Journal of Industrial Engineering International

, Volume 13, Issue 3, pp 393–416

# A single-vendor and a single-buyer integrated inventory model with ordering cost reduction dependent on lead time

• M. Vijayashree
• R. Uthayakumar
Open Access
Original Research

## Abstract

Lead time is one of the major limits that affect planning at every stage of the supply chain system. In this paper, we study a continuous review inventory model. This paper investigates the ordering cost reductions are dependent on lead time. This study addressed two-echelon supply chain problem consisting of a single vendor and a single buyer. The main contribution of this study is that the integrated total cost of the single vendor and the single buyer integrated system is analyzed by adopting two different (linear and logarithmic) types ordering cost reductions act dependent on lead time. In both cases, we develop effective solution procedures for finding the optimal solution and then illustrative numerical examples are given to illustrate the results. The solution procedure is to determine the optimal solutions of order quantity, ordering cost, lead time and the number of deliveries from the single vendor and the single buyer in one production run, so that the integrated total cost incurred has the minimum value. Ordering cost reduction is the main aspect of the proposed model. A numerical example is given to validate the model. Numerical example solved by using Matlab software. The mathematical model is solved analytically by minimizing the integrated total cost. Furthermore, the sensitivity analysis is included and the numerical examples are given to illustrate the results. The results obtained in this paper are illustrated with the help of numerical examples. The sensitivity of the proposed model has been checked with respect to the various major parameters of the system. Results reveal that the proposed integrated inventory model is more applicable for the supply chain manufacturing system. For each case, an algorithm procedure of finding the optimal solution is developed. Finally, the graphical representation is presented to illustrate the proposed model and also include the computer flowchart in each model.

## Keywords

Operations research Inventory model Continuous review system Ordering cost reduction Lead time crashing cost

90B05

## Introduction

Operations Research (OR) is a term which stands for an approach to problem solving characterized by a system orientation, an interdisciplinary philosophy, a focus on the qualification of the relevant aspects of the situation into a model and the manipulation of this model through the use of mathematical, statistical and computer methodologies to develop decisions, plans and policies. As the Operations Management (OM ) and Supply Chain Management (SCM) field has developed, a better importance on services has appeared. Inventory is important role in the Operations Research. The inventory system is taking an important part of cost controlling in business and organization.

VelMurugan and Uthayakumar (2015) discussed the definition of inventory; inventory consists of usable but idle resources which are materials and goods. The amount of material, a company has in stock at a specific time is known as inventory or in terms of money it can be defined as the total capital investment over all the materials stocked in the company at any specific time. Inventory may be in the form of, raw material inventory, in process inventory, finished goods inventory, etc. Inventory management is a key component in any production environment. This has been recognized not only in the chemical engineering literature but also in the operations research and industrial engineering domains. Inventory Control (IC) is the supervision of supply, storage and accessibility of items to ensure an adequate supply without excessive oversupply.

Inventory is an important part of our manufacturing, distribution and retail infrastructure where demand plays an important role in choosing the best inventory policy. To meet the needs of customers timely, businesses must maintain higher inventory levels to avoid shortages. However, high inventory levels are often associated with high inventory costs, many companies strive to reduce production or lead time cycle, and thus have a corresponding reduction in inventory. Continuous review is the main aspect of the inventory system. In this system the record of the inventory level is checked continuously until a specified point is reached where a new order is placed. This system is also called fixed order quantity system.

The integrated inventory management system is a common practice in the global markets and provides economic advantages for both the vendor and the buyer. In recent years, most integrated inventory management systems have focused on the integration between vendor and buyer. Once the form a strategic alliance to minimize their own cost or maximize their own profit, then trading parties can collaborate and share information to achieve improved benefits. Nowadays, companies can no longer compete solely as individual entities in the constantly changing business world. Globalization of market and increased competition force organizations to rely on effective supply chains to improve their overall performance.

The goal of many research efforts related to the SCM is to present models to reduce operational costs. The SCM has enabled numerous firms to enjoy advantages by integrating all activities associated with the raw material supplier, finished goods manufacturer, retailers, wholesalers, buyers/consumers etc. who are responsible for converting the raw material into a finished good and make them available to customers to satisfy their demand in time at least possible cost. Successful SCM requires a change from managing distinct function to integrating activities into key supply chain processes. Integration between two different business entities is an important way to gain competitive advantages as it lowers supply chain cost. The benefits of a properly managed supply include reduced costs, faster product delivery, greater efficiency and low costs for both the business and its customer. In the increasingly fierce competitive environment in today’s global markets, the supply chain coordination is becoming a key component. If no coordination exists, the supply chain members act independently to maximize their own profits or minimize the costs.

Lead Time (LT) is the time that elapses between the placing of an order (either a purchase order or a production order issued to the shop or the factory floor) and actually receiving the goods ordered. If a supplier (an external firm or an internal department or plant) cannot supply the required goods on demand, then the client firm must keep an inventory of the needed goods. The longer the lead time, the larger the quantity of goods the firm must carry in inventory. In general, the time of order receiving, order handling, order processing, manufacturing, assembly, distribution and delivery time to the customer includes in a lead time. Since the customer perspective is very important. LT has been counted until products or goods arrive to the customer. Hence the lead time measurement can be done from the customer points of view. The customers can be varies by different meanings. For instance, suppliers deliver components or parts to the main manufacturer where manufacturer is the customer from supplier’s perspective. LT can be also measured from the manufacturer points of view. Manufacturer also measures the lead from starting of the processing, fabrication and assembly up-to ready the product for shipment. This can be said as internal lead time. Whereas the external lead time can define by includes shipping, logistics and distribution time.

Inventory models incorporating lead time as a decision variable were developed by several researchers. Liao and Shyu (1991) presented a probabilistic model in which the order quantity was predetermined and lead-time was a unique decision variable. Later, Ben-Daya and Raouf (1994) extended Liao and Shyu’s (1991) model by considering both lead-time and the order quantity as decision variables where shortages were neglected. Ouyang et al. (1996) allowed shortages and extended Ben-Daya and Raouf’s (1994) model by adding the stockout cost. In addition, the total amount of stockout was considered a mixture of backorders and lost sales during the stockout period. Moon and Choi (1998) and Hariga and Ben-Daya (1999) improved the model of Ouyang et al. (1996) by simultaneously optimizing the order quantity, the reorder point and lead-time. Ouyang et al. (1999) incorporated ordering cost reduction into the model of Moon and Choi (1998), where the ordering cost can be reduced by capital investment.

LT plays an important role and has been a topic of interest for many authors in inventory management [see, for example, Das (1975), Foote et al. (1988) and Magson (1979)]. In most of the early literature dealing with inventory problems, in both deterministic and probabilistic models, lead time is viewed as a prescribed constant or a stochastic variable, which, therefore, is not subject to control [see, e.g., Naddor (1966), Silver and Peterson (1985)]. In 1983, Monden (1983) studied the Toyota production system and pointed out that shortening lead time is a crux of elevating productivity.

Ordering Cost (OC) is the costs of ordering a new batch of raw materials. These include cost of placing a purchase order, costs of inspection of received batches, documentation costs, etc. Ordering costs vary inversely with carrying costs. It means that the more orders a business places with its suppliers, the higher will be the ordering costs. However, more orders mean smaller average inventory levels and hence lower carrying costs. It is important for a business to minimize the sum of these costs which it does by applying the economic order quantity model. All the aforementioned integrated vendor–buyer inventory systems treat the ordering cost and/or lead time as constants. However, in the practical market, ordering cost and lead time can be controlled and reduced in various ways. For example, lead time can be reduced at an added crashing cost; ordering cost reduction can be attained through worker training, procedural changes, and specialized equipment acquisitions; in other words, the lead time is controllable, and the ordering cost can be reduced through further investment. It has been a trend by shortening the lead time and reducing ordering cost; we can lower the safety stock, reduce the stockout loss, and improve the service level to the customer.

