Application of hybrid binary tournament-based quantum-behaved particle swarm optimization on an imperfect production inventory problem

Abstract

Nowadays, use of various types of hybrid metaheuristic algorithms attracts the researchers to optimize the average profit or cost of an inventory system to avoid the local optimality due to high nonlinearity of the corresponding optimization problem. This paper deals with an application of binary tournament-based quantum-behaved particle swarm optimization algorithms on an imperfect production inventory problem with shortages. In order to reduce the production of defective items, modern/improvement technology has been incorporated in the production system. Also, the demand of the product is assumed to be dependent on its warranty period and selling price. The main objective of this study is to optimize the production rate, production period, selling price of the product, manufacturer’s improvement technology level and maximum shortage level as well as maximize the average profit of the production system. For this purpose, three hybrid metaheuristic algorithms based on binary tournamenting and different variants of quantum-behaved PSO techniques have been developed. Then to examine the validity of the proposed model, three numerical examples have been solved. Considering each example, nonparametric statistical tests have been performed by using four different methods to analyze the performance of the used algorithms. Finally, sensitivity analyses have been performed to investigate the effects of different parameters on optimal policy.

1 Introduction

In any manufacturing firm, all the produced items are not perfect due to imperfect production process or other factors. Considering this realistic situation of production, several researchers have developed various production models and reported in the existing literature. In 1986, Rosenblatt and Lee (1986) first introduced the concept of imperfect production in manufacturing system. Salameh and Jaber (2000) formulated an imperfect production problem. Sana et al. (2007) uplifted a production problem with volume flexible cost under imperfect production system. Sarker and Moon (2011) presented a model corresponding to the imperfect production system with development cost investment. Some other interesting research works related to imperfect production process were found in in the works of Chiu (2003), Goyal and Cárdenas-Barrón (2005), Modak et al. (2015), Das et al. (2017), Manna et al. (2017a), Mallick et al. (2018), among others. Jain et al. (2018) introduced repairing concept in the imperfect production system under fuzzy environment. Taleizadeh et al. (2019) examined product quality and returns in an imperfect production system under two warranty policies. Manna et al. (2019) established two-plant production model for two quality items under fuzzy environment. Rahman et al. (2020) developed a production inventory model with credit-linked demand in interval environment. They also assumed the produced products which deteriorate with time. Shaikh et al. (2020) developed an EPQ model with partial trade credit policy- and price-dependent demand for deteriorating items. Mishra et al. (2021) proposed the concept of preservation technology in a production inventory model to protect the items from deterioration.

Sometimes, it is observed that stock-out situation arises in integrated production system due to uncertain demand, offering of discount facility, deterioration effect of produced items, etc. As a result, in stock-out situation, manufacturer cannot able to fulfill the demand of customers/retailers. During stock-out period, two situations may occur: (i) all the customers are willing to wait for receiving the product and (ii) a part of the customers only are willing to wait for receiving the product. The second case of situation is known as partial backlogging. Abad (2000) developed a lot-size problem with partial backordering for perishable items. An inventory model for deteriorating items was developed by Giri et al. (2003). Ouyang and Chang (2013) proposed an optimal production model with complete backlogging and permissible delay in payments. In interval environment, a partially integrated production model with variable demand and partial backordering was introduced by Bhunia et al. (2017). Shaikh et al. (2017) established an inventory model for non-instantaneous deteriorating items with price- and stock-dependent demand under partially backlogged situation. Tiwari et al. (2018) investigated a green production problem with partial backordering for multi-items. Das et al. (2020) proposed an inventory model with partially backlogging and price-dependent demand for deteriorating items considering preservation facilities. Later, Das et al. (2021) developed an inventory model with partial backlogged shortages and trade credit financing under preservation technology for deteriorating items via particle swarm optimization. Apart from the earlier mentioned works, several researchers, viz. Jamal et al. (1997), Chiu (2003), Chen and Lo (2006), Chakraborty et al. (2013), studied different inventory models with complete backlogging/ partial backlogging.

The classical inventory model was developed under the assumption constant demand. After that, a number of researchers reported various types of customer’s/retailer’s demand dependent on several factors such as Shaikh et al. (2019)(stock-dependent demand), Giri et al. (2003) (ramp-type demand), Jain et al. (2018) (time-dependent demand), Jaggi et al. (2017) (price discount demand), Lee and Yao (1998) (fuzzy demand), Manna et al. (2017a) (advertisement-dependent demand), Dye and Yang (2015) (credit-linked demand), etc. However, it is very difficult to estimate the market demand due to the lack of historical data. The warranty period is an vital issue to take the customers’ decision for purchasing the product. Yeh et al. (2005) proposed warranty policy for repairable items. Wu et al. (2009) optimized price, warranty length, production rate in a production inventory model. Wang and Sheu (2003) considered free warranty policy in their production model. Chung (2013) considered production model where the demand is dependent on warranty period of the product. Taleizadeh et al. (2017) introduced warranty policy in a supply chain model. Recently, Manna et al. (2020) investigated the effects of warranty period and selling price of the product on customers’ demand in a manufacturing system. In Table 1, a comprehensive review of related articles reported in the literature is presented.

Table 1 Summary of the literature, based on various assumptions related to the proposed model

To optimize the average profit/cost of an inventory model, different methods can be applied such as,

• (i) Direct search method

• (ii) Gradient-based method

• (iii) Metaheuristic method

However, in the proposed work, the optimization (maximization) problem corresponding to the proposed production inventory model is highly nonlinear in nature and non-concave. So, this optimization problem cannot be solved by traditional direct and gradient-based optimization methods. These methods have some limitations. Among these limitations, one is that the traditional nonlinear optimization methods very often stuck to the local optimum. So, authors are bound to choose metaheuristic methods. All metaheuristic algorithms have been developed from the activities of the social organisms, properties of environments, properties of some instruments of physics, etc. Over the previous few decades, various nature-inspired algorithms have been proposed such as genetic algorithm (Goldberg 2006), particle swarm optimization (Eberhart and Kennedy 1995; Clerc and Kennedy 2002; Sun et al. 2005, 2011; Xi et al. 2008; Coelho 2010), krill herd algorithm (Abualigah 2019), grasshopper optimization algorithm (Abualigah and Diabat 2020), arithmetic optimization algorithm (Abualigah et al. 2021a), sine cosine algorithm (Abualigah and Diabat 2021b), differential evolution algorithm (Storn and Price 1997), tournament differential evolution algorithm (Akhtar et al. 2020). Besides, some modified versions of these algorithms are proposed (Duary et al. 2020; Kumar et al. 2019, 2020, 2021a, b). Surprisingly, PSO and its modified versions have been confoundedly used to optimize the average profit/cost of an inventory model in the recent years.

