Optimizing a supply chain problem with nonlinear penalty costs for early and late delivery under generalized lead time distribution

https://doi.org/10.1016/j.cie.2021.107536Get rights and content

Highlights

  • Single-vendor and single-buyer coordination problem of non-perishable product.

  • A generalized multivariable mixed-integer nonlinear programming problem is proposed.

  • Stochastic lead time with early or late delivery penalty cost plays vital decision.

  • Optimizes for reorder point, number and size of deliveries and its tolerance range.

  • Generalization of the model are proved to conform to other special cases as well.

Abstract

This article presents a tangible vendor–buyer cooperative strategy that benefits both, the vendor and the buyer where the demand is deterministic constant, and the delivery lead time follows a general distribution. To build a realistic coordination mechanism, a delivery tolerance time range is specified beyond which two different types of nonlinear penalty costs termed as an early delivery penalty cost and late delivery penalty cost are assessed. Shortages are allowed for a short time-span. The penalty costs are taken as a product of a linear function of delivery lead time and a nonlinear function of the delivery lot size. The problem is formulated as a multi-variable mixed-integer nonlinear programming (MINLP) problem and the objective of this research is to achieve the minimum integrated expected cost where decision variables are: reorder point, delivery lot size, number of deliveries, and delivery time thresholds. Since closed-form solutions are not immediately obtained, different search procedures are employed to resolve issues relating to an integer solution. Numerical results are provided for uniform, exponential and normal distributions of delivery lead time to establish the general model.

Introduction

The efficiency of a supply chain network is greatly influenced by the reliability of the supply process. The success of a supply chain lies beneath the proper timing of delivery of goods to the intermediate parties. This research work has adopted the integrated vendor–buyer optimization policy together with the idea of generalized lead time and nonlinear penalty cost for early and late delivery of shipments. This research contributes an improved delivery timing strategy to enhance over all supply chain performance. When a lot is delivered within a delivery tolerance period, then no penalty cost is assessed to the vendor, which gives latitude to both the vendor and the buyer to cope with the uncertainty of delivery mechanism and transportation time.

A real-world example of the investigated problem is the motivation of this problem under consideration and it is explained here. In Asian countries, the labor charge is significantly low for which the production cost also becomes low. Many industries such as textile, electronic goods in diverse Asian countries manufacture their product at a low price and export them to Europe, America and Australia. They prefer the naval route as the transportation mode for transporting their products to overseas due to low transportation cost. Although the transportation cost in a naval route is low; but the chance of delivery delay is high which may cause significant loss to the buyer (importer). The buyer incurs the expense for keeping the storehouse ready before receiving the ordered lot and loses both market goodwill and the potential profit during market peak time if they are not delivered in time. These losses are increased with an increasing number of units. Another practical situation in such import–export business is also explained here. Usually, the large-scale industries such as textile industries, steel industries, electronic goods industries and automobile industries maintain their owned or rented warehouses. They bear the expense to maintain the infrastructure and good storage environment (temperature, humidity, etc.) of the warehouses. If the ordered lot reaches too early, then sometimes a space problem may arise. Such space problem is increased with an increase in the number of units. To keep the early delivery lot, the buyers must rent another storehouse and bear the burden of maintaining it, which increases the storage cost, and hence the total cost. To discourage the practice of such a costly early and late delivery, the buyer imposes a significantly high penalty charge to the vendor for both early and late arrival of ordered items. Such types of penalty cost scenarios are usually found in international business of garments and other luxury goods.

Plenty of research works are available including a lead-time consideration. A brief literature survey is given in the following subsection.

