Mathematical Problems in Engineering / 2009 / Article / Tab 2

Research Article

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 ( E A C ξ…ž in weeks and Ξ· = 0.7, Ξ΄ = 0).

E A C ξ…ž 𝑛 ( 𝑄 𝑛 , 𝐴 𝑛 , πœ‹ π‘₯ 𝑛 , 𝐿 𝑛 ) E A C 𝑛 E A C ξ…ž 𝑛 / E A C 𝑛 𝑝 = 0 . 0 ∞ EVAI 𝑝 = 0 . 2

∞
0(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
0.5(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
1(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
10(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
20(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
40(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
80(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
100(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
∞ (148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133

∞
0(148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220
0.5(148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220
1(148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220
10(148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220
20(148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220
40(148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220
80(148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220
100(148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220
∞ (148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220

p = 0.4
0(148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237
0.5(148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237
1(148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237
10(148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237
20(148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237
40(148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237
80(148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237
100(148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237
∞ (148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237

p = 0.6
0(148,143, 77.463, 3)3833.2413590.772(160,154, 77.659, 3)3583.1597.6131.00212
0.5(148,143, 77.463, 3)3833.2413590.772(160,154, 77.659, 3)3583.1597.6131.00212
1(148,143, 77.463, 3)3833.2413590.772(160,154, 77.659, 3)3583.1597.6131.00212
10(148,143, 77.463, 3)3833.2413590.772(160,154., 77.659, 3)3583.1597.6131.00212
20(148,143, 77.463, 3)3833.2413590.772(160,154, 77.659, 3)3583.1597.6131.00212
40(148,143, 77.463, 3)3833.2413590.772(160,154, 77.659, 3)3583.1597.6131.00212
80(148,143, 77.463, 3)3833.2413590.772(160,154, 77.659, 3)3583.1597.6131.00212
100(148,143, 77.463, 3)3833.2413590.772(160,154, 77.659, 3)3583.1597.6131.00212
∞ (148,143, 77.463, 3)3833.2413590.772(160,154, 77.659, 3)3583.1597.6131.00212

p = 0.8
0(148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171
0.5(148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171
1(148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171
10(148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171
20(148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171
40(148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171
80(148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171
100(148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171
∞ (148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171
p = 1.0
0(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
0.5(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
1(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
10(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
20(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
40(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
80(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
100(148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133
∞ (148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133

p = 0.0
0(145,140, 77.420, 3)3731.3883430.876(151,146, 77.521, 3)3428.8302.0461.00060
0.5(145,141, 77.424, 3)3765.7603475.731(154,149, 77.562, 3)3471.9103.8211.00110
1(146,141, 77.431, 3)3781.2843494.414(155,150, 77.580, 3)3490.0194.3951.00126
10(148,143, 77.460, 3)3816.6673532.005(157,152, 77.615, 3)3527.2634.7411.00134
20(148,143, 77.464, 3)3820.2273535.434(157,152, 77.619, 3)3530.7094.7251.00134
40(148,143, 77.466, 3)3822.1253537.240(157,152, 77.620, 3)3532.5244.7161.00134
80(148,143, 77.467, 3)3823.1053538.166(157,152, 77.621, 3)3533.4564.7101.00133
100(148,143, 77.467, 3)3823.3033538.354(157,152, 77.621, 3)3533.6444.7101.00133
∞ (148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133

p = 0.2
0(145,140, 77.415, 3)3739.2223477.626(154,148, 77.559, 3)3473.4904.1361.00119
0.5(145,140, 77.421, 3)3773.2823526.203(156,151, 77.603, 3)3519.5576.6461.00189
1(146,141, 77.428, 3)3788.7243545.816(157,152, 77.621, 3)3538.3477.4691.00211
10(147,142, 77.457, 3)3824.0423583.914(159,154, 77.656, 3)3575.9737.9401.00222
20(148,143, 77.460, 3)3827.6063587.301(160,154, 77.660, 3)3579.3867.9141.00221
40(148,143, 77.462, 3)3829.5053589.078(160,154, 77.661, 3)3581.1807.8981.00221
80(148,143, 77.463, 3)3830.4873589.989(160,154, 77.662, 3)3582.1007.8891.00220
100(148,143, 77.463, 3)3830.6863590.173(160,154, 77.662, 3)3582.2867.8871.00220
∞ (148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220

