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Price and profit structuring for single manufacturer multi-buyer integrated inventory supply chain under price-sensitive demand condition
Journal Pre-proofs Price and profit structuring for single manufacturer multi-buyer integrated inventory supply chain under price-sensitive demand condition Anil Kumar Agrawal, Susheel Yadav PII: DOI: Reference:
S0360-8352(19)30677-1 https://doi.org/10.1016/j.cie.2019.106208 CAIE 106208
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
5 June 2018 20 November 2019 27 November 2019
Please cite this article as: Agrawal, A.K., Yadav, S., Price and profit structuring for single manufacturer multibuyer integrated inventory supply chain under price-sensitive demand condition, Computers & Industrial Engineering (2019), doi: https://doi.org/10.1016/j.cie.2019.106208
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Price and profit structuring for single manufacturer multi-buyer integrated inventory supply chain under price-sensitive demand condition Anil Kumar Agrawal Department of Mechanical Engineering, Indian Institute of Technology (Banaras Hindu University), Varanasi, India- 221005
Email: [email protected]
Susheel Yadav* Department of Mechanical Engineering, Indian Institute of Technology (Banaras Hindu University), Varanasi, India- 221005 Email: [email protected]. Phone No.: +91-8409969147
*Corresponding Author
Highlights An integrated production-inventory model for a single manufacturer-multiple buyers.
Issue of profit division among the partners of a supply chain is discussed.
Four different models of profit sharing have been proposed.
Profit sharing based on Shapley function values turns to be fair and supportive.
GA and TLBO meta-heuristics have also been proposed.
Price and profit structuring for single manufacturer multi-buyer integrated inventory supply chain under price-sensitive demand condition
Abstract This paper undertakes the study of an integrated production-inventory and pricing decision problem for a single manufacturer-multiple buyers supply chain where each buyer faces pricedependent demand. The item manufactured at a finite production rate is shipped to the buyers in multiple equal-sized shipments. Earlier researchers have not studied this problem for maximizing the overall supply chain profit and its allocation among the chain partners. For the allocation of the maximized profit, four different schemes are being proposed, each for a different level of understanding and faith among the partners. Demand, taken as price-sensitive, affects the costs to the manufacturer as the same is dependent upon the sales volume. Since the manufacturer fixes its sales price depending upon the sales volume affected cost, the volume will affect the buyer in terms of its unit purchase cost, and the holding cost which is charged as a percentage of the unit cost. This reality is attempted in this paper in addition to maximization of the chain profit and its allocation among the chain partners for integrated inventory management. Associated problems, formulated as mixed-integer non-linear programming problems, are computationally very hard, and thus evolutionary heuristics as Genetic Algorithm and Teaching-Learning Based Optimization are proposed. The strength of various proposed schemes for profit distributions has been analyzed using numerical experimentation. It is found that a forward contract between manufacturer and buyers will help to generate the maximum profit. It has also been shown that the distribution of profit based on Shapley function values is most fair and logical.
Keywords: price-sensitive demand; integrated inventory; single manufacturer multiple buyers; profit-sharing; pricing; genetic algorithm ______________________________________________________________________________ 1. Introduction In supply chain management (SCM), efforts are made to provide a win-win strategy for all the chain partners, right from buyers to supplier's vendors (Lummus and Vokurka, 1997). It would require a lot of coordination among the chain partners (Power, 2005; Simatupang et al., 2002). Out of the several important factors for the coordination, Thomas and Griffin, (1996) identified the following important issues: the selection of batch size, choice of transportation mode, and choice of production quantity. These issues are suggested to be resolved by having operational coordination — Buyer-Vendor coordination, Production-Distribution coordination, and InventoryDistribution coordination. Arshinder et al. (2011) also emphasized on coordination in the supply chain. They urged that the members of the supply chain should behave as parts of a unified system and coordinate with each other to improve the overall performance of the supply chain. This coordination can be at an operational level in the form of synchronized cycle times of all the buyers (Lu, 1995; Hoque, 2008; Chen and Sarker, 2017) and also at the financial level through a revenuesharing contract (Cachon, 2003; Govindan et al., 2013; Arani et al., 2016; Hu and Feng, 2017). In the absence of proper coordination, the supply chain has to face several operational and tactical issues. If a partner considers its inventory management problem independently of the other chain partners, it is found to bring in cost inefficiency and coordination issues (Bagchi et al., 2005). In order to avoid such situations, the focus is on adopting the Joint Economic Lot Sizing (JELS) framework right from buyers to the suppliers' suppliers. JELS models deal with the problems where the total relevant cost for the inventory is considered across the whole supply chain and not
separately and independently for individual partners. Working on this principle helps in resulting in a net saving in the total cost of the supply chain (Ben-Daya and Hariga, 2004; Hill and Omar, 2006; Shyu et al., 2006; Chung, 2008; Ben-Daya et al., 2008). Pando et al. (2012) asserted that, in real life, the main objective of inventory management is to maximize profit. Accordingly, the revenue from the sale of items should be considered in inventory models along with ordering, setup and holding costs. Since supply chain members are primarily concerned with their profits, it is natural to look into the joint inventory problem from the total profit perspective rather than simply and only considering the total cost. This has prompted some researchers to analyze the problem from the total supply chain profit perspective (Giri and Roy, 2015; Chen and Sarker, 2017). In the early years, integrated production-inventory management researches focused on the manufacturer's production capacity assumption and delivery methods. A review of the researches related to joint economic lot-sizing problems can be had from the work of Glock (2012). The concept of JELS for the first time came from Goyal (1976). He considered the integrated inventory management problem consisting of a single vendor and a single buyer where the vendor has an infinite production rate capability and meets the demand of the purchaser using a lot-for-lot shipment policy. Later, Banerjee (1986) relaxed the unrealistic assumption on the vendor's infinite production rate capability and instead considered a finite production rate for the vendor. The idea of the lot-for-lot shipments was scrapped by Goyal (1988) in his further work. The profit potential of a supply chain was found to depend upon the supply policy: a single batch against an order or multiple ones. Instead of meeting the whole order from one supply, he considered the production lot to be shipped in equal-sized sub-batches, but only when the production for the entire lot was over. Agrawal and Raju (1996) took this concept of sub-batching in which a lot can be divided
into many equal-sized sub-batches. However, deliveries of these sub-batches were to be made even during the production of the lot. Models with different shipment policies consisting of unequal sub-batch and combination of equal and unequal sub-batches have also been studied by several other researchers (Goyal and Gupta, 1989; Hill, 1997; Viswanathan, 1998; Hill, 1999; Goyal, 2000; Goyal and Nebebe, 2000). A single manufacturer supplying a product to a single buyer (SMSB) is hard to find in today’s business environment (Sarmah et al., 2008), but SMSB will always be the basic building block for such problems. In today's business world, manufacturers usually sell the product through several retailers rather than relying on a single retailer. On realizing this fact, Joglekar and Tharthare (1990) proposed another JELS model considering multiple homogeneous buyers who can also be viewed as retailers. However, these retailers need not be identical and may have different operating characteristics (Banerjee and Burton, 1994). Incorporation of such considerations into JELS models will make it more realistic. A Common Replacement Epoch policy was considered by Viswanathan and Piplani (2001) for a case where a vendor supplies to its buyers at the same time. It naturally reduces the flexibility available to the buyers and increases their cost as well. To compensate for this increased cost, the vendor offers a price discount to the buyers. Their proposed model takes into consideration these factors and accordingly determines the optimal common replacement period and the price discount. A case of single vendor multiple buyer supply-chain, where all the partners are interested in creating a strategic alliance for higher profits, was studied by Woo et al. (2001). The effect of investments was evaluated in terms of ordering cost reduction on the total cost reduction through electronic data interchange (EDI). Siajadi et al. (2006) studied a single vendor multiple buyer (SVMB) supply chain model and proposed a shipment policy where the buyers can have a varying number of equal-sized shipments. In their model, the hierarchy of
