# An Automated Smart Production with System Reliability Under a Leader-Follower Strategy of Supply Chain Management

Mitali Sarkar, Biswajit Sarkar, Alexandre Dolgui

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Mitali Sarkar, Biswajit Sarkar, Alexandre Dolgui. An Automated Smart Production with System Reliability Under a Leader-Follower Strategy of Supply Chain Management. IFIP International Conference on Advances in Production Management Systems (APMS), Sep 2022, Gyeongju, South Korea. pp.459-467, 10.1007/978-3-031-16407-1_54. hal-03981929

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# An Automated Smart Production with System Reliability Under a Leader-Follower Strategy of Supply Chain Management

Mitali Sarkar<fnref n="1" />, Biswajit Sarkar<fnref n="2" />,<fnref n="3" />(✉) and Alexandre Dolgui<fnref n="4" />

**Abstract.** This study deals with the absorption of an automated system in a smart production system, and how the industry tackles several problems. The incorporation of smart machines within smart manufacturing is not only a matter of investments and smart labor but also a matter of tackling them together with the proper trade-off between them. The automated system basically depends on the product's development or product design. In this direction, this research finds that the development of products itself depends on the automated system. A mathematical representation of the automated smart production is provided with a basic aim to reduce the total cost without any specific goal of the industry. A smart supply chain is found here to tackle this automated smart production with a single retailer, even though the players are competitive within them. A Stackelberg game strategy is utilized to solve the model to obtain the optimum total cost and the optimum values of the smart production quantities, and the smart production rate. Due to automated costs, the reduction of setup costs plays an important role. The numerical studies have been conducted to validate this study and obtain that the automated system works properly even though the players compete with each other.

This research was supported by the Yonsei University Research Fund of 2021 (Project Number 2021-22-0305).

<footnote n="1">¹ Information Technology Research Center, Chung-Ang University, 06974 Seoul, South Korea</footnote>

<footnote n="2">² Department of Industrial Engineering, Yonsei University, 50 Yonsei-ro, Sinchon-dong, Seodaemun-gu, 03722 Seoul, South Korea</footnote>

<footnote n="3">³ Center for Transdisciplinary Research (CFTR), Saveetha Dental College, Saveetha Institute of Medical and Technical Sciences, Saveetha University, 162, Poonamallee High Road, Velappanchavadi, Chennai, Tamil Nadu 600077, India
bsbiswajitsarkar@gmail.com</footnote>

<footnote n="4">⁴ Automation, Production and Computer Sciences Department, IMT Atlantique, La Chanterie, France
alexandre.dolgui@imt-atlantique.fr</footnote>
**Keywords:** Smart products · Smart production · Supply chain management · Automated system design · Setup cost reduction · Investments

# 1 Introduction

The necessity of technological development is increasing for any production system to make it automated and smart. Thus, the industry is also moving towards smart utilization of machines to save the cost, reduction of defective items, and human effort. But always all smart machines may not produce perfect items always. Thus, a research is needed to redesign the smart machines to control defective items. Alavian et al. [1] discussed a smart production system. They developed an automatic system to decide on the manufacturing system. A flexible manufacturing remanufacturing system was developed by Sarkar and Bhuniya [2] for a sustainable supply chain model. They used an investment to improve their manufacturing system.

Sana [3] explained the reliability and production rate. Thus, the production cost increases with the increasing amount of development cost. Jiang et al. [4] studied a remanufacturing system. They represented reliability by the failure rate. The setup cost of a smart manufacturing system is very high.

Sarkar and Moon [5] developed an integrated inventory model (IIM) with setup cost reduction and quality improvement. They considered variable backorder and random demand during lead time. They used a distribution-free approach to solve the model. Al Durgam et al. [6] proposed an IIM with random lead time demand. They used lead time crashing cost to reduce the lead time. Dey et al. [7] used a discrete setup cost reduction policy for their IIM. Table 1 shows the research gap of this study.

