Numerical example

Crisp environment

The input data of different parameters for Case 1 and Case 2 are shown in Table 1, and the expected optimum value of the total profit is given in Table 2.

Table 1 Input data of different parameters for Case 1 and Case 2

Parameters c s c m c r c r 1 h s h m T s p m n r ρ id s id m id r I e A s A m A r m i p D C D R
Case 1 7 14 25 35 0.15 0.8 10 16 4 4 0.4 1 2 3 1 25 40 45 20 16 55 100
Case 2 10 12 26 25 0.17 0.9 12 25 17 5 0.3 1.5 2.5 3.5 1 30 28 52 24 45 120 130

Table 2 Optimum results for objective functions and other parameters

Parameter Case 1 Case 2
ATP∗ 1,039.45 1,108.93
p*{*}_s 70.49
79.23
T 2.85 1.59
APS∗ 246.51 253.56
APM∗ 468.25 463.18
APR∗ 321.43 390.47

Sensitivity analysis

The major contribution of the supply chain is mainly in the inclusion of the manufacturer. We consider product reworking of defective items which are reworked just after regular production, with a different holding cost for good and defective items in the three-layer supply chain. An integrated production-inventory model is presented for the supplier, manufacturer, and retailer supply chain under conditionally permissible delay in payments which has developed in both crisp and uncertain environments (Table 3).

Table 3 Optimum value changes due to parametric changes

  Parameter name
  Parametric value
 T^{*}  P^{*}_s   ATP^{*}
  Case 1
  Case 2
  Case 1
  Case 2
  Case 1
  Case 2
c s 1.5 1.92 1.86 79.18 78.31 10.89 1,529.43
  2.5 1.61 1.76 46.76 78.31 10.14 1,526.25
  3.5 1.12 1.45 18.18 78.31 10.06 1,522.47
c m 1.5 1.82 1.82 56.14 78.31 10.56 1,528.36
  2.5 1.75 1.65 69.76 78.31 10.46 1,525.64
  3.5 1.10 1.33 64.15 78.31 10.15 1,520.25
c r 1.5 1.75 1.96 65.18 78.31 10.56 1,528.19
  2.5 1.52 1.69 62.17 78.31 10.56 1,522.58
  3.5 1.06 1.14 58.14 78.31 10.56 1,520.43
h s 1.5 1.86 1.76 69.18 78.49 10.56 1,536.45
  2.5 1.64 1.54 64.76 78.31 10.14 1,533.25
  3.5 1.25 1.19 60.19 78.04 10.56 1,520.43
h m 1.5 1.09 1.76 88.76 78.74 10.56 1,530.52
  2.5 1.21 1.52 76.78 78.54 10.18 1,525.19
  3.5 1.13 1.01 55.45 78.17 10.56 1,523.57
h r 1.5 1.75 1.89 69.76 78.31 10.19 1,520.43
  2.5 1.25 1.15 54.76 78.01 10.56 1,518.29
  3.5 1.12 1.06 48.16 78.00 10.04 1,518.21

Uncertain environment

Input data of different crisp parameters for Case 1 and Case 2 are given in Table 1; the remaining uncertain parameters are depicted in Table 4. Optimal values of objective and decision variables are given in Table 5 and graphically represented in Figure 5.

Table 4 Input data of different zigzag parameters for Case 1 and Case 2

Parameters in uncertain environments \widetilde {id}_s \widetilde {id}_m \widetilde {id}_r \widetilde {id}_e
Case 1 L(0.8,1.2,1.4) L(1.5,2.0,2.5) L(1.4,2,2.3) L(0.04,0.06,0.08)
Case 2 L(1.4,1.8,2.4) L(2,2.3,2.9) L(1.4,2.1,2.5) L(0.06,0.08,0.1)


Table 5 Optimal values of objective and decision variables

Parameter Case 1 Case 2
ATP∗  1,056.43
 1,520.43
p*{*}_s 69.76
 78.31
T 1.75 1.76
APS∗ 248.71 276.76
APM∗ 463.43 467.35
APR∗ 324.43 393.73

Figure 5 Average profit versus supply rate of suppliers in Case 1 and Case 2.