1 Introduction
2 Literature review
3 Model description and analysis
3.1 Inventory policy
\(\lambda \) | Poisson demand rate of the retailer. |
\(\mu \) | Exponential rate of on period length, expected length \(=\mu ^{-1}\). |
\(\nu \) | Exponential rate of off period length, expected length \(=\nu ^{-1}\). |
\(S_1\) | Order-up-to level for regular replenishment. |
\(s_1\) | Reorder level for regular replenishment. |
\(S_2\) | Order-up-to level for emergency replenishment. |
\(s_2\) | Reorder level for emergency replenishment. |
\(L_o\) | Regular replenishment lead time. |
b | Unit lost sales cost of the buyer. |
h | Unit holding cost of the buyer. |
\(K_o\) | Fixed cost of regular ordering. |
\(K_e\) | Fixed cost of emergency ordering. |
\(c_o\) | Unit purchasing cost of a regular order. |
\(c_e\) | Unit purchasing cost of an emergency order. |
\(\mathcal {P}_{(i,k)}\) | The fraction of time in the long run that the inventory system is in state (i, k), |
\(i=\{0,1,2,\ldots ,\max (S_1,S_2)\}\) and \(k=\{F,N\}\) | |
(F denotes off mode and N denotes on mode). | |
\(\pi _i\) | Steady-state probability that the inventory level of the retailer is i, |
\(\pi _i=\mathcal {P}_{(i,N)}+\mathcal {P}_{(i,F)}\), \(i=\{0,1,2,\ldots ,\max (S_1,S_2)\}\). | |
\(\mathbb {E}[RO]\) | Expected regular ordering cost. |
\(\mathbb {E}[EO]\) | Expected emergency ordering cost. |
\(\mathbb {E}[OH]\) | Expected on-hand inventory. |
\(\mathbb {E}[LS]\) | Expected number of lost sales. |
\(\mathbb {E}[C]\) | Expected total cost per unit time. |
3.2 CTMC model of \((s_1,S_1,s_2,S_2)\) policy with zero lead time
Type | From | To | Rate | Range |
---|---|---|---|---|
1 | (i, F) | \((S_1,N)\) | \(\nu\) | \(i=0,1,\ldots ,s_1\) |
2 | (i, F) | \((i-1,F)\) | \(\lambda\) | \(i=1,2,\ldots ,S_1\) |
3 | (i, N) | \((i-1,N)\) | \(\lambda\) | \(i=s_1+2,\ldots ,S_1\) |
4 | (i, F) | (i, N) | \(\nu\) | \(i=s_1+1,\ldots ,S_1\) |
5 | (i, N) | (i, F) | \(\mu\) | \(i=s_2+1,\ldots ,S_1\) |
6 | (i, N) | \((S_2,F)\) | \(\mu\) | \(i=s_1+1,\ldots ,s_2\) |
7 | \((s_1+1,N)\) | \((S_1,N)\) | \(\lambda\) | – |
3.3 Operating characteristics
3.4 Solution approach
4 CTMC model of \((s_1,S_1,s_2,S_2)\) policy with exponentially distributed regular replenishment lead time
From | To | Rate | Range | |
---|---|---|---|---|
1 | (i, F) | (i, N) | \(\nu\) | \(i=0,1,\ldots ,S_1\) |
2 | (i, N) | (i, F) | \(\mu\) | \(i=0,1,\ldots ,s_1\) and \(i=s_2+1,\ldots ,S_1\) |
3 | (i, N) | \((S_2,F)\) | \(\mu\) | \(i=s_1+1,\ldots ,s_2\) |
4 | (i, F) | \((i-1,F)\) | \(\lambda\) | \(i=1,2,\ldots ,S_1\) |
5 | (i, N) | \((i-1,N)\) | \(\lambda\) | \(i=1,2,\ldots ,S_1\) |
6 | (i, N) | \((S_1-s_1+i,N)\) | \(\ell\) | \(i=0,1,\ldots ,s_1\) |
5 Numerical study
Parameter | Notation | Value |
---|---|---|
Fixed cost of regular ordering | \(K_o\) | 10 |
Fixed cost of emergency ordering | \(K_e/K_o\) | 3 |
Regular ordering cost/unit | \(c_o\) | 5 |
Emergency ordering cost/unit | \(c_e\) | 5 |
Inventory holding cost | h | 1 |
Lost sales cost | b | 10 |
Demand rate | \(\lambda\) | 5 |
Mean length of off period | \(1/\nu\) | 10, 1, 0.5, 0.25, 0.1 |
Ratio of mean length of off period to and on period | \(\mu /\nu\) | 1, 0.8, 0.5, 0.25, 0.1, 0.05, 0.01 |
5.1 Numerical results of zero regular lead time case
-
Unit lost sales cost plays a key role in determining the policy parameters. The optimal policy parameters, \(s_1^*,S_1^*,s_2^*,S_2^*,s^*,S^*\), are non-decreasing and the associated expected costs of both policies show increasing behavior as the unit lost sales cost increases.
-
\(K_o\) is another important factor affecting policy parameters. Higher fixed cost enforces the retailer to order less frequently. Thus, order-up-to levels, \(S_1^{*}\) and \(S^{*}\), increase as \(K_o\) increases from 10 to 100. A high value of \(K_o\) corresponds to a high \(K_e\) value. As \(K_e\) increases, \(S_2^*\) increases as well if emergency ordering is being utilized. Expected costs of the policies also increase with \(K_o\) as expected.
-
\(1/\nu\) and \(\mu /\nu\) characterize the distributions of on and off periods’ lengths. Changing these parameters has mixed impacts on the policy parameters. However, the expected cost of exercising both policies increases consistently as the supplier’s availability decreases (i.e., \(\mu /\nu\) increases) for a given expected length of the off period.
-
For the instances that use emergency orders, \(s_1^*\) and \(S_1^*\) are below \(s^*\) and \(S^*\), respectively. When there is no emergency ordering opportunity, the only way to respond to the disruption risk is to keep more inventory and it leads to higher reorder and order-up-to levels.