p = 0.4
0(145,140, 77.414, 3)3741.9233488.279(154,149, 77.565, 3)3483.6824.5971.00132
0.5(145,140, 77.419, 3)3775.9153537.551(157,151, 77.609, 3)3530.2457.3061.00207
1(146,141, 77.427, 3)3791.3383557.216(158,152, 77.627, 3)3549.1448.0721.00227
10(147,142, 77.454, 3)3826.6413595.519(160,154, 77.663, 3)3586.8288.6911.00242
20(148,143, 77.459, 3)3830.2053598.779(160,155, 77.666, 3)3590.2378.5421.00238
40(148,143, 77.461, 3)3832.1053600.552(160,155, 77.668, 3)3592.0278.5251.00237
80(148,143, 77.462, 3)3833.0873601.460(160,155, 77.669, 3)3592.9458.5151.00237
100(148,143, 77.462, 3)3833.2863601.643(160,155, 77.669, 3)3593.1308.5131.00237
∞ (148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237

p = 0.6
0(145,140, 77.415, 3)3740.9863477.898(153,148, 77.555, 3)3473.9823.9161.00113
0.5(145,140, 77.421, 3)3775.0373526.154(156,151, 77.599, 3)3519.7696.3851.00181
1(146,141, 77.427, 3)3790.4763545.698(157,152, 77.617, 3)3538.4977.2001.00203
10(147,142, 77.456, 3)3825.7923583.761(159,154, 77.653, 3)3576.0907.6711.00215
20(148,143, 77.460, 3)3829.3563587.152(159,154, 77.656, 3)3579.5067.6461.00214
40(148,143, 77.461, 3)3831.2563588.932(159,154, 77.658, 3)3581.3027.6301.00213
80(148,143, 77.462, 3)3832.2373589.844(160,154, 77.658, 3)3582.2227.6221.00213
100(148,143, 77.463, 3)3832.4363590.028(160,154, 77.659, 3)3582.4097.6201.00213
∞ (148,143, 77.463, 3)3833.2413590.772(160,154, 77.659, 3)3583.1597.6121.00212
p = 0.8
0(145,140, 77.417, 3)3737.2853456.812(152,147, 77.539, 3)3453.8492.9631.00086
0.5(145,140, 77.422, 3)3771.4723503.469(155,150, 77.582, 3)3498.3465.1231.00146
1(146,141, 77.430, 3)3786.9483522.514(156,151, 77.599, 3)3516.7835.7311.00163
10(147,143, 77.458, 3)3822.2913560.458(158,153, 77.635, 3)3554.2166.2431.00176
20(148,143, 77.461, 3)3825.8543563.868(158,153, 77.638, 3)3557.6466.2221.00175
40(148,143, 77.463, 3)3827.7533565.660(158,153, 77.640, 3)3559.4516.2091.00174
80(148,143, 77.464, 3)3828.7343566.579(158,153, 77.641, 3)3560.3776.2021.00174
100(148,143, 77.465, 3)3828.9333566.666(158,153, 77.641, 3)3560.5646.1021.00171
∞ (148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171

p = 1.0
0(145,140, 77.420, 3)3731.3883430.876(151,146, 77.521, 3)3428.8312.0461.00060
0.5(145,141, 77.424, 3)3765.7603475.731(154,149, 77.562, 3)3471.9103.8211.00110
1(146,141, 77.431, 3)3781.2843494.414(155,150, 77.580, 3)3490.0204.3941.00126
10(148,143, 77.460, 3)3816.6673532.005(157,152, 77.615, 3)3527.2644.7411.00134
20(148,143, 77.464, 3)3820.2273535.435(157,152, 77.619, 3)3530.7094.7261.00134
40(148,143, 77.466, 3)3822.1253537.240(157,152, 77.620, 3)3532.5244.7161.00134
80(148,143, 77.467, 3)3823.1053538.167(157,152, 77.621, 3)3533.4564.7111.00133
100(148,143, 77.467, 3)3823.3033538.354(157,152, 77.621, 3)3533.6454.7091.00133
∞ (148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133