Initially, Porteus (1986) investigated the impact of capital investment in reducing ordering cost on the classical Economic Order Quantity (EOQ) model for the first time. Ouyang et al. (1999) discussed lead time and ordering cost reductions in continuous review inventory systems with partial backorders. Later, Chang et al. (2006) presented lead time and ordering cost reduction problem in the single-vendor single-buyer integrated inventory model. They considered that buyer lead time can be shortened at an extra crashing cost which depends on the lead time length to be reduced and the ordering lot size, as well buyer ordering cost can be reduced through further investment. The main important single factor which influences the decision on re-order quantity is the total of carrying cost and ordering cost. Carrying cost increase with increase in re-order quantity while ordering cost decreases with increase order quantity. Thus, carrying cost and ordering cost move in opposite directions. Material manager, in deciding the re-order quantity, endeavours to keep the total of carrying cost and ordering cost at the minimum.

In this direction, several authors has encouraged to examine setup/ordering cost reduction [e.g. Keller and Noori (1988), Nasri et al. (1990), Kim et al. (1992), Paknejad et al. (1995)]. As stated in Tersine (1994), lead time usually comprises several components, such as setup time, process time, wait time, move time and queue time. In many practical situations, lead time can be reduced using an added crashing cost. In other words, lead time is controllable. The Japanese experience of using JIT production showed that the benefits associated with lead time control are clear. Therefore, reducing lead time is both necessary and beneficial.

In the proposed model, the optimum inventory control policy in a single vendor and a single-buyer integrated inventory model with ordering cost reduction dependent on lead time. In addition, the proposed model includes an appropriate method to contain upstream members to believe the best policy. The contribution of this paper can be considered as the major aspects: the mathematical model as well as structure and concept of ordering cost is dependent on lead time. Here, the proposed model considers the two case (i) linear function case and (ii) logarithmic function case.

Specially, we modify Pan and Yang (2002) model to include the cases of the linear and logarithmic relationship between lead time and ordering cost reductions. The objective of this paper is to find out an optimal inventory strategy that can minimize the value of the integrated total cost for the single vendor and the single buyer. An algorithm is developed to determine the optimal strategy and numerical examples are taken to illustrate the solution procedure in linear case as well as logarithmic case. Finally, the graphical representation is presented to illustrate the model. Furthermore, the sensitivity analysis is incorporated and the numerical examples are given to illustrate the results.

This paper is organized as follows. In the next section “Literature review”, contains the literature review and “Notations and assumptions” section, we describe the notation and assumptions used throughout this study. We mathematical model is developed to optimize the integrated total cost for the single vendor and the single buyer where the lead time dependent on ordering cost is presented in section “Mathematical moder”. Two numerical examples are provided to illustrate the proposed models in “Numerical examples” section. In “Sensitivity analysis in linear case and logarithmic case” section, sensitivity analysis of the parameters is provided in linear case as well as logarithmic case. Managerial insights are also included in “Managerial insights” section. “Conclusion” section summarizes the paper and discusses future directions.

## Literature Review

A business viewpoint for institutional buying and vendor relationship management, or supply chain management, is a necessary functional position within any association. Organizations, large and small, have some form of a purchasing function. Even a sole-proprietor is accountable for purchasing the needed goods and services to keep their industry running. So when we consider the importance of a well-defined, well-engineered supply chain management function, the implications, organizationally, are widespread and certainly worth noting.

By tradition, inventory problems for the vendor and the buyer are treated independently. In the past, Economic Order Quantity (EOQ) and Economic Production Quantity (EPQ) was treated independently from the viewpoints of the buyer or the vendor. In most cases, the optimal solution for one player was non-optimal to the other player. In today’s competitive markets, close cooperation between the vendor and the buyer is necessary to reduce the joint inventory cost and the response time of the vendor–buyer system. The successful experiences of National Semiconductor, Wal-Mart, and Procter and Gamble have demonstrated that integrating the supply chain has significantly influenced the company’s performance and market share (Simchi-Levi et al. 2000). Other studies (Weng 1995; Li et al. 1996; Yang and Wee 2000; Chen et al. 2001) show that an integrated approach results in improved performance and increased profitability to all players in the supply chain.

Most inventory models considered to date assume just one facility (e.g., a buyer or a vendor) managing its inventory policy to minimize its own cost or maximize its own profit. This one-sided-optimal-strategy is not suitable for global markets. The issue of JIT has recently received great attention. Most JIT research has focused on the integration between vendor and buyer. Once a long-term relationship between both facilities has been developed, both parties can cooperate and share information to achieve improved benefits.

The integration between vendor and buyer for improving the performance of inventory control has received a great deal of attention and the integrated approach has been examined for years. In 1986, Banerjee (1986) assumed that the vendor manufactures at a finite rate and considered a joint economic-lot-size model in which a vendor produces to order for a buyer on a lot-for-lot basis. Goyal (1976) is among the first who analyzed an integrated inventory model for a single-buyer single- buyer system. The framework he proposed has encouraged many researchers to present various types of integrated inventory system. Banerjee (1986) modified Goyal’s (1976) model and presented a joint economic lot size model where a vendor produces for a buyer to order on a lot for lot basis. Goyal (1988) further generalized Banerjee’s (1986) model relaxing the assumption of the lot for lot policy of the vendor and suggested that the vendor’s economic production quantity should be a positive integer multiple of the buyer’s purchase quantity.

Ha and Kim (1997) further generalized Goyal’s (1988) model and presented an integrated lot splitting model of facilitating multiple shipment in small lots. Hill (1999) proposed a more general batching and shipping policy involving the successive shipment size of the first $$m$$ shipments increases by a fixed factor and remaining shipments would be equal sixed. In a recently study, Pan and Yang (2002) generalized Goyal’s (1988) model by considering lead time as a decision variable and obtained a lower joint total expected cost and shorter lead time. Yang and Pan (2004) considered variable lead time and quantity improvement investment with normal distributional demand in the model proposed in Pan and Yang (2002), Ouyang et al. (2004) extend Pan and Yang (2002) and developed a single-vendor single-buyer integrated production inventory model under the assumption that the lead time is stochastic and lead time is decision variable.

Goyal and Gupta (1989), Monden (1983), Lu (1995), Hill (1999) investigated an unequal shipment policy for the joint single-vendor single-buyer inventory problem and concluded that an optimal policy for this problem is to use shipment sizes that increase by a fixed factor in the beginning and then remaining constant after a well-specified number of shipments. Ouyang et al. (1996) extended the Ben-Daya and Raouf’s (1994) model in which shortages were allowed and the total amount of stockouts was considered as a mixture of back orders and lost sales. Hsiao and Lin (2005) investigated an economic order quantity model on Stackelberg game in supply chain; that is, a distribution channel system containing one supplier and a single retailer such that the supplier in the channel holds monopolistic status, in which he not only owns cost information about the retailer but also has the decision making right of the lead time.

Recently, some researchers investigated on the integrated vendor–buyer inventory problems with quantity discount. Weng and Wong (1993) considered the optimal pricing and replenishment policy for a general all-unit quantity discount system with multiple buyers and constant demand. Weng (1995) discussed both all-unit and incremental quantity discount policies with price-sensitive demand. On the other hand, Munson and Rosenblatt (2001) proposed a three-level supply chain system with quantity discount and a fixed demand rate. They showed that quantity discount can effectively decrease each party’s cost. Li and Liu (2006) developed a supplier–buyer supply chain system with quantity discount and probabilistic customer demand. Qin et al. (2007) Established a supply chain system consisting of a supplier and a buyer with volume discounts and price-sensitive demand. More research papers dealing with the quantity discount problem in a supply chain system can be found in Parlar and Wang (1994), Li and Huang (1995), Hofmann (2000), Yang (2004), Tsai (2007), Sheen and Tsao (2007), Burke et al. (2008), etc., and the references therein. Lin (2008) has developed minimax distribution free procedure with backorder price discount.

LT is very important things in inventory system and supply chain system. Traditional inventory models assumed that lead time is a constant or random variable which is not a controllable factor. However, in practice, lead time could be shortened by paying an additional crashing cost; in other words, it is controllable. Stated that this crashing cost could be expenditures on equipment improvement, information technology, order expedite, or special shipping and handling. By shortening lead time, buyers can lower the safety stock, reduce the out-of-stock loss, and improve the customer service level. Thus, in present supply chain and inventory management system, controllable lead time is a key to business achievement and has attracted considerable research attention.