5.2 Numerical results for positive constant regular replenishment lead time case
\(K_e/K_o\) | \(1/\nu\) | \(\mu /\nu\) | \((s_1^*,S_1^*,s_2^*,S_2^*)\) | \(\mathbb {E}[C_1]\) | \((s^*,S^*)\) | \(\mathbb {E}[C_2]\) | %Gap\(_{12}\) |
---|---|---|---|---|---|---|---|
3 | 10 | 1 | (0, 11, 4, 20) | 41.84 | (0, 11) | 41.95 | 0.26 |
0.8 | (0, 11, 4, 20) | 41.15 | (0, 11) | 41.25 | 0.24 | ||
0.5 | (0, 11, 4, 20) | 39.75 | (0, 11) | 39.82 | 0.19 | ||
0.25 | (0, 10, 4, 20) | 38.05 | (0, 10) | 38.10 | 0.13 | ||
0.1 | (0, 10, 4, 20) | 36.66 | (0, 10) | 36.68 | 0.06 | ||
0.05 | (0, 10, 3, 20) | 36.11 | (0, 10) | 36.12 | 0.03 | ||
0.01 | (0, 10, 3, 20) | 35.63 | (0, 10) | 35.63 | 0.01 | ||
1 | 1 | (0, 12, 0, 28) | 38.10 | (0, 12) | 38.10 | 0.00 | |
0.8 | (0, 12, 0, 22) | 37.86 | (0, 12) | 37.86 | 0.00 | ||
0.5 | (0, 12, 0, 22) | 37.35 | (0, 12) | 37.35 | 0.00 | ||
0.25 | (0, 11, 0, 33) | 36.65 | (0, 11) | 36.65 | 0.00 | ||
0.1 | (0, 10, 0, 34) | 36.05 | (0, 10) | 36.05 | 0.00 | ||
0.05 | (0, 10, 0, 40) | 35.79 | (0, 10) | 35.79 | 0.00 | ||
0.01 | (0, 10, 0, 31) | 35.56 | (0, 10) | 35.56 | 0.00 | ||
0.5 | 1 | (0, 12, 0, 28) | 37.02 | (0, 12) | 37.02 | 0.00 | |
0.8 | (0, 11, 0, 34) | 36.87 | (0, 11) | 36.87 | 0.00 | ||
0.5 | (0, 11, 0, 40) | 36.56 | (0, 11) | 36.56 | 0.00 | ||
0.25 | (0, 11, 0, 36) | 36.17 | (0, 11) | 36.17 | 0.00 | ||
0.1 | (0, 10, 0, 41) | 35.81 | (0, 10) | 35.81 | 0.00 | ||
0.05 | (0, 10, 0, 46) | 35.67 | (0, 10) | 35.67 | 0.00 | ||
0.01 | (0, 10, 0, 34) | 35.53 | (0, 10) | 35.53 | 0.00 | ||
0.25 | 1 | (0, 11, 0, 19) | 36.32 | (0, 11) | 36.32 | 0.00 | |
0.8 | (0, 11, 0, 15) | 36.24 | (0, 11) | 36.24 | 0.00 | ||
0.5 | (0, 11, 0, 15) | 36.07 | (0, 11) | 36.07 | 0.00 | ||
0.25 | (0, 10, 0, 20) | 35.85 | (0, 10) | 35.85 | 0.00 | ||
0.1 | (0, 10, 0, 20) | 35.66 | (0, 10) | 35.66 | 0.00 | ||
0.05 | (0, 10, 0, 17) | 35.59 | (0, 10) | 35.59 | 0.00 | ||
0.01 | (0, 10, 0, 11) | 35.52 | (0, 10) | 35.52 | 0.00 | ||
0.1 | 1 | (0, 10, 0, 22) | 35.85 | (0, 10) | 35.85 | 0.00 | |
0.8 | (0, 10, 0, 42) | 35.82 | (0, 10) | 35.82 | 0.00 | ||
0.5 | (0, 10, 0, 27) | 35.74 | (0, 10) | 35.74 | 0.00 | ||
0.25 | (0, 10, 0, 46) | 35.64 | (0, 10) | 35.64 | 0.00 | ||
0.1 | (0, 10, 0, 39) | 35.57 | (0, 10) | 35.57 | 0.00 | ||
0.05 | (0, 10, 0, 28) | 35.53 | (0, 10) | 35.53 | 0.00 | ||
0.01 | (0, 10, 0, 20) | 35.51 | (0, 10) | 35.51 | 0.00 | ||
5 | 10 | 1 | (0, 11, 0, 0) | 41.95 | (0, 11) | 41.95 | 0.00 |
0.8 | (0, 11, 0, 0) | 41.25 | (0, 11) | 41.25 | 0.00 | ||
0.5 | (0, 11, 0, 0) | 39.82 | (0, 11) | 39.82 | 0.00 | ||
0.25 | (0, 10, 0, 0) | 38.10 | (0, 10) | 38.10 | 0.00 | ||
0.1 | (0, 10, 0, 0) | 36.68 | (0, 10) | 36.68 | 0.00 | ||
0.05 | (0, 10, 0, 0) | 36.12 | (0, 10) | 36.12 | 0.00 | ||
0.01 | (0, 10, 0, 0) | 35.63 | (0, 10) | 35.63 | 0.00 | ||
1 | 1 | (0, 12, 0, 0) | 38.10 | (0, 12) | 38.10 | 0.00 | |
0.8 | (0, 12, 0, 0) | 37.86 | (0, 12) | 37.86 | 0.00 | ||
0.5 | (0, 12, 0, 0) | 37.35 | (0, 12) | 37.35 | 0.00 | ||
0.25 | (0, 11, 0, 0) | 36.65 | (0, 11) | 36.65 | 0.00 | ||
0.1 | (0, 10, 0, 0) | 36.05 | (0, 10) | 36.05 | 0.00 | ||
0.05 | (0, 10, 0, 0) | 35.79 | (0, 10) | 35.79 | 0.00 | ||
0.01 | (0, 10, 0, 0) | 35.56 | (0, 10) | 35.56 | 0.00 | ||
