نوع مقاله : مقاله پژوهشی
نویسندگان
گروه مدیریت، دانشکده علوم اداری و اقتصاد، دانشگاه اصفهان، اصفهان، ایران
چکیده
کلیدواژهها
موضوعات
عنوان مقاله [English]
نویسندگان [English]
Purpose: This study aims to investigate the influence of known price increases on the inventory model regarding both uniform and an exponential distribution of replenishment intervals with the partial backorder. It examines the optimization of inventory control decisions for deteriorating products considering a known price increase, probabilistic replenishment interval, warehouse capacity constraint, and partial backordering.
Design/methodology/approach: To obtain the specific inventory order quantity, the problem has been modeled in such a way that the total cost savings function is obtained from the differences in the optimal order policy for both special and regular orders. The two situations discussed in this study are: i) unconstrained problem modeling, and ii) constrained problem. Some computational experiments have been performed to examine the effects of various parameters on cost savings performance. For the constrained problem, Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) have been used and their results have been compared in terms of the cost savings values and computation time.
Findings: Findings indicated that for the constrained problem, GA has a better performance than PSO. Accordingly, for an unconstrained problem, by using the derivative of the profit function and performing sensitivity analysis, the influence of parameters such as demand, price, holding cost after the price increase, λ in exponential distribution, length of periods in uniform distribution, and deterioration rate on the decision variables including order quantity and the profit were obtained,
Practical implications: The model’s generated policy is more effective and profitable for retailers when demand and deterioration rate are higher and replenishment periods are decreased.
Originality/value: This study completes the previous inventory control models that were under the policy of known price increase and is closer to the real environment by utilizing deteriorating items, capacity constraints, and metaheuristic approaches.
کلیدواژهها [English]
Inflation and rising prices for some raw materials, oil, on the one hand, and the introduction of various incentive policies by some suppliers, on the other hand, have a great influence on stock decisions. Therefore, taking into account the increase in commodity prices in the future is inevitable for the supplier. When a supplier announces a price increase in the future and allows the retailer to buy a surplus of goods at the current price. Deciding whether to buy or not to buy and the amount of purchase is necessary for the retailer. Therefore, retailers must decide on their inventory based on the increased rate of product prices in the coming months, random time to the next savings, amount of deterioration items, warehouse capacity and use the optimal use of the special order provided by the supplier (Zhang, X.L & Shi., 2018; Janssen, 2018a; Tashakkor, Mirmohammadi, & Iranpoor, 2018; Li and Teng, 2018; Bounkhel et al., 2019; Soni and Suthar, 2020; Babangida and Baraya, 2020).
This study aims to determine replenishment level values in response to a price increase by maximizing the total cost saving between special and regular orders. We provide several numerical examples for both constrained and unconstrained inventory models as the simplest type of optimization is unconstrained and the unconstrainedoptimization technique is so efficient, it has been used as the point of departure for constructing a more realistic constrained inventory model (Bradley et al., 1977; Malik & Sarkar, 2018). Furthermore, a sensitivity analysis of the optimal solution is conducted to show the effects of some parameters on replenishment levels and total saving. In this paper, we will use the decision variables and expected total saving structure as is shown in Taleizadeh, Zarei, & Sarker. (2016).
According to previous research, including Taleizadeh et al. (2013a), Yang et al. (2015), KarimiNasab & Wee (2015), and Taleizadeh, Zarei, & Sarker (2016), it can be seen that they did not include realworld constraints and more attention has been paid to the increase in price over time due to a random delivery period and a motivational policy. In several studies, there are no limitations and the problem is modeled and solved with integer decision variables and linear programming (Zeballos, Seifert & ProtopappaSieke, 2013; Sarkar & Moon, 2014; Giri & Sharma, 2016; Braglia, Castellano & Frosolini, 2016; Braglia, Castellano, & Song, 2017). Therefore, in studies that focus exclusively on known price increases, the gap is quite evident when a study is aimed at getting a better understanding of the realworld conditions (Cimen & Kirkbride, 2017). Therefore, this research seeks to consider goods that do not have a stable lifespan, as well as storage space limitations and problemsolving by using metaheuristics algorithms.
The rest of the paper is organized as follows: In Section 2, a brief literature review is presented after which the problem along with assumptions is defined in section 3. In Section 4, a proposed model of the problem is devised. To do this, first, the parameters and the variables of the problem are introduced. Next, an unconstrained model with both uniform and exponential distributions is presented and the solution method is elaborated. Then, a constrained problem and the algorithms used to solve it are described. In Section 5, through the numerical examples, both constrained and unconstrained models are implemented and the results are presented in the relevant tables. A sensitivity analysis is performed and its results are shown in section 6, in section 7, the conclusions and also recommendations for future research are presented. Finally, a discussion is presented in section 8 in which the findings of this study are compared with previous research.
For many researchers and management, decisions about inventory control of deteriorating items have always been challenging due to their specific characteristics. Goyal & Giri (2001) discussed developments of deteriorating inventory from 1990 to 2001. They indicated that most of the models had been classified on the base of demand, constraint, and condition. Yang & Wee (2003) developed a mathematical multilotsize production model for a deteriorating item in which the perspective of buyers and sellers has been considered. Moon, Giri, & Ko (2005) studied the EOQ model for two kinds of products (deteriorating/ameliorating) under situations such as finite planning horizon, timedependent demand, inflation, and time value of money. Prekopa (2006) used the model which socalled Hungarian inventory control to obtain that optimal safety stock level. In his model, production was continuous without disruption. Caloieroa, Strozzia & Comenges (2008) investigated the bullwhip effect on demand in the supply chain; they focused on a single product in a serial supply chain. Another work that expanded the EOQ model is Ouyang et al. (2008) which linked permissible delay in payment to deteriorating EOQ. In recent studies, such as Amorim, Costa, & AlmadaLobo (2013), Yu et al. (2012), Abad (2008), Maihami & Karimi (2014), Chen et al. (2016), Neeraj & Kumar (2017), Jaggi, Tiwari & Goel (2017), Zhang, X.L & Shi (2018), Janssen (2018b), Tashakkor, Mirmohammadi, & Iranpoor (2018), Li & Teng (2018), Asif and Biswajit (2018), Bounkhel et al. (2019), Soni & Suthar (2020), Babangida & Baraya (2020), demand (deterministic or stochastic) has been found as a very significant factor in diversifying inventory control models for deteriorating items. To get closer to the real world, Tiwari, et al (2017) developed the model for deteriorating seasonal products with ramptype demand. They formulated their model with some considerations such: as stockdependent consumption rate and partial backordering. The main model variable was the preservation technology cost.