Most of researches in the area of lead time reduction assume that lead time is composed of n mutually dependent deterministic components where each component can be shortened by a crashing cost (Ben-Daya and Raouf 1994; Hayya et al. 2011; Pan et al. 2002). Usually, it is considered that lead time crash cost depends on the amount of lead time to be shortened. Lead time reduction falls in the field of selecting between regular and expedited shipping services which in turn is related to what transportation mode is used. Li (2013) have provided a model for designing a logistics network in which regular shipping services with long uncertain lead time could be replaced with more expensive expedited shipping services in negligible lead time.

Heydari (2014) presented lead time variation control using reliable shipment equipment: an incentive scheme for supply chain coordination. Heydari et al. (2016) developed lead time aggression: A three echelon supply chain model. Vijayashree and Uthayakumar (2015) have considered an integrated inventory model with controllable lead time involving investment for quality improvement in supply chain system. Vijayashree and Uthayakumar (2016) have presented inventory models involving lead time crashing cost as an exponential function. Vijayashree and Uthayakumar (2016) have developed an integrated vendor and buyer inventory model with investment for quality improvement and setup cost reduction. Jamshidi et al. (2015) presented flexible supply chain optimization with controllable lead time and shipping option. Heydari and Norouzinasab (2016) have developed coordination of pricing, ordering and lead time decisions in a manufacturing supply chain. Zhu (2015) have presented integration of capacity, pricing and lead time decisions in a decentralized supply chain.

Vijayashree and Uthayakumar (2014) have presented a two stage supply chain model with selling price dependent demand and investment for quality improvement. Vijayashree and Uthayakumar (2013) have discussed vendor–buyer integrated inventory model with quality improvement and negative exponential lead time crashing cost. Vijayashree and Uthayakumar (2015) have developed two-echelon supply chain inventory model with controllable lead time. Lin (2009) discussed an integrated vendor–buyer inventory model with backorder price discount and effective investment to reduce ordering cost. Vijayashree and Uthayakumar (2014) developed an integrated inventory model with controllable lead time and setup cost reduction for defective and non-defective items. Vijayashree and Uthayakumar (2015) have developed an EOQ model for time deteriorating items with infinite and finite production rate with shortage and complete backlogging. Vijayashree and Uthayakumar (2016) have considered an optimizing integrated inventory model with investment for quality improvement and Setup cost reduction. Hemapriya and Uthayakumar (2016) have developed ordering cost dependent lead time in integrated inventory model.

Pan and Yang (2002) have developed the study of an integrated inventory model with controllable lead time. In practices, the lead time and ordering cost reductions may be related closely; the reduction of lead time may accompany the reduction of ordering cost and vice versa. For example, the implementation of electronic data interchange can be reduced both the lead time and ordering cost simultaneously [see Silver and Peterson (1985), Ouyang et al. (2005), Chen et al. (2001)]. Therefore, it is more reasonable to assume that lead time and ordering cost reductions are dependent and their functional relationship may be as linear, logarithmic, exponential and the like. In the above papers (Liao and Shyu 1991; Ben-Daya and Raouf 1994; Ouyang et al. 1996; Moon and Chois 1998; Hariga and Ben-Daya 1999), which focus on deriving the benefits from lead time reduction, the ordering cost is treated as s fixed constant.

Recently, Ouyang et al. (1999) investigated the influence of ordering cost reduction on modified continuous review inventory systems involving variable lead time with partial backorders. Subsequently, Ouyang and Chang (2002) proposed a modified lot-size reorder-point inventory model with imperfect production processes to study the effects of reducing lead time and set-up cost. The optimal policies derived in these two articles are buyer focused, and the lead time and ordering/set-up cost reduction were assumed to act independently. However, an independent relationship between lead time and ordering/set-up cost is just one possibility. In some practices, lead time and ordering/set-up cost reduction might be closely related. A lead time reduction could accompany a reduction in the ordering/set-up cost, and vice versa. For example, Electronic Data Interchange (EDI) technology could simultaneously reduce both the lead time and the ordering/set-up cost. To date, little research has been done on establishing the relationship between lead times and ordering cost reduction. To provide insight and analytical tractability, as in Chiu (1998) and Chen et al. (2001), this study employed a linear function to formulate the above relationship. Pan and Yang (2002) have considered the ordering cost is fixed.

Therefore, the innovation of the proposed model ordering cost reducing lead time is both necessary and beneficial. Many researchers to reduce the ordering cost and setup cost reduction using logarithmic as well as power function, so the proposed model we have considered the ordering cost reduction dependent on lead time. An optimal solution procedure is developed by incorporating two types of investment functions such as (i) Linear case and (ii) Logarithmic case to reduce the ordering cost reduction in each mathematical model.

To the best of our knowledge, the author has developed a single vendor and single-buyer integrated inventory model with ordering cost reduction dependent on lead time. The contribution of this study is developed an effective iterative solution procedure to determine the optimal policy for a single vendor and single-buyer integrated inventory model with ordering cost reduction dependent on lead time in a supply chain system. In this study, we investigate two-echelon supply chain inventory problem consisting of a single vendor and a single buyer with controllable lead time. The purpose of this paper is to sturdy the effect of lead time reduction on continuous review inventory system with ordering cost reduction. And we consider the case where the lead time and ordering cost reductions with linear function case, and then consider the logarithmic function case relationship.

An algorithm is developed to optimize the integrated total cost for the buyer and the vendor. In addition, numerical examples and a sensitivity analysis are given to illustrate the results of the model. Furthermore, an iterative procedure is proposed to find the optimal solution. A solution procedure is developed to find the optimal solution and numerical solution is presented to illustrate the proposed model. The solution procedure is furnished to determine the optimal solution and the sensitivity analysis has been carried out to illustrate the behaviours of the proposed model. A graphical representation of the linear as well as logarithmic algorithm is represented by a flowchart.

## Notations and assumptions

To develop the proposed model, we adopt the following notations and assumptions which are similar to those used in Pang and Yang (2002). Besides, additional notations and assumptions will be given out when required.

### Notations

The notations are divided into two subsection variables and parameters are used to develop the model.

#### Variables

$$Q$$

$$L$$

$$A$$

Buyer’s ordering cost per order $$0 \le A \le A_{0}$$

$$m$$

The number of lots in which the product is delivered from the vendor to the buyer in one production cycle, a positive integer.

#### Parameters

To develop the proposed model, the following parameters are used

$$D$$

Average demand per unit time on the buyer

$$P$$

Production rate of the vendor $$\left( {P > D} \right)$$

$$S$$

Vendor’s setup cost per setup

$$c_{\text{v}}$$

Unit production cost paid by the vendor $$\left( {c_{\text{v}} < c_{\text{b}} } \right)$$

$$c_{\text{b}}$$

Unit purchase cost paid by the buyer

$$r$$

Annual inventory holding cost per dollar invested in stocks

$$R$$

$$A_{0}$$

Original ordering cost (before any investment is made)

$${\text{ITC}}$$

Integrated total cost for the single vendor and the single buyer.

### Assumptions

To develop the model, we adopt the following assumptions.
1. 1.

There is single-vendor and single-buyer for a single product in this model.

2. 2.

The buyer orders a lot of size $$Q$$ and the vendor manufactures $$mQ$$ with a finite production rate $$P$$ $$\left( {P > D} \right)$$ at one setup but ship in quantity $$Q$$ to the buyer over $$m$$ times. The vendor incurs a set up cost $$S$$ for each production run and the buyer incurs an ordering cost $$A$$ for each order of quantity $$Q$$.

3. 3.

The demand $$X$$ during lead time $$L$$ follows a normal distribution with mean $$\mu L$$ and standard deviation $$\sigma \sqrt L$$.

4. 4.

The inventory is continuously reviewed. The buyer places the order when the on hand inventory reaches the reorder point $$R$$.

5. 5.

The reorder point (ROP) equals the sum of the expected demand during lead time and the safety stock. The reorder point $$R$$ = the expected demand during lead time +safety stock, that is $$R = DL + k\sigma \sqrt L$$ where $$k$$ is safety factor.

6. 6.

The lead time $$L$$ consists of $$n$$ mutually independent components. The $$i$$th component has a normal duration $$b_{i}$$, minimum duration $$a_{i}$$, and crashing cost per unit time $$c_{i}$$. For convenience, we rearrange $$c_{i}$$ such that $$c_{1} < c_{2} < c_{3} < \cdots < c_{n}$$.