<table><caption>**Table 1.** Comparative study with the existing literature.</caption><thead><tr><th>Author(s)</th><th>Model</th><th>Imperfection</th><th>Investment</th><th>System design</th></tr></thead><tbody><tr><td>Sana [3]</td><td></td><td>*</td><td>*</td><td></td></tr><tr><td>Sarkar and Moon [5]</td><td>Production</td><td>*</td><td>*</td><td></td></tr><tr><td>Dey et al. [7]</td><td>Production</td><td>*</td><td></td><td></td></tr><tr><td>Sarkar and Chung [9]</td><td>Production</td><td>*</td><td>*</td><td></td></tr><tr><td>Huang et al. [10]</td><td>Production</td><td>*</td><td>*</td><td></td></tr><tr><td>This study</td><td>SCM</td><td>*</td><td>*</td><td>*</td></tr></tbody></table>

The following research questions can be solved by this study:

1. The setup cost of a smart automated production system is very high. How does the manufacturer control the setup cost?
2. How does the automated production system design the improvement of technology mathematically?

3. How does the lead time reduce shortages?

This study is organized as follows: Sect. 2 represents the Mathematical modelling and solution methodology, Sect. 3 consists of numerical experiments, sensitivity analysis, and managerial insights of the model. Conclusions and future works are explained in Sect. 4.

# 2 Mathematical Model

A smart manufacturing system with artificial intelligence controlled robotic system is used to produce smart electronic products. To maintain the smart production system properly a variable development cost is utilized by the manufacturer. Thus, an investment to improve the autonomated system is used here. Due to the autonomated system the setup cost increases too high which is reduced by applying some investment. The manufacturer sends the finished products using single-setup-multi-delivery (SSMD) policy to the retailer. That means the manufacturer produces all products in a single setup of production system and due to higher holding cost of retailer than manufacturer, send those products in multiple deliveries. Due to huge lead time which is random, the retailer faces shortages of products. To reduce the lead time the retailer uses lead time crashing cost. In this section mathematical model of manufacturer and retailer with their analytical solutions are explained below. The total supply chain cost are optimized analytically.

## 2.1 Manufacturer's Model

The cost using for the smart manufacturing system are explained in details below. A smart manufacturing system is utilized by the manufacturer. Thus, a flexible production rate ($P_f$) is used for the system. The unit production cost can be expressed as ($A_m + \frac{A_d}{P_f} + \gamma P_f$), where $A_m$ is the material cost, $A_d$ is the development cost, and $\gamma$ is the tool/die cost. Development cost is dependent on the system design variable ($\eta$). The development cost can be expressed as $A_d(\eta) = M + N\eta^g$, where $M$ is the fixed development cost, $N$ is scaling parameter, $g$ is the shape parameter, and $\eta$ can be defined as *system design variable* = $\frac{Number of failure}{Number of working hours}$. Thus, the manufacturing cost of the system can be written as ($A_m + \frac{M+N\eta^g}{P_f} + \gamma P_f$) $D$, where $D$ is the demand of products. The number of defective item ($\rho$) is directly proportional to the system design variable, i.e., $\rho$ is proportional to $\eta$ which indicates $\rho = \alpha\eta$. Let $E_\vartheta$ denotes the expected number of failure rate of the production system for the lot size $\vartheta$. At the beginning the system was in *in-control* state. $i$ indicates the number of lot. It is known that ($\widehat{\alpha\eta}$) = 1 - $\alpha\eta \approx$ 1 and by Taylor series expansion ($\widehat{\alpha\eta}$$^\vartheta$) = $e^{(\ln(\widehat{\alpha\eta}))\vartheta} \approx 1 + (\ln(\widehat{\alpha\eta}))\vartheta + \frac{[(\ln(\widehat{\alpha\eta}))\vartheta]^2}{2}$
(refer to Porteus, [6]). Thus, the expected number of defective can be written as $E_{\vartheta} = \alpha\eta\vartheta + (\widehat{\alpha\eta})E_{\vartheta-1} = \alpha\eta\vartheta \frac{(1-(\widehat{\alpha\eta})^i)}{\alpha\eta} - \alpha\eta \frac{(\widehat{\alpha\eta})[1+(\vartheta-1)(\widehat{\alpha\eta})^q - q(\widehat{\alpha\eta})^{\vartheta-1}]}{(\alpha\eta)^2} = \vartheta - \frac{(\widehat{\alpha\eta})(1-(\widehat{\alpha\eta})^q)}{\alpha\eta^2} = \frac{\alpha\eta\vartheta^2}{2}$. Therefore, the defective cost for $i$ shipments during imperfect production can be written as $\frac{wD_i\vartheta\alpha\eta}{2}$, where $w$ is per unit defective cost. Before start the production, the manufacturing system needs some initial setup. For the smart production system the setup cost is very high. This setup cost can be reduced by using some investment. The discrete investment to reduce the setup cost of the smart production system is $\frac{\delta D}{i\vartheta}$. The setup cost and investment of the smart production system is $\frac{S_U e^{-A\delta} D}{i\vartheta} + \frac{\delta D}{i\vartheta}$, where $S_U$ is the initial setup cost and $\delta$ is the investment to reduce setup cost per production run. The manufacturer holds the produced items before sending to retailer. Holding cost per unit item per unit time is $H_c$. The inventory holding with the manufacturer is needed to be calculated (see for reference Majumder et al. [8]). The manufacturer uses SSMD policy with a fixed transportation cost ($G$) per shipment. Thus, the total transportation cost for $i$ shipment is $\frac{GD}{i}$. Thus, the manufacturer's inventory holding and transportation cost is $\frac{H_c i \vartheta}{2} [i(1 - \frac{D}{P_f}) - 1 + \frac{2D}{P_f}] + \frac{GD}{i}$.