the buyers affects the number of shipments and saving in the total cost. Hoque (2008) proposed three policies for shipments in SVMB scenario with integrated inventory consideration: (i) equalsized shipments are to be made as soon as the manufacturer finishes the production lot, (ii) equalsized batches are to be dispatched such that they are available by the time the buyers consume the whole inventory and (iii) unequal-sized batches are to be supplied with the next shipment size increased in the ratio of the production rate to the sum of the demand rates of all the buyers. Optimal solutions are found analytically for all the policies. Sarmah et al. (2008) also studied an SVMB supply chain with two dominance conditions: (i) the dominating manufacturer ships large size lot to the buyers at a common replacement time through common carrier and compensates their increased cost by giving them a uniform credit policy, and (ii) the dominating buyers’ association offer a common replacement time proposal to the manufacturer and acceptance of the offer depends on negotiation between the association and the manufacturer. Jha and Shanker (2013) investigated the effect of lead time reduction on total supply chain cost with a service level constraint in an SVMB supply chain. Giri and Roy (2015) also studied the controllable lead time in an SVMB supply chain, where all the buyers are homogeneous, with their demand being dependent on the price of the product. All these research papers have shown that supplying the product in multiple sub-batches to the buyers helps in reducing the overall cost by saving on holding and setup costs more than the additional expenses to be made on transportation of the items from the manufacturer to the buyers. While going through the literature, it was observed that inventory management decisions are largely governed by the cost parameters involved. The holding cost constitutes an essential component of well-established inventory management models such as Economic Order Quantity (Harris, 1913; Wilson, 1934) and JELS models of (Lu, 1995; Kosadat, 2000; Mahata et al., 2005;
Yang et al., 2008; Zanoni et al., 2014). Berling (2008), in his research, has mentioned that little effort had been put in accurately determining the cost parameters. A good number of research papers related to work on JELS (Lu, 1995; Kosadat, 2000; Mahata et al., 2005; Yang et al., 2008; Zanoni et al., 2014) considered the holding cost rate as a fixed cost per unit per year. This is different from the thought of Berling (2008) as well as of Singhal and Raturi (1990), who expressed the holding cost rate as an annual rate, and the annual carrying cost was determined by relating it to the item’s value. It is only the latter case where the unit cost of the item will have a bearing on the determination of the optimal order quantity. In the case of price-sensitive demand, the unit cost will have a major role in deciding or fixing the sales price and thus will dictate the annual demand and, finally, the optimal order quantity. If the demand is not price-sensitive, the cost minimization approach will yield the same optimal inventory management policies as for profit-maximization since the revenue received from the customers is going to be fixed due to constant values of the sales price and the demand. When the demand is not fixed and is price-sensitive, it is likely to affect the costs to the manufacturer and the buyers, most significantly their holding cost. In this environment, the unit purchase cost for the buyers may also vary if the manufacturer fixes its sales price depending upon its cost incurred in manufacturing the items. This reality has not been addressed in the available research literature that focuses on the maximization of the total supply chain profit and its judicious allocation among the chain partners. The problem considered in this paper assumes holding cost rate to depend upon the unit cost of the item and the end demand to be price-sensitive. In the present paper, the focus is on maximizing the overall profit for a single vendor and multiple buyer supply-chain and, to that end, also on determining optimal values of retailers' selling prices, number and size of multiple shipments to
fulfill an order. Even though coordination and integration will lead to more of overall supply-chain profit, but the challenge will still be there in distributing the profit among the chain partners, particularly when the markup on the cost price of the upper echelon partners is not fixed. Profit allocation is an essential issue in supply chain management on which the success of the supply chain depends. Yue and You (2014) reported that there are a number of supply chain optimization models available in the literature that consider the supply chain as a whole, without taking into account the conflicting interests of its partners. This type of optimization results in a high degree of dissatisfaction among the partners due to uneven profit distribution as these models do not incorporate any fair profit division schemes. The problem of profit allocation has been addressed in several ways. Zahran et al. (2017) considered a three-level supply chain model where each partner charges a predefined profit margin from its downstream partner. The demands of the end customers were taken to be different and deterministic. The problem was solved for four different combinations of coordination and nocoordination scenarios of consignment stock policies for its partners. Jaber and Osman (2006) proposed to share the overall profit among the partners of a two-level supply chain in proportion to their incurred costs. Their model does not maximize the profit but minimizes the supply chain costs. The two objectives are equivalent as they took the demand to be pre-specified and constant. However, the demand is not necessarily pre-specified, particularly in the case of price-dependent demand. For such a demand, the minimization of the supply-chain cost will target zero demand resulting in overall supply chain cost as zero. Naturally, the maximization of overall profit will serve as the right objective in the case of price-dependent demand. For this reason, Chauhan and Proth (2005) aim to maximize the supply chain profit for the case of price-dependent demand. They also divided the profit among the partners in proportion to their investments. Revenue
sharing is a relatively new coordination contract mechanism. Under this contract, the retailer pays the supplier a wholesale price for each unit purchased and a percentage of the revenue that the retailer generates (Yao et al., 2008). Sharing of revenue can be taken as sharing of profit as the retailers will never share its revenue with the vendor beyond a particular limit so as to incur a loss. Cachon and Lariviere (2005) studied the revenue sharing contract in coordinating the supply chain of a video cassette rental industry. They showed that revenue sharing induces the retailer to choose supply-chain actions for determining the optimal quantity and selling price. In the present paper, besides the profit-sharing mechanism due to Chauhan and Proth (2005), three more schemes have also been proposed for the case of price-dependent demand while taking the holding cost of the buyers as the function of their unit purchase cost. One of the schemes minimizes the maximum opportunity loss to a supply chain partner. This scheme will be useful in situations where the level of understanding among the chain partners has not matured enough. The remaining two schemes will be more applicable for those supply chains where the understanding among the chain partners is reasonably good, and the perfect coordination is being sought by the partners in order to maximize their overall profit. For the first two schemes, the proposed mathematical programming formulations directly allocate the profit while deciding sales price at each of the supply chain partners. For the remaining two schemes, one more mathematical model has been presented to determine optimal inventory policies resulting in maximum overall supply chain profit. This optimal profit of the chain is later distributed among the chain partners using two other different schemes proposed in this paper. The organization of the paper is as follows. Section 2 describes the problem as well as assumptions and notations involved. It also presents the expressions for various elements of the cost to the manufacturer and the buyers. The four different schemes of profit-sharing and the related
mathematical models are presented in Section 3. The solution procedures are described in Section 4 in the form of two heuristics. The relevance of the four schemes is discussed in Section 5 from the perspective of their effectiveness in different scenarios of supply chain coordination. It also presents the strengths and weaknesses of each scheme by conducting experiments using an illustrative numerical example. Section 6 discusses the merits and demerits of these schemes and provides some important managerial insights. It also presents results on the comparative performance of the two proposed heuristics. Section 7 concludes the research work presented in this paper and also suggests some future research directions. 2. The problem The supply chain considered in the present paper is shown in Fig. 1. It has a single manufacturer dealing in a single item with multiple buyers. [Insert Fig. 1] Further characteristics of the problem can be best understood from the assumptions and notations described below. 2.1. Assumptions and notations
Some of the important characteristics of the problem, considered in this paper, are explained in terms of the following assumptions: 1. The buyers have equal replenishment cycle times. 2. No shortages and backordering are allowed. 3. Replenishment is instantaneous. 4. The planning horizon is considered to be infinite. 5. The manufacturer and all the buyers have enough storage capacity to accommodate the required inventory and, therefore, there is no space constraint for storing the stock of items. 6. The production capacity of the manufacturer is considered to be finite. 7. Lead times are constant. It need not be the same for all the buyers. Notations used in the description and mathematical formulation of the problem are as follows.