7. 7.

The components of lead time are crashed one at a time starting from the first component because it has the minimum unit crashing cost and then the second component, and so on.

8. 8.

Let $$L_{0} = \sum\nolimits_{i = 1}^{n} {b_{i} } ,$$ and $$L_{i}$$ be the length of lead time with components $$1,2,3, \ldots ,i$$ crashed to their minimum duration, then $$L_{i}$$ can be expressed as $$L_{i} = L_{0} - \sum\nolimits_{j = 1}^{n} {\left( {b_{j} - a_{j} } \right)} ,\quad i = 1,2, \ldots ,n;$$ and the lead time crashing cost per cycle $$R\left( L \right)$$ is given by $$R\left( L \right) = c_{i} \left( {L_{i - 1} - L} \right) + \sum\nolimits_{j = 1}^{i - 1} {c_{j} } \left( {b_{j} - a_{j} } \right),\quad L \in \left[ {L_{i} ,L{}_{i - 1}} \right]$$.

In addition, the length of lead time is equal for all shipping cycles, and the lead time crashing costs occur in each shipping cycle. The relationship between lead time and crashing cost is shown in Fig. 1. Liao and Shyu (1991), Li et al. (2012), Yang and Pan (2004), Pan and Yang (2002), Vijayashree and Uthayakumar (2014, 2016).
1. 9.

The reduction of lead time $$L$$ accompanies a reduce of ordering cost $$A$$ and $$A$$ is a firmly, concave function of $$L$$, i.e., $$A^{{\prime }} \left( L \right) > 0$$ and $$A^{{\prime \prime }} \left( L \right) < 0$$ (Ouyang et al. 2005; Chen et al. 2001).

2. 10.

If extra costs incurred by the vendor will be fully transferred to the buyer if shortened lead time is required (Pan and Yang 2002).

## Mathematical model

Under the assumptions (1–5), described above, Pan and Yang (2002), the integrated total cost, which is composed of buyer and vendor ordering cost, inventory holding cost and lead time crashing cost, is expressed by
$${\text{ITC}}\left( {Q,L,m} \right) = \frac{D}{Q}\left( {A + \frac{S}{m} + R\left( L \right)} \right) + \frac{Qr}{2}\left( {\left( {m\left( {1 - \frac{D}{P}} \right) - 1 + \frac{2D}{P}} \right)C_{\text{v}} + C_{\text{b}} } \right) + rc_{\text{b}} k\sigma \sqrt L$$
(1)

In the following two subsections, we consider the situation where shortening lead accompanies a decrease of ordering cost. Specifically, we consider the cases that the relationships between ordering cost and time is linear function and logarithmic function in two subsections.

### Linear function case

In this subsection, we assume that lead time and ordering cost reductions act dependently with the following relationship (Chen et al. 2001; Chiu 1998; Ouyang et al. 2004).
$$\frac{{L_{0} - L}}{{L_{0} }} = \omega \left( {\frac{{A_{0} - A}}{{A_{0} }}} \right)$$
(2)
where $$\omega > 0$$ is a constant scaling parameter to describe the linear relationship between percentages of reduction in lead time and ordering cost. Chen et al. (2001), Chiu (1998), Ouyang et al. (2004) utilized relationship (2) to formulate the inventory problems by treating $$Q,\,L$$ as a decision variable. In this paper, in addition to $$Q,\,L$$ and $$m$$ is also considered to be a decision variable.
By considering relationship (2), the ordering cost $$A$$ can be written as a linear function of $$L$$, that is
$$A\left( L \right) = x + yL$$
(3)
where $$x = \left( {1 - \frac{1}{\omega }} \right)A_{0}$$ and $$y = \frac{{A_{0} }}{{\omega L_{0} }}$$
Using (3) into (1), our problem is
$${\text{ITC}}\left( {Q,L,m} \right) = \frac{D}{Q}\left( {\left( {x + yL} \right) + \frac{S}{m} + R\left( L \right)} \right) + \frac{Qr}{2}\left( {\left( {m\left( {1 - \frac{D}{P}} \right) - 1 + \frac{2D}{P}} \right)C_{\text{v}} + C_{\text{b}} } \right) + rc_{\text{b}} k\sigma \sqrt L$$
(4)
for, $$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$.
To solve the nonlinear problem and try to solve the optimal solution of $${\text{ITC}}\left( {Q,L,m} \right)$$. For a fixed $$m$$, we take the first order partial derivatives of $${\text{ITC}}\left( {Q,L,m} \right)$$ with respect to $$Q$$ and $$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$, respectively, and obtain
$$\frac{{\partial {\text{ITC}}\left( {Q,L,m} \right)}}{\partial Q} = - \frac{D}{{Q^{2} }}\left( {\left( {x + yL} \right) + \frac{S}{m} + R\left( L \right)} \right) + \frac{r}{2}\left( {\left( {m\left( {1 - \frac{D}{P}} \right) - 1 + \frac{2D}{P}} \right)c_{\text{v}} + c_{\text{b}} } \right)$$
(5)
$$\frac{{\partial {\text{ITC}}\left( {Q,L,m} \right)}}{\partial L} = y\frac{D}{Q} - \frac{D}{Q}c_{i} + \frac{1}{2}rc_{\text{b}} k\sigma L^{{ - \frac{1}{2}}}$$
(6)
By examining the second-order sufficient conditions (SOSC) for a minimum value, it can be verified that $${\text{ITC}}\left( {Q,L,m} \right)$$ is not a convex function of $$\left( {Q,\,L} \right)$$. However, for a fixed $$\left( {Q,\,m} \right)$$, $${\text{ITC}}\left( {Q,L,m} \right)$$ is concave in $$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$, because
$$\frac{{\partial^{2} {\text{ITC}}\left( {Q,L,m} \right)}}{{\partial L^{2} }} = - \frac{1}{4}rc_{\text{b}} k\sigma L^{{ - \frac{3}{2}}} < 0$$
(7)

Hence for a fixed $$\left( {Q,\,L,\,m} \right)$$, the minimum total integrated cost per unit time will occur at the end points of the interval $$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$, $$i = 1,2, \ldots ,n$$.

On the other hand, for given $$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$, the minimum value of (4) will occur at the point $$Q$$ satisfying the Eq. (5), equal to zero, we obtain the resulting solution is given below
$$Q = \sqrt {\frac{{2D\left( {\left( {x + yL} \right) + \frac{S}{m} + R\left( L \right)} \right)}}{{r\left( {\left( {m\left( {1 - \frac{D}{P}} \right) - 1 + \frac{2D}{P}} \right)c_{\text{v}} + c_{\text{b}} } \right)}}}$$
(8)

For a fixed $$m$$ and $$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$, by solving Eq. (8), we obtain the values of $$Q$$(denote the value by $$Q^{*}$$). The following proposition asserts that, for fixed $$m$$ and $$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$, the point $$Q^{*}$$ is the optimal solution such that the integrated total cost has minimum value.

### Proposition 1

For a fixed $$m$$ and $$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$, the integrated total cost $${\text{ITC}}\left( {Q,L,m} \right)$$ is positive definite at point $$Q^{*}$$
$$\frac{{\partial {\text{ITC}}\left( {Q,L,m} \right)}}{{\partial Q^{2} }} = \frac{2D}{{Q^{3} }}\left( {\left( {x + yL} \right) + \frac{S}{m} + R\left( L \right)} \right) > 0$$
(9)
Next, to examine the effect of $$m$$ on the integrated total cost per unit time, we take the first and second order partial derivatives of $${\text{ITC}}\left( {Q,L,m} \right)$$ with respect to $$m$$ and obtain
$$\frac{{\partial {\text{ITC}}(Q,L,m)}}{\partial m} = - \frac{DS}{{Qm^{2} }} + \frac{Qr}{2}\left[ {c_{\text{v}} \left( {1 - \frac{D}{P}} \right)} \right]$$
(10)
and
$$\frac{{\partial^{2} {\text{ITC}}(Q,L,m)}}{{\partial^{2} m}} = \frac{2DS}{{Qm^{3} }} > 0$$
(11)

Therefore, $$ITC\left( {Q,\,L,\,m} \right)$$ is convex in $$m$$, for a fixed $$Q$$ and $$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$. As a result, the search for the optimal derivatives, $$m^{*}$$, is reduce to find a local minimum.