Particle swarm optimization (PSO) was proposed by Eberhart and Kennedy (1995). Later, Clerc and Kennedy (2002) modified the original PSO algorithm by inserting a constriction factor. Since then, the corresponding PSO is known as PSO-Co. Thereafter, Sun et al. (2004) proposed a modified PSO algorithm known as quantum-behaved PSO (QPSO) which is based on the quantum behavior of the particles. Then, to accelerate the performance of QPSO, Xu and Sun (2005) developed adaptive QPSO (AQPSO), Xi et al. (2008) developed Weighted QPSO (WQPSO), Coelho (2010) developed Gaussian QPSO (GQPSO), Kumar et al. (2019) developed AGQPSO, etc. In the current work, only three embedded QPSO algorithms, AQPSO, GQPSO and AGQPSO algorithms, are used to solve the optimization problem. Apart from these algorithms, by using these algorithms, we have developed three different algorithms based on the concept of binary tournamenting process which is followed in a game. These are called as T2-AQPSO, T2-GQPSO and T2-AGQPSO. Finally, the results are compared with each other and said PSOs.

In this paper, an imperfect production inventory model with partial backlogging and dynamic demand has been developed. Here production of defective items has been reduced by considering modern/improvement technology. Also, the partial backorder rate is dependent on the length of the waiting time of the customers. Furthermore, the demand rate of the customers is assumed to be dependent on warranty period and selling price of the product. Then, six metaheuristic optimization techniques AQPSO, GQPSO, AGQPSO, T2-AQPSO, T2-GQPSO and T2-AGQPSO have been used for solving the corresponding maximization problem (average profit) of the proposed model and compared the results obtained.

The leftover of this work is constructed as follows. The next section presents notation and assumptions for formulating the production inventory model. Mathematical formulation of the proposed model is provided in Sect. 3. Section 4 demonstrates the solution methodology to determine the optimal/best found solutions. Numerical experiments and sensitivity analyses are shown in Sects. 5 and 6, respectively. Managerial insights and conclusions are drawn in Sects. 7 and 8, respectively. Finally, limitations with future research scope of this work are presented in Sect. 9.

2 Notation and assumptions

The following notation and assumptions have been considered throughout the manuscript.

2.1 Notation

Decision Variables
P	:	Production rate
$t_p$	:	Duration of production run time
$\eta$	:	Manufacturer’s improvement technology level to reduce the production of defective units
s	:	Selling price per unit for perfect quality item
$Q_s$	:	Maximum shortage level
Dependent Variable
$T_1$	:	Time point when the inventory level reaches zero
$T_2$	:	Time point when production starts to backlogged
T	:	Time point of business period
Others
I(t)	:	Inventory level of perfect quality items
$\theta (\eta )$	:	Reduced defective production rate, a decreasing function with respect to $\eta$
$\theta _0$	:	Defective production rate without development technology
$\omega _p$	:	Warranty period of the product which is sold to the customer
$C(P, \eta )$	:	Production cost per unit
$D(\omega _p, s)$	:	Demand rate of the retailer
$\alpha$	:	Partial backorder rate
$w_p$	:	Warranty period of the product
h	:	Holding cost/ unit / unit time
A	:	Setup cost/ cycle
$c_b$	:	Backordering cost/ unit / unit time
$N_p$	:	Swarm size
$Max\_{gen}$	:	Maximum number of generations

2.2 Assumptions

• (i) The production system produces single item and time horizon is infinite.

• (ii) The production system produces perfect item at the rate $(1-\theta )P$ where $0<\theta<<1$. During the production period $(0, t_p)$, the manufacturing company will invest modern/improvement technology cost to reduce the defective production. The rate of defectiveness of produced items is defined as follows:

  $$\begin{aligned} \theta (\eta )=\theta _0 e^{-\xi \eta },&\eta \in [0, +\infty ) \end{aligned}$$

  (1)

  which is a decreasing function with respect to $\eta$ for suitable value of $\xi >0$. The graphical representation of defective rate vs modern/improvement technology is given in Fig. 1.

• (iii) The demand of an item is dependent on warranty period along with selling price, and mathematically it can be represented as follows:

  $$\begin{aligned} D(\omega _p, s)=\beta _1 +\lambda _1 \omega _p - \lambda _2 s, \end{aligned}$$

  (2)

  where $\beta _1$ is a fixed demand of customers and $\lambda _1$, $\lambda _2$ are coefficients of sensitivity of the customers about warranty period and selling price of the product.

• (iv) The unit production cost is a function of production rate and modern/improvement technology, which is given by

  $$\begin{aligned} C(P, \eta )=C_0+C_1P^\lambda +\frac{C_2}{P^\mu }+C_3\eta ^\delta +C_4\omega _p^\gamma \end{aligned}$$

  (3)

  where $C_0$ is the fixed production cost, $C_1$ and $C_2$ are coefficients of sensitivity of production cost. Also, $C_3$ and $C_4$ are coefficients of sensitivity of warranty cost and development cost, respectively. The geometrical representation of unit production cost with respect to $\eta$ and P is given in Figs. 2 and 3, respectively.

• (v) The warranty cost ($c_w$) is dependent on warranty period ($\omega _p$) and is given by

  $$\begin{aligned} c_w(\omega _p)=a+b\omega _p \end{aligned}$$

  (4)

  where a is a fixed warranty cost and b is the coefficient of sensitivity of the warranty period.

• (vi) During the stock-out period, some of the customers are willing to wait for receive the product. Here, the backlogging rate is in the form $\frac{1}{1+\alpha \Big \{T_2-t-\frac{I(t)}{(1-\theta )P}\Big \}}$, where $\alpha$ is the parameter of backlogging rate.

Fig. 1

Pictorial representation of defective rate vs $\eta$

Fig. 2

Production cost vs $\eta$

Fig. 3

Production cost vs P

3 Mathematical formulation of the proposed model

Let us assume that a manufacturing firm starts the production at time $t=0$ and continues up to the time $t=t_p$ with the production rate P. During the production time, manufacturing firm produces some defective items along with perfect ones. Here, it has been considered that the production firm produces perfect items with the rate $(1-\theta )P$ up to the time $t=t_p$ which satisfy the customers’ demand with the rate $D(\omega _p, s)$. The rest of the produced items are stored in the store room with the rate $(1-\theta )P-D(\omega _p, s)$ up to the time $t=t_p$. After that, the production process stops and the inventory level depletes gradually during the time interval $[t_p, T_1]$ due to customers’ demand only. Thereafter, shortages occur and continue up to the time $t=T_2$. At time $t=T_2$, again production process starts and fulfills the backlogged quantity after meeting up the customers’ demand.