0.5 | 1 | (0, 12, 0, 0) | 37.02 | (0, 12) | 37.02 | 0.00 | |
0.8 | (0, 11, 0, 0) | 36.87 | (0, 11) | 36.87 | 0.00 | ||
0.5 | (0, 11, 0, 0) | 36.56 | (0, 11) | 36.56 | 0.00 | ||
0.25 | (0, 11, 0, 0) | 36.17 | (0, 11) | 36.17 | 0.00 | ||
0.1 | (0, 10, 0, 0) | 35.81 | (0, 10) | 35.81 | 0.00 | ||
0.05 | (0, 10, 0, 0) | 35.67 | (0, 10) | 35.67 | 0.00 | ||
0.01 | (0, 10, 0, 0) | 35.53 | (0, 10) | 35.53 | 0.00 | ||
0.25 | 1 | (0, 11, 0, 0) | 36.32 | (0, 11) | 36.32 | 0.00 | |
0.8 | (0, 11, 0, 0) | 36.24 | (0, 11) | 36.24 | 0.00 | ||
0.5 | (0, 11, 0, 0) | 36.07 | (0, 11) | 36.07 | 0.00 | ||
0.25 | (0, 10, 0, 0) | 35.85 | (0, 10) | 35.85 | 0.00 | ||
0.1 | (0, 10, 0, 0) | 35.66 | (0, 10) | 35.66 | 0.00 | ||
0.05 | (0, 10, 0, 0) | 35.59 | (0, 10) | 35.59 | 0.00 | ||
0.01 | (0, 10, 0, 0) | 35.52 | (0, 10) | 35.52 | 0.00 | ||
0.1 | 1 | (0, 10, 0, 0) | 35.85 | (0, 10) | 35.85 | 0.00 | |
0.8 | (0, 10, 0, 0) | 35.82 | (0, 10) | 35.82 | 0.00 | ||
0.5 | (0, 10, 0, 0) | 35.74 | (0, 10) | 35.74 | 0.00 | ||
0.25 | (0, 10, 0, 0) | 35.64 | (0, 10) | 35.64 | 0.00 | ||
0.1 | (0, 10, 0, 0) | 35.57 | (0, 10) | 35.57 | 0.00 | ||
0.05 | (0, 10, 0, 0) | 35.53 | (0, 10) | 35.53 | 0.00 | ||
0.01 | (0, 10, 0, 0) | 35.51 | (0, 10) | 35.51 | 0.00 |
\(K_e/K_o\) | \(1/\nu\) | \(\mu /\nu\) | \((s_1^*,S_1^*,s_2^*,S_2^*)\) | \(\mathbb {E}[C_1]\) | \((s^*,S^*)\) | \(\mathbb {E}[C_2]\) | %Gap\(_{12}\) |
---|---|---|---|---|---|---|---|
3 | 10 | 1 | (0, 30, 0, 0) | 54.55 | (0, 30) | 54.55 | 0 |
0.8 | (0, 31, 0, 0) | 54.83 | (0, 31) | 54.83 | 0 | ||
0.5 | (0, 31, 0, 0) | 55.39 | (0, 31) | 55.39 | 0 | ||
0.25 | (0, 31, 0, 0) | 56.08 | (0, 31) | 56.08 | 0 | ||
0.1 | (0, 31, 0, 0) | 56.65 | (0, 31) | 56.65 | 0 | ||
0.05 | (0, 32, 0, 0) | 56.88 | (0, 32) | 56.88 | 0 | ||
0.01 | (0, 32, 0, 0) | 57.07 | (0, 32) | 57.07 | 0 | ||
1 | 1 | (0, 31, 0, 0) | 56.60 | (0, 31) | 56.60 | 0 | |
0.8 | (0, 31, 0, 0) | 56.65 | (0, 31) | 56.65 | 0 | ||
0.5 | (0, 31, 0, 0) | 56.77 | (0, 31) | 56.77 | 0 | ||
0.25 | (0, 31, 0, 0) | 56.91 | (0, 31) | 56.91 | 0 | ||
0.1 | (0, 32, 0, 0) | 57.03 | (0, 32) | 57.03 | 0 | ||
0.05 | (0, 32, 0, 0) | 57.07 | (0, 32) | 57.07 | 0 | ||
0.01 | (0, 32, 0, 0) | 57.11 | (0, 32) | 57.11 | 0 | ||
0.5 | 1 | (0, 31, 0, 0) | 56.85 | (0, 31) | 56.85 | 0 | |
0.8 | (0, 31, 0, 0) | 56.88 | (0, 31) | 56.88 | 0 | ||
0.5 | (0, 31, 0, 0) | 56.94 | (0, 31) | 56.94 | 0 | ||
0.25 | (0, 32, 0, 0) | 57.02 | (0, 32) | 57.02 | 0 | ||
0.1 | (0, 32, 0, 0) | 57.07 | (0, 32) | 57.07 | 0 | ||
0.05 | (0, 32, 0, 0) | 57.10 | (0, 32) | 57.10 | 0 | ||
0.01 | (0, 32, 0, 0) | 57.12 | (0, 32) | 57.12 | 0 | ||
0.25 | 1 | (0, 31, 0, 0) | 56.99 | (0, 31) | 56.99 | 0 | |
0.8 | (0, 32, 0, 0) | 57.00 | (0, 32) | 57.00 | 0 | ||
0.5 | (0, 32, 0, 0) | 57.03 | (0, 32) | 57.03 | 0 | ||
0.25 | (0, 32, 0, 0) | 57.07 | (0, 32) | 57.07 | 0 | ||
0.1 | (0, 32, 0, 0) | 57.10 | (0, 32) | 57.10 | 0 | ||
0.05 | (0, 32, 0, 0) | 57.11 | (0, 32) | 57.11 | 0 | ||
0.01 | (0, 32, 0, 0) | 57.12 | (0, 32) | 57.12 | 0 | ||
0.1 | 1 | (0, 32, 0, 0) | 57.07 | (0, 32) | 57.07 | 0 | |
0.8 | (0, 32, 0, 0) | 57.08 | (0, 32) | 57.08 | 0 | ||
0.5 | (0, 32, 0, 0) | 57.09 | (0, 32) | 57.09 | 0 | ||
