A model of an integrated supply chain with imperfect production, product deterioration, and quality inspection errors

Abstract

An integrated production-distribution model is presented in this article to determine the optimal policy for deteriorating and defective items within a single-vendor, single-buyer system. Deterioration is a widely recognized inherent characteristic that is attributed to most objects. Furthermore, the feasibility of producing products without defects is quite impractical. Nevertheless, the imperfect quality issue, as demonstrated by damaged products, can be attributed to a poorly managed manufacturing and production process. Conducting inspection from the viewpoint of both the buyer and the vendor exposes both parties to the possibility of making errors. This study estimates the optimal number of shipments and production run time by incorporating buyer demand fulfilment, inspection, transportation-related carbon emissions, deterioration and defectiveness into a single model. The study mainly intends to minimize supply chain costs to maximize the potential profit. We also investigate a production model without screening to emphasize the significance of inspection. To enhance the illustrative power of the proposed methodology, numerical example and sensitivity analysis are carried out which further assess the validity of the proposed theory.

1 Introduction

As a result of the rapid acceleration of modernization that has occurred over the course of the previous few years, a great number of manufacturers are contending with fierce rivalry in the market. Buyers, these days, are more knowledgeable about where and how to get products at reduced prices, higher efficiencies, and faster delivery times. Inspection is a significant aspect of quality control for many organisations. In recent times, with the establishment of novel inventory models, there has been an increased focus placed on two specific issues namely item deterioration and screening error. Shah Y.K. and Jaiswal M.C. [1] were of the opinion that the pace of deterioration is constant. Covert R.P. and Philip G.C. [2] created a model for commodities with variable deterioration rates that may be used in economic order quantity calculations. Misra R.B. [3] determined that a Weibull distribution with two parameters provide the best fit for the decay rate. The model that Deb M and Chaudhuri KS [4] suggested features variable rates of deterioration, which make it possible for shortages to occur. Gupta R. and Vrat P. [5] took into consideration a model of stock-dependent consumption rate while conducting their research. Mandal B.A. and Phaujdar S. [6] developed an inventory model that includes products deteriorating with time and a consumption rate dependent upon stock levels.

Options for integrated supply networks and cooperative efforts between different supply chains are widely required in order to realize an overall improved performance. In point of fact, integrated supply chain planning and coordination are now far more attainable than they ever were before, because of the significant progress in technology for information and communication. In this context, the application of operations management methodologies and mathematical models provides a solid platform for the making of decisions that are efficient and integrated throughout the supply chain. When it comes to inventory and lot sizing, Economic Production Quantity (EPQ) models are extensively employed by both practitioners and researchers. Harris F [24] is credited for creating the first Economic Order Quantity (EOQ) model, which has since been extended to include a number of real-world scenarios namely back-ordering, quantity discounts, multiplicity of stages. However, this type of EOQ/EPQ model just provides an exchange between both the holding costs and ordering costs incurred by the buyer. As a result, the conventional EOQ/EPQ model has a number of drawbacks. For instance, although flawed product quality is more likely and rational, classic EOQ/EPQ models assume ideal product quality.

These irrational presumptions opened the door for many researchers to begin taking quality into account when choosing lot sizes. Porteus EL [28], Rosenblatt and Rosenblatt M.J. and Lee H.L. [30], and Silver E [32] are a few illustrative instances of such literature. Additionally, Ahmed ARN. [33], Ben-Daya M. and Rahim A [18], Ben-Daya M., Darwish M.A. and Rahim A. [19], and Ben-Daya M. and Rahim A [20] have conducted important research on making decisions about the lot size for a manufacturing process with imperfect performance and unsatisfactory quality inspection. These lot sizing models are more accurate in embracing the notion of screening and quality assurance as well as process upkeep, alterations, and fixes, etc.

However, the majority of the traditional EOQ/EPQ models essentially disregard the impact of variables such as inspection errors and deterioration in products. It is not practical that these variables have not been taken into account while using lot sizing models for supply chain management. The lack of study on the incorporation of inspection errors and learning in supply chain performance choices is most likely due to the complexity and unpredictability that result from interactions across stages. This is the most plausible explanation for the lack of research. On the other hand, the significance of these factors in judging how efficiently a supply chain is operating might vary widely. As a direct consequence of this, it is essential especially to expand the scope of the existing research in supply chain management by developing a theoretical framework that takes into consideration critical elements such as errors in quality inspections and production learning when making decisions about lot sizing and supply chain performance in Ben-Daya M and Rahim A [20]. This article describes a straightforward mathematical model that, despite its apparent ease of use, is helpful for finding the most effective buyer-vendor inventory strategy for a specific product. It takes into account errors that occur during quality inspection both at the buyer’s end and at the vendor’s end, as well as deterioration in the products, with the objective of reducing the overall annual cost of the supply chain.

A growing number of academicians and industry professionals are actively looking for analytical tools for efficient and comprehensive supply chain management. Emphasis should be given on the fact that Type I and II errors are a result of erroneous judgments made during the screening of items. Falsely accepting a damaged product is a Type I error. An erroneous rejection of a product that is not defective, on the other hand, is known as a Type II error. The likelihood of these errors depends on the process parameters and is typically chosen based on an inspector’s past performance.

The remainder of the article is organised as shown below: In section 2, a discussion of the pertinent research literature is presented. The construction of the mathematical model is described in section 3. A numerical example is provided in section 4 to demonstrate the scalability and efficiency of the model that has been suggested. Sensitivity analysis is presented in section 5 and in section 6, the paper draws to a close by discussing a number of intriguing topics that could be investigated in the future.