The manufacturer has to optimize the total cost with respect to the decision variables; the number of shipments ($i$), the lot size ($\vartheta$), the flexible production rate ($P_f$), system design variable ($\eta$), and investment to reduce setup cost ($\delta$). The total cost of the manufacturer can be written as

$$ \Lambda_m(i, \vartheta, P_f, \eta, \delta) = \left( A_m + \frac{M + N\eta^g}{P_f} + \gamma P_f \right) D + \frac{wD_i\vartheta\alpha\eta}{2} + \frac{S_U e^{-A\delta} D}{i\vartheta} + \frac{\delta D}{i\vartheta} \\ + \frac{H_c i \vartheta}{2} \left[ i \left( 1 - \frac{D}{P_f} \right) - 1 + \frac{2D}{P_f} \right] + \frac{GD}{\vartheta}. \quad (1) $$

## 2.2 Retailer's Model

The retailer uses $O_R$ amount of cost per order to place the order for product delivery to the manufacturer. The demand during lead time of the smart product is random. $\lambda$ is the lead time and $\kappa$ is the safety factor. If $B_h$ be the holding cost per unit per unit time, then the holding and ordering cost per cycle of the retailer can be written as $B_h[(\vartheta/2) + \kappa\sigma\sqrt{\lambda} + (1 - \beta)E(\chi - \mu)^+] + \frac{O_R D}{i} = B_h[(\vartheta/2) + \kappa\sigma\sqrt{\lambda} + (1 - \beta)\sigma\sqrt{\lambda}\psi(\kappa)] + \frac{O_R D}{i}$. Due to the long lead time the retailer faces some shortages which are backordered. Here $\nu$ is the per unit backorder cost, and $\beta$ is the backordered items. To reduce the lead time the retailer uses a crashing cost, which can be written as $\frac{DC(\lambda)}{i}$. If the expected shortages are $E(\chi - \mu)^+$, then the shortage and lead time crashing cost can be written as $\frac{(\nu + \nu_0(1 - \beta))D}{i} E(\chi - \mu)^+ + \frac{DC(\lambda)}{i} = \frac{(\nu + \nu_0(1 - \beta))D}{i} \sigma \sqrt{\lambda} \psi(\kappa) + \frac{DC(\lambda)}{i}$, where $\psi(\kappa) = \phi(\kappa) - \kappa(1 - \Phi(\kappa))$, $\phi(\kappa)$ and $\Phi(\kappa)$ are standard normal distribution and the probability density function of the normal distribution function. The retailer has to optimize the total cost with respect to the decision variable; lot size ($\vartheta$),
lead time ($\lambda$), and safety factor ($\kappa$). The total cost of the retailer ($\Lambda_r$) is