0.25 | (0, 32, 0, 0) | 57.10 | (0, 32) | 57.10 | 0 | ||
0.1 | (0, 32, 0, 0) | 57.11 | (0, 32) | 57.11 | 0 | ||
0.05 | (0, 32, 0, 0) | 57.12 | (0, 32) | 57.12 | 0 | ||
0.01 | (0, 32, 0, 0) | 57.12 | (0, 32) | 57.12 | 0 | ||
5 | 10 | 1 | (0, 30, 0, 0) | 54.55 | (0, 30) | 54.55 | 0 |
0.8 | (0, 31, 0, 0) | 54.83 | (0, 31) | 54.83 | 0 | ||
0.5 | (0, 31, 0, 0) | 55.39 | (0, 31) | 55.39 | 0 | ||
0.25 | (0, 31, 0, 0) | 56.08 | (0, 31) | 56.08 | 0 | ||
0.1 | (0, 31, 0, 0) | 56.65 | (0, 31) | 56.65 | 0 | ||
0.05 | (0, 32, 0, 0) | 56.88 | (0, 32) | 56.88 | 0 | ||
0.01 | (0, 32, 0, 0) | 57.07 | (0, 32) | 57.07 | 0 | ||
1 | 1 | (0, 31, 0, 0) | 56.60 | (0, 31) | 56.60 | 0 | |
0.8 | (0, 31, 0, 0) | 56.65 | (0, 31) | 56.65 | 0 | ||
0.5 | (0, 31, 0, 0) | 56.77 | (0, 31) | 56.77 | 0 | ||
0.25 | (0, 31, 0, 0) | 56.91 | (0, 31) | 56.91 | 0 | ||
0.1 | (0, 32, 0, 0) | 57.03 | (0, 32) | 57.03 | 0 | ||
0.05 | (0, 32, 0, 0) | 57.07 | (0, 32) | 57.07 | 0 | ||
0.01 | (0, 32, 0, 0) | 57.11 | (0, 32) | 57.11 | 0 | ||
0.5 | 1 | (0, 31, 0, 0) | 56.85 | (0, 31) | 56.85 | 0 | |
0.8 | (0, 31, 0, 0) | 56.88 | (0, 31) | 56.88 | 0 | ||
0.5 | (0, 31, 0, 0) | 56.94 | (0, 31) | 56.94 | 0 | ||
0.25 | (0, 32, 0, 0) | 57.02 | (0, 32) | 57.02 | 0 | ||
0.1 | (0, 32, 0, 0) | 57.07 | (0, 32) | 57.07 | 0 | ||
0.05 | (0, 32, 0, 0) | 57.10 | (0, 32) | 57.10 | 0 | ||
0.01 | (0, 32, 0, 0) | 57.12 | (0, 32) | 57.12 | 0 | ||
0.25 | 1 | (0, 31, 0, 0) | 56.99 | (0, 31) | 56.99 | 0 | |
0.8 | (0, 32, 0, 0) | 57.00 | (0, 32) | 57.00 | 0 | ||
0.5 | (0, 32, 0, 0) | 57.03 | (0, 32) | 57.03 | 0 | ||
0.25 | (0, 32, 0, 0) | 57.07 | (0, 32) | 57.07 | 0 | ||
0.1 | (0, 32, 0, 0) | 57.10 | (0, 32) | 57.10 | 0 | ||
0.05 | (0, 32, 0, 0) | 57.11 | (0, 32) | 57.11 | 0 | ||
0.01 | (0, 32, 0, 0) | 57.12 | (0, 32) | 57.12 | 0 | ||
0.1 | 1 | (0, 32, 0, 0) | 57.07 | (0, 32) | 57.07 | 0 | |
0.8 | (0, 32, 0, 0) | 57.08 | (0, 32) | 57.08 | 0 | ||
0.5 | (0, 32, 0, 0) | 57.09 | (0, 32) | 57.09 | 0 | ||
0.25 | (0, 32, 0, 0) | 57.10 | (0, 32) | 57.10 | 0 | ||
0.1 | (0, 32, 0, 0) | 57.11 | (0, 32) | 57.11 | 0 | ||
0.05 | (0, 32, 0, 0) | 57.12 | (0, 32) | 57.12 | 0 | ||
0.01 | (0, 32, 0, 0) | 57.12 | (0, 32) | 57.12 | 0 |
\(K_e/K_o\) | \(1/\nu\) | \(\mu /\nu\) | \((s_1^*,S_1^*,s_2^*,S_2^*)\) | \(\mathbb {E}[C_1]\) | \((s^*,S^*)\) | \(\mathbb {E}[C_2]\) | %Gap\(_{12}\) |
---|---|---|---|---|---|---|---|
3 | 10 | 1 | (0, 10, 84, 97) | 103.60 | (70, 95) | 135.48 | 23.53 |
0.8 | (0, 10, 83, 96) | 96.71 | (64, 90) | 132.40 | 26.96 | ||
0.5 | (0, 10, 81, 93) | 82.23 | (49, 76) | 123.58 | 33.46 | ||
0.25 | (0, 10, 80, 91) | 63.99 | (22, 51) | 104.63 | 38.84 | ||
0.1 | (0, 10, 79, 90) | 48.58 | (0, 18) | 72.28 | 32.79 | ||
0.05 | (0, 10, 78, 89) | 42.37 | (0, 13) | 55.14 | 23.15 | ||
0.01 | (0, 10, 78, 89) | 36.93 | (0, 10) | 39.63 | 6.81 | ||
1 | 1 | (14, 28, 0, 0) | 53.28 | (14, 28) | 53.28 | 0.00 | |
0.8 | (0, 26, 13, 27) | 52.16 | (14, 27) | 52.94 | 1.47 | ||
0.5 | (0, 23, 12, 25) | 49.27 | (12, 26) | 52.00 | 5.25 | ||
0.25 | (0, 9, 12, 23) | 44.54 | (9, 24) | 49.97 | 10.86 | ||
0.1 | (0, 10, 12, 21) | 39.83 | (5, 20) | 46.27 | 13.93 | ||