Fig. 4

Pictorial representation of the inventory level

Therefore, the inventory level satisfies the governing differential equations as follows:

$$\begin{aligned} \frac{dI(t)}{dt}= & {} (1-\theta )P-D(\omega _p, s), \quad 0\le t \le t_p \end{aligned}$$

(5)

$$\begin{aligned} \frac{dI(t)}{dt}= & {} -D(\omega _p, s), \quad t_p< t\le T_1 \end{aligned}$$

(6)

$$\begin{aligned} \frac{dI(t)}{dt}= & {} -\frac{D(\omega _p, s)}{1+\alpha \Big \{T_2-t-\frac{I(t)}{(1-\theta )P}\Big \}}, \quad T_1< t\le T_2 \end{aligned}$$

(7)

$$\begin{aligned} \frac{dI(t)}{dt}= & {} (1-\theta )P-\frac{D(\omega _p, s)}{1-\frac{\alpha I(t)}{(1-\theta )P}}, \quad T_2< t\le T \end{aligned}$$

(8)

subject to the conditions that $I(0)=0$, $I(T_1)=0$, $I(T_2)=-Q_s$ and $I(T)=0$.

Using the conditions $I(0)=0$, $I(T_1)=0$ and $I(T)=0$, the solutions of the equations $(5) - (8)$ are, respectively, as follows:

$$\begin{aligned} I(t)= & {} \Big \{(1-\theta )P-D(\omega _p, s)\Big \}t, \quad 0\le t \le t_p \end{aligned}$$

(9)

$$\begin{aligned} I(t)= & {} D(\omega _p, s)(T_1-t), \quad t_p< t\le T_1 \end{aligned}$$

(10)

$$\begin{aligned}&\alpha I(t) + \Big [(1-\theta )P\Big \{1+\alpha (T_2-T_1)\Big \}-D(\omega _p, s)\Big ]\nonumber \\&exp\Big \{\frac{\alpha I(t)}{D(\omega _p, s)}\Big \}=\Big [(1-\theta )P\Big \{1+\alpha (T_2-t)\Big \}\nonumber \\&\quad D(\omega _p, s)\Big ],\,\,\,\,T_1< t\le T_2 \end{aligned}$$

(11)

$$\begin{aligned} I(t)- & {} \frac{D(\omega _p, s)}{\alpha }log\Big |\frac{\Big \{(1-\theta )P-D(\omega _p, s)-\alpha I(t)\Big \}}{\Big \{(1-\theta )P-D(\omega _p, s)\Big \}}\Big |\nonumber \\= & {} (1-\theta )P(t-T), \quad T_2< t\le T \end{aligned}$$

(12)

The continuity condition of I(t) at $t=t_p$ implies

$$\begin{aligned} T_1=\frac{P}{D(\omega _p, s)}(1-\theta )t_p \end{aligned}$$

(13)

Again, the condition $I(T_2)=-Q_s$ implies

$$\begin{aligned} T_2= & {} T_1+\frac{1}{\alpha (1-\theta )P}\Big [\Big \{(1-\theta )P-D(\omega _p, s)+\alpha Q_s\Big \}\nonumber \\&exp\Big \{\frac{\alpha Q_s}{D(\omega _p, s)}\Big \}-(1-\theta )P+D(\omega _p, s)\Big ] \end{aligned}$$

(14)

The continuity condition of I(t) at $t=T_2$ implies

$$\begin{aligned}&\!\!\!T=T_2+\frac{1}{(1-\theta )P}\left[ Q_s+\frac{D(\omega _p, s)}{\alpha }\right. \nonumber \\&\!\!\!\quad \left. log\Big |\frac{\Big \{(1-\theta )P-D(\omega _p, s)+\alpha Q_s\Big \}}{\Big \{(1-\theta )P-D(\omega _p, s)\Big \}}\Big |\right] \end{aligned}$$

(15)

$$\begin{aligned}&\!\!\!\text{ Production } \text{ cost } \text{(PC) }=C(\eta , \omega _p)\Big [\int _0^{t_p}P\,dt + \int _{T_2}^TP\,dt\Big ]\nonumber \\&\!\!\!=\Big (C_0+C_1P^\lambda +\frac{C_2}{P^\mu }+C_3\eta ^\delta +C_4\omega _p^\gamma \Big ) P(t_p+T-T_2)\nonumber \\&\!\!\!\text{ Holding } \text{ cost } \text{(HC) }=h\Big [\int _0^{t_p}I(t)\,dt + \int _{t_p}^{T_1}I(t)\,dt\Big ]\nonumber \\&\!\!\!=\frac{h}{2}\Big [(1-\theta )Pt_p^2+D(\omega _p, s)T_1^2-2D(\omega _p, s)t_pT_1\Big ]\nonumber \\&\!\!\! \text{ Backorder } \text{ cost } \text{(BC) } = c_b\Big [\int _{T_1}^{T_2}-I(t)\,dt+\int _{T_2}^T-I(t)\,dt\Big ]\nonumber \\&\!\!\!\text{ Warranty } \text{ cost } \text{(WC) } =(a+b\omega _p)\vartheta (1-\theta )P\nonumber \\&\!\!\! \text{ Sales } \text{ revenue } \text{(SR) }=s(1-\theta )P(t_p+T-T_2) \end{aligned}$$

(16)

The total profit of the manufacturer can be calculated as follows:

$$\begin{aligned} TP(P, t_p, \eta , s, Q_s)= & {} \text{ SR-PC-HC-BC-WC-A }\nonumber \\= & {} s(1-\theta )P(t_p+T-T_2)\nonumber \\&-\Big (C_0+C_1P^\lambda +\frac{C_2}{P^\mu }+C_3\eta ^\delta \nonumber \\&+C_4\omega _p^\gamma \Big ) P(t_p+T-T_2)\nonumber \\&-\frac{h}{2}\Big [(1-\theta )Pt_p^2+D(\omega _p, s)T_1^2\nonumber \\&-2D(\omega _p, s)t_pT_1\Big ]\nonumber \\&-(a+b\omega _p)\vartheta (1-\theta )P\nonumber \\&-c_b\Big [\int _{T_1}^{T_2}-I_3(t)\,dt\nonumber \\&+\int _{T_2}^T-I_4(t)\,dt\Big ]-A \end{aligned}$$

(17)

Therefore, the average profit of the manufacturer is given by

$$\begin{aligned} \Pi (P, t_p, \eta , s, Q_s)= & {} \frac{TP(P, t_p, \eta , s, Q_s)}{T} \end{aligned}$$

(18)

Hence, the objective is to determine the optimal production period ($t_p^*$), production rate ($P^*$), selling price ($s^*$), maximum shortage level ($Q^*_s$) and development technology level ($\eta ^*$) by maximizing the manufacturer’s average profit $\Pi (\eta , \omega _p, t_p, P, Q_s)$.