𝑐
Profit margin charged by the manufacturer on the unit cost (fraction of the unit cost price) Cost of one unit of the item after it has been manufactured
𝑖
Index for a buyer
𝑚
Number of sub-batches during the production cycle of the manufacturer
𝑛
Number of equal-sized sub-batches per order
𝑡
Common replenishment cycle time for all the buyers in year
𝑣
Overall unit cost of the manufacturer
𝑤
Unit purchase price to buyers
𝑥
Ratio of annual production rate to annual demand rate
𝐷
Annual demand for the manufacturer being equal to cumulative demand of all the buyers
𝑁
Total number of buyers
𝑃
Annual production rate of the manufacturer
𝑄
Production lot size of the manufacturer
𝑇
Total annual supply chain profit
𝑍
Maximum of the opportunity loss to a supply chain partner
𝐶
Total annual cost of the manufacturer
𝑃
Total annual profit to the manufacturer
𝑃
Total annual profit of all the buyers
𝑎𝑖
Maximum demand experienced by buyer 𝑖 in units
𝑏𝑖
Demand-price elasticity coefficient for buyer 𝑖
𝑝𝑖
Unit selling price of 𝑖𝑡ℎ buyer
𝑟𝑖
Annual inventory holding cost rate for 𝑖𝑡ℎ buyer (charged as a fraction on per unit monetary value of the inventory stock)
𝑟𝑚
Annual inventory holding rate for the manufacturer (charged as a fraction on per unit monetary value of the inventory stock)
𝐴𝑖
Ordering cost per order of 𝑖𝑡ℎ buyer
𝐴𝑚
Cost per setup of the manufacturer
𝐵𝑖
Best profit for buyer i
𝐵𝑚
Best profit level of the manufacturer
𝛼
𝐶𝑖
Total annual cost of buyer i
𝐶𝑚
Area of one inventory cycle of the manufacturer
𝐷𝑖
Annual demand of 𝑖𝑡ℎ buyer as a function of its selling price
𝐻𝑖
Annual inventory holding cost to 𝑖𝑡ℎ buyer
𝐻𝑚
Annual holding cost of the manufacturer
𝐼𝑚
Average inventory of the manufacturer
𝑀𝑖
Annual ordering cost to 𝑖𝑡ℎ buyer
𝑂𝑖
Opportunity cost for buyer 𝑖
𝑂𝑚
Opportunity cost for the manufacturer
𝑃𝑖
Annual purchase cost to 𝑖𝑡ℎ buyer
𝑃𝑖
Total annual profit of 𝑖𝑡ℎ buyer
𝑃𝑚
Annual cost of manufactured units of the item
𝑅𝑖
Annual revenue of 𝑖𝑡ℎ buyer
𝑆𝑚
Annual setup cost to the manufacturer
𝑇𝑖
Transportation cost per shipment of 𝑖𝑡ℎ buyer
𝑇𝑖
Annual transportation cost to 𝑖𝑡ℎ buyer
2.2. Expressions for cost elements
The buyers face demand from customers, and the same depends upon their sales price for the item. The relationship considered between price and demand (Chen and Sarker, 2017) is: 𝐷𝑖 = 𝑎𝑖 ― 𝑏𝑖𝑝𝑖
∀ 𝑖.
(1)
Fig. 2 depicts the integration of demand from the buyers and the supply from the manufacturer. In this figure, on-hand inventory is not shown on the same scale for the members of the supply chain. The manufacturer produces the item at a constant and finite production rate and completes a lot of size 𝑄 units in one setup. These units are sent in 𝑛 number of equal sub-batches. From a sub-batch
of size 𝑞, 𝑖𝑡ℎ buyer gets 𝑞𝑖 units in proportion of the ratio of its annual demand (𝐷𝑖) to the total 𝐷𝑖
𝑁
annual demand (𝐷 = ∑𝑖 = 1𝐷𝑖) of all the buyers. Thus 𝑖𝑡ℎ buyer gets a quantity 𝑞𝑖 = 𝐷 𝑞 (with 𝑞 = 𝑁
∑𝑖 = 1𝑞𝑖 ) and consumes it in time 𝑡. Hence
𝑞1
𝑞2
𝑞𝑁
𝐷1 = 𝐷2 = … = 𝐷𝑁 = 𝑡. From Fig. 2, it can be observed
that 𝑄
(𝑄 ― 𝑛 ) 𝐷
= (𝑛 ― 1)𝑡.
(2)
From the above, the following relationship will emerge. 𝑄
𝑡 = 𝑛𝐷
(3)
[Insert Fig. 2] Using the defined notations and the distribution framework provided in Fig. 2, the mathematical expressions for the various costs are determined. They are presented in subsequent sub-sections. 2.2.1 Manufacturer’s total cost
The total cost for the manufacturer will be the amount of money that the manufacturer will spend on procuring the items, in processing the same, and also in stocking the finished items. Annual cost on manufactured products without holding and setup costs
Per unit cost 𝑐 to the manufacturer is assumed to include all the related costs such as unit raw material purchase cost, unit raw material holding cost, unit processing cost, etc. In order to satisfy the cumulative annual demand 𝐷 of all the buyers, the manufacturer will bring the units of the item to finished form by incurring the total annual expense, as given below. 𝑃𝑚 = 𝑐𝐷 Annual inventory holding cost
(4)
This cost is taken in proportion to the average inventory that can be determined by dividing the manufacturer’s inventory cycle area by the cycle length. From Fig. 2, the area under the inventory cycle curve of the manufacturer (𝐶𝑚) can be determined as 𝐶𝑚
𝑃 𝑄 ― (𝑚 ― 1)𝑞 ― 𝑞 𝑞𝑞 𝑞𝑃 𝑞 𝑃 𝑞 𝑞𝑃 𝑞 𝑃 𝑞 𝐷 + +𝑞 ―1 + 𝑞 ― 1 (𝑚 ― 1) + = +…+ 2 2𝑃 2𝐷𝐷 𝐷 𝐷 2𝐷𝐷 𝐷 𝐷
{
( )}
{
}
( )
(
)
𝑃 𝑃 𝑄 ― (𝑚 ― 1)𝑞 ― 𝑞 𝑄 ― (𝑚 ― 1)𝑞 ― 𝑞 𝐷 𝐷 𝑃 𝑞 + 𝑄 ― (𝑚 ― 1)𝑞 ― 𝑞 ― 𝑃 𝑃 𝐷 𝐷
(
){
}
[
{
}
( )]
𝑞 + {1 + 2 + 3 + … + (𝑛 ― 𝑚 ― 1)} 𝑞 . 𝐷
The average inventory of the manufacturer (𝐼𝑚) can be calculated by dividing the area under the inventory cycle curve of the manufacturer (𝐶𝑚) by the cycle time (𝑄 𝐷). Hence, the average inventory of the manufacturer will be 𝐼𝑚 =
[
𝑞𝑞 2𝑃
+
(
𝑞𝑃 𝑞 2𝐷𝐷
(
+𝑞
𝑃 𝐷
) )+…+(
―1
𝑞𝑃 𝑞 2𝐷𝐷
𝑞 𝐷
(
+𝑞
𝑃 𝐷
)
― 1 (𝑚 ― 𝐷 𝑄
𝑞 1)𝐷
)+(
.