In the literature, many studies have focused on the announcement of a price increase problem. Naddor (1966) was one of the first researchers who considered the price increase in the future. He modeled an EOQ (economic order quantity) model that highlighted the rise in prices and offered a chance to buy to the buyer. Ghosh (2003) and Huang, Kulkarni & Swaminathan (2003) considered the effect of the infinite horizon on the increase of known price problems. In their studies, buyers could have spatial order before the price increase. In inventory management literature, few studies consider the constant price change. Yang (2006) developed a twowarehouse inventory model for deterioration items with different rates and linear demands under inflationary conditions. Sarker & Kindi (2006) developed economic order quantity (EOQ) models with a discounted price. In their work, they attempt to obtain the order value in five different cases: a) coincidence of sale period with replenishment time, b) noncoincidence of sale period with replenishment time, c) sale period longer than a cycle, d) discounted price as a function of the special ordering quantity, and (e) incremental discount. Sharma (2009) proposed a composite model for the environment with fractional backordering. Hsu & Yu (2011) developed an EOQ model for imperfect quality items under an announced price increase where a 100% screening process was performed; then defectives were screened out, and at the end of the inspection process, the defectives were sold as a single batch. They obtained optimal ordering policies under this situation and by some examples, illustrated their proposed model. Taleizadeh, Akhavan Niaki & Makui (2012) described an economic order quantity model in which there were costs in advance and divided the prepayment into multiple equalsize parts during a fixed lead time. Taleizadeh, et al (2013a) formulated and modeled the multiple partial prepayments of the EOQ problem with partial backordering. They considered the level of inventory at the time of special order and provided scenarios to explain it. Then, Taleizadeh (2014), Wang et al. (2015), Tsao & Linh (2016), Diabat, Taleizadeh, & Lashgari (2017), Lashgari, Taleizadeh & Sadjadi (2018), Tiwari et al. (2018), and Taleizadeh et al. (2020), developed another EOQ models in which they consider partial backordering and prepayment policy. For inflation and the time value of money or deteriorating items, Singh, Kumar & Kumari (2011) developed a twowarehouse model. Ouyang (2016), Palanivel, Uthayakumar & Finite (2015), Herbon (2017), Banerjee & Agrawal (2017), Herbon & Khmelnitsky (2017), Jaggi, Tiwari & Goel (2017), and Kaya & Ghahroodi (2018), considered various situations to obtain optimal order quantity for deteriorating items under changing prices.
In many studies, the consideration of probabilistic replenishment intervals is very common (Rabbani, Pourmohammad, & Rafiei, 2016; Chen et al., 2016; Pal, Bardhan & Giri, 2018; Palak, Sioglu, & Geunes, 2018; Janssen, 2018b). For example, Sazavar et al. (2016), investigated multiperiod with single item model with restricted order size. The model included a multiperiod/multiproduct optimal ordering problem considering the expiry date. Pan (2017), investigated a medical resource inventory model for emergency preparation with uncertain demand and stochastic occurrence time, considering different risk preferences.
Metahubristic algorithms such as Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) can be used in inventory control to obtain optimal reorder points (Dye, 2012; Mousavi et al., 2014; Buhnia, Shaikh & Gupta, 2015; Bhunia & Shaikh, 2015; Vandani, Niaki, & Aslanzade, 2017; Akbari Kaasgari, Imani, & Mahmoodjanloo, 2017; Azadeh, 2017; Hiassat, Diabat & Rahwan, 2017; Tiwari, 2017). In general, scholars suggest that hybrid metaheuristic algorithms have gained considerable attention for their capability to solve difficult problems in different fields of science especially to solve the inventory problems and due to the nonlinearity of the proposed model of this study, particle swarm optimization (PSO) and genetic algorithm (GA), are implemented as optimizing solvers instead of analytical methods (Talezadeh et al., 2013b; AlejoReyes, et al., 2020).
Yao & Chu (2008) developed an improved inventory control model with the GA approach to obtain the optimal value of replenishment cycles. They minimized maximum warehouse space and allowed the warehouse to replenish at any time. Hong & Kim (2009) used a Genetic algorithm to optimize a joint replenishment model. Chiang (2013) completed previous work and considered partial backordering and fix–costordering for replenishment. Taleizadeh et al. (2013b) assumed that in several products, the time between two replenishments is the same and follows random variables. They also considered shortages including backorder and emergency orders. Muthuraman, Seshadri, & Wu (2014) modeled an inventory system with stochastic demand and stochastic delivery lags as QuasiVariation Inequality (QVI). Their model had an infinitedimensional statespace and was intractable. Bischak et al. (2014) developed an analytical model to obtain optimum inventory levels under lead time constraints. They summed that crossover orders occur. Shu, Huang, & Fu (2015) developed a production–delivery lotsizing model in which, the time between two delivery was stochastic. KarimiNasab & Wee (2015) formulated an inventory model with stochastic replenishment intervals and deterministic sale offers in which replenishment intervals had an exponential distribution and shortage was partially backordered.
Consider a periodic inventory control model in which a supplier announces a price increase for all items in the future at or before the next scheduled ordering time of the buyer.
This paper developed and formulated an inventory control problem in which:
The following notation is used to model the problem:
For i=1, 2,..., n, the parameters and the variables of the model are defined as
Parameters:
Demand market rate of the i^{th} product (units/year) 