$$ \Lambda_r(\vartheta, \lambda, \kappa) = \left[ \frac{O_R D}{\vartheta} + \frac{DC(\lambda)}{\vartheta} + B_h \left[ \left(\frac{\vartheta}{2}\right) + \kappa \sigma \sqrt{\lambda} + (1-\beta) \sigma \sqrt{\lambda} \psi(\kappa) \right] + \frac{(\nu + \nu_0(1-\beta))D}{\vartheta} \sigma \sqrt{\lambda} \psi(\kappa) \right]. \quad (2) $$

## 2.3 Solution Methodology

Due to the unequal power of supply chain players, Stackelberg game policy is used to solve the model. Even though the leader-follower strategy is there, but both the manufacturer and relater are fair enough for their dealings. The model is solved in two cases: I) Retailer follower and manufacturer leader and II) Manufacturer follower and retailer leader.

### 2.3.1 Case I Retailer as Follower and Manufacturer as Leader

The optimum values of the decision variables can be obtained as $\Phi^*(k) = 1 - \frac{B_h \vartheta}{B_h(1-\beta)\vartheta + (\nu + \nu_0(1-\beta))D}$, $\vartheta^* = \sqrt{\frac{2[O_R D + DC(\lambda) + (\nu + \nu_0(1-\beta))D \sigma \sqrt{\lambda} \psi(k)]}{B_h}}$, and $\frac{\partial^2 \Lambda_r}{\partial \lambda^2} = -\frac{\sigma}{4\sigma^{3/2}} \left[ \frac{(\nu + \nu_0(1-\beta))\psi(k)}{\vartheta} + B_h k + (1-\beta)\psi(k) \right] < 0$ and thus $\Lambda_r$ is concave with respect to $\lambda$.

Using the optimum values of retailer, the optimum values of manufacturer's decision variables except the common decision variable, can be obtained as $P_f^* = \sqrt{\frac{2(M+N\eta^2)-H_c\ell\vartheta^*(\ell-2)}{2\gamma}}$, $\eta^* = -\left(\frac{w\ell\vartheta^*\alpha P_f}{2gN}\right)^{\frac{1}{\ell-1}}$, and $\delta^* = \frac{1}{A} \ln(AS_U)$.

### 2.3.2 Case II Manufacturer as Follower and Retailer as Leader

For this case manufacturer is follower and retailer is leader. The optimum values of the decision variables are $\vartheta^* = \sqrt{\frac{2D(S_U e^{-A\delta} + \delta + G\ell)}{wD\ell^* \alpha\eta + H_c\ell^* [\ell(1-\frac{D}{P_f}) - 1 + \frac{2D}{P_f}]}}$, $P_f^* = \sqrt{\frac{2(M+N\eta^2)-H_c\ell\vartheta^*(\ell-2)}{2\gamma}}$, $\eta^* = -\left(\frac{w\ell\vartheta^*\alpha P_f}{2gN}\right)^{\frac{1}{\ell-1}}$, and $\delta^* = \frac{1}{A} \ln(AS_U)$.

Using decision variables of manufacturer, the optimum values of retailer except the common decision variable can be written as $\Phi^*(k) = 1 - \frac{B_h \vartheta}{B_h(1-\beta)\vartheta + (\nu + \nu_0(1-\beta))D}$, and $\frac{\partial^2 \Lambda_r}{\partial \lambda^2} = -\frac{\sigma}{4\sigma^{3/2}} \left[ \frac{(\nu + \nu_0(1-\beta))\psi(k)}{\vartheta} + B_h k + (1-\beta)\psi(k) \right] < 0$. $\Lambda_r$ is concave with respect to $\lambda$.