0.05 | (0, 10, 12, 21) | 37.81 | (1, 17) | 42.98 | 12.05 | ||
0.01 | (0, 10, 12, 20) | 35.99 | (0, 11) | 37.35 | 3.66 | ||
0.5 | 1 | (8, 20, 0, 0) | 45.48 | (8, 20) | 45.48 | 0.00 | |
0.8 | (8, 20, 0, 0) | 45.29 | (8, 20) | 45.29 | 0.00 | ||
0.5 | (7, 19, 0, 0) | 44.75 | (7, 19) | 44.75 | 0.00 | ||
0.25 | (0, 17, 6, 17) | 42.44 | (5, 18) | 43.63 | 2.73 | ||
0.1 | (0, 13, 6, 15) | 39.14 | (3, 16) | 41.58 | 5.87 | ||
0.05 | (0, 12, 5, 15) | 37.54 | (1, 14) | 39.79 | 5.64 | ||
0.01 | (0, 10, 5, 14) | 35.96 | (0, 11) | 36.56 | 1.66 | ||
0.25 | 1 | (5, 16, 0, 0) | 41.27 | (5, 16) | 41.27 | 0.00 | |
0.8 | (4, 16, 0, 0) | 41.15 | (4, 16) | 41.15 | 0.00 | ||
0.5 | (4, 15, 0, 0) | 40.77 | (4, 15) | 40.77 | 0.00 | ||
0.25 | (3, 15, 0, 0) | 40.09 | (3, 15) | 40.09 | 0.00 | ||
0.1 | (0, 13, 2, 13) | 38.34 | (2, 13) | 38.93 | 1.51 | ||
0.05 | (0, 12, 2, 12) | 37.09 | (1, 12) | 37.90 | 2.14 | ||
0.01 | (0, 10, 2, 11) | 35.86 | (0, 11) | 36.07 | 0.58 | ||
0.1 | 1 | (2, 13, 0, 0) | 38.47 | (2, 13) | 38.47 | 0.00 | |
0.8 | (2, 13, 0, 0) | 38.37 | (2, 13) | 38.37 | 0.00 | ||
0.5 | (2, 13, 0, 0) | 38.17 | (2, 13) | 38.17 | 0.00 | ||
0.25 | (1, 12, 0, 0) | 37.87 | (1, 12) | 37.87 | 0.00 | ||
0.1 | (1, 12, 0, 0) | 37.15 | (1, 12) | 37.15 | 0.00 | ||
0.05 | (0, 11, 0, 0) | 36.55 | (0, 11) | 36.55 | 0.00 | ||
0.01 | (0, 10, 0, 0) | 35.73 | (0, 10) | 35.73 | 0.00 | ||
5 | 10 | 1 | (0, 10, 81, 98) | 104.57 | (70, 95) | 135.48 | 22.82 |
0.8 | (0, 10, 80, 96) | 97.57 | (64, 90) | 132.40 | 26.30 | ||
0.5 | (0, 10, 78, 94) | 82.89 | (49, 76) | 123.58 | 32.93 | ||
0.25 | (0, 10, 77, 91) | 64.38 | (22, 51) | 104.63 | 38.46 | ||
0.1 | (0, 10, 76, 90) | 48.76 | (0, 18) | 72.28 | 32.54 | ||
0.05 | (0, 10, 75, 89) | 42.47 | (0, 13) | 55.14 | 22.97 | ||
0.01 | (0, 10, 75, 89) | 36.95 | (0, 10) | 39.63 | 6.76 | ||
1 | 1 | (14, 28, 0, 0) | 53.28 | (14, 28) | 53.28 | 0.00 | |
0.8 | (14, 27, 0, 0) | 52.94 | (14, 27) | 52.94 | 0.00 | ||
0.5 | (12, 26, 0, 0) | 52.00 | (12, 26) | 52.00 | 0.00 | ||
0.25 | (0, 20, 10, 24) | 47.06 | (9, 24) | 49.97 | 5.81 | ||
0.1 | (0, 10, 10, 22) | 41.45 | (5, 20) | 46.27 | 10.43 | ||
0.05 | (0, 10, 10, 21) | 38.70 | (1, 17) | 42.98 | 9.96 | ||
0.01 | (0, 10, 10, 21) | 36.18 | (0, 11) | 37.35 | 3.13 | ||
0.5 | 1 | (8, 20, 0, 0) | 45.48 | (8, 20) | 45.48 | 0.00 | |
0.8 | (8, 20, 0, 0) | 45.29 | (8, 20) | 45.29 | 0.00 | ||
0.5 | (7, 19, 0, 0) | 44.75 | (7, 19) | 44.75 | 0.00 | ||
0.25 | (5, 18, 0, 0) | 43.63 | (5, 18) | 43.63 | 0.00 | ||
0.1 | (0, 15, 4, 16) | 40.32 | (3, 16) | 41.58 | 3.03 | ||
0.05 | (0, 13, 4, 15) | 38.26 | (1, 14) | 39.79 | 3.84 | ||
0.01 | (0, 11, 4, 14) | 36.13 | (0, 11) | 36.56 | 1.17 | ||
0.25 | 1 | (5, 16, 0, 0) | 41.27 | (5, 16) | 41.27 | 0.00 | |
0.8 | (4, 16, 0, 0) | 41.15 | (4, 16) | 41.15 | 0.00 | ||
0.5 | (4, 15, 0, 0) | 40.77 | (4, 15) | 40.77 | 0.00 | ||
0.25 | (3, 15, 0, 0) | 40.09 | (3, 15) | 40.09 | 0.00 | ||
0.1 | (2, 13, 0, 0) | 38.93 | (2, 13) | 38.93 | 0.00 | ||
0.05 | (0, 12, 1, 12) | 37.60 | (1, 12) | 37.90 | 0.80 | ||
0.01 | (0, 10, 1, 10) | 35.98 | (0, 11) | 36.07 | 0.23 | ||
0.1 | 1 | (2, 13, 0, 0) | 38.47 | (2, 13) | 38.47 | 0.00 | |
0.8 | (2, 13, 0, 0) | 38.37 | (2, 13) | 38.37 | 0.00 | ||