𝑃
)(
𝑃
) + {𝑄 ― (𝑚 ― 1)𝑞
𝑄 ― (𝑚 ― 1)𝑞𝐷 ― 𝑞
𝑄 ― (𝑚 ― 1)𝑞𝐷 ― 𝑞
2
𝑃
𝑃 𝐷
(5)
After simplification, 𝑄
[
𝐼𝑚 = 2𝑛2
1
𝑃 𝐷 + (𝑛 ― 𝑚 ― 1)(𝑛 ― 𝑚) + 𝑚(2𝑛 ― 𝑚 ― 1) ―
].
(𝑛 ― 1)2 𝑃𝐷
(6)
Therefore, the annual inventory holding cost of the manufacturer (Hm) can be calculated by multiplying the average inventory by inventory holding cost rate (rm) and per unit cost (c). Thus,
𝐻𝑚 = 𝑟𝑚𝑐𝐼𝑚
}{
―𝑞
𝑞 𝐷
―
𝑄 ― (𝑚 ― 1)𝑞 𝑃
After putting the value of 𝐼𝑚 from equation (6) in the above equation, we would get, 𝑄
[
𝐻𝑚 = 2𝑛2
1
𝑃 𝐷 + (𝑛 ― 𝑚 ― 1)(𝑛 ― 𝑚) + 𝑚(2𝑛 ― 𝑚 ― 1) ―
]𝑟 𝑐.
(𝑛 ― 1)2
(7)
𝑚
𝑃𝐷
Annual setup cost
Besides incurring the cost for bringing the item to the finished level and also on holding them in stock, the other important cost to be borne by the manufacturer is the setup cost. The annual setup cost of the manufacturer can be calculated by multiplying the number of setups per year with the cost incurred in making one setup. 𝐷
Thus,
(8)
𝑆𝑚 = 𝑄𝐴𝑚.
So, the total annual cost of the manufacturer (𝐶) will be the sum of the annual cost of manufactured products (Pm), the annual inventory holding cost (Hm) and the annual setup cost (Sm). Thus,
𝐶 = 𝑃𝑚+ 𝐻𝑚+ 𝑆𝑚.
Substituting the values of 𝑃𝑚, 𝐻𝑚 and 𝑆𝑚 into the above equation from their respective equations (4), (7) and (8), we get 𝑄
[
𝐶 = 𝑐𝐷 + 2𝑛2
1 𝑃𝐷
+ (𝑛 ― 𝑚 ― 1)(𝑛 ― 𝑚) + 𝑚(2𝑛 ― 𝑚 ― 1) ―
]𝑐 𝑐 +
(𝑛 ― 1)2
𝑚
𝑃𝐷
𝐷
𝑄 𝐴𝑚.
Per unit overall cost of the manufacturer (𝑣) can be found by dividing the total annual cost by the total annual demand. Therefore, 1
[
𝑄
{
𝑣 = 𝐷 𝑐𝐷 + 2𝑛2
2.2.2. Buyers’ total cost
1 𝑃𝐷
+ (𝑛 ― 𝑚 ― 1)(𝑛 ― 𝑚) + 𝑚(2𝑛 ― 𝑚 ― 1) ―
} 𝑐 𝑐 + 𝐴 ].
(𝑛 ― 1)2 𝑃𝐷
𝑚
𝐷 𝑄 𝑚
(9)
Each of the buyers will spend money on purchasing the item from the manufacturer, placing the order to the manufacturer and also for holding the item in stock. Annual purchase cost to 𝒕𝒉𝒆 𝒊𝒕𝒉 buyer
Since the manufacturer may have a markup on its unit overall cost price (𝑣), the unit purchase cost to the buyers (𝑤) need not be equal to the unit overall cost of the manufacturer (𝑣). The annual purchase cost for 𝑖𝑡ℎ buyer will be 𝑃𝑖 = 𝐷𝑖𝑤.
(10)
Annual inventory holding cost
Buyer 𝑖 incurs a cost to hold the item in its stock, and it depends upon the lot size (𝑞𝑖) and the inventory holding cost rate (ri). Thus, the annual inventory holding cost to 𝑡ℎ𝑒 𝑖𝑡ℎ buyer (𝐻𝑖) will be 𝐻𝑖 =
𝑞𝑖
( 𝑟 𝑤) = 2 𝑖
𝑄
2𝑛𝐷𝐷𝑖𝑟𝑖𝑤.
(11)
Annual ordering cost
Annual ordering cost to the 𝑖𝑡ℎ buyer can be determined by multiplying the ordering cost per order (𝐴𝑖) by the number of orders per year Thus,
(𝐷𝑄). 𝐷
𝑀𝑖 = 𝑄 𝐴 𝑖 .
(12)
Annual transportation cost
Annual transportation cost to the 𝑖𝑡ℎ buyer will be 𝐷
𝑇𝑖 = 𝑄𝑛𝑇𝑖 .
(13)
So, the total annual cost of the buyer i, (𝐶𝑖) will be the sum of the annual purchase cost (Pi), the annual inventory holding cost (Hi), the annual ordering cost (𝑀𝑖) and the annual transportation cost (𝑇𝑖). Thus,
𝐶𝑖 = 𝑃𝑖 + 𝐻𝑖+ 𝑀𝑖 + Ti.
(14)
3. Profit-sharing considerations Four different proposed schemes for profit sharing among the chain partners proposed are detailed and discussed below. 3.1. Scheme I: minimization of the maximum opportunity cost
In this case, it is assumed that the various supply chain partners may not have perfect and true collaborative thinking for overall profit maximization. This kind of scenario is not uncommon when the various partners, managing their inventory problem independently so far, begin to initiate to coordinate and integrate themselves to form the supply chain. The traditional mindset of working for the best for themselves is not seen to change so drastically to what is really required in profit maximization as a whole. The best policy for an individual is not necessarily the best policy for the other partners. A deviation from their best position will naturally be viewed by themselves as an opportunity loss. Therefore, while in negotiation with each other, every chain partner will try to keep such an opportunity loss as low as may be possible. These buyers will earn the maximum profit when the manufacturer sells these products at its cost price to them, i.e. when 𝛼 = 0. The moment the manufacturer will go with a markup on its cost price, the unit purchase cost of the buyers will increase and push their total cost upwards. So they will not be in a position to make their best profit. Since the manufacturer is also in business for profit-making and not for only buyers to make the profit, it will try to push up its selling price. In this process, the opportunity
loss to the manufacturer will reduce, but it will increase the same for the buyers. Going beyond a particular limit, the opportunity loss to the manufacturer will become less as compared to some buyers. Such buyers are quite likely not to accept such an increase in the sales price of the manufacturer, which will make them suffer a loss more than that for the manufacturer. In such a situation, ultimate negotiation would converge on the philosophy of minimization of opportunity loss to every partner. A relevant, pragmatic and logical strategy in such an ecosystem will have to minimize the maximum opportunity loss experienced by any of the supply chain partners. This philosophy can be further understood from the following elaborations. If the manufacturer charges a profit margin (𝛼) on its incurred overall unit cost (𝑣), then unit selling price to the buyers (𝑤) would be 𝑤 = (1 + 𝛼)𝑣 .
