4.1. Numerical Analysis
A numerical analysis was performed to illustrate the results of simulating the system of a limited number of waiting places such that only N customers are admitted to the system simultaneously with fixed-order quantity Q.
We compare the proposed framework on the performance measures at specific values of fixed-order quantity Q, the average rate of arrival customers , the service rate , and the time required to deliver the new order quantity , where it is assumed to be a case wherein the order quantity is fixed and the delivery is fixed.
4.1.1. The Effect of on Performance Measures for Different Q Values
In the following
Table 2,
Table 3,
Table 4 and
Table 5,
Q = 15, 25, 33 and 50.
= 11, 15, 25 and 40. The fixation of
= 45,
= 3, and the size of the system to accommodate the customers
N = 40. It was noticed that the average stock level
decreases as it increases
for each
Q. For
, it was noticed that it increases as
increases because it measures the rate of reordering per unit of time and for
, which is the rate of arrival of customers for each time unit that increases with an increase
. The average number of lost sales and the average number of lost sales in each order cycle (
,
) increase with the increase in the flow rate. The service level
also decreases by increasing
. The average number of customers in the whole system and the average number of customers in a queue increase (
,
) with increasing
. The average customer waiting time and average stay increase (
,
) with the increase in the arrival rate and the cost
F increases.
4.1.2. The Effect of on Performance Measures for Different Q Values
In the following
Table 6,
Table 7,
Table 8 and
Table 9, it was assumed in this case that
Q = 15, 25, 33 and 50.
= 10, 16, 20 and 35. The fixation of
= 9,
= 3 and the size of the system to accommodate the customers
N = 40. It was noticed that the average stock level
decreases as it increases
for each
Q. For
, it was noticed that it increases as
increases because it measures the rate of reordering per unit of time and for
, which is the rate of arrival of customers for each time unit that increases with an increase
. The average number of lost sales and the average number of lost sales in each order cycle (
,
) increase with the increase in the flow rate. The service level
also decreases by increasing
. The average number of customers in the whole system and the average number of customers in a queue increase (
,
) with increasing
. The average customer waiting time and average customer length of stay increase (
,
) with the increase in the service rate. Moreover, the cost
F decreases. It can be observed that after
= 16,
,
,
,
,
and
seem to be constant.
In order to check the effect of on performance measures for different Q values, it was assumed that Q = 15, 25, 33 and 50. = 3, 10, 25 and 40. Fixation of = 14, = 15, and the size of the system to accommodate the customers N = 40. It was noticed that the average stock level decreases as it increases for each Q. For , it was noticed that it increases as increases because it measures the rate of reordering per unit of time and for , which is the rate of arrival of customers for each time unit that increases with an increase . Moreover, the average number of lost sales and the average number of lost sales in each order cycle (, ) increase with the increase in flow rate. The service level also decreases by increasing . The average number of customers in the whole system and the average number of customers in a queue increase (, ) with increasing . The average customer waiting time and the average stay of the customer increase (, ) with the increase in the lead time and the cost F decreases. In general, there are some measures that change their behavior depending on Q values, such as (, , and F); for example, when and , these measures decrease, while for and , they increase.
4.2. M/M/1/N-System with Deterioration under Deterministic and Uniformly Distributed Order Size
In this section, the results were obtained from simulating a MATLAB program designed to model a limited inventory system with lost sales and a deterioration parameter under deterministic order size probabilities. The maximum system size for the customer (N) is chosen to be equal to 10. Every time a new order is placed (Q), the order quantity is chosen to have different values from the range of 5–35 with step 5. The average arrival rate of customers () was chosen to have different values from the range 5–35 with step 5. The average lead time (time needed to deliver the new order quantity) () was chosen to be equal to 1. The average service rate (number of customers served per unit of time) () was chosen to be equal to 55. The deterioration parameter was chosen to have different values from the range 0.5–4.5 with step 2. The distribution of the delivered quantity (P) was chosen to be deterministic fixed-order size.
For all values of Q and , it was noticed that the probability that the inventory is empty, , which increases as the demand () increases. The probability that we do not have customers in our system, , decreases as the demand () increases. The probability that the inventory level is less than the customers in our system, , increases as the demand () increases. The probability that our system is full of customers, , increases as the demand () increases. The expectation of the inventory position, , is decreased as the demand () increases. The mean number of replenishments per unit of time or reorder rate, , increases as the demand () increases. The mean number of customers arriving per unit of time, , increases as the demand () increases. The average number of lost sales incurred per unit of time, , increases as the demand () increases. The expected number of lost sales per cycle, , increases as the demand () increases. The service level, , decreases as the demand () increases. The average number of customers in the whole system and the average number of customers in a queue increase (, ) with increased . The customer mean sojourn time, , increases as the demand () increases. The customer mean sojourn time (waiting), , increases as the demand () increases.
In terms of deterministic order size, the effect of adding the deterioration parameter
on
, referred to in
Figure 3a,b, shows that for a constant
Q value, the probability that the inventory being empty takes less value when the deterioration happens and vice versa makes sense as deterioration affects the availability of the items or the services for the customers. Furthermore, as can be seen, this measure increases as the customers arrive at the system and request a service or an item, as the number of items will decrease as it is given to the customer who requested the item or service. It was also noted that when the
Q value increases, this measure decreases, which also makes sense as when the order quantity increases, the probability of the empty inventory decreases.
It was noticed that the dependency of , and on the deterioration parameter is increased as the value of Q increased. The relation between the parameters , Q, and the mentioned measures have the same behavior as in the deterministic order size, as the only change was in the probability of delivering the items and services to the customers.
The visual representation of the effect of
on different measures with different
values under uniform order size is given in
Figure 4a,b.
In this research, a limited number of waiting places of the integrated inventory-queuing model with deterioration was studied under uniformly distributed and deterministic order fixed quantity Q. In the proposed model, deterioration means falling from a higher to a lower level in quantity.
In the first instance, the Schwarz model was studied, modeled and analyzed under deterministic and uniformly distributed order quantity, depending on their mathematical approach in their article. The resulting performance measures found with the cost function have been presented. The same model that has been represented by the proposed method depends on finding a solution for the linear system found from generating the balance equations obtained from the drawn states of the desired system. After that, the generated model was analyzed and compared to the results obtained from examining the Schwarz model to ensure system validity. By analyzing the results, it was noticed that when the value of increased, and for different values of Q, the performance measures such as , , , , , , , , , , and F will be increased. There is no observed change in the values of the following measures, such as , , and . There is a critical value of at which the cost function F changes its behavior, and it is approximately 23 in deterministic and 17 in uniformly distributed order size. At that point, the cost function values reversed from maximum to minimum and vice versa depending on the Q value.
A novel approach was used to analyze the performance measures of a limited integrated queuing-inventory model with deterioration parameter , which is exponentially distributed and has a value greater than Q under deterministic and uniform distributed order quantity. It was observed that the value of increased, and for different values of , the performance measures such as , , , and expanded. Furthermore, it was noticed that the value of increased, and for different values of , the performance measures such as , , and diminished. There was no observed change in , , , , and .