The fixed ordering cost of i^{th} product (in dollars) 

The selling price of the i^{th} product (unit /year) 

Regular purchasing price per unit of the i^{th} product (in dollars) 

Purchasing price in future per unit of the i^{th} product (in dollars) 

The fraction of shortages that will be backordered of the i^{th} product (Percent) 

Unit backorder cost of the i^{th} product ($/unit/year) 

Lost sale cost in normal price per unit of the i^{th} product ($/unit) 

Lost sale cost in increased price per unit of the i^{th} product ($/unit) 

Inventory holding cost per unit of the i^{th} product ($/unit/year) 

Inventory holding cost when pricing increases per unit of the i^{th} product ($/unit/year) 

Deteriorating rate of the i^{th} product 

The time between two consecutive replenishments of the i^{th} product 

The maximum amount of time between two consecutive replenishments in uniform distribution of the i^{th} product 

The minimum amount of time between two consecutive replenishments in uniform distribution of the i^{th} product 

Mean number of replenishments per year for the exponential probability distribution function of the i^{th} product 

Probability distribution function (pdf) of L of the i^{th} product 

The cumulative distribution function of L of the i^{th} product 

The required warehouse space per unit of the i^{th} product 

Total available warehouse space 

Variables:
The replenishupto level in the regular order of the i^{th} product (unit) 

The replenishupto level when the price increase of the i^{th} product (unit) 

The replenishupto level in the special sale of the i^{th} product (unit) 

Expected number of units replenished per cycle in the special sale before price increases of the i^{th} product (Unit/order) 

Expected number of units sold per cycle in the special sale before price increases of the i^{th} product (unit) 

Expected backordered quantity per cycle in the special sale of the i^{th} product, (unit) 

Expected lost sale quantity per cycle in the special sale of the i^{th} product, (unit) 

Expected inventory per cycle in the special sale of the i^{th} product (unit) 

Expected number of units replenished per cycle when the price increased of the i^{th} product (Unit/order) 

Expected number of units sold per cycle in special sale when the price increased of the i^{th} product (Unit) 

Expected backordered quantity per cycle of the i^{th} product when the price increased (unit) 

Expected lost sale quantity per cycle of the i^{th} product when the price increased (unit) 

Expected inventory per cycle of the i^{th} product when the price increased (unit) 

Expected number of units replenished per cycle in the special sale before price increases of the i^{th} product (Unit/order) 