## 3 Numerical Experiment

Two numerical experiments are taken place with same data. The flexibility of a smart production system can reduce the possibility to move to out-of-control state which can reduce the production of defective items. The contribution of flexible production rate is taken from Sarkar and Chung [9]. To make a production system smart and flexible the setup cost of the manufacturer becomes
very high which can be reduced by using some investments. The idea of discrete setup cost reduction for the manufacturer is taken from Huang et al. [10]. Instead of using flexible rate the smart production system produces some defective items for which shortages arise. The shortage cost is included in this study from Sarkar and Moon [5]. For numerical analysis the modified data from Sarkar and Moon [5], Sarkar and Chung [9], and Huang et al. [10] are considered to test the validity of the proposed model. The experiments are done by MATLAB R2020a. Data is enlisted as $D = 600$ units, $\sigma = 7$ units/week, $S_U = \$800$/setup, $\alpha = 0.2$, $w = \$75$/item, $A_m = \$50$, $M = \$70$, $N = 90$, $g = 0.2$, $\gamma = \$0.02$/unit, $A = 3$, $G = \$.1$/shipment, $H_c = \$.08$/unit/year, $O_R = \$200$/order, $C(\lambda) = 22.4$, $B_h = \$55$/unit/year, $\nu = \$0.4$/unit, $\nu_0 = \$9$/unit, $\beta = 0.87$, $\lambda = 3$ weeks. For Case I, the optimum values of retailer's decision variables are $\vartheta = 706.45$ units, $\Phi(k) = 0.61$, $k = 0.28$, and the total cost of the retailer is \$390.67/cycle. Using the same amount of $\vartheta$, the optimum value of other decision variables of the manufacturer are $P_f = 2.75$ units/cycle, $\eta = 0.000001$, $\delta = 2.59$, and the total cost of manufacturer is \$30,248.50/cycle. The total supply chain cost is \$30,639.17/cycle. For Case II, the optimum values of manufacturer's decision variables are $\vartheta = 10.27$ units, $P_f = 63.35$ units/cycle, $\eta = 0.000001$, $\delta = 2.59$, and the total cost of manufacturer is \$31,507.86/cycle. Using the same amount of $\vartheta$, the optimum value of other decision variables of the retailer are $\Phi(k) = 0.99$, $k = 2.51$, and the total cost of the retailer is \$11,324.79/cycle. The total supply chain cost for Case II is \$42,832.65/cycle.

Comparing both cases, it can be concluded that the Case I, i.e., manufacturer is a leader and retailer is a follower gives the better result than the Case II. The total cost of both the players as well as the supply chain total cost is less for Case I than the Case II. The comparison of the decision variables with the existing research [11,16, and 17], it is found that the proposed model provides better results of values of decision variables comparing with them.

## 3.1 Sensitivity Analysis

Sensitivity analysis is done only for Case I, because it has the lowest supply chain cost. The changing of total cost of each player corresponding to key parameters are given Table 2 and 3.

* The most sensitive parameter is material cost of the manufacturer among all the cost parameters. The total cost of manufacturer changes almost 50% at both negative and positive direction for 50% changes of the material cost at direction.

* The next sensitive parameter is the retailer's holding cost. The total cost of retailer decreases almost 30% for decreasing of holding cost 50% and increases about 23% for 50% increasing of the holding cost. Whereas the holding cost of the manufacturer fluctuates in the reverse direction. Though the changes are less compare to the retailer's holding cost.