(15)
Overall profit to the manufacturer (𝑃) will be 𝑃 = 𝛼𝑣𝐷.
(16)
Annual revenue of 𝑖𝑡ℎ buyer will be 𝑅𝑖 = 𝐷𝑖𝑝𝑖. The above equation, after substituting for 𝐷𝑖 from equation (1), will become 𝑅𝑖 = (𝑎𝑖 ― 𝑏𝑖𝑝𝑖)𝑝𝑖.
(17)
Annual profit of 𝑖𝑡ℎ buyer will be its annual revenue minus its annual expenses. Thus, 𝑃𝑖 = 𝑅𝑖 ― (𝑃𝑖 + 𝐻𝑖 + 𝑀𝑖 + 𝑇𝑖).
(18)
After substituting the expressions for variables from equations (11), (12), (13) and (14), the above equation can be rewritten as
(
𝑃𝑖 = (𝑎𝑖 ― 𝑏𝑖𝑝𝑖)𝑝𝑖 ― 𝐷𝑖(1 + 𝛼)𝑣 +
)
𝑄 𝐷 𝐷 𝐷𝑖𝑟𝑖(1 + 𝛼)𝑣 + 𝐴𝑖 + 𝑛𝑇𝑖 . 2𝑛𝐷 𝑄 𝑄
It is obvious that the maximum value of 𝑃𝑖, denoted as 𝐵𝑖, will occur only when 𝛼 = 0. It has been mentioned earlier that the increase in the value of 𝛼 may bring more profit to the manufacturer at a higher cost to the buyer and, ultimately, a lower level of demand. With the increase in the value of 𝛼, the profit to the manufacturer can be thus seen to increase initially to reach the peak and then to reduce thereafter. The opportunity costs to the chain partners can be mathematically expressed as 𝑂𝑚 = 𝐵𝑚 ― 𝑃, and
(19) ∀ 𝑖.
𝑂 𝑖 = 𝐵 𝑖 ― 𝑃𝑖
(20)
The profit maximization philosophy discussed herein can be summarized in the form of the following mathematical model.
{𝑂 }, 𝑂 ) (maximum 𝑖
minimize 𝑍 = maximum
𝑖
𝑚
subject to equations (1), (3), (9), (10), (11), (12), (13), (15), (16), (17), (18), (19), (20) and 𝑁
(21)
𝐷 = ∑𝑖 = 1𝐷𝑖 𝑎𝑖
𝑝𝑖 ≤ 𝑏𝑖 𝛼,𝑡,𝑄,𝐷,𝑣,𝑤,𝑃 ≥ 0
𝑛,𝑚 ≥ 0 𝑎𝑛𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
∀𝑖
(22) (23) (24)
𝐷𝑖, 𝐻𝑖, 𝑂𝑖, 𝑝𝑖, 𝑃𝑖, 𝑅𝑖, 𝑃𝑖, 𝑇𝑖 ≥ 0
(25)
∀𝑖
3.2. Scheme II: all partners have the same markup on their unit overall cost
Scheme I, of the profit division among the chain partners, will be more suitable when the partners have less amount of faith for each other and have not developed a reasonable level of confidence in the sustenance of coordination among themselves. At the next level of understanding, the partners may not be so self-centered and may perceive a win-win situation for themselves in terms of having earned the same profit margin. Thus, as compared to Scheme I, Scheme II will represent the comparatively less conservative position of the partners as they agree to have the same markup on their unit overall cost. In view of this, the selling price (𝑤) of the manufacturer will be the same as given by equation (15); but the selling price of the 𝑖𝑡ℎ buyer (𝑝𝑖) will change and will be
𝑝𝑖 =
or,
𝑝𝑖 =
(1 + 𝛼) (𝑃𝑖 + 𝐻𝑖 + 𝑂𝑖 + 𝑇𝑖) 𝐷𝑖
(𝑤 +
(1 + 𝛼) 𝐷𝑖
𝑄
2𝑛𝐷𝐷𝑖𝑟𝑖𝑤
𝐷
𝐷
)
+ 𝑄( 𝐴𝑖) + 𝑄𝑛𝑇𝑖 .
(26)
Now, the problem of profit distribution can be expressed by the following mathematical model. 𝑁
maximize T = 𝑃 + ∑𝑖 = 1 𝑃𝑖 subject to equations (1), (3), (9), (10), (11), (12), (13), (15), (16), (17), (18), (21), (22), (23), (24), (25) and (26). In this case, the chain partners enjoy the same profit margin. The manufacturer supplies the units to the retailers treating this profit margin as its markup on its overall unit cost price. Since the unit
purchase cost to the buyers is going to increase, their selling price is expected to increase while causing a fall in their annual demand. It is going to reduce the overall chain profit and, thus, it will not be in the interest of the chain naturally. Having understood the reason for a lower level of profit earning due to a low level of coordination, the buyers will become interested in going in with a forward contract with the manufacturer to share their profits with the manufacturer while receiving the item from the manufacturer at the cost price. The following two schemes are based on this concept. However, profit distribution mechanisms are different. In the two schemes, the manufacturer will supply the item to the buyers without any markup and will share the profits of the buyers under a forward contract. In such a case, the thought of Ingene and Parry (2005) can be employed for having a forward contract by a manufacturer with its buyers. The forward contract has to ensure that the buyers share their profit with the manufacturer in a logical and rational manner. 3.3. Scheme III: profit distribution in the proportion of incurred costs
This scheme aims to distribute the resulting maximum overall profit among the chain partners in proportion to their incurred costs. In order to find the maximum overall supply chain profit, the following mathematical programming model is being presented. maximize
𝑁
𝑇 = 𝑃 + ∑ 𝑖 = 1 𝑃𝑖
(27)
subject to equations (1), (3), (9), (10), (11), (12), (13), (15), (16), (17), (18), (21), (22), (23), (24) and (25) The distribution of the supply chain profit among the partners is to take place in proportion to their incurred costs. That is,
𝐶 𝑁
𝐶 + ∑𝑖 = 1𝐶𝑖
=
𝐶1 𝑁
𝐶 + ∑𝑖 = 1𝐶𝑖
=
𝐶2 𝑁
𝐶 + ∑𝑖 = 1𝐶𝑖
=… =
𝐶𝑁
. 𝑁 𝐶 + ∑𝑖 = 1𝐶𝑖
(28)
The above mathematical programming model will not allow any amount of profit to the manufacturer requiring it to supply all the units to the buyers at its cost price. Because of the reduction in the cost price, the holding cost of the buyers will also come down as it is charged in proportion to the unit purchase cost of the buyers. As a result, the overall cost to a buyer will reduce. The retailers can now think of reducing the sales price to target more of demand, even keeping the profit margin from each sold item unchanged. Thus the overall profit is expected to increase. 3.4. Scheme IV: profit distribution in proportion to contribution indices
Here, for the distribution of the profit, a methodology due to Shapley (1953) is employed. In his proposed approach, the contribution of each chain partner has to be evaluated. Based on the input as the contribution made by a chain partner, its contribution index is worked out. The overall supply chain profit is shared by the chain partners in proportion to these contribution indices. The use of the Shapley function for distribution of profit would require solving of mathematical programming formulation of Scheme III for several times, with the manufacturer in common and the buyers in various combinations. Since there are ‘N’ buyers, the buyers are to be associated in 𝑁
𝐶1 + 𝑁𝐶2+ 𝑁𝐶3+ … +𝑁𝐶𝑁 combinations. In case N is large, the mathematical programming
formulation of Scheme III has to be used for a very large number of combinations before profit distribution can be worked out. Total combinations will be 7 for the number of buyers as 3, will be 15 for 4, 31 for 5 and will increase exponentially with the number of the buyers. In other cases, where either the manufacturer alone is taken or only the buyers in any combination, the overall contribution is taken to be zero. Even though this process of profit distribution may be time-
consuming, but it is worth trying as it will have more of practicality in terms of its acceptance by the chain partners. The utility of the schemes, proposed above, is also elaborated later by taking a numerical example. 4. Proposed heuristic approaches 4.1. Genetic Algorithm (GA) as a solution methodology