Therefore, the corresponding optimization problem is as follows:

$$\begin{aligned}&\text{ Maximize }\,\, \Pi (P, t_p, \eta , s, Q_s)\nonumber \\&\text{ subject } \text{ to }\,\, P>D(\omega _p, s),\,t_p>0,\,s>0,\,Q_s>0,\,\eta >0\nonumber \\ \end{aligned}$$

(19)

This is a highly nonlinear constrained maximization problem.

4 Solution methodology

Considering the second strategy (situation) taken by binary tournamenting process, three hybrid algorithms have been developed to solve the optimization problem (19). These algorithms are called as T2-AQPSO, T2-GQPSO and T2-AGQPSO. As the hybrid algorithms are depending on AQPSO, GQPSO and AGQPSO and also the tournamenting process, thus before discussing the hybrid algorithm, it is required to illustrate PSO, QPSO, AQPSO, GQPSO and AGQPSO algorithms and tournamenting process. The brief descriptions of these are given in the following subsections.

4.1 Particle Swarm optimization (PSO)

Particle swarm optimization is a prominent and efficient algorithm based on the observations of the social behavior of animals, such as fishes and birds. Here each solution of the swarm is considered as ‘bird’ or ‘fish’-like volume free particle in activities. All the particles of the swarm fly throughout the search space aim to find the position of food (optimal position). At each iteration, particles in the swarm update their position by their personal experience and experience of the entire particles of the swarm. As a result, each particle has a memory to maintain its earlier best positions called ‘personal best positions’ with their own fitness. The position of the particle of the entire swarm which has highest fitness is called ‘global best position’. Assume that such swarm of size $N_p$ is moving in $n_v$-dimensional space. Let $u_i^K=(u_{i1}^K, u_{i2}^K,...,u_{in_v}^K)$, $v_i^K=(v_{i1}^K, v_{i2}^K,...,v_{in_v}^K)$, $p_i^K=(p_{i1}^K, p_{i2}^K,...,p_{in_v}^K)$, $p_g^K=(p_{g1}^K, p_{g2}^K,...,p_{gn_v}^K)$ be the current position, current velocity, personal best position and global best position, respectively, in the K-th iteration of the swarm. The velocity and position of particles are updated by the following rules:

$$\begin{aligned}&v_{ij}^{K+1}=v_{ij}^K+c_1 r_{1j}^K (p_{ij}^K-u_{ij}^K)+c_2 r_{2j}^K (p_{gj}^K-u_{ij}^K) \end{aligned}$$

(20)

$$\begin{aligned}&u_{ij}^{K+1}=u_{ij}^K+v_{ij}^{K+1}\nonumber \\&\text{ for }\,i=1,2,...,N_p;\nonumber \\&j=1,2,...,n_v; K=1,2,...,Max\_{IT} \end{aligned}$$

(21)

where $c_1>0$, $c_2>0$ are two constants, termed as the acceleration coefficients, and $r_{ij}^K$, $r_{2j}^K$ are random numbers which follows uniform distribution in (0, 1).

4.2 Quantum-behaved PSO (QPSO)

Sometimes, traditional PSO algorithm is trapped by local optimum value and therefore cannot reach the global optimum position. To overcome these difficulties, Sun et al. (2005) proposed quantum-behaved PSO based on the quantum behavior of the particles. In quantum space, Newton’s laws of motion is totally invalid because the position and velocity cannot be determined simultaneously according to Heisenberg’s uncertainty principle. Hence, PSO algorithm is needed to design in terms of wave function model.

While a particle of mass M is moving in quantum space, the wave function $\psi (u, t)$ satisfies the Schrodinger wave equation in $\delta$-potential well,

$$\begin{aligned} \frac{d^2\psi }{du^{2}}+\frac{2M}{\hbar ^2}[E+\gamma _1\delta (u-p)]\psi =0 \end{aligned}$$

(22)

where E is the total energy of the particle; p is the center of potential of the particle; h and $\hbar =\frac{h}{2\pi }$ are the Planck’s constant and modified Planck’s constant; and $\gamma _1$ is a positive constant which is chosen in such a way that it is always proportional to depth of the potential well.

Let $v=u-p$ then (22) reduces to,

$$\begin{aligned} \frac{d^2\psi }{dv^{2}}+\frac{2M}{\hbar ^2}[E+\gamma _1\delta (v)]\psi =0 \end{aligned}$$

(23)

Thus, from (23) normalized wave function can be represented as

$$\begin{aligned} \psi (v)=\frac{1}{\sqrt{L}}e^{-\frac{|v-p|}{L}} \end{aligned}$$

(24)

Thus, probability density function of the wave function is calculated as

$$\begin{aligned} Q=|\psi (v)|^2=\frac{1}{L}e^{-\frac{2|v-p|}{L}} \end{aligned}$$

(25)

From this equation, the probability of any particle appears at a certain position relative to p is found. But to get the fitness value, it is needed to determine the exact position of the particles. Therefore, to get the exact position, quantum state of the particle is to be collapsed into classical state. Monte Carlo simulation is used to measure this.

Monte Carlo simulation Since the value $\frac{1}{L}e^{-\frac{2|v-p|}{L}}$ always lies in the interval $\left( 0, \frac{1}{L}\right)$, let us consider a random number in $\left( 0, \frac{1}{L}\right)$ as $\frac{\xi }{L}$, where $\xi$ is a random number in (0, 1). Now replace the value $\frac{\xi }{L}$ in place of Q in (25),

$$\begin{aligned}&\frac{\xi }{L}=\frac{1}{L}e^{-\frac{2|v-p|}{L}} \end{aligned}$$

(26)

$$\begin{aligned}&\text{ Hence, }\,\,v=\pm \frac{L}{2}log(\frac{1}{\xi }) \end{aligned}$$

(27)

Thus, the equation (27) is going to be equal to

$$\begin{aligned} u=p\pm \frac{L}{2}log(\frac{1}{\xi }) \end{aligned}$$

(28)

In the analysis of PSO, it is proved that p is the local attractor $\Gamma$ of the particles. Hence, the equation (28) reduces to

$$\begin{aligned} u=\Gamma \pm \frac{L}{2}log(\frac{1}{\xi }) \end{aligned}$$

(29)

Thus, in $K-$th iteration $j-$th component of $i-$th particle is updated as follows

$$\begin{aligned} u_{ij}^{K+1}=\Gamma _{ij}^K\pm \frac{L_{ij}^K}{2} log(\frac{1}{\xi _{ij}^{K+1}}) \end{aligned}$$

(30)