Realizing the necessity of compact coordination in a supply chain system, a significant number of researchers have incorporated several papers in multi-dimensional directions including variable lead time. Pan and Yang (2002) have developed a supply chain model with controllable lead time and stochastic lead time demand. Ouyang et al. (2004) have extended the work of Pan and Yang (2002) by introducing the concept of stochastic demand in conjunction with controllable lead time while Hoque and Goyal (2006) have extended Pan and Yang (2002) model by incorporating the shipments of equal or unequal sized batches together with controllable lead time. A reduction function for lead time is considered by Hsu and Huang (2009) whereas Li et al. (2012) have developed a supply chain coordination system for multi-product with controllable lead time. Braglia et al. (2014) have investigated a two-stage supply chain with a safety stock management and consignment stock agreement where they have considered the demand and lead time both are random in nature. An operational consignment stock policy for normally distributed demand is stated by Yi and Sarker (2014) where buyers’ space limitation and controllable lead time are considered. Besides these, many articles are incorporated by considering fixed as well as variable lead time. A few of them are Glock, 2012, Yi and Sarker, 2013, Rodrigues and Yoneyama, 2020, and Das Roy and Sana (2021).

In management and other scientific researches, the normal distribution is the most commonly used probability distribution among the others. Several researchers have developed inventory models with the consideration of normal demand. Dey and Chakraborty (2012) have framed an inventory model where the demand rate is assumed to be a normally distributed fuzzy random variable. They have considered both non-truncated and the truncated normal distributions of demand. The concept of truncated normal distribution is also discussed by Thomopoulos (2015). Hossain et al. (2017) have considered a supply chain system with a non-truncated normal distribution of lead time while an integrated supply chain model with normally distributed lead time demand is investigated by Das Roy and Sana (2020). The present study has discussed three types of lead time distributions. Non-truncated normal distribution of lead time is one of them.

The occurrence of a stock-out situation in an inventory management system is very common. Shortages can be backlogged in two ways: Partially or completely. Many authors [Ng et al., 2001, Das Roy et al., 2012, Das Roy et al., 2014, Hossain et al., 2017, San-Joséa et al., 2019] have included backlogging in their studies. The proposed article has also considered backlogging. Any article that has addressed the concept of nonlinear early delivery penalty cost together with nonlinear late delivery penalty cost in a supply chain having random delivery lead time seldom follows a general distribution. Recently, researchers have focused on the consequences when the lead time is stochastic in nature and follows some known probability distributions. Lee et al. (2007) have investigated an integrated inventory model with stochastic lead time, ordering cost reduction and backorder discount whereas a multi-supplier and single buyer supply chain coordination system with a milk-run delivery network is presented by Zhou et al. (2012). They have included stochastic lead time and capacity constraints in their study. A supply chain model with stochastic lead time is also discussed by Lin, 2016, Hossain et al., 2017.

The concept of penalty cost for delivery lateness is addressed by many authors. Guiffrida and Jaber (2008) have introduced penalty cost for early and late delivery in a supply chain to study the managerial and economic impacts of reducing delivery variance while an optimal position of supply chain delivery window which minimizes the expected penalty cost for delivery earliness and lateness is determined by Bushuev and Guiffrida (2012). Zhu (2015) has incorporated a decentralized supply chain where the penalty cost is considered in terms of compensation to the customer for delivery lateness. Hossain et al. (2017) have discussed a vendor–buyer cooperative policy in an integrated supply chain model with a general distribution of lead time. They have introduced the concept of penalty cost for delivery lateness. A cost base delivery performance model is developed by Bushuev (2018) where he has considered an expected penalty cost for delivery earliness and tardiness. The present study introduces nonlinear penalty costs for early delivery as well as for late delivery of the ordered lot size. Biswas and Sarker (2020) have developed an operational planning of supply chains in a production and distribution center with just-in-time delivery policy. Lin et al. (2021) have showed how to reduce optimally the setup cost and lot size for economic production quantity model with imperfect quality and quantity discounts. A comparison between the contributions of previous works with the present study is presented in a tabular form (see Table 1).

