Optimal Inventory Policy Involving Ordering Cost Reduction, Back-Order Discounts, and Variable Lead Time Demand by Minimax Criterion
Table 2
Summary of the optimal solution procedure ( in weeks and η = 0.7, δ = 0).
EVAI
0
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
0.5
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
1
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
10
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
20
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
40
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
80
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
100
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
0
(148,143, 77.464, 3)
3831.490
3590.915
(160,154, 77.663, 3)
3583.035
7.880
1.00220
0.5
(148,143, 77.464, 3)
3831.490
3590.915
(160,154, 77.663, 3)
3583.035
7.880
1.00220
1
(148,143, 77.464, 3)
3831.490
3590.915
(160,154, 77.663, 3)
3583.035
7.880
1.00220
10
(148,143, 77.464, 3)
3831.490
3590.915
(160,154, 77.663, 3)
3583.035
7.880
1.00220
20
(148,143, 77.464, 3)
3831.490
3590.915
(160,154, 77.663, 3)
3583.035
7.880
1.00220
40
(148,143, 77.464, 3)
3831.490
3590.915
(160,154, 77.663, 3)
3583.035
7.880
1.00220
80
(148,143, 77.464, 3)
3831.490
3590.915
(160,154, 77.663, 3)
3583.035
7.880
1.00220
100
(148,143, 77.464, 3)
3831.490
3590.915
(160,154, 77.663, 3)
3583.035
7.880
1.00220
(148,143, 77.464, 3)
3831.490
3590.915
(160,154, 77.663, 3)
3583.035
7.880
1.00220
p = 0.4
0
(148,143, 77.463, 3)
3834.091
3602.383
(160,155, 77.670, 3)
3593.878
8.505
1.00237
0.5
(148,143, 77.463, 3)
3834.091
3602.383
(160,155, 77.670, 3)
3593.878
8.505
1.00237
1
(148,143, 77.463, 3)
3834.091
3602.383
(160,155, 77.670, 3)
3593.878
8.505
1.00237
10
(148,143, 77.463, 3)
3834.091
3602.383
(160,155, 77.670, 3)
3593.878
8.505
1.00237
20
(148,143, 77.463, 3)
3834.091
3602.383
(160,155, 77.670, 3)
3593.878
8.505
1.00237
40
(148,143, 77.463, 3)
3834.091
3602.383
(160,155, 77.670, 3)
3593.878
8.505
1.00237
80
(148,143, 77.463, 3)
3834.091
3602.383
(160,155, 77.670, 3)
3593.878
8.505
1.00237
100
(148,143, 77.463, 3)
3834.091
3602.383
(160,155, 77.670, 3)
3593.878
8.505
1.00237
(148,143, 77.463, 3)
3834.091
3602.383
(160,155, 77.670, 3)
3593.878
8.505
1.00237
p = 0.6
0
(148,143, 77.463, 3)
3833.241
3590.772
(160,154, 77.659, 3)
3583.159
7.613
1.00212
0.5
(148,143, 77.463, 3)
3833.241
3590.772
(160,154, 77.659, 3)
3583.159
7.613
1.00212
1
(148,143, 77.463, 3)
3833.241
3590.772
(160,154, 77.659, 3)
3583.159
7.613
1.00212
10
(148,143, 77.463, 3)
3833.241
3590.772
(160,154., 77.659, 3)
3583.159
7.613
1.00212
20
(148,143, 77.463, 3)
3833.241
3590.772
(160,154, 77.659, 3)
3583.159
7.613
1.00212
40
(148,143, 77.463, 3)
3833.241
3590.772
(160,154, 77.659, 3)
3583.159
7.613
1.00212
80
(148,143, 77.463, 3)
3833.241
3590.772