2 Literature survey

The idea of coordination has been around for quite some time. The work of Goyal S.K. [7] is one of the earliest models that solves the problem of determining a combined economic lot size for a system which includes both a vendor and a buyer. He made two assumptions: the first being unlimited production rate, and the second is that the vendor ships the goods in bunches. Banerjee A. [8] relaxed the restriction on the manufacturing pace and asserted that following the lot-for-lot strategy, the vendor manufactures the item in the required amount as soon as an order is received, and after the required batch is prepared, the vendor ships the full lot to the buyer. After the complete production lot has been made using the same scenario with one vendor and one buyer, Goyal SK [9] suggested sending it in several equal-sized shipments. This could be done after the lot has been completed. Lu L.[10] extended this concept with the assumption that shipping could take place even while production was in progress. In a circumstance quite similar to this one, Goyal SK [11] proposed a different approach to the transportation of the lots. He recommended that the succeeding deliveries within a production lot should increase by a factor that is equivalent to the product of the production and demand rates. Hill RM [12] extended the aforementioned technique by employing a factor of successive growth as the choice variable. He reasoned that the proportional unit holding costs of both the buyer and the vendor should be taken into consideration when determining whether or not a batch should be distributed in lots of similar size. Sarmah S.P., Acharya D. and Goyal S.K. [37] presented the buyer vendor cooperation in supply chain management. As they classed various models after using quantity discount as a coordination mechanism in a deterministic setting. Khan M, Jaber MY, Zanoni S and Zavanella L [35] provided a model for a consignment stock agreement used for vendor-managed inventory, in situations such as a supply chain considering defective goods. In a recent study, Rout C, Paul A, Kumar RS, Chakraborty D and Goswami A [36] established a model for flawed production and deteriorating items in a cooperative sustainable supply chain. The model takes into account various carbon emission regulations. The topic of economic ordering and production size is one that is still considered essential, and academicians are continuously looking for new paths to investigate in this field. Zhang Q., Tsao YC and Chen TH [13] established a current optimal inventory method for the situation in which payments are made prior to the delivery of the goods . They provided evidence to show that the duration of the advance payment does not have any impact on the inventory policy. Nevertheless, this is due to the fact that there is a discount associated to early payment. This results in a longer replenishment cycle. Considering a scenario where a buyer is offered with a vendor’s one-time-only discount, Zhou Y, Chen J, Wu Y and Zhou W [14] developed EPQ models in order to evaluate which decisions would be most advantageous for the buyer. They presented an example that highlights how the date and the price are both factors influencing their decision. Shu MH, Huang JC and Fu YC [15] made the assumption that the lead time for transportation is exponentially distributed, and that the cost of transportation is a function of lot size, in the context of a vendor-buyer supply chain. They established that these hypotheses result in large cost reductions by comparing them to traditional coordination models.

Recent research conducted by Rahdar M. and Nookabadi A.S. [16] explores the collaboration that occurs between a single vendor and several buyers when inventory levels drop as a result of items deteriorating over time. They devised a method that enables buyers to coordinate their production schedules and delivary times with those of the vendors. Moreover, they suggested that as an additional incentive for cooperation, a profit-sharing plan be offered to the buyer. According to Lee S and Kim D [17], the deterioration of products that takes place along a vendor-buyer supply chain is regarded to be an endogenous trait. In order to discriminate between the effects of quality and deterioration in a supply chain that comprise of two stages, authors carried out a sensitivity analysis as well.

The research that has been done on inventory and supply chain management has to pay adequate attention to the extremely important problem of quality inspection errors. Bennett G.K., Case K.E. and Schmidt JW [21], conducted an investigation into the influence that inspection errors have on attribute sampling plan. Ahmed ARN [33] and Ben-Daya M and Rahim A [20], incorporated inspection errors in combined production, qualitative, and maintenance options in two-stage and multistage flawed inventory production contexts, respectively. Raouf A, Jain JK and Sathe PT [29] devised a model to predict the appropriate number of repeat inspections for multi-characteristic components in situations, when both Type I and Type II errors are present throughout the quality inspection process. The model was then extended by Duffuaa SO and Khan M. [22] considering some added features like scrap and rework for flawed objects. A new path has been paved in the research on inventory management as a result of the work done by Salameh MK and Jaber MY [31]. They addressed an EOQ model that could be used for items having manufacturing defects. An overview of the models that extend the work of Salameh MK and Jaber MY [31] is presented by Khan M, Jaber MY, Guiffrida AL and Zolfaghari S [27].

These are some of the oldest techniques to indicate how well a vendor and a buyer coordinate, and they consist of establishing a combined economic lot size model, which produces on a lot-by-lot basis. These models were extended by Ahmed ARN [33], Ben-Daya M. and Rahim A. [18], Ben-Daya M., Darwish M.A. and Rahim A. [19], and Ben-Daya M. and Rahim A. [20] to encompass integrated production, quality inspections, and maintenance choices. Their extensions, on the other hand, did not take into account essential human aspects like learning. Goyal S.K. [9] proposed a economic lot size model by relaxing the lot-for-lot assumption that was presented by Banerjee A. [8]. A model for a supply chain with two levels was developed by Salameh M.K. and Jaber MY [31] and extended by Huang CK [25]. They used the same shipment size policy regardless of whether they were dealing with damaged items or not. One may claim that integrated inventory management has recently captured the attention of a significant number of scholars. It is recommended that readers follow the articles by Jaber MY and Zolfaghari S [26] and Glock CH [23] for a more in-depth review. Khan M, Jaber MY and Ahmad AR [34] explored an integrated supply chain network including errors in quality inspection at the buyers’ end and learning in production. Juman ZAMS, M’Hallah R, Lokuhetti R and Battaïa O [38] investigated a model that integrates production and inventory for many vendors and buyers and includes synchronised delivery of batches of varying sizes.

The level of coordination between a vendor and a buyer has been successfully assessed by numerous researchers till date. An inventory system with an integrated vendor-customer approach was developed by Darma Wangsa I and Wee HM [40], in which transportation costs are considered along with stochastic demand pattern. Omar, M. and Zulkipli H [41] formulated an integrated single-vendor and multi-buyers’ production-inventory system considering stock-dependent demand rate. A comprehensive vendor-buyer supply chain network incorporating price-dependent demand and discount on backorders was presented by Malleeswaran B and Uthayakumar R [44]. Besides, their model also takes into account service level restrictions and carbon emission costs. Gharaei, A., Hoseini Shekarabi SA and Karimi M [42] offered an outer approximation for the optimal lot-sizing of an integrated EPQ model that considers reworkable items under partial backordering. Also Gharaei A, Amjadian A, Amjadian A, Shavandi A, Hashemi A, Taher M and Mohamadi N [43] proposed a null-space technique for the implementation of an integrated lot-sizing policy in the inventory management of limited multi-level supply chains. Wangsa ID, Vanany I and Siswanto N [45] investigated a sustainable supply chain network which includes investments in green technology and electric equipment.