The value of $L_{ij}^K$ is calculated as

$$\begin{aligned} 2\beta |u_{ij}^K-m_j^K| \end{aligned}$$

(31)

where $\beta$ is the contraction parameter (which plays an important role to control the convergence speed of the algorithm) and $m_j^K$ is the mean best position defined by averages of the pbest positions of all the particles and

$$\begin{aligned} \text{ Now, }\,\,m^K= & {} (m_1^K, m_2^K,...,m_{n_v}^K)\nonumber \\= & {} \left( \frac{1}{N_p} \sum _{i=1}^{N_p}p_{i1}^K, \frac{1}{N_p} \sum _{i=1}^{N_p}p_{i2}^K,\ldots ,\frac{1}{N_p} \sum _{i=1}^{N_p}p_{in_v}^K\right) \nonumber \\ \end{aligned}$$

(32)

also $\Gamma _{ij}^K$ is defined as

$$\begin{aligned} \Gamma _{ij}^K=\phi _j p_{ij}^K+(1-\phi _j)p_{gj}^K \end{aligned}$$

(33)

It is proved in Sun et al. (2005) if $rand \ge 0.5$ then updating formula is

$$\begin{aligned} u_{ij}^{K+1}=\Gamma _{ij}^K+\frac{L_{ij}^K}{2}log\Bigg (\frac{1}{\xi _{ij}^{K+1}}\Bigg ) \end{aligned}$$

(34)

and if $rand < 0.5$ then updating formula is

$$\begin{aligned} u_{ij}^{K+1}=\Gamma _{ij}^K-\frac{L_{ij}^K}{2}log\left( \frac{1}{\xi _{ij}^{K+1}}\right) \end{aligned}$$

(35)

Thus, the updating formula for QPSO algorithm is as follows

$$\begin{aligned} u_{ij}^{K+1}=\Gamma _{ij}^K+\beta |u_{ij}^K-m_j^K|log\left( \frac{1}{\varsigma _{ij}^{K+1}}\right) , \text{ if } \text{ r }\,\ge 0.5 \end{aligned}$$

(36)

$$\begin{aligned} u_{ij}^{K+1}=\Gamma _{ij}^K-\beta |u_{ij}^K-m_j^K|log\left( \frac{1}{\varsigma _{ij}^{K+1}}\right) , \text{ if } \text{ r }\,< 0.5 \end{aligned}$$

(37)

where $\xi _{ij}^{K+1}$ is a random number in (0, 1), $\phi _j \sim U(0, 1)$ and r is the random number in (0, 1).

4.3 Gaussian quantum-behaved PSO (GQPSO)

To avoid the premature convergence Coelho (2010) proposed GQPSO algorithm. In this version of PSO, particles of the swarm are more volatile and diversify. Here, the QPSO attractor $\Gamma _{ij}^K=\phi _j p_{ij}^K+(1-\phi _j)p_{gj}^K$ is replaced by $\Gamma _{ij}^K=\frac{G_1p_{ij}^K+G_2p_{gj}^K}{G_1+G_2}$, $j=1, 2,...,n_v$ and the random number $\varsigma _{ij}^{K+1}$ is replaced by the Gaussian random numbers $G_{ij}^{K+1}$ in the QPSO algorithm, where $G_1$ and $G_2$ are the random numbers generated by Gaussian probability distribution with zero mean and unit variance.

4.4 Adaptive quantum-behaved PSO (AQPSO)

In AQPSO (Xu and Sun 2005), the analysis of control parameters is studied in detail, which was not discussed in primary QPSO so far. Here, the parameter $\beta$ (known as creativity coefficient) plays an important role and it is dependent on attraction and repulsion phases. When the particles are in attraction phase, the diversity of the particles increases and in this situation one has to assign $\beta =\beta _a$, where $\beta _a\le 1$ as far as in repulsion phase one has to $\beta =\beta _r$, where $\beta _r>1$. Diversity of the swarm at K-th iteration is defined as

$$\begin{aligned} d=\frac{1}{N_p|L|} \sum _{i=1}^{N_p}\sqrt{\sum _{j=1}^{n_v}(p_{ij}^K-m_{j}^K)^2} \end{aligned}$$

(38)

where |L| is the longest diagonal of the search space.

The position of the particles at the K-th iteration is updated by using the following rules

$$\begin{aligned} u_{ij}^{K+1}=\Gamma _{ij}^K+\beta |u_{ij}^K-m_j^K|log \left( \frac{1}{\varsigma _{ij}^{K+1}}\right) , \text{ if } \text{ r }\,\ge 0.5 \end{aligned}$$

(39)

$$\begin{aligned} u_{ij}^{K+1}=\Gamma _{ij}^K-\beta |u_{ij}^K-m_j^K|log \left( \frac{1}{\varsigma _{ij}^{K+1}}\right) , \text{ if } \text{ r }\,< 0.5 \end{aligned}$$

(40)

where $\beta$ runs through the $\beta _r$ to $\beta _a$ of the swarm and if $d<dlow$ assign $\beta =\beta _a$ and if $d> dhigh$ assign $\beta =\beta _r$.

4.5 Adaptive Gaussian quantum-behaved PSO (AGQPSO)

On inspiring to get the advantages of AQPSO and GQPSO algorithms, Kumar et al. (2019) proposed AGQPSO algorithm. Here, to avoid the premature convergence, Gaussian attractor and Gaussian random number have been used. Over and above, to make less susceptible to stuck by stagnation fault, parameter controls have been made. Thus, proposed AGQPSO technique can be called as a modification of AQPSO technique. In AQPSO technique, QPSO attractor $\Gamma _{ij}^K=\phi _j p_{ij}^K+(1-\phi _j)p_{gj}^K$ has been used for compelling the particles towards the global optimum position. But in AGQPSO, Gaussian attractor $\Gamma _{ij}^K=\frac{G_1p_{ij}^K+G_2p_{gj}^K}{G_1+G_2}$, $j=1, 2,...,n_v$ has been used in place of the QPSO attractor $\Gamma _{ij}^K=\phi _j p_{ij}^K+(1-\phi _j)p_{gj}^K$. Also, Gaussian random numbers $G_{ij}^{K+1}$ have been used. The pseudo-code AGQPSO is as follows:

4.6 Pseudo-code

4.7 Tournamenting QPSOs

In genetic algorithm, sometimes researchers have to run the program several times by taking different populations in different runs to get the best solutions from them. For smaller search space, this process gives better outputs but for broad solution space, and for non-convex/non-concave problem, it becomes arduous. To overcome this type of difficulties, Bhunia and Samanta (2014) proposed an algorithm, known as tournament genetic algorithm as an alternative technique.