(160,154, 77.659, 3)
3583.159
7.613
1.00212
100
(148,143, 77.463, 3)
3833.241
3590.772
(160,154, 77.659, 3)
3583.159
7.613
1.00212
(148,143, 77.463, 3)
3833.241
3590.772
(160,154, 77.659, 3)
3583.159
7.613
1.00212
p = 0.8
0
(148,143, 77.466, 3)
3829.737
3567.416
(159,153, 77.642, 3)
3561.319
6.096
1.00171
0.5
(148,143, 77.466, 3)
3829.737
3567.416
(159,153, 77.642, 3)
3561.319
6.096
1.00171
1
(148,143, 77.466, 3)
3829.737
3567.416
(159,153, 77.642, 3)
3561.319
6.096
1.00171
10
(148,143, 77.466, 3)
3829.737
3567.416
(159,153, 77.642, 3)
3561.319
6.096
1.00171
20
(148,143, 77.466, 3)
3829.737
3567.416
(159,153, 77.642, 3)
3561.319
6.096
1.00171
40
(148,143, 77.466, 3)
3829.737
3567.416
(159,153, 77.642, 3)
3561.319
6.096
1.00171
80
(148,143, 77.466, 3)
3829.737
3567.416
(159,153, 77.642, 3)
3561.319
6.096
1.00171
100
(148,143, 77.466, 3)
3829.737
3567.416
(159,153, 77.642, 3)
3561.319
6.096
1.00171
(148,143, 77.466, 3)
3829.737
3567.416
(159,153, 77.642, 3)
3561.319
6.096
1.00171
p = 1.0
0
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
0.5
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
1
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
10
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
20
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
40
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
80
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
100
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
p = 0.0
0
(145,140, 77.420, 3)
3731.388
3430.876
(151,146, 77.521, 3)
3428.830
2.046
1.00060
0.5
(145,141, 77.424, 3)
3765.760
3475.731
(154,149, 77.562, 3)
3471.910
3.821
1.00110
1
(146,141, 77.431, 3)
3781.284
3494.414
(155,150, 77.580, 3)
3490.019
4.395
1.00126
10
(148,143, 77.460, 3)
3816.667
3532.005
(157,152, 77.615, 3)
3527.263
4.741
1.00134
20
(148,143, 77.464, 3)
3820.227
3535.434
(157,152, 77.619, 3)
3530.709
4.725
1.00134
40
(148,143, 77.466, 3)
3822.125
3537.240
(157,152, 77.620, 3)
3532.524
4.716
1.00134
80
(148,143, 77.467, 3)
3823.105
3538.166
(157,152, 77.621, 3)
3533.456
4.710
1.00133
100
(148,143, 77.467, 3)
3823.303
3538.354
(157,152, 77.621, 3)
3533.644
4.710
1.00133
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
p = 0.2
0
(145,140, 77.415, 3)
3739.222
3477.626
(154,148, 77.559, 3)
3473.490
4.136
1.00119
0.5
(145,140, 77.421, 3)
3773.282
3526.203
(156,151, 77.603, 3)
3519.557
6.646
1.00189
1
(146,141, 77.428, 3)
3788.724
3545.816
(157,152, 77.621, 3)
3538.347
7.469
1.00211
10
(147,142, 77.457, 3)
3824.042
3583.914
(159,154, 77.656, 3)
3575.973
7.940
1.00222
20
(148,143, 77.460, 3)
3827.606
3587.301
(160,154, 77.660, 3)
3579.386
7.914
1.00221
40
(148,143, 77.462, 3)
3829.505
3589.078
(160,154, 77.661, 3)
3581.180
7.898
1.00221
80
(148,143, 77.463, 3)
3830.487
3589.989
(160,154, 77.662, 3)