Expected number of units sold per cycle in the special sale before price increases of the i^{th} product (Unit) 

Expected backordered quantity per cycle in the special sale of the i^{th} product (unit) 

Expected lost sale quantity per cycle in the special sale of the i^{th} product (unit) 

Expected inventory per cycle in the special sale of the i^{th} product (unit) 

Expected cyclic profit without ordering a special sale ($) 

Expected cyclic profit when special order is placed ($) 

Expected total saving function 
TS 
Unconstrained Modeling
If the retailer or buyer decided to place a special order, the total cost of the order is computed as follows. Taleizadeh, Zarei, & Sarker (2016) proposed the following total cost function when an order is placed:
(1) 

If the special order is not placed:
(2) 

To calculate the optimal size of the replenishment level, the difference in total costs must be maximized:
(3) 

(4) 
}{

By simplifying the equation (4):
(5) 

Where
(6) 

(7) 

(8) 

(9) 

(10) 

(11) 

Fig 1 The inventory system scheme for one product
Considering that both normal price period and increased price period are probabilistic, are computed as below.
Uniform distribution
When the time follows a uniform distribution, Isi, and total cost saving would be:
(12) 

(13) 

(14) 

(15) 


(16) 


(17) 



The value of are
(18) 

(19) 

(20) 

(21) 

(22) 

Then :

(23) 



(24) 



(25) 



(26) 



(27) 





In the total saving equation described above, the are considered constant and do not have any effect on the concavity of the functions. TS is a quadratic equation and its second derivative is negative (see Appendix). Therefore, the concavity of profit function is proven. To obtain the optimal value of replenishment level value, the firstorder derivative of TS must be equal to zero, therefore:
(28) 

Exponential distribution
Similar to what was done above, when time follows exponential distribution with (replenishments/period) the , in total cost saving are computed as below:
(29) 

(30) 

(31) 

(32) 

(33) 

(34) 

(35) 

(36) 

(37) 

(38) 

(39) 

(40) 

(41) 

(42) 

(43) 

(44) 

Similar to a uniform distribution, the are constant and do not have any effect on the concavity. It is proved that this function is concave (see appendix). Therefore, the optimal value of the function is given by the first derivative equal to zero.
(45) 

Solution method
Similar to the approach of Taleizadeh, Zarei, & Sarker (2016), KarimiNasab & Wee (2015), the following lemma is used to obtain decision variables:
Constrained model
As the total available warehouse space is M, the space required for each unit of product is M_{i}, and the upper limits for inventory in various periods are . In summary, the complete mathematical model would be:

(46) 
Maximize TS



(47) 



(49) 


(50) 


Note that the TS (above) varies in two distributions and follows equations (17) and (44) for uniform and exponential distributions.
Solution method
Constrained nonlinear optimization is a key form of the problem in the fields of economics, management, and engineering. Mathematical programming and metaheuristic methods are two common ways to solve these types of problems. Mathematical programming can obtain solutions with higher accuracy as compared to the metaheuristic method but in large scale and NPhard problems, consume a lot of time So during the last decades, a wide variety of metaheuristic algorithms have been designed and applied to solve the constrained nonlinear optimization problems. GA and PSO are typical examples of these algorithms that have strengths and weaknesses (Talezadeh et al., 2013b; AlejoReyes, 2020). In this paper, GA and PSO algorithms are used to solve the problem, and then, their accuracy, performance, and timeconsuming are compared.
GA initial population
GA chromosomes or candidate solutions for the i^{th} product are the maximum inventory levels in three periods, i.e., normal price, the announcement of price increase after the price increased. Therefore, one chromosome or string for 10 items is a10*3 matrixes. The positive real numbers are randomly generated in each matrix to meet constraints. Hence, N chromosomes are generated for the initial population:




i=1 
R_{1} 
R_{S1} 
R_{k1} 
i=2 
R_{2} 
R_{S2} 
R_{k2} 
i=3 
. 
. 
. 
i=4 
. 
. 
. 
i=5 
. 
. 
. 
i=6 
. 
. 
. 
i=7 


. 
i=8 



i=9 



i=10 



Fig 2 The structure of a chromosome
GA crossover operation
To perform crossover operation, there are two common algorithms including singlepoint crossover and multiple crossovers point like in real organisms. Therefore, a submatrix from the parent (1) is randomly selected and then, the permutation is copied from the chromosome of the first paren. Finally, the chromosome of the second parent is scanned and if the number is not yet in the offspring, it would be added. Repeatedly, the second child is also made by using the same procedure just as for the first child. (Fig. 3)





