Result Discussion

NOMENCLATURE

6.1 Result Discussion

At first, the Nelder Mead Downhill simplex method is used to find the optimal values of the decision variables which results in minimized average cost for a given set of input parameters. The numerical results obtained from three of the Nelder Mead Simplex runs are summarized in Table 6.1. Although the Nelder Mead method does not guarantee convergence to the global optimal solution, but in this case it is apparent that the cost values resulting from implementations with different initial points are close to each other. In the table 6.1 three of such implementations’ results are summarized and it is observed that the results are more or less similar for the cost values and also for the decision variables.

Table 6.1 Optimization using Nelder-Mead downhill simplex method

𝒒_𝟎^′ 𝒒_𝟎^′′ 𝒒_𝟏 𝒒_𝟐 𝒓 Cost/unit time

0.760 0.777 9.288 11.932 1.174 56.282 0.780 0.725 9.534 11.940 1.092 56.291 0.766 0.759 9.227 11.778 1.14 56.278

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Fig.6.1 Convergence path of points in the domain in Nelder-Mead Method

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So it can be concluded that as an optimization technique Nelder-Mead algorithm inhibited good convergence. And the Convergence paths of points in the domain in Nelder-Mead Method are shown in the Fig 6.1. From Fig 6.1 it can be observed that though the initial parameters are set different and therefore the simplex of the Nelder Mead graphs start form a different domain but the destinations of the paths or the convergence points are nearly same and found on the similar places of the graphs showed. So it can be certainly concluded that the solutions might reach to global optimal value.

Moreover, to validate the effectiveness of the result obtained in this approach another technique genetic algorithm approach is also used to have the optimal values of decision variables that minimize the total cost of inventory model per unit time. It has been observed that, the results are almost similar to the results of Nelder-Mead method. There are three results summarized in table 6.2 obtained using GA. Stall generation (G) and mutation rate (m) have been changed to have better view of the result. The Convergence paths of points in the domain in GA are shown in the Fig 6.2.

From the Fig.6.2 it has been observed that for the first graph when stall generation or iteration was set to 100 the convergence path was stepping decreased one. And the fine or smooth portion was not reached quickly. On the contrary in the second and third graph when the stall generation was set to 500 the fine or smoother convergent portion was attained earlier. So it can be concluded that with the increase of stall generation or iteration the chance of having optimal result can be ensured effectively.

Besides, in the third graph when mutation rate was increased that mean the variation of variable parameter can form in a large domain then the path or results can drastically fall and go toward the convergent result very soon. So increasing the mutation rate is also good for having the global optimal solution.

Table 6.2 Optimization using Genetic Algorithm

G m 𝒒_𝟎^′ 𝒒_𝟎^′′ 𝒒_𝟏 𝒒_𝟐 𝒓 Cost/unit time 100 0.01 ^{0.774 0.736 9.142 11.077 1.242 56.300}

500 0.01 ^{0.747 0.754 9.02 11.636 1.116 56.295} 500 0.05 ^{0.761 0.783 9.116 11.25 1.197 56.290}

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Fig. 6.2.Minimization of cost (Best cost and population average) with number of generations in GA

0 10 20 30 40 50 60 70 80 90 100

40 60 80 100 120 140 160 180 200 220

generation

cost

Best

Population average

Best

Population average

0 50 100 150 200 250 300 350 400 450 500

40 60 80 100 120 140 160 180 200 220

generation

cost

Best

Population average

Best

Population average

0 50 100 150 200 250 300 350 400 450 500

40 60 80 100 120 140 160 180 200 220

generation

cost

Best

Population average

Best

Population average

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To observe the effect of the decision variables on the total cost, 30 data points are plotted for each independent variable on X-axis keeping all other variables constant and the resulting average cost values are plotted on the Y-axis. For five variables five separate graphs are generated and these are shown in Fig. 6.3 to 6.7. These graphs provide a clear understanding about the relationship or effect of the variable change on the average cost. All of these graphs reveal the fact that at first average cost decreases with increase of values of the decision variables. But after reaching an optimum value the cost started to increase with the increase of the values of the decision variables. So for every decision variables there is an optimum point where the summation of ordering, holding and backordering cost is minimum. In fact these graphs resemble the ideology of standard EOQ graph.

