and Lenovo are famous for products that reach relatively low prices and features. Consumer’s preference for higher prices to quality will influence himself to buy Apple products which are regarded suitable for their requirements. By preferring features and prices, end-customers’ accommodate on product quality and buy Samsung or Lenovo products. The vendors need both advanced manufacturing processes and good quality raw materials if they want to produce high reliability products. To assemble laptops, Lenovo uses low cost raw-materials to reduce overall costs whereas Apple uses very high quality ingredients in MacBooks and hence overall costs increases. To optimize a firm’s profit, a balance needs to be struck between time, price and reliability of its product. Integrated supply chain
models have been developed in the literature based on some limited assumptions.
One of these assumptions is that the vendor produces products of perfect
quality. In fact, there can be a few imperfect items in any production lot due
to poor control of processes, non-adherence to plans, inappropriate operating
guidelines, and so on. If the vendor has to pay extra cost for each defective
item produced then it is profitable to reduce the number of defective items in
the production process. The rate of defective items produced by the vendor
affects other critical decisions such as the vendor’s production lot size and
reliability of the product. Further, a vendor has a reputation for making more
reliable product which is preferable for a buyer to place an order. To improve
the quality of a product, investment can be made to reduce errors in the
vendor’s production process. In an integrated supply chain system, when
non-conformable items are produced, it is most likely that some kind of
supervision/inspection activity needs to be performed by the buyer before
selling the goods to the end customers. This article develops an
integrated single-vendor single-buyer supply chain model with time, price and
reliability dependent market demand. The vendor’s production process is
imperfect and it rejects all the non-conformable items produced during a
production run. The buyer screens all the items before selling to
end-customers. The defective items are sold in the secondary market with a
discount. The vendor plans for a lot-for-lot production policy to meet the
buyer’s demand. The primary objectives of this article are to find the response
of the following queries: 1)
How much time
will be taken by the vendor and the buyer to produce a lot and sell to
customers? 2)
How much time
will be delayed by the vendor to produce items ordered by the buyer? 3)
What will be the
selling price of each good item from the buyer’s side? 4)
What will be the
reliability of a product produced by the vendor? The rest of the paper is
arranged as follows: In the next section, the related literatures are reviewed.
Section 3 presents assumptions and notations for developing the proposed model.
Section 4 discusses the mathematical model and solution procedure. A numerical
example is provided in Section 5. The optimal results are analyzed in Section
6. Section 7 concludes the paper and indicates some future research
directions. 2.
LITERATURE REVIEW In reality, the market
demand of certain products may not remain constant always; it may change with
the passage of time. Hariga and Benkherouf (1994) presented a heuristic inventory model in which the
market demand changes exponentially in time over a finite planning horizon. Hariga (1996) developed an inventory lot-sizing model with time-varying demand for
deteriorating items. An inventory model with Weibull deterioration, time
proportional demand rate and effects of inflation was developed by Chen (1998). Khanra and Chaudhuri (2003) proposed an inventory model with quadratic time dependent demand where
the on-hand inventory deteriorates with time. Ghosh and
Chaudhuri (2006) developed this model by considering shortages in inventory. Actually, a
large volume of research papers on time dependent demand are available in the
literature Giri and Maiti
(2012), Chowdhury et al. (2014), Samanta
et al. (2018) Now-a-days the
customer’s demand depends not only on time but also on other factors such as
product price, after sales service, advertisement, product quality, etc. Price
of a product plays an important role in customer’s mind. So, it is more
realistic to include price sensitive demand. Burwell et al.
(1991) determined the optimal lot size and selling price when a supplier offers
all-unit quantity discounts by considering price-dependent demand and allowing
for shortages. A finite period system was considered by Datta and Paul (2001) under multi-replenishment scenario, where the demand rate is influenced
by both displayed stock level and selling price. An economic production
quantity (EPQ) model for deteriorating items was developed by Teng and Chang (2005) where the demand rate depends on the selling price and display stock
level with limited display space consideration. You (2005) investigated a supply chain model in which a leading member of the supply
chain gets the scope to settle value of the product to impress demand and more
revenues. Avinadav et al. (2013) formulated a model for finding the optimal pricing, order quantity and
replenishment period for deteriorating items with price- and time-dependent
demand. Yang et
al. (2013) studied a piecewise production-inventory model for a deteriorating item
with time-varying and price-sensitive demand to optimize the vendor’s total
profit. Herbon
and Khmelnitsky (2017) considered a dynamic pricing policy for perishable products, attracting
customers to buy less-fresh products due to expiry, potentially increasing
revenue and eliminating waste. Numerous works in this direction could be found
in the literature You and Hsieh
(2007), Chen et
al. (2010), Ghosh et
al. (2011), Kim et
al. (2011), Bhunia and Shaikh
(2014), Maiti
and Giri (2015), Giri and
Roy (2015), Maiti
and Giri (2017), Chan (2019), Roy and
Giri (2020). When end-customers buy
some goods from buyers, it is the outcome of the endeavors of several members
of supply chains. But, the main credit goes to the vendor as the customer
prefer that product for his reliability. So, the balance between price and
reliability is an important factor in inventory/supply chain management.
Therefore, the reliability of a product must be taken into consideration. An
EPQ model with a flexible and imperfect production process was proposed by Cheng (1989) under reliability consideration. Sadjadi et al. (2009) considered a production-marketing problem where the reliability of the
production process assumed to be imperfect and the inventory and the setup
costs per production cycle are not known in advance. An inventory model with
imperfect production process was developed by Shah and Shah (2014) for time-declining demand pattern where reliability of the production
process was considered as a decision variable. Shah and Vaghela (2018) analysed EPQ model with time and advertisement sensitive demand with the
effect of inflation and reliability. The above works
considered reliability of the product and its effect on the optimal results.