4.7.1 Hybrid binary tournament

In computational optimization, to make more efficient than original algorithm/algorithms, hybrid algorithms are proposed. Hybrid algorithm refers to the combination of two or more algorithms or embedding an algorithm in terms of different fashion (like tournament fashion, league fashion, chaotic mapping fashion). As a result, the new formed hybrid algorithm is better than the original algorithm/algorithms.

Actually in any game to select the best team among all the teams is arranged through tournament. This tournament can be designed in different fashions, e.g., binary tournament, league tournament, etc. In binary tournament, in each game one team is selected out of two teams in every round of the tournament. The whole tournament is performed in different rounds dependent on the number of teams which take part in the game.

4.7.2 Hybrid binary tournament-based QPSOs

In this work, AQPSO, GQPSO and AGQPSO techniques have been applied to elevate the swarm of particles in each round of tournament process. Here four teams’ tournamenting process has been considered. Firstly, two swarms $S_1$, $S_2$ have been updated into $S'_1$, $S'_2$ and finally taken 50% particles from $S'_1$, $S'_2$. This swarm is renamed as $S'_{12}$. 50% particles may be considered from the improved swarms $S'_1$ and $S'_2$ with the help of the following strategies:

• Strategy-1: Alliance of the best 50% from each of the modified swarms $S'_1$, $S'_2$.

• Strategy-2: Chosen of the best 50% from the alliance of two swarms $S'_1$, $S'_2$.

• Strategy-3: Creation of a new swarm by choosing a better individual by making a comparison between randomly

• taken particles from each of the swarms $S'_1$, $S'_2$.

• Strategy-4: Chosen of better swarm which carries the best particles between two swarms $S'_1$, $S'_2$.

• Strategy-5: Chosen of better swarm which carries better average fitness value.

• Strategy-6: Chosen of better swarm which carries finer standard deviation of the fitness values of the particles.

It should be noted that in a real game, the better one among two teams is selected for the next round. However, in simulation process, different strategies may occur for the computational optimization.

Fig. 5

Pictorial representation of second strategy of tournament (T2) with four teams

In the current work, using three rounds, all embedded/hybridized PSO algorithms (AQPSO, GQPSO and WQPSO) are designed in the form of second strategy-based tournament fashion using four teams. As a result, the said algorithms are hybridized in the form of binary tournament fashion. The details of binary tournament-based algorithms and their different scenarios are found in Kumar et al. (2019), Kumar et al. (2020), Akhtar et al. (2020).

4.8 Pseudo-code of binary tournamenting QPSOs

5 Numerical experiment

In this section, different types of numerical experiments have been considered. In subsect. 5.1, three numerical examples are considered to check the validation of the proposed model as well as the robustness of the hybrid algorithms. Here, the best found solutions, worst found solutions, statistical results and convergence history for each of the problems have been provided. In subsect. 5.2, the discussion of the obtained results is performed. The four different nonparametric statistical tests are performed in subsect. 5.3.

Table 2 Parameters of different algorithms

Table 3 Best found results of Example 1

Table 4 Worst found results of Example 1

Table 5 Statistical results of Example 1 using different algorithms

Table 6 Best found results of Example 2

Table 7 Worst found results of Example 2

Table 8 Statistical results of Example 2 using different algorithms

Table 9 Best found results of Example 3

Table 10 Worst found results of Example 3

Table 11 Statistical results of Example 3 using different algorithms

5.1 Numerical illustration

To validate the proposed model, three different numerical examples have been considered which are as follows.

Example 1: In this example the values of different parameters are taken as $C_0=Rs.\,80$ per unit, $C_1=Rs.\,0.01$ per unit, $C_2=Rs.\,500$ per unit, $C_3=Rs.\,2$ per unit, $C_4=Rs.\,2.5$ per unit, $\alpha =1.25 \in R^+$, $\beta _1=310$ unit, $\lambda _1=2.0$ unit, $\lambda _2=0.6$ unit, $\gamma =0.5 \in R^+$, $\delta =2 \in R^+$, $w_p=0.5$ year, $\lambda =0.3 \in R^+$, $\mu =0.5 \in R^+$, $\theta _0=0.1 \in (0, 1)$, $\vartheta =0.09 \in R^+$, $\xi =0.5 \in R^+$, $h=Rs.\,6.0$ per unit/ unit time, $a=Rs.\,20.0$ per unit, $b=Rs.\,5$ per unit, $c_b=Rs.\,20.0$ per unit, $A=Rs.\,300$ per cycle.

Example 2: In this example the values of different parameters are taken as $C_0=Rs.\,80$ per unit, $C_1=Rs.\,0.03$ per unit, $C_2=Rs.\,490$ per unit, $C_3=Rs.\,2.2$ per unit, $C_4=Rs.\,2.9$ per unit, $\alpha =1.15 \in R^+$, $\beta _1=300$ unit, $\lambda _1=1.5$ unit, $\lambda _2=0.5$ unit, $\gamma =0.4 \in R^+$, $\delta =4.2 \in R^+$, $w_p=0.7$ year, $\lambda =0.6 \in R^+$, $\mu =0.4 \in R^+$, $\theta _0=0.2 \in (0, 1)$, $\vartheta =0.04 \in R^+$, $\xi =1.5 \in R^+$, $h=Rs.\,6.0$ per unit/ unit time, $a=Rs.\,14.0$ per unit, $b=Rs.\,14.0$ per unit, $c_b=Rs.\,5.5$ per unit, $A=Rs.\,305$ per unit.

Example 3: In this example the values of different parameters are taken as $C_0=Rs.\,82$ per unit, $C_1=Rs.\,0.02$ per unit, $C_2=Rs.\,510$ per unit, $C_3=Rs.\,1.5$ per unit, $C_4=Rs.\,2.3$ per unit, $\alpha =1.10 \in R^+$, $\beta _1=305$ unit, $\lambda _1=2.4$ unit, $\lambda _2=0.8$ unit, $\gamma =0.6 \in R^+$, $\delta =3.5 \in R^+$, $w_p=0.6$ year, $\lambda =0.2 \in R^+$, $\mu =0.8 \in R^+$, $\theta _0=0.15 \in (0, 1)$, $\vartheta =0.1 \in R^+$, $\xi =0.9 \in R^+$, $h=Rs.\,5.0$ per unit/ unit time, $a=Rs.\,16.0$ per unit, $b=Rs.\,4.0$ per unit, $c_b=Rs.\,16.0$ per unit, $A=Rs.\,300$ per unit.

The best found average profit has been obtained using six different algorithms including hybrid tournamenting algorithms. Since all the algorithms are probabilistic in nature, so 30 independent runs have been performed for each of the algorithms. Each algorithm has been coded by using C++ software, and all runs have been done on a Laptop core i3-7020U CPU, 7th generations, 2.30 GHz processors in LINUX environment. Simulation parameters of different algorithms are provided in Table 2.