p = 0.0
0(142,137, 77.367, 3)3630.3183319.556(142,137, 77.364, 4)3306.32913.2271.00400
0.5(143,138, 77.379, 3)3705.6593411.048(150,145, 77.501, 3)3408.0193.0291.00089
1(144,139, 77.394, 3)3737.6913449.035(152,147, 77.536, 3)3444.9404.0951.00119
10(147,142, 77.452, 3)3809.2043524.967(157,151, 77.608, 3)3520.1044.8621.00138
20(148,143, 77.460, 3)3816.3423531.755(157,152, 77.615, 3)3527.0094.7461.00135
40(148,143, 77.464, 3)3820.1413535.369(157,152, 77.618, 3)3530.6424.7271.00134
80(148,143, 77.466, 3)3822.1033537.223(157,152, 77.620, 3)3532.5074.7161.00134
100(148,143, 77.466, 3)3822.5003537.598(157,152, 77.621, 3)3532.8844.7141.00133
∞ (148,143, 77.468, 3)3824.1073539.109(157,152, 77.622, 3)3534.4054.7051.00133

p = 0.2
0(142,137, 77.362, 3)3638.7033361.079(144,139, 77.405, 4)3354.2456.8341.00204
0.5(142,138, 77.374, 3)3713.3243460.282(152,147, 77.541, 3)3454.6685.6131.00162
1(143,139, 77.391, 3)3745.1873500.043(155,149, 77.577, 3)3492.9717.0721.00202
10(147,142, 77.448, 3)3816.5713577.010(159,154, 77.650, 3)3568.8948.1161.00227
20(147,142, 77.455, 3)3823.7153583.795(159,154, 77.656, 3)3575.7338.0621.00225
40(148,143, 77.460, 3)3827.5193587.240(160,154, 77.660, 3)3579.3247.9161.00221
80(148,143, 77.462, 3)3829.4833589.063(160,154, 77.661, 3)3581.1647.8981.00221
100(148,143, 77.462, 3)3829.8813589.431(160,154, 77.662, 3)3581.5367.8951.00220
∞ (148,143, 77.464, 3)3831.4903590.915(160,154, 77.663, 3)3583.0357.8801.00220

p = 0.4
0(142,137, 77.362, 3)3641.5203370.810(145,140, 77.411, 4)3365.7245.0851.00151
0.5(142,138, 77.374, 3)3715.9883471.297(153,148, 77.547, 3)3465.2006.0971.00176
1(143,139, 77.390, 3)3747.8133511.377(155,150, 77.584, 3)3503.7247.6531.00218
10(147,142, 77.447, 3)3819.1673588.520(159,154, 77.657, 3)3579.7628.7581.00245
20(147,142, 77.454, 3)3826.3133595.290(160,154, 77.663, 3)3586.5918.6991.00243
40(148,143, 77.459, 3)3830.1183598.719(160,155, 77.666, 3)3590.1748.5451.00238
80(148,143, 77.461, 3)3832.0833600.536(160,155, 77.668, 3)3592.0118.5261.00237
100(148,143, 77.461, 3)3832.4813600.903(160,155, 77.668, 3)3592.3828.5221.00237
∞ (148,143, 77.463, 3)3834.0913602.383(160,155, 77.670, 3)3593.8788.5051.00237

p = 0.6
0(142,137, 77.362, 3)3640.4733361.860(144,139, 77.401, 4)3355.7856.0751.00181
0.5(142,138, 77.375, 3)3715.0833460.331(152,147, 77.537, 3)3454.9705.3611.00155
1(143,139, 77.390, 3)3746.9403499.948(154,149, 77.574, 3)3493.1476.8021.00195
10(147,142, 77.447, 3)3818.3203576.848(159,153, 77.646, 3)3569.0037.8451.00220
20(147,142, 77.454, 3)3825.4653583.641(159,154, 77.652, 3)3575.8487.7921.00218
40(148,143, 77.459, 3)3829.2693587.091(159,154, 77.656, 3)3579.4437.6481.00214
80(148,143, 77.461, 3)3831.2333588.916(159,154, 77.658, 3)3581.2857.6311.00213
100(148,143, 77.462, 3)3831.6313589.285(159,154, 77.658, 3)3581.6587.6271.00213
( 𝑄 𝑛 , 𝐴 𝑛 , πœ‹ π‘₯ 𝑛 , 𝐿 𝑛 ) (148,143, 77.463, 3)3833.2403590.772(160,154, 77.659, 3)3583.1597.6131.00212