In our proposed work, a two-level supply chain model is investigated by taking into account a more realistic approach that considers imperfect production, product deterioration, carbon emissions due to transportation, and screening at both the vendor and buyer ends of the supply chain. To the best of our knowledge, no other model in the current literature has all of the aforesaid characteristics and practical factors. Furthermore, we explore the issue of fulfilling the demands of the consumer by manufacturing supplementary goods during the production time. With the purpose of carrying out a comparative study, we also take into consideration a model in which there are no inspections performed. At this point in time, it is accepted that neither the buyer nor the vendor will conduct any kind of inspection whatsoever during the duration of the contract.

3 Model illustration

In this proposed approach, a lot size policy has been devised for a co-ordinated supply chain network consisting of vendors and buyers. A single type of product is produced by the manufacturer in accordance with an EPQ policy. We designed the model so that the vendor can begin manufacturing immediately based on the demand of the buyer. The vendor fulfills the order with several similar-sized shipments. We assume that some products are not shipped by the vendor due to deterioration and imperfect production. As a result, in order to meet the demand, the vendor produces more goods than the order required according to the rate of defective items. The inspection process is carried out by both the vendor and the buyer. Inspection at both ends result in Type I and Type II errors. After initiating the screening process on the vendor side with both types of screening errors, the model is enlarged. Carbon emissions from transportation are a significant factor in the decision-making process thereby we consider carbon emission cost for transportation.

The various cost components involved in the system include ordering (setup) costs, screening costs, transportation costs, carbon emission cost and production costs. The buyer is responsible for the fixed and variable costs related to the shipment transportation. The principal objective of the proposed model lies in minimizing the overall supply chain costs.

3.1 Notations

n:

Number of shipments sent out by the vendor to the buyer per cycle (decision variable)

$T_1$:

Production period (decision variable)

$\mu$:

% of items that are deteriorated and defective shipped by the vendor

$\mu _e$:

% of deteriorated and defective goods screened by the buyer

Q:

The size of every shipment

$d_b$:

The cost of screening at the buyer side per unit $(\$)$

$d_v$:

The cost of screening at the vendor side per unit $(\$)$

T:

The time gap between two consecutive shipments (yrs)

X:

Screening rate of buyer (units/yr)

$m_1$:

Stochastic variable(%) describing Type I error at buyer

$m_2$:

Stochastic variable(%) describing Type II error at buyer

$\theta _v$:

Rate of item deterioration over time at the vendor’s location

$\theta _b$:

Rate of item deterioration over time at the buyer’s location

$f(m_1)$:

Probability distribution function of $m_1$

$f(m_2)$:

Probability distribution function of $m_2$

$M_1$:

Stochastic variable(%) describing Type I error at vendor

$M_2$:

Stochastic variable(%) describing Type II error at vendor

$f(M_{1})$:

Probability distribution function of $M_{1}$

$f(M_{2})$:

Probability distribution function of $M_{2}$

$\alpha$:

Ratio of defective items to good quality items

D:

Buyer’s demand (units/cycle)

P:

Production rate of vendor (units/year)

c:

Production cost of vendor per unit time ($\$$/year)

F:

Fixed cost of transportation ($\$$)

$\nu$:

Variable cost of transportation ($\$$/unit)

$A_v$:

Fixed setup cost of vendor ($\$$/setup)

$A_b$:

Fixed ordering cost of buyer($\$$/order)

$h_v$:

Holding cost for the vendor ($\$$/unit/year)

$h_b$:

Holding cost for the buyer ($\$$/unit/year)

$c_r$:

Cost of Type I error of buyer

$c_a$:

Cost of Type II error of buyer

S:

Selling price per unit of product at the buyer end($\$$/unit)

$I_{v_1}(t_1)$:

Serviceable instantaneous inventory level during production

$I_{v_2}(t_1)$:

Serviceable instantaneous inventory level during non-production

$I_{b}(t)$:

Serviceable instantaneous inventory level at the buyer end

$c_q$:

Emission cost ($/Ton)

$c_t$:

Carbon emission by vehicle(Ton/km)

d:

Distance between the vendor and the buyer (km)

3.2 Description and presumptions about the situation

An integrated supply chain model is examined in this study. The model includes a single buyer and a single vendor who interact with one another over an indefinite time. The vendor obtains the raw materials from a supplier (one that is not a part of this integrated supply chain) and then initiates the production process in order to completely fulfil the buyer’s requirement. The rate of output from the vendor is consistent. Inevitably, during the manufacturing process, some of the products will be erroneous. Therefore, the vendor needs to manufacture a greater quantity of goods so that the rate at which the product is being manufactured is higher than the steady demand.

This article takes into account carbon emissions generated by various modes of transportation. The inspections are carried out from the buyer’s perspective as well as the vendor’s, which result in errors being made on both sides.

During constructing the model, the relevant presumptions that are taken into account are as follows:

• 1. A single vendor-buyer integrated model is taken into consideration.

• 2. Over an infinite planning horizon, only one item type is taken into account.

• 3. Instant restocking is available.

• 4. Shortages are not allowed.

• 5. The rate of production is higher than the demand rate.

• 6. During the inspection process, both the buyer and the vendor are prone to errors.

• 7. Carbon emission due to transportation is considered.

In order to define the optimization problem, we make use of the afore-mentioned notations and assumptions.

3.3 Inventory Illustrations

The inventory of vendor is depicted in figure 1, whereas buyer’s inventory is shown in figure 2(a). The vendor sends Q items as soon as more than $Q/(1-\mu _e)$ items are manufactured. Despite the fact that there is deterioration and defective in goods, the vendor keeps on producing goods until a period of optimal production is reached. This is because the vendor is attempting to satisfy the whole demand placed by the buyer. When production is stopped; nonetheless, the supplier will continue to send out the same shipments throughout this time. Now, the vendor begins screening prior to each shipment, and as shown in figure 1, they discard any goods that are damaged at the completion of the cycle.