Fig.6.3 Relationship between average cost and order quantity used for supplier 1 in order up to policy when both suppliers are available

55 56 57 58 59 60 61 62 63 64 65

0 0.5 1 1.5 2 2.5

Average Cost (AC)

q

⁰

'

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Fig.6.4 Relationship between average cost and order quantity used for supplier 2 in order up to policy when both suppliers are available

Fig.6.5 Relationship between average cost and order quantity used for supplier 1 in order up to policy when only supplier 1 is available

55 56 57 58 59 60 61 62 63 64 65

0 0.5 1 1.5 2 2.5

Average Cost (AC)

q

⁰

''

0 10 20 30 40 50 60 70 80 90 100

0 5 10 15 20 25 30

Average Cost (AC)

q

¹

70

Fig.6.6 Relationship between average cost and order quantity used for supplier 2 in order up to policy when only supplier 2 is available

Fig.6.7 Relationship between average cost and reorder point

56 56.5 57 57.5 58 58.5 59 59.5 60

0 5 10 15 20 25 30 35 40

Average Cost (AC)

q

²

56 57 58 59 60 61 62

0 0.5 1 1.5 2 2.5 3 3.5

Average Cost (AC)

r

71 6.2 Sensitivity Analysis

In order to study the effects of the model parameters, a sensitivity analysis is performed with the illustrative example described in chapter 5. In Table.6.3, level 1 is the basic level with the parameter values which are used to solve the example in Section 6.1. Level 2 and 3 represent the values of these parameters at +20 and +40%

of the basic level respectively.

The optimum value of the decision variables and the corresponding values of the objective function at different levels of the model parameters are shown in Table.6.4.

Case 1 shows the basic model results, and the all other cases show the results at level 2 and 3 of each of the eleven model parameters given in Table 6.3. Thus, a total of 23 cases are solved to study the sensitivity of model parameters as given in Table 6.4.

The following observations are made from the analysis.

Table 6.3 Experimental data set for sensitivity analysis

Case 2 and 3 reveals that if holding cost increases optimum reorder point and all the order quantity decreases. It suggests that when holding cost is high it is not justified to hold too much item as safety stock and it is beneficial to go for lower order quantity

Parameter Basic(1) Level 2 (+20%) Level 3 (+40%)

h 5 6 7

𝑘 10 12 14

𝜋 250 300 350

𝜋̂ 25 30 35

𝜆₁ 0.25 0.3 0.35

𝜇₁ 2. 5 3 3. 5

𝜆₂ 1 1.2 1.4

𝜇₂ 0.5 0.6 0.7

𝜃₁ 0.025 0.03 0.035

𝜃₂ 0.050 0.06 0.07

𝛼 5 6 7

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and order more frequently. Final average cost is highly sensitive to the increase of holding cost (11% and 22% increase at 20% and 40% increase of holding cost). Case 4 and 5 suggests that if ordering cost is high it will be sensible to order more at each ordering time. Case 6, 7, 8 and 9 suggests that when the components of backorder cost are high one should maintain a high reorder point to reduce the chance of backordering.