3582.100
7.889
1.00220
100
(148,143, 77.463, 3)
3830.686
3590.173
(160,154, 77.662, 3)
3582.286
7.887
1.00220
(148,143, 77.464, 3)
3831.490
3590.915
(160,154, 77.663, 3)
3583.035
7.880
1.00220
p = 0.4
0
(145,140, 77.414, 3)
3741.923
3488.279
(154,149, 77.565, 3)
3483.682
4.597
1.00132
0.5
(145,140, 77.419, 3)
3775.915
3537.551
(157,151, 77.609, 3)
3530.245
7.306
1.00207
1
(146,141, 77.427, 3)
3791.338
3557.216
(158,152, 77.627, 3)
3549.144
8.072
1.00227
10
(147,142, 77.454, 3)
3826.641
3595.519
(160,154, 77.663, 3)
3586.828
8.691
1.00242
20
(148,143, 77.459, 3)
3830.205
3598.779
(160,155, 77.666, 3)
3590.237
8.542
1.00238
40
(148,143, 77.461, 3)
3832.105
3600.552
(160,155, 77.668, 3)
3592.027
8.525
1.00237
80
(148,143, 77.462, 3)
3833.087
3601.460
(160,155, 77.669, 3)
3592.945
8.515
1.00237
100
(148,143, 77.462, 3)
3833.286
3601.643
(160,155, 77.669, 3)
3593.130
8.513
1.00237
(148,143, 77.463, 3)
3834.091
3602.383
(160,155, 77.670, 3)
3593.878
8.505
1.00237
p = 0.6
0
(145,140, 77.415, 3)
3740.986
3477.898
(153,148, 77.555, 3)
3473.982
3.916
1.00113
0.5
(145,140, 77.421, 3)
3775.037
3526.154
(156,151, 77.599, 3)
3519.769
6.385
1.00181
1
(146,141, 77.427, 3)
3790.476
3545.698
(157,152, 77.617, 3)
3538.497
7.200
1.00203
10
(147,142, 77.456, 3)
3825.792
3583.761
(159,154, 77.653, 3)
3576.090
7.671
1.00215
20
(148,143, 77.460, 3)
3829.356
3587.152
(159,154, 77.656, 3)
3579.506
7.646
1.00214
40
(148,143, 77.461, 3)
3831.256
3588.932
(159,154, 77.658, 3)
3581.302
7.630
1.00213
80
(148,143, 77.462, 3)
3832.237
3589.844
(160,154, 77.658, 3)
3582.222
7.622
1.00213
100
(148,143, 77.463, 3)
3832.436
3590.028
(160,154, 77.659, 3)
3582.409
7.620
1.00213
(148,143, 77.463, 3)
3833.241
3590.772
(160,154, 77.659, 3)
3583.159
7.612
1.00212
p = 0.8
0
(145,140, 77.417, 3)
3737.285
3456.812
(152,147, 77.539, 3)
3453.849
2.963
1.00086
0.5
(145,140, 77.422, 3)
3771.472
3503.469
(155,150, 77.582, 3)
3498.346
5.123
1.00146
1
(146,141, 77.430, 3)
3786.948
3522.514
(156,151, 77.599, 3)
3516.783
5.731
1.00163
10
(147,143, 77.458, 3)
3822.291
3560.458
(158,153, 77.635, 3)
3554.216
6.243
1.00176
20
(148,143, 77.461, 3)
3825.854
3563.868
(158,153, 77.638, 3)
3557.646
6.222
1.00175
40
(148,143, 77.463, 3)
3827.753
3565.660
(158,153, 77.640, 3)
3559.451
6.209
1.00174
80
(148,143, 77.464, 3)
3828.734
3566.579
(158,153, 77.641, 3)
3560.377
6.202
1.00174
100
(148,143, 77.465, 3)
3828.933
3566.666
(158,153, 77.641, 3)
3560.564
6.102
1.00171
(148,143, 77.466, 3)
3829.737
3567.416
(159,153, 77.642, 3)
3561.319
6.096
1.00171
p = 1.0
0
(145,140, 77.420, 3)
3731.388
3430.876
(151,146, 77.521, 3)
3428.831
2.046
1.00060
0.5
(145,141, 77.424, 3)
3765.760
3475.731
(154,149, 77.562, 3)
3471.910
3.821
1.00110
1
(146,141, 77.431, 3)
3781.284
3494.414
(155,150, 77.580, 3)
3490.020
4.394
1.00126