From Eq. (8) requires knowledge of the value of others; we can prove the convergence of the procedure by adopting a graphical technique similar to that used in Hadley and Whitin (1963). Further, based on the convexity behaviour of the objective function with respect to the decision variable, the following linear function case algorithm is designed to find the optimal values for order quantity, lead time, ordering cost and the number of deliveries in one production cycle. Linear function case algorithm describes the computer flowchart shown in Fig. (2).
Algorithm for Linear Function Case
• Set 1 Set $$m = 1$$

• Set 2 For each $$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$ perform (2.1)–(2.2),$$i = 1,2, \ldots ,n$$.
• 2.1 Compute $$Q_{i}$$ from Eq. (8).

• 2.2 Compute the corresponding $${\text{ITC}}\left( {Q_{i}^{*} ,\theta_{i}^{*} ,L_{i} ,m} \right)$$, by putting $$Q_{i}$$ in Eq. (4).

• Step 3. Find $$\min_{i = 1,2,3, \ldots n} {\text{ITC}}\left( {Q_{i}^{*} ,\theta_{i}^{*} ,L_{i} ,m} \right)$$. Let $${\text{ITC}}\left( {Q_{m}^{*} ,\theta_{m}^{*} ,L_{m} ,m} \right) =$$

• $$\min_{i = 1,2,3, \ldots n} {\text{ITC}}\left( {Q_{i}^{*} ,\theta_{i}^{*} ,L_{i} ,m} \right)$$, then $$\left( {Q_{m}^{*} ,\,\,\theta_{m}^{*} ,\,L_{m} } \right)$$ is the optimal solution for a fixed $$m$$.

• Step 4. Set $$m = m + 1$$ and repeat steps (2)–(3) to get $${\text{ITC}}\left( {Q_{m}^{*} ,L_{m} ,m} \right)$$.

• Step 5. If $${\text{ITC}}\left( {Q_{m}^{*} ,L_{m} ,m} \right) \le {\text{ITC}}\left( {Q_{m - 1}^{*} ,L_{m - 1} ,m - 1} \right)$$; go to step 4, otherwise go to step 6.

• Step 6. Set $${\text{ITC}}\left( {Q^{*} ,m^{*} ,L^{*} } \right) = {\text{IITC}}\left( {Q_{m - 1}^{*} ,L_{m - 1} ,m - 1} \right)$$, then $$\left( {Q^{*} ,\,L^{*} ,\,m^{*} } \right)$$ is the optimal solution. The optimal ordering cost $$A\left( L \right) = x + yL$$(for linear case) follow.

### Logarithmic function case

In this subsection, we assume that the lead time and ordering cost reductions act dependently with the following relationship (Chen et al. 2001)
$$\tau \ln \left( {\frac{L}{{L_{0} }}} \right) = \frac{{A_{0} - A}}{{A_{0} }}$$
(12)
where $$\delta < 0$$ is a constant scaling parameter to describe the logarithmic relationship between percentages of reductions in lead time and ordering cost. In this case, the ordering cost $$A$$ can be written as
$$A\left( L \right) = a + b\ln L$$
(13)
where $$a = A_{0} + \tau A_{0} \ln L_{0}$$ and $$b = - \delta A_{0}$$
Using (13) into (1), our problem is
\begin{aligned} {\text{ITC}}\left( {Q,L,m} \right) & = \frac{D}{Q}\left( {\left( {a + b\ln L} \right) + \frac{S}{m} + R\left( L \right)} \right)\\&\quad + \frac{Qr}{2}\left( {\left( {m\left( {1 - \frac{D}{P}} \right) - 1 + \frac{2D}{P}} \right)C_{v} + C_{b} } \right) \\ &\quad + rc_{b} k\sigma \sqrt L \\ \end{aligned}
(14)
$$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$.
To solve the nonlinear problem and try to solve the optimal solution of $${\text{ITC}}\left( {Q,L,m} \right)$$. For a fixed $$m$$, we take the first order partial derivatives of $${\text{ITC}}\left( {Q,L,m} \right)$$ with respect to $$Q$$ and $$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$, respectively, and obtain
$$\frac{{\partial {\text{ITC}}\left( {Q,L,m} \right)}}{\partial Q} = - \frac{D}{{Q^{2} }}\left( {\left( {a + b\ln L} \right) + \frac{S}{m} + R\left( L \right)} \right) + \frac{r}{2}\left( {\left( {m\left( {1 - \frac{D}{P}} \right) - 1 + \frac{2D}{P}} \right)c_{\text{v}} + c_{\text{b}} } \right)$$
(15)
$$\frac{{\partial {\text{ITC}}\left( {Q,L,m} \right)}}{\partial L} = \frac{b}{L}\frac{D}{Q} - \frac{D}{Q}c_{i} + \frac{1}{2}rc_{\text{b}} k\sigma L^{{ - \frac{1}{2}}}$$
(16)
By examining the second-order sufficient conditions (SOSC) for a minimum value, it can be verified that $${\text{ITC}}\left( {Q,L,m} \right)$$ is not a convex function of $$\left( {Q,\,L} \right)$$. However, for a fixed $$\left( {Q,\,m} \right)$$, $${\text{ITC}}\left( {Q,L,m} \right)$$ is concave in $$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$, because
$$\frac{{\partial^{2} {\text{ITC}}\left( {Q,L,m} \right)}}{{\partial L^{2} }} = - \frac{b}{{L^{2} }} - \frac{1}{4}rc_{b} k\sigma L^{{ - \frac{3}{2}}} < 0$$
(17)

Hence for a fixed $$\left( {Q,\,L,\,m} \right)$$, the minimum total integrated cost per unit time will occur at the end points of the interval $$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$, $$i = 1,2, \ldots ,n$$.

On the other hand, for given $$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$,the minimum value of (14) will occur at the point $$Q$$ satisfying the Eq. (15), equal to zero, we obtain the resulting solution is given below,
$$Q = \sqrt {\frac{{2D\left( {\left( {a + b\ln L} \right) + \frac{S}{m} + R\left( L \right)} \right)}}{{r\left( {\left( {m\left( {1 - \frac{D}{P}} \right) - 1 + \frac{2D}{P}} \right)c_{\text{v}} + c_{\text{b}} } \right)}}}$$
(18)

For a fixed $$m$$ and $$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$, by solving Eq. (18), we obtain the values of $$Q$$ (denote the value by $$Q^{*}$$). The following proposition asserts that, for fixed $$m$$ and $$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$, the point $$Q^{*}$$ is the optimal solution such that the integrated total cost has minimum value.

### Proposition 1

For a fixed $$m$$ and $$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$, the integrated total cost $${\text{ITC}}\left( {Q,L,m} \right)$$ is positive definite at point $$Q^{*}$$
$$\frac{{\partial {\text{ITC}}\left( {Q,L,m} \right)}}{{\partial Q^{2} }} = \frac{2D}{{Q^{3} }}\left( {\left( {a + b\ln L} \right) + \frac{S}{m} + R\left( L \right)} \right) > 0$$
(19)
Next, to examine the effect of $$m$$ on the integrated total cost per unit time, we take the first and second order partial derivatives of $${\text{ITC}}\left( {Q,L,m} \right)$$ with respect to $$m$$ and obtain loagrithmic function case
$$\frac{{\partial {\text{ITC}}(Q,L,m)}}{\partial m} = - \frac{DS}{{Qm^{2} }} + \frac{Qr}{2}\left[ {c_{\text{v}} \left( {1 - \frac{D}{P}} \right)} \right]$$
(20)
and
$$\frac{{\partial^{2} {\text{ITC}}(Q,L,m)}}{{\partial^{2} m}} = \frac{2DS}{{Qm^{3} }} > 0$$
(21)

Therefore, $${\text{ITC}}\left( {Q,L,m} \right)$$ is convex in $$m$$, for a fixed $$Q$$ and $$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$. As a result, the search for the optimal derivatives, $$m^{*}$$, is reduce to find a local minimum.

