1 s v cB cL βπ Ξ³ b L
50000 30000 100 87600 25 50 20 10 50 5 0.04 0.32 0.82 units/
yr
units/
yr $/cycle $/yr $/unit $/unit $/unit $/unit/
yr $/unit $/unit/
yr - - yr
Lost Sales Backorders
ππβ Annual Profit ππβ Annual Profit Partial
Forgetting 2201 1222394 2112 1222757 Total
Learning 1701 1228448 8085 1228516
Total
Forgetting 3625 1216192 2923 1218890
To enhance the above example, the effect of the change in learning exponent was studied on the three scenarios discussed, for lost sales and backorders, at a fixed percentage of defectives. The average of the expected annual profit from the first ten cycles of learning was obtained at different values of the learning exponent. The results are shown in Figures 4.3 and 4.4. It can be seen that the annual profit in both the cases, in all the scenarios of learning, tends to be increasing with the learning exponent except only for total forgetting in case of lost sales.
That is, learning in screening makes the buyer order less and less. This saves him some of the screening cost and the holding cost, and causes increase in the annual profit. The unusual behavior in case of lost sales with total forgetting tells us that at a very high learning exponent, the buyer tends to pile inventory with him, i.e. an increase in the holding cost which is not there in other scenarios of learning. This counters the increase in the annual profit. In general one can say that the more the knowledge in screening is retained the more is the annual profit.
Furthermore, total transfer of learning remains to be the best of the three scenarios discussed, both for lost sales and backorders, in terms of annual profit.
105 Figure 4.3 Annual profit w.r.t. b for lost sales at Ξ³ = 0.04
Figure 4.4 Annual profit w.r.t. b for backorders at Ξ³= 0.04
Similarly, to understand the effect of the shortage costs, annual profit was obtained for varying lost sales and backorder costs, for the case of partial transfer of learning. Again, an average of the annual profit from ten consecutive cycles of learning was obtained at different
1205000 1210000 1215000 1220000 1225000 1230000 1235000 1240000
0.32 0.36 0.40 0.44 0.48 0.52 0.56 0.60 0.64 0.68 0.72 0.76 0.80
Expected Annual Profit
Learning Exponent
Total Learning Partial Forgetting Total Forgetting
1205000 1210000 1215000 1220000 1225000 1230000 1235000 1240000
0.32 0.36 0.40 0.44 0.48 0.52 0.56 0.60 0.64 0.68 0.72 0.76 0.80
Expected Annual Profit
Learning Exponent
Total Learning Partial Forgetting Total Forgetting
106 unit-costs of lost sales, at a fixed value of the learning exponent. Figure 4.5 shows that, as intuitively expected, the annual profit tends to decline with the increasing unit lost sales cost.
That is, the lost sales cost offsets the annual profit margin of the buyer. On the other hand, the annual profit for the backorder case showed almost no variation when the unit backorder cost was varied from twice to four-times the unit holding cost. This indicates that this much change in the unit backorder cost is just not enough to hint a clear difference in the annual profit, especially when the learning rate is fixed. It should be noted that in the backorder case, the demand is eventually met and hence profit is not lost. However, the only additional cost is that of back orders, which is more than the holding cost, for the quantity backordered. In order to investigate the effect of defective items on the annual profit, the sensitivity of the model to πΎ, was tested for the lost sales case with partial transfer of learning. Figure 4.6 shows that at a fixed learning exponent, the annual profit tends to be decreasing with the percentage of defectives. This is an expected result as the imperfect items are likely to slash the profit obtained from the non- defective items.
Figure 4.5 Annual profit w.r.t. cL for lost sales with partial forgetting 1222150
1222200 1222250 1222300 1222350 1222400 1222450
50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88
Expected Annual Profit
Unit Cost of Lost Sales
107 Figure 4.6 Annual profit w.r.t. Ξ³ for lost sales with partial forgetting
To investigate the sensitivity of the time for total forgetting, the response of the model to varying L was tested for the lost sales case with partial transfer of learning. The three curves in Figure 4.7 indicate that as the worker retains his/her skills for a longer period, the increase in the profit by virtue of learning ends up at a higher level than for the case where he loses his experience in shorter spells of time. As shown in Table 4.1, it can also be seen here (Figure 4.7) that the most of the benefit from learning shows in the earlier cycles (e.g., from 1 to 2).
Figure 4.7 Annual profit w.r.t. L for lost sales with partial forgetting 1210000
1215000 1220000 1225000 1230000 1235000
0.010 0.012 0.014 0.016 0.018 0.020 0.022 0.024 0.026 0.028 0.030 0.032 0.034 0.036 0.038 0.040 0.042 0.044 0.046 0.048
Expected Annual Profit
Percentage of Defectives
1210000 1212000 1214000 1216000 1218000 1220000 1222000 1224000 1226000 1228000
1 2 3 4 5 6 7 8 9 10
Expected Annual Profit
Cycle
L = 60 days L = 180 days L = 360 days
108 Finally, the sensitivity of the model to the unit inspection cost is explored. The response of the annual profit for the lost sales case, with total transfer of learning, is plotted in Figure 4.8, at different levels of learning. That is, the unit inspection cost is varied from $0.5 to $2.5 for b between 0.05 and 0.4. It should be noted here that the unit inspection cost in all the above analyses was $0.5. Figure 9 indicates that the annual profit drops by a great extent by varying the unit inspection cost at lower levels of b (i.e. when the learning is slow). This difference in the annual profit starts diminishing as learning becomes faster (higher values of b). This indicates that the unit inspection cost affects the annual profit more at the higher values of screening time.
One should notice that the screening time per cycle gets shorter and shorter with learning which tends to increase the annual profit. Besides, this screening time has an inverse relation to the learning exponent b.
Figure 4.8 Annual profit w.r.t. d and b for lost sales with total transfer of learning
109