0.5 | (2, 13, 0, 0) | 38.17 | (2, 13) | 38.17 | 0.00 | ||
0.25 | (1, 12, 0, 0) | 37.87 | (1, 12) | 37.87 | 0.00 | ||
0.1 | (1, 12, 0, 0) | 37.15 | (1, 12) | 37.15 | 0.00 | ||
0.05 | (0, 11, 0, 0) | 36.55 | (0, 11) | 36.55 | 0.00 | ||
0.01 | (0, 10, 0, 0) | 35.73 | (0, 10) | 35.73 | 0.00 |
\(K_e/K_o\) | \(1/\nu\) | \(\mu /\nu\) | \((s_1^*,S_1^*,s_2^*,S_2^*)\) | \(\mathbb {E}[C_1]\) | \((s^*,S^*)\) | \(\mathbb {E}[C_2]\) | %Gap\(_{12}\) |
---|---|---|---|---|---|---|---|
3 | 10 | 1 | (0, 29, 66, 107) | 121.38 | (55, 110) | 142.77 | 14.99 |
0.8 | (0, 29, 65, 105) | 114.75 | (49, 105) | 139.85 | 17.95 | ||
0.5 | (0, 30, 64, 102) | 100.94 | (34, 94) | 131.37 | 23.17 | ||
0.25 | (0, 31, 63, 99) | 83.71 | (7, 70) | 112.79 | 25.78 | ||
0.1 | (0, 31, 63, 98) | 69.29 | (0, 44) | 85.26 | 18.73 | ||
0.05 | (0, 31, 62, 97) | 63.51 | (0, 37) | 72.28 | 12.13 | ||
0.01 | (0, 32, 62, 97) | 58.46 | (0, 33) | 60.34 | 3.12 | ||
1 | 1 | (10, 44, 0, 0) | 69.97 | (10, 44) | 69.97 | 0.00 | |
0.8 | (9, 44, 0, 0) | 69.61 | (9, 44) | 69.61 | 0.00 | ||
0.5 | (7, 43, 0, 0) | 68.66 | (7, 43) | 68.66 | 0.00 | ||
0.25 | (5, 41, 0, 0) | 66.59 | (5, 41) | 66.59 | 0.00 | ||
0.1 | (0, 37, 3, 37) | 62.19 | (0, 37) | 62.88 | 1.10 | ||
0.05 | (0, 34, 3, 35) | 59.88 | (0, 35) | 60.27 | 0.65 | ||
0.01 | (0, 32, 3, 33) | 57.72 | (0, 32) | 57.81 | 0.16 | ||
0.5 | 1 | (5, 39, 0, 0) | 64.12 | (5, 39) | 64.12 | 0.00 | |
0.8 | (5, 38, 0, 0) | 63.91 | (5, 38) | 63.91 | 0.00 | ||
0.5 | (4, 38, 0, 0) | 63.35 | (4, 38) | 63.35 | 0.00 | ||
0.25 | (3, 37, 0, 0) | 62.22 | (3, 37) | 62.22 | 0.00 | ||
0.1 | (0, 35, 0, 0) | 60.14 | (0, 35) | 60.14 | 0.00 | ||
0.05 | (0, 33, 0, 0) | 58.74 | (0, 33) | 58.74 | 0.00 | ||
0.01 | (0, 32, 0, 0) | 57.47 | (0, 32) | 57.47 | 0.00 | ||
0.25 | 1 | (3, 36, 0, 0) | 61.05 | (3, 36) | 61.05 | 0.00 | |
0.8 | (3, 35, 0, 0) | 60.95 | (3, 35) | 60.95 | 0.00 | ||
0.5 | (2, 35, 0, 0) | 60.57 | (2, 35) | 60.57 | 0.00 | ||
0.25 | (1, 34, 0, 0) | 59.89 | (1, 34) | 59.89 | 0.00 | ||
0.1 | (0, 33, 0, 0) | 58.67 | (0, 33) | 58.67 | 0.00 | ||
0.05 | (0, 32, 0, 0) | 57.95 | (0, 32) | 57.95 | 0.00 | ||
0.01 | (0, 32, 0, 0) | 57.30 | (0, 32) | 57.30 | 0.00 | ||
0.1 | 1 | (1, 34, 0, 0) | 59.10 | (1, 34) | 59.10 | 0.00 | |
0.8 | (1, 33, 0, 0) | 58.99 | (1, 33) | 58.99 | 0.00 | ||
0.5 | (1, 33, 0, 0) | 58.78 | (1, 33) | 58.78 | 0.00 | ||
0.25 | (1, 33, 0, 0) | 58.49 | (0, 33) | 58.49 | 0.00 | ||
0.1 | (1, 32, 0, 0) | 57.75 | (0, 32) | 57.75 | 0.00 | ||
0.05 | (1, 32, 0, 0) | 57.45 | (0, 32) | 57.45 | 0.00 | ||
0.01 | (1, 32, 0, 0) | 57.19 | (0, 32) | 57.19 | 0.00 | ||
5 | 10 | 1 | (0, 29, 58, 112) | 129.27 | (55, 110) | 142.77 | 9.46 |
0.8 | (0, 29, 57, 109) | 122.09 | (49, 105) | 139.85 | 12.71 | ||
0.5 | (0, 30, 55, 104) | 106.84 | (34, 94) | 131.37 | 18.67 | ||
0.25 | (0, 31, 54, 100) | 87.47 | (7, 70) | 112.79 | 22.45 | ||
0.1 | (0, 31, 54, 98) | 71.07 | (0, 44) | 85.26 | 16.65 | ||
0.05 | (0, 31, 53, 97) | 64.45 | (0, 37) | 72.28 | 10.83 | ||
0.01 | (0, 32, 53, 97) | 58.65 | (0, 33) | 60.34 | 2.79 | ||
1 | 1 | (10, 44, 0, 0) | 69.97 | (10, 44) | 69.97 | 0.00 | |
0.8 | (9, 44, 0, 0) | 69.61 | (9, 44) | 69.61 | 0.00 | ||