From Eq. (18) requires knowledge of the value of others; we can prove the convergence of the procedure by adopting a graphical technique similar to that used in Hadley and Whitin (1963). Further, based on the convexity behaviour of the objective function with respect to the decision variable, the following logarithmic function case algorithm is designed to find the optimal values for order quantity, lead time, and the number of deliveries in one production cycle. Loagrithmic function case algortithm describes the computer flowchart shown in Fig. (3).
Algorithm for Logarithmic Function Case
• Set 1 Set $$m = 1$$

• Set 2 For each $$L \in \left[ {L_{i} ,\,L{}_{i - 1}} \right]$$ perform (2.1)–(2.2),$$i = 1,2, \ldots ,n$$.
• 2.1. Compute $$Q_{i}$$ from Eq. (18).

• 2.2. Compute the corresponding $${\text{ITC}}\left( {Q_{i}^{*} ,\theta_{i}^{*} ,L_{i} ,m} \right)$$, by putting $$Q_{i}$$ in Eq. (14).

• Step 3. Let $${\text{ITC}}\left( {Q_{m}^{*} ,L_{m} ,m} \right)$$ = minimum of $${\text{ITC}}\left( {Q_{i}^{*} ,L_{i} ,m} \right)$$, then $$\left( {Q_{m}^{*} ,\,\,L_{m} } \right)$$ is the optimal solution for a fixed $$m$$.

• Step 4. Set $$m = m + 1$$ and repeat steps (2)–(3) to get $${\text{ITC}}\left( {Q_{m}^{*} ,L_{m} ,m} \right)$$.

• Step 5. If $${\text{ITC}}\left( {Q_{m}^{*} ,L_{m} ,m} \right) \le {\text{ITC}}\left( {Q_{m - 1}^{*} ,L_{m - 1} ,m - 1} \right)$$; go to step 4, otherwise go to step 6.

• Step 6. Set $${\text{ITC}}\left( {Q^{*} ,m^{*} ,L^{*} } \right) = {\text{IITC}}\left( {Q_{m - 1}^{*} ,L_{m - 1} ,m - 1} \right)$$, then $$\left( {Q^{*} ,\,L^{*} ,\,m^{*} } \right)$$ is the optimal solution. The optimal ordering cost $$A\left( L \right) = a + b\ln L$$(for logarithmic case).

## Numerical examples

To illustrate the above solution procedure, let us consider an inventory system with the data used in Pan and Yang (2002) $$D = 1000\,{\text{units/year}},$$ $$P = 3200\,{\text{units/year,}}$$ $$k = 2.33,$$ $$c_{\text{v}} = 20/{\text{units}},$$ $$r = 0.2,$$ $$A_{\text{o}} = \ 25/{\text{order}},$$ $$S = \ 400/{\text{setup}},$$ $$c_{\text{b}} = 25/{\text{units}},$$ $$\sigma = 7\,{\text{units}}/{\text{week}}$$ and lead time has three components with data shown in Table 1 and summarized lead time data shown in Table 2.
Table 1

Lead time component $$\left( i \right)$$

Normal duration $$b_{i}$$ (days)

Minimum duration $$a_{i}$$ (days)

Unit crashing cost $$c_{i}$$ (days)

1

20

6

0.1

2

20

6

1.2

3

16

9

5.0

Table 2

R (L)

8

0

6

1.4

4

18.2

3

53.2

### Example 1 (Linear case)

We consider the case that the relationship between lead time and ordering cost is linear. We solve the case when $$\omega = 5.00$$.

Applying the algorithm in subsection, the results of the solution procedures are summarized in Table 3.
Table 3

The solution procedures for linear case

$$\omega$$

$$A\left( L \right) = x + yL$$ where $$x = \left( {1 - \frac{1}{\omega }} \right)A_{0}$$ and $$y = \frac{{A_{0} }}{{\omega L_{0} }}$$

$$m$$

$$Q$$

$${\text{ITC}}$$

5.00

25.00

1

369

2536

2

224

2160

3

164

2160

4

131

2135

5

110

2134

6

96

2146

7

85

2164

8

77

2186

9

70

2212

10

65

2238

23.75

1

369

2505

2

224

2213

3

164

2130

4

131

2105

5

110

2104

6

96

2116

7

85

2135

8

77

2158

9

70

2183

10

65

2210

22.50

1

376

2510

2

231

2245

3

172

2185

4

139

2183

5

118

2204

6

104

2236

7

93

2273

8

84

2314

9

78

2356

10

72

2400

21.87

1

390

2578

2

247

2366

3

188

2354

4

155

2395

5

134

2454

6

119

2522

7

108

2594

8

99

2667

9

92

2740

10

86

2812

A graphical representation is presented to show the convexity of $${\text{ITC}}\left( {Q^{*} ,L^{*} ,m} \right)$$ in Fig. 4 and the graphical representation of the integrated total cost for different number of deliveries $$m$$ is shown in Fig. 5.
The optimal solutions from Table 4, can be read off as lead time $$L^{*} = 6\,{\text{weeks}},$$ order quantity $$Q^{*} = 110\,{\text{units}}$$, ordering cost $$A^{*} = 23.75$$ number of deliveries $$m^{*} = 5$$ and the corresponding integrated total cost $${\text{ITC}}^{*} = 2104$$. Plotted the optimal solution for ordering cost dependent on lead time in Fig. 6.
Table 4

Optimal solution for linear case

$$\omega$$

$$L$$

$$A\left( L \right) = x + yL$$ where $$x = \left( {1 - \frac{1}{\omega }} \right)A_{0}$$ and $$y = \frac{{A_{0} }}{{\omega L_{0} }}$$

$$m$$

$$Q$$

$${\text{ITC}}$$

5.00

3

21.87

3

188

2354

4

22.50

4

139

2183

6

23.75

5

110

2104

8

25.00

5

110

2134

### Example 2 (Logarithmic case)

We consider the case that the relationship between lead time and ordering cost is linear. We solve the case when $$\delta = - 0.5$$.

Applying the algorithm in subsection, the results of the solution procedures are summarized in Table 5.
Table 5

The solution procedures for logarithmic case

$$\delta$$

$$A\left( L \right) = a + b\ln L$$ where $$a = A_{0} + \tau A_{0} \ln L_{0}$$ and $$b = - \delta A_{0}$$

$$m$$

$$Q$$

$${\text{ITC}}$$

−0.5

25.00

1

369

2536

2

224

2160

3

164

2160

4

131

2135

5

110

2134

6

96

2146

7

85

2164

8

77

2186

9

70

2212

10

65

2238

21.40

1

368

2499

2

223

2202

3

163

2115

4

130

2087

5

109

2083

6

95

2083

7

84

2107

8

76

2127

9

69

2149

10

64

2173

16.33

1

373

2494

2

228

2218

3

169

2149

4

136

2138

5

115

2151

6

100

2175

7

90

2206

8

81

2240

9

75

2276

10

69

2313

12.73

1

386

2555

2

243

2329

3

184

2305

4

151

2335

5

130

2385

6

115

2444

7

104

2508

8

95

2573

9

95

2639

10

83

2704

A graphical representation is presented to show the convexity of $${\text{ITC}}\left( {Q^{*} ,L^{*} ,m} \right)$$ in Fig. 7 and the graphical representation of the integrated total cost for different number of deliveries $$m$$ is shown in Fig. 8.
The optimal solutions from Table 6, can be read off as lead time $$L^{*} = 6\,\,{\text{weeks}},$$ order quantity $$Q^{*} = 109\,{\text{units}}$$, number of deliveries $$m^{*} = 5$$ and the corresponding integrated total cost $${\text{ITC}}^{*} = 2083$$. Plotted the optimal solution for ordering cost dependent on lead time in Fig. 9.
Table 6

optimal solution for logarithmic case

$$\delta$$

$$L$$

$$A\left( L \right) = a + b\ln L$$ where $$a = A_{0} + \tau A_{0} \ln L_{0}$$ and $$b = - \delta A_{0}$$

$$m$$

$$Q$$

$${\text{ITC}}$$

−0.5

3

12.73

3

184

2305

4

16.33

4

136

2138

6

21.40

5

109

2083

8

25.00

5

110

2134

## Sensitivity analysis in linear case and logarithmic case

We now study the effects of changes in the system parameters demand, production rate, vendor’s setup cost, purchase cost and production cost on the optimal order quantity $$Q,$$ lead time $$L$$, ordering cost $$A$$ and the total number of deliveries $$m$$ in order to minimize the integrated total cost $${\text{ITC}}$$ of the given example.

