1.   Introduction

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 unconstrained-optimization 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), Karimi-Nasab & Wee (2015), and Taleizadeh, Zarei, & Sarker (2016), it can be seen that they did not include real-world 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 & Protopappa-Sieke, 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 real-world 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 problem-solving by using meta-heuristics 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.

2.   Literature review

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 multi-lot-size 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, time-dependent demand, inflation, and time value of money. Prekopa (2006) used the model which so-called 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, & Almada-Lobo (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 ramp-type demand. They formulated their model with some considerations such: as stock-dependent 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 two-warehouse 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) non-coincidence 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 back-ordering. 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 equal-size parts during a fixed lead time. Taleizadeh, et al (2013a) formulated and modeled the multiple partial prepayments of the EOQ problem with partial back-ordering. 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 back-ordering and prepayment policy. For inflation and the time value of money or deteriorating items, Singh, Kumar & Kumari (2011) developed a two-warehouse 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 multi-period with single item model with restricted order size. The model included a multi-period/multi-product 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.

Meta-hubristic 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 meta-heuristic 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 non-linearity 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; Alejo-Reyes, 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 back-ordering and fix–cost-ordering 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 back-order and emergency orders. Muthuraman, Seshadri, & Wu (2014) modeled an inventory system with stochastic demand and stochastic delivery lags as Quasi-Variation Inequality (QVI). Their model had an infinite-dimensional state-space 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 lot-sizing model in which, the time between two delivery was stochastic. Karimi-Nasab & 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.

3.   Problem definition and assumptions

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 time between two consecutive visits is stochastic and follows exponential and uniform distributions
• There are several products
• The level of inventory at the beginning of the spatial period is zero
• The supplier offers special sales
• The price increase is not temporary
• The shortage is in the partial back-ordering form
• Goods are deteriorating items and the deteriorating rate is constant
• The demand rate is constant
• There is a warehouse space restriction
• The holding cost per unit will increase after the price is increased
• There are holding, shortage, and purchase costs
• Only one provider exists

4.   Mathematical modeling

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:

Variables:

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:

If the special order is not placed:

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

By simplifying the equation (4):

Where

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:

The value of  are

Then :

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 first-order derivative of TS must be equal to zero, therefore:

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:

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.

Solution method

Similar to the approach of Taleizadeh, Zarei, & Sarker (2016), Karimi-Nasab & Wee (2015), the following lemma is used to obtain decision variables:

• for uniform and exponential distribution (see Appendix)
• calculate  for two distributions
• calculate
• calculate an optimal value for

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:

Note that the TS (above) varies in two distributions and follows equations (17) and (44) for uniform and exponential distributions.

Solution method

Constrained non-linear optimization is a key form of the problem in the fields of economics, management, and engineering. Mathematical programming and meta-heuristic methods are two common ways to solve these types of problems. Mathematical programming can obtain solutions with higher accuracy as compared to the meta-heuristic method but in large scale and NP-hard problems, consume a lot of time So during the last decades, a wide variety of meta-heuristic algorithms have been designed and applied to solve the constrained non-linear optimization problems. GA and PSO are typical examples of these algorithms that have strengths and weaknesses (Talezadeh et al., 2013b; Alejo-Reyes, 2020). In this paper, GA and PSO algorithms are used to solve the problem, and then, their accuracy, performance, and time-consuming 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:

Fig 2- The structure of a chromosome

GA crossover operation

To perform crossover operation, there are two common algorithms including single-point crossover and multiple crossovers point like in real organisms. Therefore, a sub-matrix 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)

1. Define Fitness F
2. Initialize population
3. Initialize probabilities of crossover (pc) and mutation (pm)
4. Do
5. Generate new solutions by crossover and mutation
6. Apply mutation and cross-over to each chromosome
7. Accept the new solution if its fitness increases
8. Select the current best for the next generation
9. While maximum iterations or minimum error criteria are not attained*

*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:

Where ‘t’ is the previous iteration, c₁ and c₂ are the individuals and global learning rates, r₁ and r₂ are uniformly random numbers in ranges U= [0 1], and  is the inertia weight. Commonly, the values of c₁ and c₂ are set equal to 2. Then, two multiples by r₁, 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

1. Objective function: f(x), x = (x₁, x₂, ..., x_n);
2. Initialize particle position and velocity for each particle and set k = 1.
3. Initialize the particle’s best-known position to its initial position
4. Do
5. Update the best-known position of each particle and swarm’s best-known position
6. Calculate particle velocity according to the velocity equation
7. Update particle position according to the position equation
8. While maximum iterations or minimum error criteria are not attained*

*Note that the algorithm stops until a maximum number of 500 iterations is reached.

5. Numerical example

Unconstraint problem

To illustrate the application of the above-mentioned solution procedure, we will use numerical examples. The parameters of examples are addressed in Table 1.

Table 1- Parameters of the model

*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

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

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

Fig. 4- Comparison of optimal solutions by GA and PSO for (a)uniform distribution and (b)exponential distribution

6. Discussion and sensitivity analysis

The selection of parameters is a significant issue in the decision-making 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

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

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 special-order policy when the orders include high-deterioration rate products.

6.1 Theoretical implications

According to previous research, including Taleizadeh et al. (2013c), Yang et al. (2015), Karimi-Nasab & Wee (2015), and Taleizadeh, Zarei, & Sarker (2016), it can be found that they did not include real-world 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 & Protopappa-Sieke, 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 real-world 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 problem-solving by using meta-heuristics algorithms.

Meta-hubristic 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 meta-heuristic algorithms have gained considerable attention for their capability to solve difficult problems in different fields of science especially solving inventory problems. Due to the non-linearity 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; Alejo-Reyes et al., 2020).

7. Conclusions

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 cost-effective 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 real-world 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, pre-payment 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 meta-heuristic algorithms, hybrid algorithms, considering fuzzy parameters, and adding more constraints to the model including limitations on order quantities and also financial constraints.