p = 0.8
0(142,137, 77.364, 3)3636.5343343.016(143,138, 77.384, 4)3334.1118.9051.00267
0.5(143,138, 77.377, 3)3711.4563438.123(151,146, 77.520, 3)3433.9704.1531.00121
1(143,139, 77.391, 3)3743.3883477.029(153,148, 77.556, 3)3471.5575.4721.00158
10(147,142, 77.449, 3)3814.8243553.487(158,152, 77.628, 3)3547.0956.3921.00180
20(147,143, 77.457, 3)3821.9653560.217(158,153, 77.635, 3)3553.9686.2491.00176
40(148,143, 77.461, 3)3825.7673563.804(158,153, 77.638, 3)3557.5816.2231.00175
80(148,143, 77.463, 3)3827.7313565.644(158,153, 77.640, 3)3559.4346.2091.00174
100(148,143, 77.463, 3)3828.1283566.015(158,153, 77.640, 3)3559.8086.2071.00174
𝐹 βˆ— (148,143, 77.466, 3)3829.7373567.416(159,153, 77.642, 3)3561.3196.0961.00171

p = 1.0
0(142,137, 77.367, 3)3630.3183319.556(142,137, 77.364, 4)3306.32913.2271.00400
0.5(143,138, 77.379, 3)3705.6593411.049(150,145, 77.501, 3)3408.0193.0291.00089
1(144,139, 77.394, 3)3737.6913449.036(152,147, 77.536, 3)3444.9404.0951.00119
10(147,142, 77.452, 3)3809.2043524.967(157,151, 77.608, 3)3520.1054.8621.00138
20(148,143, 77.460, 3)3816.3423531.755(157,152, 77.615, 3)3527.0094.7461.00135
40(148,143, 77.464, 3)3820.1413535.369(157,152, 77.618, 3)3530.6424.7271.00134
80(148,143, 77.466, 3)3822.1023537.223(157,152, 77.620, 3)3532.5074.7161.00134
100(148,143, 77.466, 3)3822.5003537.598(157,152, 77.621, 3)3532.8844.7141.00133
E A C 𝑛 ( 𝑄 𝑛 , 𝐴 𝑛 , πœ‹ π‘₯ 𝑛 , 𝐿 𝑛 ) (148,143, 77.468, 3)3824.1073539.110(157,152, 77.622, 3)3534.4054.7051.00133

Note: we obtain the optimal ( 𝑄 βˆ— , 𝐴 βˆ— , πœ‹ βˆ— π‘₯ , 𝐿 βˆ— ) by the standard procedure, E A C ( 𝑄 βˆ— , 𝐴 βˆ— , πœ‹ βˆ— π‘₯ , 𝐿 βˆ— ) is mixture of normal distribution, and incur an expected annual cost ( 𝑄 βˆ— , 𝐴 βˆ— , πœ‹ βˆ— π‘₯ , 𝐿 βˆ— ) . ( 𝑄 𝑛 , 𝐴 𝑛 , πœ‹ π‘₯ 𝑛 , 𝐿 𝑛 ) 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; E A C 𝑛 ( 𝑄 βˆ— , 𝐴 βˆ— , πœ‹ βˆ— π‘₯ , 𝐿 βˆ— ) is the minimum total expected annual cost. We use E A C ξ…ž = E A C ( 𝑄 βˆ— , 𝐴 βˆ— , πœ‹ βˆ— π‘₯ , 𝐿 βˆ— ) instead of the optimal E A C ξ…ž 𝑛 = E A C 𝑛 ( 𝑄 βˆ— , 𝐴 βˆ— , πœ‹ βˆ— π‘₯ , 𝐿 βˆ— ) , for E A C 𝑛 = E A C 𝑛 ( 𝑄 𝑛 , 𝐴 𝑛 , πœ‹ π‘₯ 𝑛 , 𝐿 𝑛 ) . In other word, 𝑝 , 𝛿 = 0 . 5 , 1 . 0 and πœ€ ↑ .

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