Figure 2(a) shows a representation of the buyer’s inventory that employs the same format as was done by Salameh MK and Jaber MY [31]. Now, the buyer starts screening at the beginning of the cycle, and once this screening operation is through, the buyer either rejects up on things that have been damaged or rescues them.

Figure 2(b) presents the level of inventory held by the vendor, which is proportional to the total of all of the areas denoted by the triangles ABG and BCEG in the squared-off rectangle BCEG. On the other side, the opaque area shows the quantity of inventory that is transferred from the vendor to the buyer. To the contrary, the amount of stock held by the buyer is, in fact, calculated based on figure 2(a).

Figure 1

Instantaneous inventory level of the vendor throughout a complete cycle.

Figure 2

Graphical plots for (a) One cycle of the buyer’s inventory in the vendor’s cycle (b) Accumulation and supply of vendor’s inventory in a cycle.

3.4 Mathematical model

Differential equations can be employed to illustrate how production, deterioration, and demand rate influence the changes that are made to the buyer’s and vendor’s inventory levels. The instantaneous states at the vendor end can be explained by the differential equations that are listed below:

$$\begin{aligned}&\frac{d I_{v_1}\left( t_1\right) }{d t_1}=(1-\alpha ) P-D-\theta _v I_{v_1}\left( t_1\right) , \quad 0 \le t_1 \le T_1, \\&\quad I_{v_1}(0)=0\\&\quad \frac{d I_{v_2}\left( t_1\right) }{d t_1}=-D-\theta _v I_{v_2}\left( t_1\right) , \quad T_1 \le t_1 \le t_3,\\&\quad I_{v_2}(T_1) = I_{v_1}(T_1). \end{aligned}$$

The solutions to the differential equations are provided as follows:

$$\begin{aligned}&I_{v_1}(t_1)=\frac{(1-\alpha ) P-D}{\theta _v}\left( 1-e^{-\theta _v t_1}\right) , \quad 0 \le t_1 \le T_1\\&\quad I_{v_2}\left( t_1\right) =-\frac{D}{\theta _v}+\left[ e^{\theta _vT_1}\left( I_{v_1}(T_1)+\frac{D}{\theta _v}\right) \right] e^{-\theta _vt_1} , \\&\quad T_1 \le t_1 \le t_3. \end{aligned}$$

Total serviceable inventory in a cycle at the vendor’s end is

$$\begin{aligned} (1-M_1)DT_1+I_{v_1}(T1)-(t_2-t_3)D \end{aligned}$$

where

$$\begin{aligned} t_2=\frac{I_{v_1}(T_1)}{D} \end{aligned}$$

and

$$\begin{aligned} t_3=\frac{1}{\theta _v}ln\left[ \frac{(I_{v_1}(T_1)+\frac{D}{\theta _v})e^{\theta _vT_1}}{\frac{D}{\theta _v}}\right] -T_1. \end{aligned}$$

Total Production = $PT_1.$

Total (deteriorated + defective) items = $PT_1- (1-M_1)DT_1-I_{v_1}(T1)+(t_2-t_3)D$

Total number of items sent by the vendor in a cycle is

$$\begin{aligned} DT_1+(1-M_1)\left[ I_{v_1}(T_1)-(t_2-t_3)D\right] +M_2\left[ PT_1-DT_1-I_{v_1}(T_1)+(t_2-t_3)D\right] \end{aligned}$$

So,

$$\begin{aligned} Q=\frac{DT_1+(1-M_1)\left[ I_{v_1}(T_1)-(t_2-t_3)D\right] +M_2\left[ PT_1-DT_1-I_{v_1}(T_1)+(t_2-t_3)D\right] }{n} \end{aligned}$$

Ratio of non-serviceable items to serviceable items is given by

$$\begin{aligned} \mu =\frac{M_2\left[ PT_1-I_{v_1}(T_1)+(t_2-t_3)D\right] }{nQ}. \end{aligned}$$

According to the buyer’s inspection, the percentage of unworkable units would be

$$\begin{aligned} \mu _e=(1-\mu )m_1+\mu (1-m_2). \end{aligned}$$

For each cycle, the initial number of serviceable items at the buyer’s end is

$$\begin{aligned} I_b(0)=(1-\mu )(1-m_1)Q. \end{aligned}$$

The following is a representation of the buyer’s instantaneous serviceable inventory level:

$$\begin{aligned} \frac{dI_b(t)}{dt}=-\frac{D(1-\mu )(1-m_1)}{(1-\mu _e)}-\theta _bI_b(t). \end{aligned}$$

Solving the above, we obtain:

$$\begin{aligned} I_b(t)=\frac{D(1-\mu )(1-m_1)}{\theta _b(1-\mu _e)}\left[ e^{-\theta _bt}-1\right] +I_b(0)e^{-\theta _bt}. \end{aligned}$$

The total number of items which unserviceable that are sold by the buyer is

$$\begin{aligned} n\left[ (1-\mu _e)Q-I_b(0)+(t_4-t_5)D\right] \end{aligned}$$

where

$$\begin{aligned} t_4=\frac{I_b(0)(1-\mu _e)}{D(1-\mu )(1-m_1)} \end{aligned}$$

and

$$\begin{aligned} t_5=\frac{1}{\theta _b}ln\left[ 1+\frac{I_b(0)(1-\mu _e)\theta _b}{D(1-\mu )(1-m_1)}\right] . \end{aligned}$$

In a cycle, the entire inventory with the vendor equals the sum of the areas of the region ABG and BCEG in figure 3.

$$\begin{aligned} \text{ Area } _{A B G}&= \frac{1}{2}(PT_1)(P)=\frac{1}{2}PT_1^2 \\ \text{ Area } _{B C E G}&= PT_1\left[ (n-1)T+\frac{T_1}{n}-T_1\right] \\ &= \frac{PT_1}{n}\left[ (n-1)(nT-T_1)\right] \end{aligned}$$