Fig 3 The crossover operation
GA mutation operation
A mutation is performed by making minor changes in the mutated chromosomes. To do this, a random number RN between (0,1) is generated for each gene. If RN is less than a predetermined mutation probability Pm, then the mutation occurs in the gene. Otherwise, it does not. In this research, 0 and10, are chosen as the values of Pm. Note that infeasible chromosomes resulting from this operation do not move to the new population.
GA
Objective function: f(x)
*Note that the algorithm stops until a maximum number of 500 iterations is reached.
Particle Swarm Optimization (PSO)
PSO is a technique based on swarm (population) and particles. In this method, each particle is a possible solution to the problem; and moves around in a multidimensional search space. Each particle changes its position according to its location and the position of its neighboring particles. PSO tries to find the optimal solution by moving particles and evaluating the fitness of their new position.
Initial population
The initial population in PSO is generated by creating a particle. In this research, the particles are similar to chromosomes in the GA algorithm. It means that they are from a 3*10martix for 10 products. Every particle has its position and velocity.
PSO fitness function
Fitness functions in PSO algorithms are considered objective functions.
Velocity and Position
After the initialization stage, every particle must be updated by its best local position and also its best global position:
(51) 


(52) 


Where ‘t’ is the previous iteration, c_{1} and c_{2} are the individuals and global learning rates, r_{1} and r_{2} are uniformly random numbers in ranges U= [0 1], and is the inertia weight. Commonly, the values of c_{1} and c_{2} are set equal to 2. Then, two multiples by r_{1}, and r2 contribute to the social and personal experience equal to each particle. Intertie weight is a mechanism for controlling exploration and exploitation, i.e., the contribution of the previous velocity. At the first stage, its value is close to 1 form more exploration and other iterations reduced for more exploitation:
PSO algorithm
*Note that the algorithm stops until a maximum number of 500 iterations is reached.
Unconstraint problem
To illustrate the application of the abovementioned solution procedure, we will use numerical examples. The parameters of examples are addressed in Table 1.
Table 1 Parameters of the model
i 
D 
H 
π 
π' 
C 
C_{K} 
π'_{K} 
H_{K} 
P 
α 
l_{min} 
l_{max} 
λ 
1 
70 
8.00 
1.01 
29.00 
40.00 
50.000 
19.000 
10.000 
68.00 
0.7 
0.1 
0.4 
5 
2 
76 
7.22 
1.03 
26.27 
36.10 
45.125 
17.245 
9.025 
61.37 
0.8 
0.2 
0.4 
5 
3 
82 
6.44 
1.05 
23.54 
32.20 
40.250 
15.490 
8.050 
54.74 
0.8 
0.2 
0.4 
5 
4 
88 
5.66 
1.07 
20.81 
28.30 
35.375 
13.735 
7.075 
48.11 
0.7 
0.2 
0.4 
10 
5 
94 
4.88 
1.09 
18.08 
24.40 
30.500 
11.980 
6.100 
41.48 
0.7 
0.2 
0.4 
5 
6 
100 
4.10 
1.11 
15.35 
20.50 
25.625 
10.225 
5.125 
34.85 
0.5 
0.1 
0.4 
10 
7 
106 
3.32 
1.13 
12.62 
16.60 
20.750 
8.470 
4.150 
28.22 
0.7 
0.1 
0.2 
5 
8 
112 
2.54 
1.15 
9.89 
12.70 
15.875 
6.715 
3.175 
21.59 
0.6 
0.1 
0.2 
10 
9 
118 
1.76 
1.17 
7.16 
8.80 
11.000 
4.960 
2.200 
14.96 
0.7 
0.1 
0.2 
5 
10 
124 
0.98 
1.19 
4.43 
4.90 
6.125 
3.205 
1.225 
8.33 
0.5 
0.1 
0.3 
10 
*Note that: θ=0.1، A=50
The optimal values for uniform and exponential distributions are addressed in Table 2.
Table 2 The optimal solutions of the unconstrained model
Uniform 
Exponential 