0.5 | (7, 43, 0, 0) | 68.66 | (7, 43) | 68.66 | 0.00 | ||
0.25 | (5, 41, 0, 0) | 66.59 | (5, 41) | 66.59 | 0.00 | ||
0.1 | (0, 37, 0, 0) | 62.88 | (0, 37) | 62.88 | 0.00 | ||
0.05 | (0, 35, 0, 0) | 60.27 | (0, 35) | 60.27 | 0.00 | ||
0.01 | (0, 32, 0, 0) | 57.81 | (0, 32) | 57.81 | 0.00 | ||
0.5 | 1 | (5, 39, 0, 0) | 64.12 | (5, 39) | 64.12 | 0.00 | |
0.8 | (5, 38, 0, 0) | 63.91 | (5, 38) | 63.91 | 0.00 | ||
0.5 | (4, 38, 0, 0) | 63.35 | (4, 38) | 63.35 | 0.00 | ||
0.25 | (3, 37, 0, 0) | 62.22 | (3, 37) | 62.22 | 0.00 | ||
0.1 | (0, 35, 0, 0) | 60.14 | (0, 35) | 60.14 | 0.00 | ||
0.05 | (0, 33, 0, 0) | 58.74 | (0, 33) | 58.74 | 0.00 | ||
0.01 | (0, 32, 0, 0) | 57.47 | (0, 32) | 57.47 | 0.00 | ||
0.25 | 1 | (3, 36, 0, 0) | 61.05 | (3, 36) | 61.05 | 0.00 | |
0.8 | (3, 35, 0, 0) | 60.95 | (3, 35) | 60.95 | 0.00 | ||
0.5 | (2, 35, 0, 0) | 60.57 | (2, 35) | 60.57 | 0.00 | ||
0.25 | (1, 34, 0, 0) | 59.89 | (1, 34) | 59.89 | 0.00 | ||
0.1 | (0, 33, 0, 0) | 58.67 | (0, 33) | 58.67 | 0.00 | ||
0.05 | (0, 32, 0, 0) | 57.95 | (0, 32) | 57.95 | 0.00 | ||
0.01 | (0, 32, 0, 0) | 57.30 | (0, 32) | 57.30 | 0.00 | ||
0.1 | 1 | (1, 34, 0, 0) | 59.10 | (1, 34) | 59.10 | 0.00 | |
0.8 | (1, 33, 0, 0) | 58.99 | (1, 33) | 58.99 | 0.00 | ||
0.5 | (1, 33, 0, 0) | 58.78 | (1, 33) | 58.78 | 0.00 | ||
0.25 | (0, 33, 0, 0) | 58.49 | (0, 33) | 58.49 | 0.00 | ||
0.1 | (0, 32, 0, 0) | 57.75 | (0, 32) | 57.75 | 0.00 | ||
0.05 | (0, 32, 0, 0) | 57.45 | (0, 32) | 57.45 | 0.00 | ||
0.01 | (0, 32, 0, 0) | 57.19 | (0, 32) | 57.19 | 0.00 |
\(K_e/K_o\) | \(1/\nu\) | \(\mu /\nu\) | \((s_1^*,S_1^*,s_2^*,S_2^*)\) | \(\mathbb {E}[C_1]\) | \((s^*,S^*)\) | \(\mathbb {E}[C_2]\) | %Gap\(_{12}\) |
---|---|---|---|---|---|---|---|
3 | 10 | 1 | (10, 21, 88, 101) | 107.79 | (73, 103) | 135.05 | 20.19 |
0.8 | (10, 21, 76, 104) | 102.49 | (67, 94) | 133.28 | 23.10 | ||
0.5 | (10, 20, 80, 103) | 90.43 | (55, 84) | 127.38 | 29.01 | ||
0.25 | (10, 21, 84, 94) | 68.56 | (22, 58) | 102.56 | 33.15 | ||
0.1 | (10, 21, 92, 93) | 54.20 | (11, 28) | 75.17 | 27.89 | ||
0.05 | (10, 21, 89, 97) | 48.49 | (10, 23) | 58.10 | 16.53 | ||
0.01 | (10, 21, 83, 84) | 42.36 | (10, 21) | 44.50 | 4.81 | ||
1 | 1 | (19, 34, 0, 0) | 53.76 | (19, 34) | 53.76 | 0.00 | |
0.8 | (18, 35, 0, 0) | 53.51 | (18, 35) | 53.51 | 0.00 | ||
0.5 | (17, 32, 0, 0) | 52.79 | (17, 32) | 52.79 | 0.00 | ||
0.25 | (15, 29, 0, 0) | 50.32 | (15, 29) | 50.32 | 0.00 | ||
0.1 | (11, 25, 0, 0) | 46.94 | (11, 25) | 46.94 | 0.00 | ||
0.05 | (10, 23, 16, 26) | 43.33 | (11, 23) | 44.58 | 2.79 | ||
0.01 | (10, 21, 17, 25) | 41.82 | (10, 21) | 42.07 | 0.59 | ||
0.5 | 1 | (14, 28, 0, 0) | 46.53 | (14, 28) | 46.53 | 0.00 | |
0.8 | (14, 26, 0, 0) | 46.18 | (14, 26) | 46.18 | 0.00 | ||
0.5 | (13, 25, 0, 0) | 45.73 | (13, 25) | 45.73 | 0.00 | ||
0.25 | (12, 25, 0, 0) | 44.78 | (12, 25) | 44.78 | 0.00 | ||
0.1 | (11, 23, 0, 0) | 43.34 | (11, 23) | 43.34 | 0.00 | ||
0.05 | (10, 22, 0, 0) | 42.43 | (10, 22) | 42.43 | 0.00 | ||
0.01 | (10, 21, 0, 0) | 41.59 | (10, 21) | 41.59 | 0.00 | ||
0.25 | 1 | (11, 23, 0, 0) | 43.06 | (11, 23) | 43.06 | 0.00 | |
0.8 | (11, 24, 0, 0) | 43.11 | (11, 24) | 43.11 | 0.00 | ||
0.5 | (11, 22, 0, 0) | 42.67 | (11, 22) | 42.67 | 0.00 | ||