As a consequence of this, the total amount of stock that is transferred during the course of a cycle, from the vendor to the buyer is denoted by $n(n-1)Q^2/(2D)$. So, the cumulative inventory of the vendor during a cycle is

$$\begin{aligned} I_{V}(T_1, n)&=\frac{1}{2}PT_1^2+\frac{PT_1}{n}\left[ (n-1)(nT-T_1)\right] -\frac{n(n-1)Q^{2}}{2 D} \end{aligned}$$

The cost of a cycle for the vendor is the sum of the expenditures associated with setting up, carrying, screening, and producing the good or service.

$$\begin{aligned} C_{v}(T_1,n)= A_{v}+h_vI_v(T_1,n) +cT_1+PT_1d_v \end{aligned}$$

The expenses of the buyer in a cycle include ordering, transportation, screening and shipping costs as provided below:

$$\begin{aligned} C_{b}(T_1, n)&=A_{b}+n h_{b}\left\{ \frac{Q(1-\mu _e) T}{2}+\frac{\mu _e Q^{2}}{X}\right\} \\& \quad +n d_{b} Q+n(F+\nu Q). \end{aligned}$$

Now the cost of mis-classification, deterioration and carbon emission due to transportation is

$$\begin{aligned} C_t(T_1,n)&= c_an\left[ (1-\mu _e)Q-I_b(0)+(t_4-t_5)D\right] \\ &\quad +c_r(1-\mu )m_1nQ+2ndc_{q}c_{t}. \end{aligned}$$

In a cycle, the overall supply chain (vendor-buyer) cost is

$$\begin{aligned} TC(T_1,n)=C_{v}(T_1, n)+C_{b}(T_1, n)+C_t(T_1,n). \end{aligned}$$

Taking expectation of all the above functions we get, total expected serviceable inventory in a cycle at the vendor’s end is

$$\begin{aligned} (1-E[M_1])DT_1+I_{v_1}(T1)-(t_2-t_3)D. \end{aligned}$$

Expected(deteriorated items + defective items) is

$$\begin{aligned} PT_1- (1-E[M_1])DE[T]_1+I_{v_1}(T1)-(t_2-t_3)D. \end{aligned}$$

Expected number of items sent by the vendor is

$$\begin{aligned}&DT_1+(1-E[M_1])\left[ I_{v_1}(T_1)-(t_2-t_3)D\right] \\&\qquad +E[M_2]\left[ PT_1-DT_1-I_{v_1}(T_1)+(t_2-t_3)D\right] . \end{aligned}$$

So,

$$\begin{aligned} E[Q]=\frac{DT_1+(1-E[M_1])\left[ I_{v_1}(T_1)-(t_2-t_3)D\right] +E[M_2]\left[ PT_1-DT_1-I_{v_1}(T_1)+(t_2-t_3)D\right] }{n}. \end{aligned}$$

Expected ratio of non serviceable items to serviceable items is given by

$$\begin{aligned} E[\mu ]=\frac{E[M_2]\left[ PT_1-DT_1-I_{v_1}(T_1)+(t_2-t_3)D\right] }{nE[Q]}. \end{aligned}$$

The expected fraction of non serviceable units as perceived by the inspector of the buyer would be

$$\begin{aligned} E[\mu _e]=(1-E[\mu ])E[m_1]+E[\mu ](1-E[m_2]). \end{aligned}$$

The amount of serviceable goods that are expected to be available at the buyer’s end for each cycle initially is

$$\begin{aligned} E[I_b(0)]=(1-E[\mu ])(1-E[m_1])E[Q]. \end{aligned}$$

The following is a representation of the buyer’s estimated serviceable inventory variation:

$$\begin{aligned} E[I_b(t)]&=\frac{D(1-E[\mu ])(1-E[m_1])}{\theta _b(1-E[\mu _e])}\left[ e^{-\theta _bt}-1\right] \\ &\quad +E[I_b(0)]e^{-\theta _bt}. \end{aligned}$$

Expected number of non-serviceable items sold by the buyer is

$$\begin{aligned} n\left[ (1-E[\mu _e])E[Q]-E[I_b(0)]+(E[t_4]-E[t_5])D\right] \end{aligned}$$

where

$$\begin{aligned} E[t_4]=\frac{E[I_b(0)](1-E[\mu _e])}{D(1-E[\mu ])(1-E[m_1])} \end{aligned}$$

and

$$\begin{aligned} E[t_5]&=\frac{1}{\theta _b}ln\left[ 1+\frac{E[I_b(0)](1-E[\mu _e])\theta _b}{D(1-E[\mu ])(1-E[m_1])}\right] .\\ E[I_{V}(T_1, n)]&=\frac{1}{2}PT_1^2+\frac{PT_1}{n}\left[ (n-1)(nE[T]-T_1)\right] \\&\quad -\frac{n(n-1)E[Q]^{2}}{2 D} \end{aligned}$$

The vendor’s anticipated expense is given by

$$\begin{aligned} E[C_{v}(T_1,n)]= A_{v}+h_vE[I_v(T_1,n)] +cT_1+PT_1d_v. \end{aligned}$$

The expected cost for the buyer is

$$\begin{aligned}{} & {} E[C_{b}(T_1, n)]=A_{b}+n h_{b}\left\{ \frac{E[Q](1-E[\mu _e]) E[T]}{2}\right. \\{} & {} \quad \left. +\frac{E[\mu _e]E[Q]^{2}}{X}\right\} +n d_{b} E[Q]+n(F+\nu E[Q]). \end{aligned}$$

Now the expected cost of mis-classification, deterioration and carbon emission for transportation is

$$\begin{aligned}{} {} E[C_t(T_1,n)]&= c_an\left[ (1-E[\mu _e])E[Q]-E[I_b(0)]\right. \\{} & {} \quad \left. +(E[t_4]-E[t_5])D\right] \\{} & {} \quad +c_r(1-E[\mu ])E[m_1]nE[Q]+2ndc_{q}c_{t}. \end{aligned}$$

In a cycle, the expected supply chain (vendor-buyer) cost is given by

$$\begin{aligned} E[TC(T_1,n)]=E[C_{v}(T_1, n)]+E[C_{b}(T_1, n)]+E[C_t(T_1,n)]. \end{aligned}$$

As a response, the interval between subsequent deliveries takes the form

$$\begin{aligned} E[T]=\frac{(1-E[\mu _{e}])E[Q]}{D}. \end{aligned}$$

As we have seen, $E[T]=(1-E[\mu _e]) E[Q] / D$, using nE[T], the overall duration of the cycle, the expected annual cost is

$$\begin{aligned} E[TCU(T_1,n)]=\frac{E[C_{v}(T_1, n)]+E[C_{b}(T_1, n)]+E[C_t(T_1,n)]}{nE[T]}. \end{aligned}$$

The objective is to minimize the function $E[TCU(T_1,n)]$ by considering the constraint of demand fulfilment $D\le nE[Q]$, in order to calculate the optimal production time($T^*$) time and the optimal number of shipments($n^{*}$).