* Ordering cost of the retailer is another sensitive parameter. Retailer's total cost decreases about 25% for 50% decreasing of ordering cost and increases almost 20% for 50% increasing of the cost.
<table><caption>**Table 2.** Sensitivity of key parameters for manufacturer.</caption><thead><tr><th>Parameters</th><th>Changes (%)</th><th>Changes Λ<sub>m</sub></th><th>Parameters</th><th>Changes (%)</th><th>Changes in Λ<sub>m</sub></th></tr></thead><tbody><tr><td rowspan="4">S<sub>U</sub></td><td>-50</td><td>-0.00022</td><td rowspan="4">A<sub>m</sub></td><td>-50</td><td>-49.59</td></tr><tr><td>-25</td><td>-0.00009</td><td>-25</td><td>-24.79</td></tr><tr><td>+25</td><td>+0.00007</td><td>+25</td><td>+24.79</td></tr><tr><td>+50</td><td>+0.00013</td><td>+50</td><td>+49.59</td></tr><tr><td rowspan="4">w</td><td>-50</td><td>+0.089</td><td rowspan="4">M</td><td>-50</td><td>-0.17</td></tr><tr><td>-25</td><td>+0.032</td><td>-25</td><td>-0.14</td></tr><tr><td>+25</td><td>-0.022</td><td>+25</td><td>+1.35</td></tr><tr><td>+50</td><td>-0.037</td><td>+50</td><td>+2.53</td></tr><tr><td rowspan="4">γ</td><td>-50</td><td>-0.09</td><td rowspan="4">H<sub>c</sub></td><td>-50</td><td>+2.63</td></tr><tr><td>-25</td><td>-0.04</td><td>-25</td><td>+1.51</td></tr><tr><td>+25</td><td>+0.04</td><td>+25</td><td>-0.0064</td></tr><tr><td>+50</td><td>+0.08</td><td>+50</td><td>+0.102</td></tr></tbody></table>

<table><caption>**Table 3.** Sensitivity of key parameters for retailer.</caption><thead><tr><th>Parameters</th><th>Changes (%)</th><th>Changes Λ<sub>r</sub></th><th>Parameters</th><th>Changes (%)</th><th>Changes in Λ<sub>r</sub></th></tr></thead><tbody><tr><td rowspan="4">O<sub>R</sub></td><td>-50</td><td>-25.36</td><td rowspan="4">B<sub>h</sub></td><td>-50</td><td>-29.74</td></tr><tr><td>-25</td><td>-11.69</td><td>-25</td><td>-13.61</td></tr><tr><td>+25</td><td>+10.41</td><td>+25</td><td>+11.97</td></tr><tr><td>+50</td><td>+19.87</td><td>+50</td><td>+22.78</td></tr></tbody></table>

## 3.2 Managerial Insights

* It is common among almost all the production industries that the flexible production rate is always better than the constant production rate. But to move in the direction of the flexibility there is a huge hazard for system design change and corresponding increasing value of setup cost. The industry manager has to decide that they can tackle this hazard for improving system design or may maintain a traditional production system without any hazard of system design. But utilizing traditional production system cannot produce more degree of accuracy of products properly along with there are huge cost for human labor, which ultimately affects the increasing value of the per unit production cost. Based on this proposed strategy the industry managers can decide which type of production system they need for their product.

* A common finding of any equal and unequal powers of supply chain players is that the total cost must be reduced in case of equal power with joint total cost rather than the unequal power with joint font supply chain cost. However, this study obtains a major total cost reduction for both cases, whoever is
the leader or follower. Therefore, the study of equal power joint total profit was not conducted. In the industry, if the players are unequal in power, still they can get reduced total cost utilizing the recommended strategy of design improvement of the flexible production system.

# 4 Conclusions

This is the first pioneer attempt to prove the effectiveness of automated production system under a supply chain management. It was proved that flexibility in production system and system designing made a significant effect on the production system to reduce per unit production cost of any smart product. Due to the improvement of the technological development of manufacturer's system design the setup cost was increased, thus, the setup cost reduction strategy was utilized and found a huge amount of savings to control the production cost of the system. Numerically, the savings for this purpose is about 10%. A comparative study was conducted to show the effectiveness of flexible production rate instead of constant production rate and found 5% of savings due to flexibility in the production system. As the demand of these smart products during lead time was a random variable and followed a normal distribution, lead time crashing cost was used which saved 2% of the total cost. Though the research was conducted for the development of the manufacturing system design and its improvement under a supply chain management, but those SCM players are not equal in power, they thought about their own profit only. Even though SCM players do not follow each other, using Stackleburg game policy it was found that both the players were gainer at their optimum level of decisions. This study can be extended with several constraints, which came to improve the machine design and the optimization procedure will be constrained optimization rather than the unconstrained optimization technique. It will be a potential extension if a comparative study can be done within the SCM players having equal and unequal powers with a multi-stage production system.

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