### Effect of demand $$D$$ on the optimal solution (linear case)

To study how various demand $$D$$ affect the optimal solution of the model, the demand sensitivity analysis is performed by changing the values of parameter $$D$$ by +50, +25, −50, −25% and keeping the remaining parameters unchanged. The results of the demand analysis are shown in Table 7 and the corresponding curve representing of the minimum integrated total cost is plotted in Fig. 10 as well.
Table 7

Effect of demand on the optimal solution

$$\omega$$

$$D$$

$$L$$

$$A\left( L \right) = x + yL$$ where $$x = \left( {1 - \frac{1}{\omega }} \right)A_{0}$$ and $$y = \frac{{A_{0} }}{{\omega L_{0} }}$$

$$m$$

$$Q$$

$${\text{ITC}}$$

5.00

+50% (1500)

8

25.00

6

125

2424

6

23.75

6

125

2395

4

22.50

5

153

2523

3

21.87

4

199

2779

+25% (1250)

8

25.00

5

127

2300

6

23.75

5

127

2271

4

22.50

4

159

2372

3

21.87

4

178

2606

−25% (750)

8

25.00

4

111

1915

6

23.75

3

111

1885

4

22.50

3

147

1938

3

21.87

2

214

2068

−50% (500)

8

25.00

3

113

1630

6

23.75

3

113

1600

4

22.50

3

119

1631

3

21.87

2

175

1715

### Effect of production rate on the optimal solution (linear case)

To study how various production rate $$P$$ affect the optimal solution of the model, the demand sensitivity analysis is performed by changing the values of parameter $$P$$ by +50, +25, −50, −25% and keeping the remaining parameters unchanged. The results of the production rate analysis are shown in Table 8 and the corresponding curve representing of the minimum integrated total cost is plotted in Fig. 11 as well.
Table 8

Effect of production rate $$P$$ on the optimal solution

$$\omega$$

$$P$$

$$L$$

$$A\left( L \right) = x + yL$$ where $$x = \left( {1 - \frac{1}{\omega }} \right)A_{0}$$ and $$y = \frac{{A_{0} }}{{\omega L_{0} }}$$

$$m$$

$$Q$$

$${\text{ITC}}$$

5.00

+50% (4800)

8

25.00

4

128

2189

6

23.75

4

128

2159

4

22.50

3

169

2221

3

21.87

2

247

2366

+25% (4000)

8

25.00

4

129

2167

6

23.75

4

129

2137

4

22.50

3

170

2207

3

21.87

2

247

2366

−25% (2400)

8

25.00

6

100

2064

6

23.75

5

115

2034

4

22.50

4

143

2124

3

21.87

3

192

2315

−50% (1600)

8

25.00

8

91

1874

6

23.75

8

91

1845

4

22.50

6

120

1958

3

21.87

5

152

2188

### Effect of setup cost $$S$$ on the optimal solution (linear case)

To study how various setup cost $$S$$ affect the optimal solution of the model, the demand sensitivity analysis is performed by changing the values of parameter $$S$$ by +50, +25, −50, −25% and keeping the remaining parameters unchanged. The results of the setup cost analysis are shown in Table 9 and the corresponding curve representing of the minimum integrated total cost is plotted in Fig. 12 as well.
Table 9

Effect of setup cost on the optimal solution

$$\omega$$

$$S$$

$$L$$

$$A\left( L \right) = x + yL$$ where $$x = \left( {1 - \frac{1}{\omega }} \right)A_{0}$$ and $$y = \frac{{A_{0} }}{{\omega L_{0} }}$$

$$m$$

$$Q$$

$${\text{ITC}}$$

5.00

+50% (600)

8

25.00

6

112

2467

6

23.75

6

112

2437

4

22.50

4

162

2493

3

21.87

3

216

2684

+25% (500)

8

25.00

5

120

2307

6

23.75

5

120

2278

4

22.50

4

151

2355

3

21.87

3

203

2525

−25% (300)

8

25.00

4

117

1934

6

23.75

4

118

1904

4

22.50

3

155

1982

3

21.87

2

224

2154

−50% (200)

8

25.00

3

125

1698

6

23.75

3

125

1669

4

22.50

3

135

1752

3

21.87

2

197

1917

### Effect of purchase cost $$c_{\text{b}}$$ and production cost $$c_{\text{v}}$$ on the optimal solution (linear case)

To study how various purchase cost $$c_{\text{b}}$$ and production cost $$c_{\text{v}}$$ affect the optimal solution of the model, the demand sensitivity analysis is performed by changing the values of parameter $$c_{\text{b}} \;{\text{and}}\;c_{\text{v}}$$ by +50, +25, −50, −25% and keeping the remaining parameters unchanged. The results of the purchase cost $$c_{\text{b}}$$ and production cost $$c_{\text{v}}$$ rate analysis are shown in Table 10 and the corresponding curve representing of the minimum integrated total cost is plotted in Fig. 13 as well.
Table 10

Effect of purchase cost $$c_{\text{b}}$$ and production cost $$c_{\text{v}}$$ on the optimal solution

$$\omega$$

$$c_{\text{b}} \;{\text{and}}\;c_{\text{v}}$$

$$L$$

$$A\left( L \right) = x + yL$$ where $$x = \left( {1 - \frac{1}{\omega }} \right)A_{0}$$ and $$y = \frac{{A_{0} }}{{\omega L_{0} }}$$

$$m$$

$$Q$$

$${\text{ITC}}$$

5.00

+50% (37.5, 30)

8

25.00

5

90

2677

6

23.75

5

90

2632

4

22.50

4

114

2719

3

21.87

4

127

2971

+25% (31.25, 25)

8

25.00

5

99

2416

6

23.75

5

99

2379

4

22.50

4

125

2462

3

21.87

3

168

2651

−25% (18.75, 15)

8

25.00

5

127

1821

6

23.75

5

127

1799

4

22.50

4

161

1872

3

21.87

3

217

2023

−50% (12.5, 10)

8

25.00

5

156

1461

6

23.75

5

156

1447

4

22.50

4

197

1510

3

21.87

3

266

1636

### Effect of demand $$D$$ on the optimal solution (logarithmic case)

To study how various demand $$D$$ affect the optimal solution of the model, the demand sensitivity analysis is performed by changing the values of parameter $$D$$ by +50, +25, −50, −25% and keeping the remaining parameters unchanged. The results of the demand analysis are shown in Table 11 and the corresponding curve representing of the minimum integrated total cost is plotted in Fig. 14 as well.
Table 11

Effect of demand on the optimal solution

$$\delta$$

$$D$$

$$L$$

$$A\left( L \right) = a + b\ln L$$ where $$a = A_{0} + \tau A_{0} \ln L_{0}$$ and $$b = - \delta A_{0}$$

$$m$$

$$Q$$

$${\text{ITC}}$$

−0.5

+50% (1500)

8

25.00

6

125

2424

6

21.40

6

124

2367

4

16.33

5

149

2417

3

12.73

3

232

2702

+25% (1250)

8

25.00

5

127

2300

6

21.40

5

126

2247

4

16.33

4

156

2323

3

12.73

3

209

2528

−25% (750)

8

25.00

4

111

1915

6

21.40

4

110

1869

4

16.33

3

144

1906

3

12.73

2

211

2036

−50% (500)

8

25.00

3

113

1630

6

21.40

3

112

1590

4

16.33

3

116

1604

3

12.73

2

172

1688

### Effect of production rate $$P$$ on the optimal solution (logarithmic case)

To study how various production rate $$P$$ affect the optimal solution of the model, the demand sensitivity analysis is performed by changing the values of parameter $$P$$ by +50, +25, −50, −25% and keeping the remaining parameters unchanged. The results of the production rate analysis are shown in Table 12 and the corresponding curve representing of the minimum integrated total cost is plotted in Fig. 15 as well.
Table 12