R_{S} 
Q_{1S} 
TS 
R_{S} 
Q_{1S} 
T_{S} 

1 
25.8518 
17.7291 
210.1595 
40.8695 
20.7984 
181.3497 
2 
28.8247 
23.0648 
267.1384 
44.4957 
35.3571 
215.7841 
3 
31.106 
24.8912 
255.525 
48.0696 
49.923 
206.0894 
4 
33.366 
26.8079 
214.727 
31.6774 
31.8496 
73.3584 
5 
35.6519 
28.6414 
197.0898 
55.2122 
79.0533 
145.7149 
6 
36.9305 
25.4855 
138.0554 
35.9423 
46.2197 
52.5992 
7 
20.6394 
16.0665 
69.8089 
62.7084 
108.1936 
109.4345 
8 
21.8083 
16.9869 
55.6206 
40.7645 
60.6251 
38.0602 
9 
23.0089 
17.893 
40.897 
71.1347 
137.3088 
61.4744 
10 
35.4927 
25.2401 
32.0431 
46.7342 
75.0152 
15.2866 
Constrained problem
In this section, the same parameters in the previous section are considered and both GA and PSO methods are used to obtain the optimal solution. The specific parameters of those algorithms are presented in Table 3. All of these parameters are obtained by trial and error.
Table 3 parameters of the algorithm
Values 
PSO Parameters 
Values 
GA Parameters 
100 
Number of initial particles 
100 
Initial population size 
The decrease from 0.9 to 0.3 
ω 
0.9 
Pc 
2.0 
C1 
0.1 
Pm 
2.0 
C2 
Reach maximum iteration 
Stop criteria 
Reach maximum iteration 
Stop criteria 
In this section, multiple problems are designed and solved in different sizes, which can be classified into small, medium, and large categories. The first category includes 10 and 20 products; the second category includes 80 and 100 products, and the third category includes 400 and 500 products. The following table shows the differences between time and cost for both uniform and exponential distributions. It should be noted that all values in the tables are written after 10 iterations of each problem (each problem has 500 iteration loops) and its best mode are displayed. In Table 4 and Figure 4, the results are compared.
Table 4 The optimal solutions of the constrained model
Uniform distribution 

Number of products 
GA 
PSO 

TS($) 
Time(s) 
TS($) 
Time(s) 

10 
1478.9625 
14.0776 
994.9977 
35.7438 
20 
1509.8552 
15.3666 
1071.4022 
36.0521 
80 
1658.9988 
17.2976 
1481.4763 
39.0854 
100 
1924.2439 
17.5966 
1744.0593 
39.3118 
400 
31649.2948 
26.0723 
30956.226 
49.4003 
500 
41441.7108 
28.5222 
41099.5035 
54.1311 
Exponential distribution 

Number of products 
GA 
PSO 

TS($) 
Time(s) 
TS($) 
Time(s) 

10 
3348.9605 
16.1923 
1750.0724 
36.3512 
20 
5938.2014 
16.9589 
3604.1214 
36.9368 
80 
13166.0403 
21.6333 
7319.1813 
42.1105 
100 
13536.1344 
23.4927 
8768.8806 
44.1671 
400 
82371.4125 
46.2719 
59179.1205 
70.2324 
500 
104280.3503 
53.6850 
76239.5825 
77.8652 


(a) 
(b) 
Fig. 4 Comparison of optimal solutions by GA and PSO for (a)uniform distribution and (b)exponential distribution
The selection of parameters is a significant issue in the decisionmaking context. Thus, to analyze the effects of changes on the maximum value of inventory levels, the order quantity, and total profit, some sensitivity analysis is performed and the results are shown in Tables 4 and 5. The values of each parameter are changed from +75% to 75% for a single product, regardless of the space constraints. Better displaying parameter changes and their effects are shown in Figures 5 and 6.
Table 5 The results of sensitivity analysis for uniform distribution
Parameters 
Change (%) 
Values 
Change (%) 

Q_{S} 
R_{S} 
T(R^{*}_{S}) 
Q_{S} 
R_{S} 
T(R^{*}_{S}) 

D 
75 
18.6 
23.8 
195.8 
0.754716981 
0.75 
0.7497766 
50 
15.9 
20.4 
167.8 
0.500000000 
0.50 
0.4995532 

25 
13.2 
17.0 
139.9 
0.245283019 
0.25 
0.2502234 

0 
10.6 
13.6 
111.9 
0.000000000 
0.00 
0.0000000 

25 
7.9 
10.2 
83.9 
0.254716981 
0.25 
0.2502230 

50 
5.3 
6.8 
55.9 
0.500000000 
0.50 
0.5004470 

75 
2.6 
3.4 
27.9 
0.754716981 
0.75 
0.750670 

P 
75 
10.6 
13.6 
108.7 
0.000000000 
0.00 
0.0285970 
50 
10.6 
13.6 
108.8 
0.000000000 
0.00 
0.0277030 

25 
10.6 
13.6 
109.1 
0.000000000 
0.00 
0.0250220 

0 
10.6 
13.6 
111.9 
0.000000000 
0.00 
0.0000000 

25 
 
 
 
 
 
 

50 
 
 
 
 
 
 

75 
 
 
 
 
 
 

h_{k} 
75 
10.6 
13.6 
125.1 
0 
0 
0.1179625 
50 
10.6 
13.6 
120.6 
0 
0 
0.0777480 

25 
10.6 
13.6 
116.2 
0 
0 
0.0384272 

0 
10.6 
13.6 
111.9 
0 
0 
0.0000000 

25 
10.6 
13.6 
107.6 
0 
0 
0.0384270 

50 
10.6 
13.6 
103.5 
0 
0 
0.0750670 

75 
10.6 
13.6 
99.5 
0 
0 
0.1108130 

[lmin lmax] 
75 
13.3 
18.3 
146.4 
0.254716981 
0.3455882 
0.3083110 
50 
12.4 
16.7 
134.5 
0.169811321 
0.2279412 
0.2019660 