However, none of these works would consider the market demand as a function of
reliability of the product. Khara et al. (2017) considered a model that deals with an imperfect production process, where
both perfect and imperfect quality items are produced and demand depends on
selling price and reliability of the product. Later, Khara et al. (2019) developed that model by considering demand as a function of selling
price, reliability of the product and advertisement cost. Shah and Naik (2020) investigated an inventory model with imperfect production process and
reliability-dependent demand. Chung
and Wee (2008) developed an integrated production-inventory deteriorating model
considering imperfect production, inspection planning and warranty-period-and
stock-level-dependant demand. Jauhari (2016) proposed a vendor-buyer model where the lot transferred from the vendor
to the buyer contains some defective items and the buyer conducts an imperfect
inspection process to classify the quality of the items. Jauhari et al.
(2016) developed an imperfect production-inventory model where the buyer uses
periodic review policy to manage his inventory. The demand on the buyer side
was assumed to be normally distributed, and the shortage was assumed to be
fully backordered and the defective rate of the items was assumed to be fixed. In this article, we
consider the market demand as a function of time, selling price and reliability
of the product. The production rate is not constant but depends on the market
demand, as considered by Giri and
Maiti (2012). The variable production rate was also considered by Jauhari et al.
(2016). In the literature, unit production cost is considered as a fixed. But in
reality, it should depends on order quantity to be produced by the vendor. More
production implies less unit production cost and less production implies
expensive production cost. On the other hand, if a vendor prefers to produce an
item with more reliable to keep/increase his reputation in market, then (s)he
has to use raw material which are also more reliable. Thus the material cost
depends on reliability of the product. The demand may change at any time during
production process. In that case, to maintain the on-time delivery to the
buyer, the vendor’s production rate has to be changed. Therefore, we consider
the unit production cost as a function of material cost and production rate.
Variable unit production cost was also considered in different forms by Khara et al. (2017). 3. MODEL ASSUMPTIONS AND NOTATIONS The notations used
throughout the paper are as follows:
The following
assumptions are made to develop the proposed integrated vendor-buyer inventory
model: ·
The supply chain
consists of a single-vendor and a single-buyer who stocks and sells a single
product. ·
The demand for a
product depends on time selling price as well as the
reliability of the product . We assume that the demand rate and are real
constants. This type of demand was considered by Khara et al. (2017). ·
The vendor
follows the lot-for-lot policy for replenishment made to the buyer. ·
The buyer
receives the first order from the vendor at time and (s)he
receives order from the vendor in every time interval. ·
Shortages are not
allowed in the buyer’s inventory. ·
As the
reliability of the product depends not only on the manufacturing system but
also on the quality of the raw material of the product, we assume that the
material cost is an
increasing function of the reliability () of the product such that where and . ·
The production
rate of the vendor varies with the demand rate. Also, the production rate is
greater than the demand rate. We take the production rate as where . ·
As the vendor’s
production rate is greater than the buyer’s demand rate, the vendor may start
production with a time delay in the n-th
production cycle. ·
The production
cost not only depends on the material cost but also on
tool or die cost, which is proportional to the vendor’s production rate.
Therefore, the unit production cost is assumed as where . ·
The vendor’s
production process is not perfectly reliable. During a production run, it may
produce some defective (non-conforming) items. ·
The buyer starts
error-free screening after received products from vendor. We assume that the
number of perfect units is at least equal to the demand during the screening
time. ·
Product quality
may be imperfect. In other words, only of all produced
items meet the demand while of items are
defective. It is apparent that the maximum reliability of the production
process cannot exceed . This type of assumption was also considered by Sadjadi et al.
(2009). ·
The vendor
produced quantity in
total during 4. MODEL FORMULATION The graphical
presentation of the vendor-buyer model is shown in Figure 1. We suppose that is the length
of each cycle. For the -th cycle, the vendor starts his/her production at
time and the buyer
receives his/her order of quantity from the vendor
at time , and meets the
market demand for period . The buyer starts screening at a rate of units per unit
time immediately after receiving the products from the buyer. The buyer’s
screening is completed at time . We assume that only of received
products are acceptable as good products to meet the customer demand. The
customer’s demand rate at time is where and are real
constants. Therefore, the total
demand during the period is given by (1)
The quantity produced by the
vendor in the time interval is given by (2) 4.1. DECENTRALISED MODEL 4.1.1. VENDOR’S PERSPECTIVE Let be the vendor’s
inventory level at any time . Then the instantaneous states of the vendor’s
inventory level can be described by the differential equation: (3) Solving (3), we get (4) At time , we have The vendor’s holding cost per unit time for
the period The vendor’s production cost per unit time in
that period As the vendor’s sales revenue = set-up cost , discount cost for defective items per unit time = , therefore, the vendor’s total profit per unit time
is given by
(5) 4.1.2. BUYER’S PERSPECTIVE The differential
equation governing the buyer’s inventory level at any time is given by Solving, we get From (6), the buyer’s
inventory level at the time pointis given by (7) Also, we have (8) From (7) and (8), we
have (9) Which is a quadratic
equation in with
discriminant . Hence there always
exists a positive (real) production lot size of the vendor
in any time interval , for all . Now, the buyer’s holding
cost per unit time Also, sales revenue per
unit time = , purchase cost per unit time =, transportation cost per unit time = , screening cost per unit time = and ordering
cost per unit time = . Therefore, the buyer’s total profit per unit time is
given by
(10)
provided that
It is clear from the
above that provided that which gives (say) which gives or, (say) or, or, [since ] or, |