Here, two different types of swarm size and maximum generations have been considered in two types of algorithms. Since for small swarm size and generations, functions evaluations are large in binary tournament-based algorithms, to compare these two types of algorithms, the swarm size and maximum generations have to set in such a way that difference of function evaluations in two types of algorithms is minimum. On considering swarm size 15 and maximum generation 30 in tournament-based PSOs, the total number of function evaluations is $7\times 15\times 30+4\times 15=3210$, while in general PSOs the total number of function evaluations is $25\times 130+25=3275$. Therefore, the difference of function evaluations is not so large. However, function evaluations are small in tournament-based PSOs. Besides, other parameters of PSOs are considered as per the directions of Coelho (2010) and Sun et al. (2005). To test the efficiency of the hybrid tournament PSOs, best found results, worst results and statistical measurements of the average profit are shown in Tables 3, 4, 5, 6, 7, 8, 9, 10, 11 for each of the problems.

From the results, it is observed that the hybrid tournament PSOs give better results than its general algorithm with minimum function evaluations and CPU times for each problem.

5.2 Results discussion

The best found solutions of Example 1 are the same for all the algorithms. However, the worst found solutions are different for all the algorithms. The difference of best found solutions and worst found solutions is minimum for T2-AQPSO algorithms. Also from Tables 3, 4 and 5, it is observed that T2-AQPSO algorithm performs better with respect to best found objective value, worst objective value, mean objective value and standard deviation of the objective values with minimum CPU times as well as function evaluations. Though AGQPSO algorithm does not perform so well, it has satisfactory performance in tournament hybridization.

The best found solutions of Example 2 be the same for all the algorithms except for T2-AGQPSO. However, it is same up to eight decimal places. Here also, the difference between the found solutions and worst found solutions is minimum for T2-AQPSO algorithms. Here T2-AQPSO algorithm performs better with respect to the best objective value, worst objective value and standard deviation. Here, mean and median of the objective values are better for AQPSO algorithm, but it has larger CPU times and function evaluations. So, overall T2-AQPSO algorithm performs better for Example 2 also.

In discussions of Example 3, it is observed from Tables 9, 10 and 11 that the best found solutions be the same for all the algorithms. The differences between best found solutions and worst found solutions are very less for T2-AQPSO and AQPSO algorithms. From the statistical data, it is observed that their performances be the same except standard deviations. The standard deviation is less for T2-AQPSO algorithm. Also, it should be noted that T2-AQPSO algorithm produces the best found result with minimum CPU times and function evaluations. Here also, AGQPSO and its tournament hybridized version perform well than other hybrid algorithms.

So in overall discussions, it does not have any hesitance to say that T2-AQPSO is the best algorithm than others in performance for this profit optimization problem. Of course, it should be noted that the computational results obtained in Tables Tables 9, 10 and 11 do not establish that T2- AQPSO algorithm performs well for all types of optimization problems (Wolpert and Macready 1997). Hence, this hybrid T2- AQPSO algorithm has the highest performance for the optimization problem (19) only.

The convergence history of the best found solutions obtained from AGQPSO, AQPSO and GQPSO algorithms is shown in Figures 6, 7 and 8. However, the convergence graphs of different hybrid algorithms cannot be drawn as the parent metaheuristic algorithms like AQPSO, AGQPSO and GQPSO are applied in different rounds of the tournament. From the convergence history, it is observed that AQPSO performs better for Example 1 and Example 3, whereas AGQPSO performs better for Example 2.

5.3 Nonparametric statistical test

From the earlier mentioned statistical results, it is observed that T2-AQPSO algorithm performs better than all other algorithms (T2-AGQPSO, T2-GQPSO, AGQPSO, AQPSO and GQPSO) in overall comparison. But it is needed to perform some statistical tests due to randomness of the metaheuristic algorithm to check the significance of the results. In this section, four nonparametric statistical tests, viz. Wilcoxon rank-sum test (Derrac et al. 2011; Duary et al. 2020; García et al. 2009), Friedman test (Derrac et al. 2011), Wilcoxon signed rank test (García et al. 2009) and Iman and Davenport test (Derrac et al. 2011), have been performed on the 30 objective values (average profit values) obtained from 30 runs to compare with other used algorithms for each of the examples. By considering T2-AQPSO as the control algorithm, the p values of the different methods are presented in Tables 12, 13, 14. All p values are calculated by using Microsoft Office excel 2007.

Fig. 6

Convergence history of the best found solution of different algorithms for Example 1

Fig. 7

Convergence history of the best found solution of different algorithms for Example 2

Fig. 8

Convergence history of the best found solution of different algorithms for Example 3

From Table 12, it is seen that the superiorities of T2-AQPSO algorithm are statistically significant for T2-GQPSO, AGQPSO and AQPSO algorithms for Example 1 as the p values are less than 0.05. However, T2-AGQPSO and GQPSO perform very similar with T2-AQPSO algorithm.

Also from Table 13, it is noticed that the superiorities of T2-AQPSO algorithm are statistically significant for T2-AGQPSO,T2-GQPSO, AGQPSO and AQPSO algorithms for Example 2 as the p values are less than 0.05. However, the performance of GQPSO algorithm is very comparative with T2-AQPSO algorithm.

In Table 14, it is seen that the superiorities of T2-AQPSO algorithm are statistically significant for T2-GQPSO and AGQPSO algorithms for Example 3 at $5\%$ level of significance. Here, the performance of T2-AGQPSO, AQPSO and GQPSO algorithms is very similar with T2-AQPSO algorithm.

6 Sensitivity analysis

For the proposed model, the sensitivity analyses by changing the values of known important model parameters A, $c_b$, $C_0$, $\beta _1$, $\theta _0$, $w_p$, h and $\alpha$ from -20% to 20% have been demonstrated. Also, the effects of P, $t_p$, s, $\eta$, $Q_s$, $T_1$, $T_2$, T, $\Pi$ with respect to each parameter have been analyzed and the results are shown graphically in Figs. 9, 10, 11, 12, 13, 14, 15 and 16.

The effectiveness of the parameters A, $c_b$, $C_0$, $\beta _1$, $\theta _0$, $w_p$, h and $\alpha$ on the best found solution of maximization problem (17) (P, $t_p$, s, $\eta$, $Q_s$, $T_1$, $T_2$, T, $\Pi$) is measured by the following scales:

• (i) The model parameters are known as highly sensitive directly (reversely) with respect to the best found solution of maximization problem (19) if the said solutions change from less than -20% (greater than +20%) to greater than +20% (less than -20%) with the changes of that parameters from -20% to +20%.

• (ii) The model parameters are known as equally sensitive with directly (reversely) with respect to the best found solution of maximization problem (19) if the said solutions change from -20% (+20%) to +20% (-20%) with the changes of that parameters from -20% to +20%.