For the integrated multi-echelon supply chains, GAs have been found to yield solutions close to optimal ones by Nachiappan and Jawahar (2007), and also by Jawahar and Balaji (2009). With the motivation from their findings, Genetic Algorithm (GA) is being proposed to solve the mathematical model stated in this paper mainly due to non-linearity in the model with integer value requirement on some of the variables. GAs have been reported to lead to the best solution if the fitness functions are properly defined (Yang et al., 2018; Prakash et al., 2011). Holland (1975) introduced the Genetic Algorithms (GAs). These are population-based adaptive heuristic search algorithms. GAs are based on the Darwinism, the theory of biological evolution developed by Charles Darwin in his book "On the Origin of Species" in 1859. GAs have been theoretically and empirically proven to be a robust search technique (Goldberg, 1989; Behzadi, et al. 2008). The available research, over the last few decades, has demonstrated the enormous success of GAs in terms of their ability to handle complex, nonlinear optimization and real-world problems (Ali et al., 2016). Large number of researchers have used GAs to solve the problems from different fields such as, inventory management (Dania, 2010; Priya and Iyakutti, 2011; Maiti and Maiti, 2008), supply chain management (Han and Damrongwongsiri, 2005), group technology (Moon et al., 2006), machine layout (Kovačič et al.,
2013), advertising display optimization (Miralles-Pechuán, et al., 2018), airspace configuration (Sergeeva et al., 2017), optimal land uses (Palogos et al., 2017), etc. GA begins with the initial population, which is a set of arbitrarily initialized solutions that evolves toward better and better regions of the search space by means of randomized processes of selection, cross-over (recombination) and mutation (Bäck and Schwefel, 1993). The selection operator chooses the “best” parents in the population for mating, the crossover operator picks two parents and generates offsprings by interchanging the segments of the parent genes, and the mutation operator alters some of the genes of a chromosome to create a new chromosome. The crossover operator has to maintain a proper balance between exploitation and exploration that leads towards the success or failure of the search process (Gallard and Esquivel, 2001). The exploitation strategy focuses on local search in a narrow optimistic region consisting of solutions that have already been achieved. Whereas, the exploration strategy is used in carrying out a global search over a wider region in the hope of a better solution. The proposed GA has been used for finding the maximum total supply chain profit and corresponding profits of individual partners for the mathematical formulation given in Section 3.3. Various steps of the algorithm are as described in Fig. 3. [Insert Fig. 3] Real-coded chromosomes are taken to define the population of the solutions with the genes showing the real values of the decision variables. The variables included in the problem are: demands of the buyers (𝐷𝑖), replenishment time(𝑡), the profit margin of manufacturer (𝛼) and number of the shipments (𝑛). Since demands and the sales prices of the buyers are interrelated,
the demands are taken as the variables with the prices are calculated using the price-demand relationship. The solution procedure consists of the following steps. (a) Representation of individual chromosome (solution): Each individual chromosome consists of genes (variables) that are real values. For a single manufacturer and three buyers problem, the total number of genes is 6 (3 for each buyer’s demand; and 1 each for replenishment time (𝑡), the profit margin of the manufacturer (α) and number of the shipments (𝑛)). A solution chromosome is generated by randomly choosing values of the variables, where the choice on the values depends on some realistic criteria. For example, the demand values (𝐷𝑖) are generated randomly anywhere between minimum demand (0) to maximum demand (𝑎𝑖), and replenishment time value(𝑡) between
1 365
to 1 year. Similarly, for the manufacturer's profit
margin (𝛼) and the number of shipments (𝑛), respective values between 0.00 to 0.80 and from 1 to 10 are generated randomly. A sample chromosome is shown in Fig. 4 for a problem with a single manufacturer and three buyers. [Insert Fig. 4] Further steps of the proposed GA are clarified using this single manufacturer and three buyers chain. (b) Generation of the initial random population: An initial population of solutions (chromosomes) is generated by randomly choosing the variable values. In the implementation of GA for numerical experimentation, the population size is taken as 20. (c) Evaluation of fitness function: A specific function, using which the strength of each generated solution is determined, is known as the fitness function. This fitness function can be identical to the objective function (Fahimnia et al., 2018).
The objective function for the profit, defined in Scheme III and represented by expression (27) that determines the value of total supply chain profit (𝑇), serves as the fitness function. (d) Genetic operations: Once the fitness function value for each of the population solutions is determined, then control is handed to genetic operators: selection, crossover and mutation. Selection: Selection is the first operation to be performed. It selects the chromosomes from the present generation for further operations of the GA. A number of selection operations are available in the GA literature, viz., Tournament Selection, Roulette Wheel Selection, Proportionate Selection, Rank Selection, etc. The roulette wheel selection process is applied to the selection process in the present work. Roulette wheel selection is a stochastic selection process where fittest individuals have a higher chance of survival than weaker ones (Pasala et al., 2013). Crossover: In the traditional crossover, a cross-site will be used to decide a string of continuous genes to be interchanged over the two parent chromosomes. In the other case, only the genes at the cross-site are interchanged. A crossover rate of 20% is applied in the present work. The strategy chosen for the crossover is 1-point crossover, in which case a single cross-site is randomly chosen. Another random number would be used to decide whether conventional crossover will be followed or not. This random number for the present work is kept as 0.50. It means that only one gene will be interchanged in 50% of the descendants, and it can be even more than this in the remaining 50% cases. This philosophy strikes a balance between the solutions with modified values of one to many parameters to visualize their effect in a more rational manner. The working of the proposed crossover is explained in Fig. 5.
Fig. 5(b) depicts the case of crossover of one gene and Fig. 5(c) for the crossover of multiple genes. [Insert Fig. 5] Mutation: After the crossover, the mutation operation is carried out. The mutation prevents the algorithm from being trapped in a local minimum. It plays the role of recovering the lost genetic materials as well as for randomly disturbing genetic information. It is an insurance policy against the irreversible loss of genetic material (Abdoun et al., 2012). A creep mutation with 5% mutation rate is carried out in the present work. Creep mutation selects a gene randomly and changes the value with a random value lying between lower and upper bounds (Sivanandam and Deepa, 2007). The process of mutation has been explained in Fig. 6, where the second gene value after mutation becomes 150 from its original value of 250.