10
(148,143, 77.460, 3)
3816.667
3532.005
(157,152, 77.615, 3)
3527.264
4.741
1.00134
20
(148,143, 77.464, 3)
3820.227
3535.435
(157,152, 77.619, 3)
3530.709
4.726
1.00134
40
(148,143, 77.466, 3)
3822.125
3537.240
(157,152, 77.620, 3)
3532.524
4.716
1.00134
80
(148,143, 77.467, 3)
3823.105
3538.167
(157,152, 77.621, 3)
3533.456
4.711
1.00133
100
(148,143, 77.467, 3)
3823.303
3538.354
(157,152, 77.621, 3)
3533.645
4.709
1.00133
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
p = 0.0
0
(142,137, 77.367, 3)
3630.318
3319.556
(142,137, 77.364, 4)
3306.329
13.227
1.00400
0.5
(143,138, 77.379, 3)
3705.659
3411.048
(150,145, 77.501, 3)
3408.019
3.029
1.00089
1
(144,139, 77.394, 3)
3737.691
3449.035
(152,147, 77.536, 3)
3444.940
4.095
1.00119
10
(147,142, 77.452, 3)
3809.204
3524.967
(157,151, 77.608, 3)
3520.104
4.862
1.00138
20
(148,143, 77.460, 3)
3816.342
3531.755
(157,152, 77.615, 3)
3527.009
4.746
1.00135
40
(148,143, 77.464, 3)
3820.141
3535.369
(157,152, 77.618, 3)
3530.642
4.727
1.00134
80
(148,143, 77.466, 3)
3822.103
3537.223
(157,152, 77.620, 3)
3532.507
4.716
1.00134
100
(148,143, 77.466, 3)
3822.500
3537.598
(157,152, 77.621, 3)
3532.884
4.714
1.00133
(148,143, 77.468, 3)
3824.107
3539.109
(157,152, 77.622, 3)
3534.405
4.705
1.00133
p = 0.2
0
(142,137, 77.362, 3)
3638.703
3361.079
(144,139, 77.405, 4)
3354.245
6.834
1.00204
0.5
(142,138, 77.374, 3)
3713.324
3460.282
(152,147, 77.541, 3)
3454.668
5.613
1.00162
1
(143,139, 77.391, 3)
3745.187
3500.043
(155,149, 77.577, 3)
3492.971
7.072
1.00202
10
(147,142, 77.448, 3)
3816.571
3577.010
(159,154, 77.650, 3)
3568.894
8.116
1.00227
20
(147,142, 77.455, 3)
3823.715
3583.795
(159,154, 77.656, 3)
3575.733
8.062
1.00225
40
(148,143, 77.460, 3)
3827.519
3587.240
(160,154, 77.660, 3)
3579.324
7.916
1.00221
80
(148,143, 77.462, 3)
3829.483
3589.063
(160,154, 77.661, 3)
3581.164
7.898
1.00221
100
(148,143, 77.462, 3)
3829.881
3589.431
(160,154, 77.662, 3)
3581.536
7.895
1.00220
(148,143, 77.464, 3)
3831.490
3590.915
(160,154, 77.663, 3)
3583.035
7.880
1.00220
p = 0.4
0
(142,137, 77.362, 3)
3641.520
3370.810
(145,140, 77.411, 4)
3365.724
5.085
1.00151
0.5
(142,138, 77.374, 3)
3715.988
3471.297
(153,148, 77.547, 3)
3465.200
6.097
1.00176
1
(143,139, 77.390, 3)
3747.813
3511.377
(155,150, 77.584, 3)
3503.724
7.653
1.00218
10
(147,142, 77.447, 3)
3819.167
3588.520
(159,154, 77.657, 3)
3579.762
8.758
1.00245
20
(147,142, 77.454, 3)
3826.313
3595.290
(160,154, 77.663, 3)
3586.591
8.699
1.00243
40
(148,143, 77.459, 3)
3830.118
3598.719
(160,155, 77.666, 3)
3590.174
8.545
1.00238
80
(148,143, 77.461, 3)
3832.083
3600.536
(160,155, 77.668, 3)
3592.011
8.526
1.00237
100
(148,143, 77.461, 3)
3832.481
3600.903
(160,155, 77.668, 3)
3592.382
8.522
1.00237
(148,143, 77.463, 3)
3834.091
3602.383
(160,155, 77.670, 3)