25 
11.5 
15.2 
123 
0.084905660 
0.1176471 
0.0991957 

0 
10.6 
13.6 
111.9 
0.000000000 
0.0000000 
0.0000000 

25 
9.7 
11.9 
101.1 
0.084905660 
0.1250000 
0.0965150 

50 
8.8 
10.3 
90.7 
0.169811321 
0.2426470 
0.1894550 

75 
7.9 
8.6 
80.7 
0.254716981 
0.3676470 
0.2788200 

θ 
75 
10.7 
13.5 
114 
0.009433962 
0.0073530 
0.0187668 
50 
10.6 
13.5 
113.3 
0.000000000 
0.0073530 
0.0125112 

25 
10.6 
13.5 
112.6 
0.000000000 
0.0073530 
0.0062556 

0 
10.6 
13.6 
111.9 
0.000000000 
0.0000000 
0.0000000 

25 
10.5 
13.6 
111.2 
0.009433962 
0.0000000 
0.0062560 

50 
10.5 
13.6 
110.5 
0.009433962 
0.0000000 
0.0125110 

75 
10.5 
13.7 
109.9 
0.009433962 
0.0073529 
0.0178730 
In figure 5, a comparison is made for the sensitivity results of changing demand rates in uniform distribution.
Fig. 5 The sensitivity results in changes in the Demand rate
Table 6 The results of sensitivity analysis for exponential distribution
Parameters 
Change (%) 
Values 
Change (%) 

Q_{S} 
R_{S} 
T(R^{*}_{S}) 
Q_{S} 
R_{S} 
T(R^{*}_{S}) 

D 
75 
25.1 
71.4 
317.50 
0.755245 
0.750000 
0.751241 
50 
21.5 
61.2 
272.10 
0.503497 
0.500000 
0.500827 

25 
17.9 
51.0 
226.70 
0.251748 
0.250000 
0.250414 

0 
14.3 
40.8 
181.30 
0.000000 
0.000000 
0.000000 

25 
10.7 
30.6 
135.90 
0.251750 
0.250000 
0.250410 

50 
7.1 
20.4 
90.50 
0.503500 
0.500000 
0.500830 

75 
3.5 
10.2 
45.10 
0.755240 
0.750000 
0.751240 

p 
75 
14.3 
40.8 
135.60 
0.000000 
0.000000 
0.252070 
50 
14.3 
40.9 
141.20 
0.000000 
0.002451 
0.221180 

25 
14.3 
40.9 
152.10 
0.000000 
0.002451 
0.161060 

0 
14.3 
40.8 
181.30 
0.000000 
0.000000 
0.000000 

25 
 
 
 
 
 
 

50 
 
 
 
 
 
 

75 
 
 
 
 
 
 

h_{k} 
75 
14.3 
40.9 
225.30 
0.000000 
0.002451 
0.242692 
50 
14.3 
40.9 
211.70 
0.000000 
0.002451 
0.167678 

25 
14.3 
40.9 
197.10 
0.000000 
0.002451 
0.087148 

0 
14.3 
40.8 
181.30 
0.000000 
0.000000 
0.000000 

25 
14.3 
40.8 
164.20 
0.000000 
0.000000 
0.094320 

50 
14.3 
40.7 
145.70 
0.000000 
0.002450 
0.196360 

75 
14.3 
40.6 
125.40 
0.000000 
0.004900 
0.308330 

λ 
75 
8.1 
27.6 
96.560 
0.433570 
0.323530 
0.467400 
50 
9.5 
30.8 
114.81 
0.335660 
0.245100 
0.366740 

25 
11.4 
35.0 
140.99 
0.202800 
0.142160 
0.222340 

0 
14.3 
40.8 
181.35 
0.000000 
0.000000 
0.000276 

25 
19.1 
49.4 
250.46 
0.335664 
0.210784 
0.381467 

50 
28.7 
63.6 
390.73 
1.006993 
0.558824 
1.155157 

75 
56.9 
93.6 
784.20 
2.979021 
1.294118 
3.325427 

θ 
75 
14.5 
37.2 
185.30 
0.013986 
0.088240 
0.022063 
50 
14.5 
38.3 
184.10 
0.013986 
0.061270 
0.015444 