[Insert Fig. 6]
(e) Stopping criteria: After the mutation operation, one generation of the GA is complete. The best value, obtained so far, is recorded. The process iterates until some fixed number of generations evolve or some fitter chromosomes are not obtained. In the present work, the number of generations as 25 times the number of supply-chain partners is selected as a stopping criterion. 4.2. Teaching-Learning Based Optimization (TLBO) as a solution methodology
In solving the proposed mathematical models, Teaching-Learning Based Optimization (TLBO) meta-heuristic developed by Rao et al. (2011) is also being proposed. Among several metaheuristic techniques, TLBO is a relatively new technique with an edge over the other techniques. Since it does not require any algorithm parameters to be tuned, the implementation of TLBO is simpler (Rao et al., 2012). In this algorithm, a random population of the solutions is generated. A population of 20 learners is generated randomly in the implementation of the heuristic in the present paper. The population size represents the number of the students in a class and each design variable to represent a subject taught. In encoding the problem, a learner, with D1, D2, D3, t, α and n as its subjects with the marks in the respective subjects as 400, 250, 705, 0.25, 0.20 and 4, is shown in Fig. 7, with the subject values being generated randomly within the range as described below. [Insert Fig. 7]
The choice for the values will depend upon some realistic criteria. For example, the demand values
(𝐷𝑖) are generated randomly anywhere between the minimum demand value (0) to the maximum demand value (𝑎𝑖), replenishment time value (𝑡) between
1 365
year to 1 year, the manufacturer’s
profit margin value (𝛼) between 0.00 to 0.80, and the number of shipments value (𝑛) from 1 to 10. The algorithm is divided into two major steps: (i) Teacher’s phase and (ii) Students’ phase.
Teacher’s phase: In this phase, each of the randomly generated solutions will lead to an objective function value that is analogous to the result or the overall grade of an individual student. The solution (student) with the highest objective function value (the highest grade) is declared as a
teacher. The teacher is considered to be a highly learned person who tries to improve the knowledge of the students. The mean result of the students in each of the subjects is calculated. A factor namely “difference_ meani” is calculated for subject i using the following expression difference_ meani = ri(Xi_teacher – TFXi_learner) where ri is a uniformly distributed random number between 0 and 1, Xi__teacher is marks in the 𝑖𝑡ℎ subject of the teacher, TF is taken to denote “Teaching Factor” with its value being either equal to 0 or 1, and Xi_learner is the mean value of the marks of the students in 𝑖𝑡ℎ subject. At the subsequent stages of learning, the students are supposed to improve their learning by having the subject knowledge from the teachers. The change brings the students to a different level. The same is expressed as Xi_learner = Xi_learner + difference_ meani A student’s new learning level is accepted only when it results in a better value of the objective function compared to the old one. In this way, the whole population is updated. The updated population serves as input to the students’ phase.
Students’ phase: This phase mimics the learning of students through discussion among them. In this phase, two students are selected randomly. The one, with the higher knowledge, is assumed to improve the knowledge of the other. Let X1 and X2 (being vectors) represent marks in different subjects of some two students and f(Xi) to represent objective function value (grade) for the ith student. Since the amount of learning is going to be random, the same is controlled by a uniformly distributed random number rn taking values in the range zero to one in the following manner:
and
X1 =X1 + rn (X2 - X1)
if f(X2 ) > f(X1),
X2 =X2+rn (X1 - X2)
if f(X1) > f(X2 ).
If the new solution is better compared to the corresponding old solution, then it is updated; otherwise, the old one is retained. In the present work, the number of unique pairs of students to be identified for learning under this phase is taken to be equal to the number of the students, i.e. the size of the class or the population. After the two phases are over, the current iteration gets completed. This procedure is being iterated for 25 times the number of chain partners in the present paper. The proposed GA and TLBO have been implemented on the platform of MATLAB (R2015a) and are run on a computer system with Intel (R) Core (TM) i7-6700 CPU with 3.40 GHz processor. 5. Numerical experimentation The proposed profit distribution schemes are further described using an illustrative example. The supply chain considered consists of a single manufacturer and three retailers (𝑁 = 3). This problem takes most of the data from the numerical example of Jokar and Sajadieh (2009) and Jha and Shanker (2013). The manufacturer related data are: P = 12,000, c = 60, 𝐴𝑚 = 1,000 and 𝑟𝑚= 0.10. The buyer’s related data are given in Table 1. [Insert Table 1] The solutions to the example problem from the proposed schemes have been obtained using a commercially available software LINGO. The solutions, along with some insights, are been presented below. 5.1. On the application of Scheme I
The example problem is solved for different values of 𝛼 and the results are presented in Tables 2 and 3. Fig. 8 shows the variation in the profit to chain partners and maximum opportunity loss with
change in the value of α. A careful observation of these tables and the figure will provide the following relevant information. i)
As the profit margin (𝛼) increases, the total supply chain profit (T) decreases. It happens because of the increase in the unit purchasing cost to the buyers with the increase in the value of 𝛼. As a result, the sales price would increase in order to maximize the profit and thus to cause demand value to go down (Table 3).
ii)
Profit to the buyers is the maximum for 𝛼 = 0 or when the manufacturer sells the item with no markup on its unit overall cost. On the other hand, when the profit to the manufacturer is the maximum (𝛼 = 66%), the buyers make no profit. These two situations are extreme ones. The optimal solution has to naturally lie in between these two extremes (Table 2).
iii)
The least possible profit for any of the supply chain partners is zero because the scheme denies the possibility of a partner being in the chain even when it makes a loss.
iv)
Increasing 𝛼 value beyond 66%,
the buyers are not making any profit and the
manufacturer’s profit moves downwards instead of increasing. This clearly indicates that the manufacturer cannot afford to charge an exorbitantly high profit-margin in order to earn a very high profit (Table 2). v)
From Table 3, it can be noticed that, with the increase in the value of α upto 66%, the change in the sales price and the demand is either not there or is much less compared to when α is increased beyond 66%.
From Table 2 and Figure 8, it can also be noticed that when the manufacturer charges a profit margin of 0% (or no profit for itself), the buyers earn the respective maximum profit as 𝐵1 = 18,491, 𝐵2 = 22,284 and 𝐵3 = 11,008. Any deviation from this position is seen as a loss of
opportunity (𝑂𝑖 for buyer i). For example, 𝑂1 = 2,795 for 𝛼 = 10%. It is obtained by subtracting the resulted profit to buyer 1 as 15,696 from its maximum profit value of 18,491. The manufacturer, to make itself better-off, will go for a higher value of 𝛼. As the value of 𝛼 increases, the profit to manufacturer also increases, but to a certain limit (α = 66%). Increasing 𝛼 beyond 66% causes the profit to the manufacturer to decrease. This value of 𝛼 as 66% corresponds to the best profit to the manufacturer, thus 𝐵𝑚 = 49,698. Any deviation from this profit level will be seen as a loss of opportunity by the manufacturer for earning the maximum possible profit. The value of 𝑂𝑚 for the manufacturer is calculated by subtracting the current profit from this maximum possible profit. For example, for 𝛼 = 10%, 𝑂𝑚 = 49,698 ― 8,319 = 41,380. Opportunity loss from the best position for each supply chain partner is calculated, and the maximum of the opportunity loss to a supply chain partner is recorded in the last column of Table 2. The minimum of this maximum opportunity loss is found to be 14,355 for the corresponding value of 𝛼 as 43% with the values of some of the decision variables given in Table 4. Thus, the profit distribution philosophy of this scheme yields an overall supply chain profit as 51,403 for 𝛼 = 0.43. Out of this supply chain profit, the manufacturer makes a profit of 35,753 while the buyers 1, 2 and 3 make respective profits of 6,498, 7,929 and 1,222. While the manufacturer receives a markup of 43% on its incurred cost, the buyers 1, 2 and 3 earn respective profit margins as 16.18%, 16.49% and 3.73% on their incurred costs. Since the profit percentage on the incurred cost of buyer 3 is very small as compared to the other two buyers and significantly smaller compared to that for the manufacturer. Hence this scheme of profit distribution is quite likely not to be appreciated by buyer 3 because of the high level of disparity involved. In fact, every chain partner would like to have the same profit margin. With this very reason, profit distribution using the philosophy built into Scheme II will make a lot of sense.
