Detection of an appropriate pharmaceutical company to get a suitable vaccine against COVID‐19 with minimum cost under the quality control process

. 2021 Apr 7;37(6):2646–2664. doi: 10.1002/qre.2881

Q_{i}

The queue number

i,

i = 1, 2, \dots, K

μ_{i}

The queue number

i

service rate

λ_{i}

The queue number

i

arrival rate

p_{n_{i}}

Steady‐state probability when the queue number

i

has

n_{i}

units

p_{0}

Steady‐state total probability when there are no units in all companies (the system)

p_{0_{i}}

Steady‐state probability when there are no units in the queue number

i

N

Total capacity of the system (the summation of all company capacities)

N_{i}

The capacity of the queue number

i,

i = 1, 2, \dots, K

n

Units number in the system,

0 \leq n \leq N

n_{i}

Units number in

Q_{i}

0 \leq n_{i} \leq N_{i}

L_{i}

The expected value of the number of units in the queue number

i,

i = 1, 2, \dots, K

L_{q_{i}}

The expected value of the waiting units to be served in the queue number

i

(i.e., the ability of introducing the service for at least one country in the queue number

i

)

L_{s_{i}} =

L_{i} - L_{q_{i}}

The countries average number that actually serves in the queue number

i

H_{i}^{*}

The minimum excepted value of the waiting time for a selected queue number

i

from the system

H_{q_{i}}^{*}

The minimum excepted value of the waiting time in the queue number

i

H_{s i}^{*}

The minimum excepted value of the service time in the queue number

i

W (Z)

The probability of detection for the suitable company under the quality control process

\tilde{W} (Z)

The complement of

W (Z)

R_{r i}

The queue number

i

average rate of reneging

R_{R i}

The queue number

i

average rate of retention

C_{s i}

The queue number

i

service cost per unit time

C_{h i}

The queue number

i

holding cost per unit time

C_{l i}

The lost country cost in the queue number

i

C_{r i}

The reneged country cost in the queue number

i

C_{R i}

Retaining a reneged country cost in the queue number

i

C_{s 1 i}

Serving a feedback country cost in the queue number

i

R_{i}

Revenue from the service provided to the queue number

i

T E C_{i}

The total expected cost per unit time of the company number

i

T E R_{i}

The total expected revenue per unit time of the company number

i

T E P_{i}

The total expected profit per unit time of the company number

i