Flexibility in lead time plays an important role in operating a coordinated system within a certain tolerance time (early or late arrivals of shipments). From Table 1, the contribution of the present paper in the literature and the comparison of the proposed research with other articles is clearly observed. Two main points are highlighted in this respect. First, most of the researchers have restricted their study by considering lead time as a deterministic variable or random variable which follows a specific probability distribution. Consideration of general distribution of lead time is rarely observed in those studies which is one of the features of this study. Secondly, very few authors have included a penalty cost for early or late delivery or both. Also, the researchers who have included penalty cost for early or late delivery, have considered linear penalty cost. In the present paper, a nonlinear penalty cost has been addressed to generalize the penalty function. The method also leaves another aspect of the contributions to capture other variants of such cost function as the system subscribes to fit the existing system.

In the present study, an integrated vendor-buyer cooperative supply chain network is proposed where the lead time is assumed to be generally distributed. The buyer provides a delivery tolerance range to the vendor. If the vendor delivers the ordered lot beyond this delivery tolerance range, he must face two types of penalty costs: early delivery penalty cost and late delivery penalty cost. If the delivery lot reaches before the lower limit of a delivery tolerance range, then it increases the holding cost of the buyer. The buyer charges a penalty cost termed as early delivery penalty cost to the vendor equivalence to this extra holding cost. Again, if the delivery lot reaches after the upper limit of the delivery tolerance period, then the vendor will be responsible for paying a penalty cost to the buyer. Here, this type of penalty cost is termed as late delivery penalty cost which is equivalent to the buyer’s loss of market goodwill and opportunity loss for unable to sell the product during this late period. The main goal of the research work is to determine the optimal values of the replenishment lot size, reorder point, number of shipments for minimizing the integrated expected cost under the environment of generalized lead time distribution of delivery to improve the supply chain performance.

The entire paper is organized into seven sections. Section 1 carries the introduction part. The notation and assumptions of this study are stated in Section 2. Section 3 describes the general model while Section 4 illustrates the solution procedure. Numerical results are provided in Section 5 and sensitivity analyses are carried out in Section 6. Section 7 presents the conclusion of the whole study.

Section snippets

Notation and assumptions

The notation and assumptions used in the present paper to develop the model are as follows.

The general model

In this section, the model is described and framed mathematically.

The solution procedure

To optimize the above integrated expected cost of the proposed supply chain model, the following theorems are to be followed.

Theorem 2

If L>R/D, then the integrated expected cost function EACI is

  • (i)

    strictly convex in n, for given Q and R,

  • (ii)

    strictly convex in Q and R, for given n.

Proof

(i) The second order partial derivative of Eq. (10) with respect to n is2EACIn2=2DCVvQ2n3>0QandR.

Hence, EACI is strictly convex in n, for given Q and R.

(ii) The second order partial derivatives of Eq. (10) with respect to Q and R

Numerical results

The following numerical examples are discussed to establish the general model. The parameter values are taken from the model of Hossain et al. (2017) except the new ones.

Example 1. Uniformly distributed lead time.

Let us consider a small-scale industry that produces toys for kids and supplies them to a retailer. The delivery lead time follows a uniform distribution. The parametric values of this system are: D=1000toys/year, CVv=$400/setup, C0b=$25/order, CHv=$4/toy/year, Chb=$5/toy/year, Cbb=$30

Sensitivity analyses

The sensitivity analyses of the optimal solutions for Examples 1, 2 and 3 are performed by changing the values of the parameters D,CVv,C0b,CHv,Chb,Cbb,CEv,CFv and m by −50%, −25%, +25% and +50% while the other parameters value remain unaltered. The effects of such changes on the optimal values of order lot size Q, reorder point R and the integrated expected cost EACI for Examples 1, 2, and 3 are reported in Table 5, Table 6, Table 7, respectively.

The scenarios observed from Table 5, Table 6,

Conclusions

This study presents a viable cooperative agreement between two parties in a supply chain network for a single product. It determines a delivery policy under some conditions which are experienced in many industrial situations where the demand is constant and the delivery lead is stochastic in character. The system allows shortages that are completely backlogged. Moreover, two types of nonlinear penalty costs have been incorporated: early delivery penalty cost and late delivery penalty cost. The

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (37)

View full text