• (iii) The model parameters are known as moderately sensitive directly (reversely) with respect to the best found solution of maximization problem (19) if the said solutions change from near -10% (+10%) to near +10% (-10%) with the changes of that parameters from -20% to +20%.

• (iv) The model parameters are known as less sensitive directly (reversely) with respect to the best found solution of maximization problem (19) if the said solutions change from greater than -10% (less than +10%) to less than +10% (greater than-10%) with the changes of that parameters from -20% to +20%.

• (v) The model parameters are known as insensitive with respect to the best found solution of maximization problem (19) if the said solution changes from greater than -1% (+1%) to less than +1% (-1%) with the changes of that parameters from -20% to +20%.

Table 12 Nonparametric statistical tests of Example 1

Table 13 Nonparametric statistical tests of Example 2

Table 14 Nonparametric statistical tests of Example 3

From Figs. 9, 10, 11, 12, 13, 14, 15 and 16, the following observations can be made:

• (i) From Fig. 9, it is observed that P, $t_p$, $Q_s$, $T_1$, $T_2$ are less sensitive directly against the changes of setup cost (A). On the other hand, s, $\eta$, T, $\Pi$ are insensitive with respect to A.

• (ii) Figure 10 shows that $Q_s$ is less sensitive directly with respect to backordering cost ($c_b$). On the other hand, P, $t_p$, $\eta$, s, $\eta$, $T_1$, T, $\Pi$ are insensitive with the changes of $c_b$.

• (iii) From Fig. 11, it is seen that s, $Q_s$, $T_1$, $T_2$, T are less sensitive directly against the changes of fixed production cost ($C_0$). Also $\eta$ has moderate impact reversely with respect to $C_0$. On the other hand, P, $t_p$ are slightly reverse sensitive and average profit ($\pi$) is moderately reverse sensitive with the changes of $C_0$.

• (iv) In view of Fig. 12, it is clear that P, $t_p$, s are moderately sensitive directly and average profit ($\Pi$) has direct impact largely with the changes of fixed demand ($\beta _1$). Furthermore, $Q_s$, $T_1$, $T_2$, T have reverse sensitive moderately and $\eta$ is insensitive with the changes of $\beta _1$.

• (v) In Fig. 13, it is obvious that P, $Q_s$ are less sensitive directly and $t_p$, $\pi$ are reverse sensitive slightly with the changes of defective rate parameter $\theta _0$. Also $\eta$ is largely sensitive directly and s, $T_1$, $T_2$, T are insensitive with the changes of $\theta _0$.

• (vi) Figure 14 exposes that $t_p$ has less sensitive directly and P has less sensitive reversely with the change of warranty period ($w_p$). On the other hand, s, $\eta$, $Q_s$, $T_1$, $T_2$, T, $\Pi$ are insensitive with the changes of $w_p$.

• (vii) From Fig. 15, it is observed that P, $T_1$ and T are equally reverse sensitive and $t_p$ is reverse sensitive moderately with the changes of holding cost (h). Also $Q_s$ is moderately sensitive directly and $\Pi$ is reverse sensitive slightly with the changes of h. On the other hand, s, $\eta$, $T_2$ are insensitive with the changes of h.

• (viii) Finally, Fig. 16 exposes that $Q_s$ has largely sensitive reversely and s, $\eta$, $Q_s$, $T_1$, $T_2$, T, $\Pi$ are insensitive with the backlogged parameter $\alpha$.

7 Managerial insights

The proposed production inventory model can be applied to any manufacturing systems where the production system produces perfect as well as defective items. The manager of any manufacturing company will invest development cost to reduce the production of defective items. The management of manufacturing company gives the product warranty to increase annual customers’ demand and profit of the company. On the other hand, the customers will be benefited due to the product warranty. Here, the customers’ demand is dependent on the selling price of the produced item and warranty period. Finally, the managers of this type of manufacturing company will investigate the optimal selling price, optimal production rate, optimal production period, optimal business period and maximum shortage level which maximize the average profit of the system. To find these optimal values, managers may use a hybrid tournamenting QPSO algorithms for solving the corresponding optimization problem (average profit function) of the manufacturing system. On the other hand, on using these algorithms managers of different manufacturing firms utilize computational cost and time with more reliable results because these algorithms give very finer results by taking lesser memory and function evaluations. Hence, managers can use these models in industry.

8 Conclusion

In this study, a production inventory problem has been investigated and formulated the corresponding model with dynamic demand and inventory level-dependent partially backlogged shortages. Due to complexity of the corresponding optimization problem, different variants of metaheuristic optimization techniques AQPSO, GQPSO, AGQPSO, T2-AQPSO, T2-GQPSO and T2-AGQPSO have been applied in order to find the maximum average profit of the system. From the sensitivity analyses, it is observed that the development cost for reducing the production of defective units has a good impact on production process as well as to increase the average profit. Hence, it is concluded that the manufacturer must use development technology to reduce the defective item as well as to increase the average profit. Moreover, after analyzing this model, it is observed that if manufacturer imposes the warranty policy of the product, then its impact directly goes to the customers’ demand as well as the average profit. The length of warranty period is dependent on the quality of the products. So, the manufacturer aims to increase the warranty period for increasing demand rate.

From the numerical experiments and results, it can be concluded that the tournament-based hybrid algorithms stabilize the results of a complicated optimization problem than its parent algorithm with lesser computational costs (i.e., memory allocation, function evaluations, CPU times, etc.)

Fig. 9

Sensitivity with respect to A

Fig. 10

Sensitivity with respect to $c_b$

Fig. 11

Sensitivity with respect to $C_0$

Fig. 12

Sensitivity with respect to $\beta _1$

Fig. 13

Sensitivity with respect to $\theta _0$

Fig. 14

Sensitivity with respect to $w_p$

Fig. 15

Sensitivity with respect to h

Fig. 16

Sensitivity with respect to $\alpha$

9 Limitations and future research scope

Firstly, the proposed methodologies are based on advanced quantum-behaved particle swarm optimization (AQPSO/AGQPSO/GQPSO) and binary tournamenting process. In these methodologies, initially 4 swarms/populations (teams) are considered. Then in different rounds, populations are updated by (AQPSO/AGQPSO/GQPSO). So, there is no scope to compare the efficiencies of the methodologies with respect to convergence diagram. Besides, these methodologies work better with suitable swarm size with suitable generations.

Further, this study can be extended by considering several realistic assumptions such as advance payment scheme, time-dependent production rate, time-dependent demand, trade credit facility, inflation, overtime production, etc. This model may also be extended to take interval-valued demand rate, defective rate and different inventory costs. Finally, anyone can extend this model by taking green production process.