3593.878
8.505
1.00237
p = 0.6
0
(142,137, 77.362, 3)
3640.473
3361.860
(144,139, 77.401, 4)
3355.785
6.075
1.00181
0.5
(142,138, 77.375, 3)
3715.083
3460.331
(152,147, 77.537, 3)
3454.970
5.361
1.00155
1
(143,139, 77.390, 3)
3746.940
3499.948
(154,149, 77.574, 3)
3493.147
6.802
1.00195
10
(147,142, 77.447, 3)
3818.320
3576.848
(159,153, 77.646, 3)
3569.003
7.845
1.00220
20
(147,142, 77.454, 3)
3825.465
3583.641
(159,154, 77.652, 3)
3575.848
7.792
1.00218
40
(148,143, 77.459, 3)
3829.269
3587.091
(159,154, 77.656, 3)
3579.443
7.648
1.00214
80
(148,143, 77.461, 3)
3831.233
3588.916
(159,154, 77.658, 3)
3581.285
7.631
1.00213
100
(148,143, 77.462, 3)
3831.631
3589.285
(159,154, 77.658, 3)
3581.658
7.627
1.00213
(148,143, 77.463, 3)
3833.240
3590.772
(160,154, 77.659, 3)
3583.159
7.613
1.00212
p = 0.8
0
(142,137, 77.364, 3)
3636.534
3343.016
(143,138, 77.384, 4)
3334.111
8.905
1.00267
0.5
(143,138, 77.377, 3)
3711.456
3438.123
(151,146, 77.520, 3)
3433.970
4.153
1.00121
1
(143,139, 77.391, 3)
3743.388
3477.029
(153,148, 77.556, 3)
3471.557
5.472
1.00158
10
(147,142, 77.449, 3)
3814.824
3553.487
(158,152, 77.628, 3)
3547.095
6.392
1.00180
20
(147,143, 77.457, 3)
3821.965
3560.217
(158,153, 77.635, 3)
3553.968
6.249
1.00176
40
(148,143, 77.461, 3)
3825.767
3563.804
(158,153, 77.638, 3)
3557.581
6.223
1.00175
80
(148,143, 77.463, 3)
3827.731
3565.644
(158,153, 77.640, 3)
3559.434
6.209
1.00174
100
(148,143, 77.463, 3)
3828.128
3566.015
(158,153, 77.640, 3)
3559.808
6.207
1.00174
(148,143, 77.466, 3)
3829.737
3567.416
(159,153, 77.642, 3)
3561.319
6.096
1.00171
p = 1.0
0
(142,137, 77.367, 3)
3630.318
3319.556
(142,137, 77.364, 4)
3306.329
13.227
1.00400
0.5
(143,138, 77.379, 3)
3705.659
3411.049
(150,145, 77.501, 3)
3408.019
3.029
1.00089
1
(144,139, 77.394, 3)
3737.691
3449.036
(152,147, 77.536, 3)
3444.940
4.095
1.00119
10
(147,142, 77.452, 3)
3809.204
3524.967
(157,151, 77.608, 3)
3520.105
4.862
1.00138
20
(148,143, 77.460, 3)
3816.342
3531.755
(157,152, 77.615, 3)
3527.009
4.746
1.00135
40
(148,143, 77.464, 3)
3820.141
3535.369
(157,152, 77.618, 3)
3530.642
4.727
1.00134
80
(148,143, 77.466, 3)
3822.102
3537.223
(157,152, 77.620, 3)
3532.507
4.716
1.00134
100
(148,143, 77.466, 3)
3822.500
3537.598
(157,152, 77.621, 3)
3532.884
4.714
1.00133
(148,143, 77.468, 3)
3824.107
3539.110
(157,152, 77.622, 3)
3534.405
4.705
1.00133
Note: we obtain the optimal by the standard procedure, is mixture of normal distribution, and incur an expected annual cost . stands for the optimal order quantity, the ordering cost, the back-order price discount, and the optimal lead time, respectively, that the demand in the lead time is mixture of free distribution; is the minimum total expected annual cost. We use instead of the optimal for . In other word, , and .