25 
14.4 
39.5 
182.80 
0.006993 
0.031860 
0.008274 

0 
14.3 
40.8 
181.30 
0.000000 
0.000000 
0.000000 

25 
14.2 
42.3 
179.70 
0.006990 
0.036765 
0.008830 

50 
14.1 
44.0 
178.00 
0.013990 
0.078431 
0.018200 

75 
14 
45.9 
176.20 
0.020980 
0.125000 
0.028130 
In Figure 6, a comparison is made for the sensitivity results of changing λ in the exponential distribution.
Fig. 6 The sensitivity results in changes in λ
According to tables 5, and 6, when the demand rate (D) increases, the replenishment level (R_{s}), total saving T(R_{s}), and order quantity (Q_{s}) increase too. In other words, the replenishment level, order quantity, and total saving are highly sensitive to the demand rate. It means that it is more profitable to place an order when the demand increases. From tables 5 and 6 we can understand that when the price increases, the replenishment level, and order quantity doesn't change but saving cost decreases slightly, so, total saving and price are to some extent sensitive to each other. It is clear that after a price increase, the holding cost does not change replenishment level and order quantity, but directly affects the total saving. It means that the replenishment level and order quantity are not sanative but the total saving value is slightly sensitive to the changes in holding cost. According to Table 6, when the λ (mean number/year) in exponential distribution increases the replenishment level, order quantity, and total saving decrease. In other words, all three items are moderately sensitive to λ. Table 5 shows direct but little interaction between the maximum and the minimum amounts of allowable time in uniform distribution and replenishment level, order quantity, and total saving. Finally, according to Table 5,6, we can see that there are positive effects of deteriorating rate on profit and order quantity level but negative effects on replenishment level. It is worth noting that in all cases, the effects are low. Therefore, it must be said that the customer should use a specialorder policy when the orders include highdeterioration rate products.
6.1 Theoretical implications
According to previous research, including Taleizadeh et al. (2013c), Yang et al. (2015), KarimiNasab & Wee (2015), and Taleizadeh, Zarei, & Sarker (2016), it can be found that they did not include realworld constraints and more attention has been paid to the increase in price over time due to a random delivery period and a motivational policy. In several studies, there are no limitations and the problem is modeled and solved with integer decision variables and linear programming (Zeballos, Seifert & ProtopappaSieke, 2013; Sarkar & Moon, 2014; Giri & Sharma, 2016; Braglia, Castellano & Frosolini, 2016; Braglia, Castellano, & Song, 2017). Therefore, in studies that focus exclusively on known price increases, the gap is quite evident when a study is aimed at getting a better understanding of the realworld conditions (Cimen & Kirkbride, 2017). Therefore, this research seeks to consider goods that do not have a stable lifespan, as well as storage space limitations and problemsolving by using metaheuristics algorithms.
Metahubristic algorithms such as Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) can be used in inventory control to obtain optimal reorder points (Dye, 2012; Mousavi et al., 2014; Buhnia, Shaikh & Gupta, 2015; Bhunia & Shaikh, 2015; Vandani, Niaki, & Aslanzade, 2017; Akbari Kaasgari, Imani, & Mahmoodjanloo, 2017; Azadeh, 2017; Hiassat, Diabat & Rahwan, 2017; Tiwari, 2017). In general, scholars suggest that hybrid metaheuristic algorithms have gained considerable attention for their capability to solve difficult problems in different fields of science especially solving inventory problems. Due to the nonlinearity of the proposed model of this study, particle swarm optimization (PSO) and genetic algorithm (GA), were implemented as optimizing solvers instead of analytical methods (Talezadeh et al., 2013b; AlejoReyes et al., 2020).
The models presented in this study were solved with consideration of the relevant assumptions along with numerical examples. The derivation method was used to solve the unconstrained problem, and both the genetic algorithm and particle swarm optimization algorithm were used for the constrained problem. Accordingly, the optimum value of the model was calculated. Changes in profits, replenishment level, and order quantities were examined for some of the parameters, including demand, purchase price, and holding cost. Sensitivity analysis indicated that an increase in deterioration and demand rates leads to increased total profits. Also, the fewer the number of replenishment periods in one year, the more is costeffective it. It has also been shown that the genetic algorithm has the best ability to converge in comparison with the particle swarm algorithm, and in less time, it becomes more desirable.
In general, based on the findings of this research, managers and retailers would be able to have more effective plans for their inventory and replenishment levels under changing circumstances. The research model is a realworld inventory control problem that has been observed in many cases, such as small supermarkets, pharmacies, grocery stores, and so on. This model helps the suppliers decide when to visit and replenish the retail inventory. Hence, suppliers can visit retailers at irregular intervals. The purpose of this study was to determine the retailer’s optimal order quantity and maximize the benefits.
7.1 Research limitations and future study agenda
In this research, some parameters such as delay in payment, prepayment policies, and also financial constraints were not considered effective variables, hence they should be noted as the limitations of this study. The proposed model of this study could be extended in several ways. It may deal with the demand rate as a function of price, time, stock, etc., considering the delayed payment and advanced payment policy. It is also possible to use other metaheuristic algorithms, hybrid algorithms, considering fuzzy parameters, and adding more constraints to the model including limitations on order quantities and also financial constraints.