Effect of production rate on the optimal solution

$$\delta$$

$$P$$

$$L$$

$$A\left( L \right) = a + b\ln L$$ where $$a = A_{0} + \tau A_{0} \ln L_{0}$$ and $$b = - \delta A_{0}$$

$$m$$

$$Q$$

$${\text{ITC}}$$

−0.5

+50% (4800)

8

25.00

4

128

2189

6

21.40

4

127

2140

4

16.33

3

166

2184

3

12.73

3

181

2343

+25% (4000)

8

25.00

4

129

2167

6

21.40

4

128

2119

4

16.33

3

167

2170

3

12.73

3

182

2328

−25% (2400)

8

25.00

6

100

2064

6

21.40

6

99

2011

4

16.33

6

96

1922

3

12.73

5

135

2302

−50% (1600)

8

25.00

8

91

1874

6

21.40

8

90

1819

4

16.33

7

105

1903

3

12.73

5

147

2126

### Effect of setup cost $$S$$ on the optimal solution (logarithmic case)

To study how various setup cost $$S$$ affect the optimal solution of the model, the demand sensitivity analysis is performed by changing the values of parameter $$S$$ by +50, +25, −50, −25% and keeping the remaining parameters unchanged. The results of the setup cost analysis are shown in Table 13 and the corresponding curve representing of the minimum integrated total cost is plotted in Fig. 16 as well.
Table 13

Effect of setup cost on the optimal solution

$$\delta$$

$$S$$

$$L$$

$$A\left( L \right) = a + b\ln L$$ where $$a = A_{0} + \tau A_{0} \ln L_{0}$$ and $$b = - \delta A_{0}$$

$$m$$

$$Q$$

$${\text{ITC}}$$

−0.5

+50% (600)

8

25.00

6

112

2467

6

21.40

6

111

2416

4

16.33

5

134

2472

3

12.73

3

213

2641

+25% (500)

8

25.00

5

120

2307

6

21.40

5

119

2258

4

16.33

4

148

2314

3

12.73

3

199

2479

−25% (300)

8

25.00

4

117

1934

6

21.40

4

116

1884

4

16.33

3

151

1941

3

12.73

2

219

2113

−50% (200)

8

25.00

3

125

1698

6

21.40

3

123

1650

4

16.33

3

131

1705

3

12.73

2

192

1870

### Effect of purchase cost $$c_{\text{b}}$$ and production cost $$c_{\text{v}}$$ on the optimal solution (logarithmic case)

To study how various purchase cost $$c_{\text{b}}$$ and production cost $$c_{\text{v}}$$ affect the optimal solution of the model, the demand sensitivity analysis is performed by changing the values of parameter $$c_{\text{b}} \;{\text{and}}\;c_{\text{v}}$$ by +50, +25, −50, −25% and keeping the remaining parameters unchanged. The results of the purchase cost $$c_{\text{b}}$$ and production cost $$c_{\text{v}}$$ rate analysis are shown in Table 14 and the corresponding curve representing of the minimum integrated total cost is plotted in Fig. 17 as well.
Table 14

Effect of purchase cost $$c_{\text{b}}$$ and production cost $$c_{\text{v}}$$ on the optimal solution

$$\delta$$

$$c_{\text{b}} \;{\text{and}}\;c_{\text{v}}$$

$$L$$

$$A\left( L \right) = a + b\ln L$$ where $$a = A_{0} + \tau A_{0} \ln L_{0}$$ and $$b = - \delta A_{0}$$

$$m$$

$$Q$$

$${\text{ITC}}$$

−0.5

+50% (37.5, 30)

8

25.00

4

117

1934

6

21.40

4

89

2606

4

16.33

4

111

2664

3

12.73

3

150

2862

+25% (31.25, 25)

8

25.00

5

99

2416

6

21.40

5

98

2355

4

16.33

4

122

2412

3

12.73

3

165

2596

−25% (18.75, 15)

8

25.00

5

156

1461

6

21.40

5

154

1432

4

16.33

4

193

1478

3

12.73

3

260

1601

−50% (12.5, 10)

8

25.00

5

127

1821

6

21.40

5

126

1781

4

16.33

4

157

1833

3

12.73

3

213

1980

## Managerial insights

The following attractive comments are made regards managerial insights in linear case as well as logarithmic case
• When tabulating (7)–(14) the optimal values for linear function case as well as logarithmic function case the increment and decrement of various parameters in demand, production rate, setup cost, purchase cost and production cost we could be able to suggest that the integrated total cost of the 6th weeks is lesser than all the other weeks.

• When tabulating (7)–(14) the optimal values for linear function case as well as logarithmic function case the increment and decrement of various parameters in demand, production rate, setup cost, purchase cost and production cost we could be able to suggest that the order quantity of the 6th weeks is lesser than all the other weeks.

• When tabulating (7)–(14) the optimal values for linear function case as well as logarithmic function case the decrease the number of delivers and ordering cost in various parameters in demand, production rate, setup cost, purchase cost and production cost we could be able to suggest that the integrated total cost.

• The proposed model can be used in industries such as aircraft, healthcare, automobiles, computers, textiles, footwear, printers, refrigerators, mobile phones, televisions, air conditioners, washing machines, tyres and bulky products such as printed circuit boards, etc.

• The proposed integrated inventory model is useful particularly for Just-in-time (JIT) inventory systems where the vendor and the buyer form a strategic alliance for profit sharing.

• The proposed integrated inventory model is more valid for the supply chain manufacturing system and vendor and buyer management.

## Conclusion

In conclusion, we formulated a single vendor and the single buyer integrated inventory model with ordering cost reduction dependent on lead time. Lead time is an imperative factor in any inventory management organization. By shortening the lead time, we can lower the safety stock, reduce the pasting caused by stockout, get better buyer service level and increase the opposition’s ability in business. In many sensible situations, lead time can be reduced by an additional crashing cost. That is, lead time is controllable. In this paper, the lead crashing cost $$R\left( L \right)$$ as the previous researches on lead time reduction [see, examples Hariga and Ben-Daya (1999), Liao and Shyu (1991), Moon and Chois (1998), Ouyang et al. (1999, 2004, 2005), Pan and Yang (2002), Vijayashree and Uthayakumar (2014, 2016)] is assumed to be a piecewise linear function.

Pan and Yang (2002) have considered the ordering cost is fixed. So, the proposed model, we have reduced the ordering cost, using the relationship between lead time and ordering cost reduction is linear and logarithmic case function. A mathematical model is employed in this study for optimizing the order quantity, lead time, ordering cost and number of deliveries in one production cycle.

An algorithm to find the optimal solutions is developed. The mathematical modelling is developed by incorporating two types of cases. The aim of our model is to reduce the ordering cost. Here the ordering cost dependent on lead time. The algorithm with the help of the software Matlab 2008 is furnished to determine the optimal solution. A graphical representation of the linear and logarithmic algorithm is represented by a flowchart.

Numerical examples are provided to illustrate the models and sensitivity analysis has been carried out to analyze the behavior of the key parameters on order quantity, lead time, ordering cost, number of deliveries from the vendor to the buyer in one production run and the integrated total cost of the proposed models.

Finally, some numerical examples are presented to illustrate the models. In future research on this, would be motivated to deal with different constraints like ordering constraints, inventory constraints etc. The model can be extended to the single-buyer, multiple-vendor and multiple-buyer, single-vendor and multiple-buyer multiple-vendor systems.

Another possible extension of this work can be done by assuming a discrete investment to reduce the vendor’s setup cost instead of continuous investment. Another possible research topic is to evaluate the impact of various types of imperfect production systems and inspection policies on integrated inventory models.

## Notes

### Acknowledgements

The authors are thankful to the anonymous reviewers and the editor for their perceptive and beneficial comments and encouraging suggestions, which have led to a most important development in the earlier revised version of the paper. The authors greatly appreciate the anonymous referees for their valuable comments and suggestions on an earlier version of the revised paper. First author research work is supported by DST INSPIRE Fellowship, Ministry of Science and Technology, Government of India under the Grant No. DST/INSPIRE Fellowship/2011/413A dated 09.09.2016 and UGC–SAP, Department of Mathematics, The Gandhigram Rural Institute—Deemed University, Gandhigram—624302, Tamilnadu, India.

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