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:

	:	time interval between successive deliveries (decision variable)
	:	time delayed by the vendor to start production (decision variable)
	:	unit selling price for the buyer(decision variable)
	:	reliability of the product (decision variable)
	:	variable time
	:	number of cycles
	:	demand rate at the buyer
	:	production rate at the vendor
	:	scaling constant for production rate
	:	set up cost per production run for the vendor
	:	ordering cost per order for the buyer
	:	unit stock-holding cost per unit per unit time for the vendor
	:	unit stock-holding cost per unit per unit time for the buyer
	:	quantity produced by the vendor during the period
	:	market demand during the period
	:	material cost
	:	price elasticity to demand
	:	reliability elasticity to demand
	:	reliability elasticity to material cost
	:	fixed material cost
	:	material cost increases the reliability of the produced item
	:	variation constant of tool/die costs
	:	transportation cost per shipment
	:	unit production cost
	:	screening rate
	:	unit screening cost
	:	unit wholesale price for the vendor
	:	discount price per defective item for the vendor
	:	buyer’s inventory level
	:	vendor’s inventory level
	:	buyer’s profit function
	:	vendor’s profit function
	:	average total profit to the whole supply chain

 

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 n-th production cycle and delivered to the buyer to meet the customer / market demand  in the next cycle.

 

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)

 

Figure 1 A schematic diagram to represent the vendor’s and the buyer’s inventory.

 

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)

 

Proposition 1  When the buyer’s selling price  is known, the profit function  is concave with respect to  for all   where

         

         

         

 provided that

Proof. Differentiating (10) twice with respect to , we get

              

It is clear from the above that  provided that

 which gives  (say)

 which gives

             or,  (say)

or,

or, [since ]

or,