0.25 | (11, 22, 0, 0) | 42.34 | (11, 22) | 42.34 | 0.00 | ||
0.1 | (10, 21, 0, 0) | 41.85 | (10, 21) | 41.85 | 0.00 | ||
0.05 | (10, 21, 0, 0) | 41.69 | (10, 21) | 41.69 | 0.00 | ||
0.01 | (10, 21, 0, 0) | 41.51 | (10, 21) | 41.51 | 0.00 | ||
0.1 | 1 | (10, 22, 0, 0) | 41.89 | (10, 22) | 41.89 | 0.00 | |
0.8 | (10, 21, 0, 0) | 41.77 | (10, 21) | 41.77 | 0.00 | ||
0.5 | (10, 21, 0, 0) | 41.69 | (10, 21) | 41.69 | 0.00 | ||
0.25 | (10, 21, 0, 0) | 41.57 | (10, 21) | 41.57 | 0.00 | ||
0.1 | (10, 21, 0, 0) | 41.55 | (10, 21) | 41.55 | 0.00 | ||
0.05 | (10, 21, 0, 0) | 41.44 | (10, 21) | 41.44 | 0.00 | ||
0.01 | (10, 21, 0, 0) | 41.43 | (10, 21) | 41.43 | 0.00 |
5.3 Numerical results of exponentially distributed regular replenishment lead time case
\(K_e/K_o\) | \(1/\nu\) | \(\mu /\nu\) | \((s_1^*,S_1^*,s_2^*,S_2^*)\) | \(\mathbb {E}[C_1]\) | \((s^*,S^*)\) | \(\mathbb {E}[C_2]\) | %Gap\(_{12}\) |
---|---|---|---|---|---|---|---|
3 | 10 | 1 | (11, 65, 101, 114) | 124.51 | (74, 124) | 146.43 | 14.97 |
0.8 | (12, 63, 99, 112) | 118.65 | (69, 114) | 141.74 | 16.29 | ||
0.5 | (14, 56, 96, 109) | 105.79 | (56, 94) | 130.59 | 18.99 | ||
0.25 | (15, 48, 92, 104) | 88.06 | (33, 64) | 110.21 | 20.10 | ||
0.1 | (16, 41, 89, 101) | 71.28 | (21, 42) | 82.90 | 14.02 | ||
0.05 | (17, 38, 88, 100) | 63.87 | (19, 38) | 69.98 | 8.74 | ||
0.01 | (18, 35, 87, 99) | 56.95 | (18, 35) | 58.19 | 2.12 | ||
1 | 1 | (0, 41, 25, 41) | 66.73 | (36, 65) | 78.71 | 15.22 | |
0.8 | (0, 42, 26, 42) | 67.68 | (33, 61) | 75.42 | 10.27 | ||
0.5 | (29, 53, 0, 0) | 69.87 | (29, 53) | 69.87 | 0.00 | ||
0.25 | (23, 39, 25, 45) | 64.14 | (24, 45) | 64.17 | 0.05 | ||
0.1 | (20, 36, 23, 39) | 59.44 | (21, 40) | 59.53 | 0.15 | ||
0.05 | (19, 34, 22, 37) | 57.42 | (19, 37) | 57.50 | 0.12 | ||
0.01 | (18, 33, 21, 35) | 55.55 | (18, 35) | 55.57 | 0.04 | ||
0.5 | 1 | (0, 33, 16, 33) | 58.34 | (33, 60) | 74.38 | 21.57 | |
0.8 | (0, 34, 17, 34) | 59.15 | (30, 56) | 71.21 | 16.94 | ||
0.5 | (0, 37, 20, 37) | 62.03 | (26, 49) | 66.07 | 6.10 | ||
0.25 | (22, 42, 0, 0) | 61.20 | (22, 42) | 61.20 | 0.00 | ||
0.1 | (20, 38, 0, 0) | 57.75 | (20, 38) | 57.75 | 0.00 | ||
0.05 | (19, 36, 0, 0) | 56.46 | (19, 36) | 56.46 | 0.00 | ||
0.01 | (18, 35, 0, 0) | 55.33 | (18, 35) | 55.33 | 0.00 | ||
0.25 | 1 | (0, 28, 10, 28) | 53.04 | (31, 58) | 72.36 | 26.71 | |
0.8 | (0, 28, 11, 28) | 53.58 | (29, 54) | 69.30 | 22.68 | ||
0.5 | (0, 30, 13, 30) | 55.60 | (25, 47) | 64.43 | 13.71 | ||
0.25 | (22, 41, 0, 0) | 60.02 | (22, 41) | 60.02 | 0.00 | ||
0.1 | (19, 37, 0, 0) | 57.13 | (19, 37) | 57.13 | 0.00 | ||
0.05 | (19, 36, 0, 0) | 56.11 | (19, 36) | 56.11 | 0.00 | ||
0.01 | (18, 35, 0, 0) | 55.25 | (18, 35) | 55.25 | 0.00 | ||
0.1 | 1 | (0, 24, 6, 24) | 49.07 | (30, 56) | 71.23 | 31.10 | |
0.8 | (0, 24, 6, 24) | 49.39 | (28, 52) | 68.24 | 27.63 | ||
0.5 | (0, 25, 7, 25) | 50.45 | (24, 46) | 63.57 | 20.64 | ||
0.25 | (0, 28, 10, 28) | 53.29 | (21, 40) | 59.44 | 10.35 | ||
0.1 | (19, 37, 0, 0) | 56.84 | (19, 37) | 56.84 | 0.00 | ||
0.05 | (18, 36, 0, 0) | 55.96 | (18, 36) | 55.96 | 0.00 | ||
0.01 | (18, 35, 0, 0) | 55.22 | (18, 35) | 55.22 | 0.00 |