As a result, the number of units sold by the buyer is $n^*E[Q]^*(1-E[\mu _e])$ and the combined profit is $n^*E[Q]^*(1-E[\mu _e])*S-E[TCU(T_1^*,n^*)]$.

3.5 Special case

We also take into consideration a model in which there are no inspections carried out so that we can bring about a comparison. It is considered that neither the buyer nor the vendor will carry out any inspection throughout this period of the contract. Differential equations can be utilized to illustrate how production, deterioration, and demand rate affect changes in the buyer’s and vendor’s inventory levels. The instantaneous states of serviceable inventory level can be explained using the differential equations that are listed below:

$$\begin{aligned}&\frac{d I_{v_1}\left( t_1\right) }{d t_1}=(1-\alpha ) P-D-\theta _v I_{v_1}\left( t_1\right) , ~~ 0 \le t_1 \le T_1, ~~ I_{v_1}(0)=0\\&\quad \frac{d I_{v_2}\left( t_1\right) }{d t_1}=-D-\theta _v I_{v_2}\left( t_1\right) ,~~ T_1 \le t_1 \le t_3,\quad I_{v_2}(T_1) = I_{v_1}(T_1). \end{aligned}$$

Following are the associated solutions:

$$\begin{aligned}&I_{v_1}(t_1)=\frac{(1-\alpha ) P-D}{\theta _v}\left( 1-e^{-\theta _v t_1}\right) ,~~ 0 \le t_1 \le T_1\\&\quad I_{v_2}\left( t_1\right) =-\frac{D}{\theta _v}+\left[ e^{\theta _vT_1}\left( I_{v_1}(T_1)+\frac{D}{\theta _v}\right) \right] e^{-\theta _vt_1},~~ T_1 \le t_1 \le t_3. \end{aligned}$$

At the completion of each cycle, the total amount of the vendor’s inventory that is usable is

$$\begin{aligned} (1-M_1)DT_1+I_{v_1}(T1)-(t_2-t_3)D \end{aligned}$$

where

$$\begin{aligned} t_2=\frac{I_{v1}(T_1)}{D} \end{aligned}$$

and

$$\begin{aligned} t_3=\frac{1}{\theta _v}ln\left[ \frac{(I_{v_1}(T_1)+\frac{D}{\theta _v})e^{\theta _vT_1}}{\frac{D}{\theta _v}}\right] -T_1. \end{aligned}$$

Total Production = $PT_1=D i.e., T_1=\frac{D}{P}$

Total (deteriorated + defective) items = $PT_1- (1-M_1)DT_1+I_{v_1}(T_1)-(t_2-t_3)D$

Total number of items sent by the vendor is D. So

$$\begin{aligned} Q=\frac{D}{n} \end{aligned}$$

and

$$\begin{aligned} T=\frac{Q}{D} \end{aligned}$$

Ratio of non serviceable items to serviceable items,

$$\begin{aligned} \mu =\frac{PT_1- (1-M_1)DT_1+I_{v_1}(T1)-(t_2-t_3)D}{nQ} \end{aligned}$$

The initial quantity of usable products available to the buyer at the beginning of each cycle is

$$\begin{aligned} I_b(0)=(1-\mu )Q \end{aligned}$$

The following is a representation of the buyer’s fluctuating serviceable inventory:

$$\begin{aligned} \frac{dI_b(t)}{dt}= -D(1-\mu )-\theta _bI_b(t) \end{aligned}$$

Solving this, we get

$$\begin{aligned} I_b(t)=\frac{D(1-\mu )}{\theta _b}\left[ e^{-\theta _bt}-1\right] +I_b(0)e^{-\theta _bt} \end{aligned}$$

The total number of items that cannot be serviced and are sold by the buyer is

$$\begin{aligned} n\left[ Q-I_b(0)+(t_4-t_5)D\right] \end{aligned}$$

where

$$\begin{aligned} t_4=\frac{I_b(0)}{D(1-\mu )} \end{aligned}$$

and

$$\begin{aligned} t_5=\frac{1}{\theta _b}ln\left[ 1+\frac{I_b(0)\theta _b}{D(1-\mu )}\right] \end{aligned}$$

In a cycle, the entire inventory with the vendor equals the sum of $\triangle$ ABG and rectangle BCEG regions in figure 3.

$$\begin{aligned} \text{ Area } _{A B G}= & {} \frac{1}{2}(PT_1)(P)=\frac{1}{2}PT_1^2\\ \text{ Area } _{B C E G}= & {} PT_1\left[ (n-1)T+\frac{T_1}{n}-T_1\right] \\= & {} \frac{PT_1}{n}\left[ (n-1)(nT-T_1)\right] \end{aligned}$$

In this instance, neither the vendor nor the buyer do any inspections. This model has been proposed using the three images.