Table 6.4 Results of sensitivity analysis

Case Parameter Level 𝒒_𝟎^′ 𝒒_𝟎^′′ 𝒒_𝟏 𝒒_𝟐 𝒓 Cost

1 Basic Model 1 0.766 0.759 9.227 11.778 1.14 56.278

2 h 2 0.691 0.699 8.6529 11.03 0.909 62.427

3 h 3 0.648 0.659 8.102 10.408 0.753 68.138

4 𝑘 2 0.807 0.895 9.768 12.380 1.095 59.352

5 𝑘 3 0.872 0.890 10.693 13.187 0.971 62.097

6 𝜋 2 0.731 0.777 9.097 11.345 1.469 57.750

7 𝜋 3 0.747 0.769 9.275 11.629 1.682 58.978

8 𝜋̂ 2 0.766 0.760 9.343 11.689 1.142 56.292

9 𝜋̂ 3 0.767 0.763 9.350 11.578 1.161 56.309

10 𝜆1 2 0.742 0.761 8.983 11.280 1.444 57.059

11 𝜆₁ 3 0.756 0.744 8.960 11.130 1.670 57.626

12 𝜇1 2 0.749 0.758 9.124 11.596 0.788 53.705

13 𝜇₁ 3 0.755 0.749 9.010 11.490 0.568 51.873

14 𝜆2 2 0.744 0.753 8.767 11.111 1.106 52.985

15 𝜆₂ 3 0.731 0.740 8.469 10.662 1.071 50.366

16 𝜇2 2 0.746 0.761 9.784 12.114 0.963 57.311

17 𝜇2 3 0.757 0.765 10.040 12.334 0.838 58.182

18 𝜃1 2 0.759 0.754 9.257 11.907 1.16 56.557

19 𝜃1 3 0.748 0.782 9.112 11.998 1.212 56.842

20 𝜃₂ 2 0.766 0.754 9.317 11.897 1.14 56.359

21 𝜃2 3 0.757 0.759 9.380 12.085 1.15 56.442

22 𝛼 2 0.709 0.712 11.147 14.142 1.383 67.489

23 𝛼 3 0.669 0.667 13.081 16.852 1.644 79.866

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Case 10, 11, 14 and 15 suggests that when the duration of ON period is low which means suppliers are not likely to available for a long period of time the value of order up to level q + r will be high due to the increased value of r. Case 12, 13, 16 and 17 reveals that when the duration of OFF period is low which means suppliers are not likely to unavailable for a long period of time reorder point should be low because the chance of finding all the supplier in OFF state will be low. Case 18, 19, 20 and 21 suggests that when chance of supplier’s delivery quantity become smaller compared to the actual quantity ordered it should be beneficial to order less from him and more from the other supplier. Case 22 and 23 suggests that when the average rate of demand increases order quantity and reorder point both increases to meet up the high demand. Also the average cost value indicates that it is too sensitive to the increase of rate of demand (19% and 41%) increase in two cases. So undoubtedly, it is the most sensitive parameters among those used in the model.

For better understanding of the result two further sensitivity analyses on the effects of 𝜆1/μ1 and ^ℎ/𝜋̂ on decision variables and average cost is carried out. At first μ1 is kept constant at its base value of 2.5 and 𝜆1 is varied so that 𝜆1/μ1 took the values of 0.1, 0.2…..to 1. Then the problem is optimized for these new values of 𝜆¹. The new values of 𝜆1/μ1 are plotted on X- axis and the new values of average cost (AC), order quantity for supplier 1 (q1), order quantity up to the level (q1+r) and reorder point (r) are plotted on Y-axis. From the figure it is found that when the value of 𝜆1 increases which means the average duration of ON period is low, the order up to level (q1+r) increases mainly due to increase of reorder point at a fast rate. Here q1 decreases with increase of 𝜆1 as q1 is smaller at high values of 𝜆1 .Cost also increases with increase of 𝜆1 but decreases after a certain point. Increase in r and q1+r is expected because reducing the average time supplier stays ON means a larger probability of having shortages; hence one should keep the order up to levels and reorder point at higher levels to avoid the shortages.

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Fig.6.8 Effects of 𝜆1/μ1 on the decision variables and average cost

Next the value of ^𝜋̂ is kept at its base level of 25 and holding cost h is increased gradually to see the effect of these on other decision variables and final average cost.

From the figure it has been clear that the increase in holding cost resulted in decrease of reorder quantity r and order up to level (q1+r, q2+ r) and order quantity for the supplier 1 and 2 ( q1 and q2) and increase of cost. The reason is when holding cost is higher than the backorder cost, to minimize the value of total average cost model chooses low value of order quantity and reorder point and hence low order up to level quantity.

56 56.5 57 57.5 58 58.5 59 59.5

0 2 4 6 8 10 12 14

0 0.2 0.4 0.6 0.8 1 1.2

Optimal AC, q1, r, q1+r

λ¹/μ¹

q1 r q1 + r AC

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Fig.6.9 Effects ^ℎ/𝜋̂ of on the decision variables and average cost

0 50 100 150 200 250

-2 0 2 4 6 8 10 12 14 16 18

0 0.5 1 1.5 2 2.5 3 3.5

Optimal AC, q1, q2, r, q1+r and q2+r

Holding cost/backorder cost

q1 q2 r q1+r q2+r AC

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CHAPTER VII

CONCLUSIONS AND RECOMMENDATIONS