As a result, $n(n-1)Q^2/(2D)$ is the total amount of stock that the vendor transmits to buyer. Therefore, the vendor’s total inventory for a cycle equals

$$\begin{aligned} I_{v}(n)&=\frac{1}{2}PT_1^2+\frac{PT_1}{n}\left[ (n-1)(nT-T_1)\right] -\frac{n(n-1)Q^{2}}{2 D} \end{aligned}$$

The overall cost of setup, carrying, screening, and manufacturing incurred by the vendor in a cycle is:

$$\begin{aligned} C_{v}(n)= A_{v}+h_vI_v(n) +cT_1 \end{aligned}$$

The expense of the buyer in the cycle includes ordering. transportation, screening and shipping costs i.e.,

$$\begin{aligned} C_{b}(n)=A_{b}+n h_{b}\left\{ \frac{Q T}{2}\right\} +n(F+\nu Q) \end{aligned}$$

Now the cost of defective and deteriorated products sold and carbon emission for transportation is given by

$$\begin{aligned} C_t(n)= c_an\left[ Q-I_b(0)+(t_4-t_5)D\right] +2ndc_{q}c_{t} \end{aligned}$$

In a cycle, the overall two-level supply chain (vendor-buyer) cost is

$$\begin{aligned} TC(n)=C_{v}(n)+C_{b}(n)+C_t(n) \end{aligned}$$

Taking expectation of all the above functions, we get,

Expected Ratio of non serviceable items to serviceable items given by:

$$\begin{aligned} E[\mu ]=\frac{PT_1- (1-E[M_1])DT_1+I_{v_1}(T1)-(t_2-t_3)D}{nQ} \end{aligned}$$

For each cycle, the initial number of expected serviceable items at the buyer’s end is

$$\begin{aligned} E[I_b(0)]=(1-E[\mu ])Q \end{aligned}$$

The buyer anticipates selling a certain amount of non-serviceable equipment:

$$\begin{aligned} n\left[ Q-E[I_b(0)]+(E[t_4]-E[t_5])D\right] \end{aligned}$$

where

$$\begin{aligned} E[t_4]=\frac{E[I_b(0)]}{D(1-E[\mu ])} \end{aligned}$$

and

$$\begin{aligned} E[t_5]=\frac{1}{\theta _b}ln\left[ 1+\frac{E[I_b(0)]\theta _b}{D(1-E[\mu ])}\right] \end{aligned}$$

Now the expected cost of defective and deteriorated products sold by the buyer and carbon emission for transportation is

$$\begin{aligned} E[C_t(n)]= c_an\left[ Q-E[I_b(0)]+(E[t_4]-E[t_5])D\right] +2ndc_{q}c_{t} \end{aligned}$$

The expected supply chain (vendor-buyer) cost is expressed as

$$\begin{aligned} E[TC(n)]=C_{v}(T_1, n)+C_{b}(T_1, n)+E[C_t(T_1,n)] \end{aligned}$$

As a response, the interval between subsequent deliveries is given by

$$\begin{aligned} T=\frac{Q}{D} \end{aligned}$$

The estimated annual cost takes the following form

$$\begin{aligned} E[TCU(n)]=\frac{C_{v}(n)+C_{b}(n)+E[C_t(n)]}{nT} \end{aligned}$$

The objective is to minimize the function $E[TCU(T_1,n)]$ by considering the constraint of demand fulfilment $D\le nE[Q]$, in order to calculate the optimal production time($T^*$) time and the optimal number of shipments($n^*$).

As a result, the number of units sold by the buyer is nQ and the combined profit is $nQS-E[TCU(n^*)]$.

3.6 Solution procedure

The model’s purpose is to reduce overall costs while simultaneously increasing overall profits as much as possible. In order to choose the best course of action, we apply the following methodology. The optimal annual cost, optimal number of shipments, and ideal size of each shipment are calculated using the following algorithm:

Algorithm:

1. set $y := INT\_MAX$, $m := 1$

2. for n := 1 to D

3. calculate $y^* := min_{T_1}\{E[~TCU(T_1,n)~]:D\le nE[Q]\}$ at $T_1=T_{1}^*$

4.        if $y^*\le y$ then set $y := y^* , m := n,T=T_{1}^*$

5. print m,T and y as optimal shipment quantity, optimal production time and expected annual cost.

4 Numerical example

An example of a combined (vendor-buyer) supply chain that mainly relies on human labour to complete activities on its terminals, such supply chains for cars or agri-fresh foods. To fulfil the buyer’s wish, the merchant produces a single item. This paper introduces two problems: product deterioration and screening errors made by both the vendor and the buyer. For a constant rate of defective production system, the rate of production in this case is set at $\alpha = 0.02$, and the selling price per unit of the items is set at $S=50$. At the vendor end, the constant rate of deterioration is $\theta _v=0.02$, and at the buyer end, it is $\theta _b=0.02$ also we set $c_q=50$ and $c_t=0.0002$. The majority of the extra input parameters used in this example were taken from Goyal SK, Huang CK and Chen KC [39], Salameh M.K. and Jaber MY [31], and Khan M., Jaber MY and Ahmad AR [34]. Table 1 displays the input criteria and output of our case. To create value during the manufacturing process, it is anticipated that the unit holding cost of the product at the end of the buyer will be greater than the unit holding cost at the end of the vendor. The following makes the assumption that Type I and Type II inspection errors made by the vendor are distributed uniformly, as are Type I and II inspection errors made by the buyer.

$$\begin{aligned} f\left( m_{1}\right) =\left\{ \begin{array}{cc} \frac{1}{0.02-0}, &{} 0 \le m_{1} \le 0.02 \\ 0, &{} \text{ otherwise } \end{array}\right. \\ f\left( m_{2}\right) =\left\{ \begin{array}{cc} \frac{1}{0.02-0}, &{} 0 \le m_{2} \le 0.02 \\ 0, &{} \text{ otherwise } \end{array}\right. \\ f\left( M_{1}\right) =\left\{ \begin{array}{cc} \frac{1}{0.02-0}, &{} 0 \le M_{1} \le 0.02 \\ 0, &{} \text{ otherwise } \end{array}\right. \\ f\left( M_{2}\right) =\left\{ \begin{array}{cc} \frac{1}{0.02-0}, &{} 0 \le M_{2} \le 0.02 \\ 0, &{} \text{ otherwise } \end{array}\right. \end{aligned}$$

Table 1 Input parameters of numerical illustration.

The outcomes of the two scenarios described in this article are shown in table 2. Scenario 1 depicts the main mathematical model with imperfect production, product deterioration, and inspection error at both the vendor’s and buyer’s ends. Scenario 2 is a variant of scenario 1 in which neither the vendor nor the buyer conducts an inspection.

Table 2 Outcomes of numerical illustration.

Scenario 1 reveals an inspection error at both the vendor and buyer ends that is $E[M_1]=E[M_2]=E[m_1]=E[m_2]=0.01$. According to Scenario 1, there is an inspection error on the parts of both the vendor and the buyer, and the rates of deterioration and defective output are constant. At this instance, there will be 11 shipments, a total estimated cost of $\$$38734 per year, a manufacturing time of 0.3231 year, and a total combined profit of $\$$10741.

In scenario 2, there are 11 shipments, a total estimated cost of $\$$41613 per year, a total production time of 0.3125 year, and overall combined profit being $\$$8387.

Therefore, it is evident that our model, considering inspection errors, is significantly superior to the model, in which there is no inspection encountered both in terms of cost and profit. In light of this, it is not difficult to assert that the model that we have provides better results than those which do not take inspection into consideration.

5 Sensitivity analysis

As a result of a variety of factors, including unpredictability and shifting market conditions, there is a possibility that the values of some parameters in a decision-making scenario will fluctuate. Because of these alterations, it is essential to investigate the associated price shifts. A sensitivity analysis, which demonstrates the consequences of varying the parameter values can be of assistance. The ideal solutions for n and $T_1$, are determined in each case by changing the value of just one parameter at a time by –50%, –25%, 25%, and 50% while keeping the values of the other parameters constant.

Table 3 demonstrates how the total annual cost rises while the overall profit for each model declines as the value of $\alpha$ increases.

Table 3 Effect rate of imperfect production ($\alpha$).

By observing figure 3(a), it is evident that the value of $\alpha$ directly effect on the system cost. Cost is observed to increase with the increment in the value of $\alpha$. However, our model demonstrates a slower increment in cost compared to the model without inspection. From figure 3(b), it is clear that when $\alpha$ increases, the profit of our model drops at a relatively slower rate compared to the model without inspection.

Figure 3

Graphical Plots for (a) Total expected cost depending upon $\alpha$ (b) Profit depending upon $\alpha$.

Table 4 demonstrates that for the two models described, the combined profit decreases as the value of the rate of deterioration rises, while the total annual cost increases.

Table 4 Effect of rate of deterioration ($\theta _v/\theta _b$).

Upon examining figure 4(a), it becomes evident that the rate of deterioration has a direct impact on the cost growth. Nevertheless, our model exhibits a less rapid increase in cost in comparison to the model without inspection. Based on figure 4(b), it is clear that when the rate of deterioration increases, the profit of our model decreases at a comparatively slower rate compared to the model without inspection.

Figure 4

Graphical Plots for (a) Total expected cost depending upon $\theta _v/\theta _b$ (b) Profit depending upon $\theta _v/\theta _b$.

As the value of $c_a$ increases, the cumulative annual cost is shown to increase in table 5, whereas the overall profit for each model indicates a decreasing trend.

Table 5 Effect of cost of Type II error ($c_a$).

The relationship between the cost of Type II error and the development of costs is clearly shown in figure 5(a). While the special case model shows a faster growth in cost, our model shows a slower increase. Figure 5(b) shows that our model’s profit declines more slowly when the cost of Type II error increases.

Figure 5

Graphical Plots for (a) Total expected cost depending upon $c_a$ (b) Profit depending upon $c_a$.

Table 6 illustrates how an increment in the value of expected inspection error results in an increase in the total annual cost of the model while simultaneously resulting in a decrease in the overall profit.

Table 6 Effect of inspection errors.

Figure 6(a) provides a clear illustration of the connection that exists between the inspection errors and the costs. The data presented in figure 6(b) demonstrates that the relationship between the inspection error and the profit.

Figure 6

Graphical Plots for (a) Total expected cost depending upon the values of expected inspection error (b) Profit depending upon the values of expected inspection error.

Additionally, we have investigated the sensitivity analysis of other factors that were utilised in the formation of our models. However, the changes in expected annual cost and profit that were brought about by changes in the parameters were so insignificant that we did not include the sensitivity analysis of these parameters in our work.

6 Conclusions

In this paper, we address an integrated supply chain comprising of a single vendor and a single buyer, in which the vendor creates a product for the buyer utilising a known proportion of defective goods and rate of deterioration. The buyer uses a method of examination that is one hundred percent thorough in order to eliminate these problematic components. The proper batch volume, ideal manufacturing time, and number of deliveries per order are all calculated using a mathematical model appropriate for this situation. Two factors, namely, inspection errors and product deterioration are taken into account in this article. Firstly, a case in which faulty items are incorrectly classified by quality inspectors at the vendor’s end as well as errors found during inspection at the buyer’s end are also taken into consideration. Secondly, when neither the vendor nor the buyer performs any inspection, in which case there is no cost involved for the process of screening.

The two elements covered in this paper are crucial for several companies that still prioritize employing people. Examples are leather product manufacturing companies, clothing factories, and sports equipment producers. Additionally, screening errors can be harmful when it comes to some important topics because the results of these errors could be deadly. To reduce the impact of screening errors, many components of nuclear reactors, space craft, aircraft equipment, and gas generated voltage are all routinely inspected. For additional information on the recurrent assessment of multi-characteristic components, see Raouf A, Jain JK and Sathe PT [29] and Duffuaa SO and Khan M [22]. This kind of integrated approach can be utilised to support large expenditures in regard to training, product innovation, operational processes, and buyer engagement.

The results indicate that inspection remains a crucial component in the discussed coordination, and two factors that have a direct impact on the overall cost of the supply chain are the proportion of defective products and the cost of admitting deteriorating products. The application of this model is possible in situations where inspection is an essential component, such as supply chains for automobiles or agri-fresh foods, aircraft, and so on.

This article investigates the interaction between a single buyer and a single vendor, which is an intriguing area for further research because it has the potential to be expanded to include a large number of buyers and a number of different vendors. In addition to this, it solves the difficulties that are involved with the management of inventory and transportation for a number of different products. Despite the fact that this model does not take into account shortages, it is possible to modify it such that shortages are allowed. Because the current plan does not take into account rework, the possibility of incorporating the process of reworking defective items into the framework for future efforts is something that should be examined.