[Insert Table 2] [Insert Table 3] [Insert Table 4] [Insert Fig. 8]
5.2. On the application of Scheme II
Scheme II of profit-sharing is based on the idea that every chain partner shall have the same profit margin, i.e., ratios of profits earned to their investment or incurred costs are going to be equal. The optimal solution to the problem is shown in Table 5, with the optimal value of 𝛼 as 26.44%. In this scheme, the manufacturer makes a profit of 22,069 while the buyers 1, 2 and 3 make profits of 10,242, 12,324 and 5,871, respectively (Table 5). [Insert Table 5] From Tables 4 and 5, the following can be observed: i) For a high value of α as 0.43, the unit selling price to the buyers (w) is 89. The same is 79 for α as 0.26. Because of the fall in the value of α, the profit to the manufacturer can be observed to decrease and that for the buyers to increase. ii) There is not much change in the cumulative demand faced by the buyers. It changes from 1,327 (Table 4) to 1,333 (Table 5). However, the demand for buyers changes. Demands to the buyers 1 and 2 increase while that to the buyer 3 decreases.
The use of rationality in profit distribution helps buyer 3 to lift profit for itself, but overall supply chain profit drops from 51,403 to 50,506. Rational distribution of profit without compromising on the total supply chain profit is obtained from the application of the philosophy of Schemes III and IV. 5.3. On the application of Scheme III
Since the manufacturer is not having any markup on its cost price to earn profit for itself, it has to sign a forward contract with its buyers for proper profit distribution. Since Scheme III believes in earning maximum overall supply chain profit, the application of the respective mathematical model, presented in Section 3.3, yields maximum profit as 51,783 for 𝛼 = 0 (Table 2). The solution for the example problem under this philosophy of profit distribution is given in Table 6. From this table and Tables 4 and 5, it can be noticed that the overall demand has changed by a very small amount. The unit purchase cost price to the buyers experiences a lot of change. From a high value of 89, it has come down to 62. In Table 6, profit to the manufacturer and to the buyers are in the proportion of their incurred costs and are obtained by distributing the overall profit as 51,783 using the relationship given by equation (28). The profit to manufacturer as 25,609 is received under a forward contract with the buyers due to which buyer 1will share an amount of 9,804 from its profit of 18,491 (Table 2) to finally have profit amount as 8,687, buyer 2 will share 11,889 and buyer 3 as 3,916 to finally retain the respective profits as 10,395 and 7,092. The distribution in the proportion of the incurred costs has inherent fallacy. More costs one incurs, more of the profit is earned by it. Thus, this philosophy has a weakness of bringing inefficiencies if some partner chooses to incur more costs in order to just earn more profit for itself. This point is further explained in Section 6.1. [Insert Table 6]
5.4. On the application of Scheme IV
As mentioned earlier, the overall chain profit obtained in Scheme III will be distributed among the chain partners using the Shapley function. The application of the Shapley function would require the determination of maximum profit when chain partners are used in different combinations, as shown in Table 7. In this table, the overall profit is zero whenever either the manufacturer is taken alone or only the buyers in any combination without it. In rest other combinations, the manufacturer is there with some of the buyers. For each of these combinations, the mathematical model of Scheme III (presented in Section 3.3) is used to determine the overall profit that is recorded in the last column of Table 7. From Table 7, the average contribution for each chain partner has been worked out. The same is detailed in Appendix-A. It is the contribution that is likely to be highly appreciated by every chain partner as their contribution to overall profit. Naturally, the profit shared by a chain partner would be equal to its average contribution. Thus the manufacturer will get a profit of 25,503; whereas buyers 1, 2 and 3 will respectively receive the profit of 9,373, 11,311 and 5,596. [Insert Table 7] [Insert Table 8] 6. Discussions 6.1 On the fairness of profit distribution
In Scheme III for the profit distribution, it was mentioned earlier that it has a weakness in giving extra profit to the inefficient partner. From this perspective, fairness in the distribution of the overall profit using Shapley function vis-à-vis philosophy of Scheme III is being analyzed by modifying the cost data of buyer 3 by taking the same as a multiple by 2 and 3 of its original cost
values given in Table 1. The overall profit and its distribution among the partners under the two schemes (Scheme III and IV) are presented in Table 9. The following observations can be made with respect to the increase in the values of the cost parameters of buyer 3. i)
The overall profit keeps decreasing. The profit margin to the chain partners also decreases under both the schemes.
ii)
Except for buyer 3, profits to the chain partners also decrease. When the cost parameters as holding cost rate, ordering cost and transportation cost are doubled of the earlier respective values (Table 1), the overall profit decreases to 51,277 from 51,783 causing the manufacturer’s share to drop down from 25,609 to 25,293, the profit of 8,687 to 8,599 for buyer 1 and 10,395 to 10,285 for buyer 2. Buyer 3, who has become costinefficient, now receives an increased profit of 7,099 instead of an earlier amount of 7,092. This substantiates that Scheme III has a weakness of favoring cost-inefficient supply chain partners and thus the philosophy of the distribution of the overall profit in proportion to the incurred cost of the chain partners (Chauhan and Proth, 2005; Jaber and Osman, 2006) is not a fair policy.
iii)
In the solutions from Scheme IV, the profit decreases for every partner. Buyer 3 is not found to be favored as opposed to that under Scheme III. Buyers 2 and 3 lose the profit by a little amount, but the manufacturer and the buyer 3 by a significant amount. This is in line with the common belief that the selection of a wrong buyer (retailer) by a manufacturer (an organization owning the item) should not affect the other buyers. Naturally, such kind of inefficiencies with the buyer has to be borne by the
manufacturer and the concerned buyer only. This shows the strength of Scheme IV in terms of distributing the profit in the fairest manner. [Insert Table 9] From Table 4, it can be noticed that Scheme I has favored the manufacturer over the buyers, with buyer 3 suffering the most. Thus buyer 3 can be said to be the weakest partner in the supply chain from the profit earning perspective. Schemes II and III (Tables 5 and 6) take a somewhat better and balanced approach. But the most balanced, democratic and fair process seems to be of Scheme IV (Table 8). It was observed that the manufacturer had to suffer because of the inefficiencies of buyer 3. Even then, buyer 3 was favored and was taken as one of the supply-chain partners. This may lead to the thought of removing such a partner from the chain so that the remaining partners are well awarded. The worthiness of such consideration is being analyzed below. When buyer 3 is taken out of the supply chain, the profit available will be 40,284 (figure available against serial number 11 in Table 7) and thus less by 11,499 from a total of 51,783 when buyer 3 was also a supply chain partner (Table 6). In Scheme IV, with all the three buyers, the share of buyer 3 was 5,596 (Table 8). Therefore, even sharing profit amount of 5,596 with buyer 3, the other supply chain partners (the manufacturer and buyers 2 and 3) receive put together an extra amount of 5,903 out of the extra profit of 11,499. This means that buyer 3 benefits the supply chain at least by an amount of 5,903 when profit is shared using Shapley function. A similar observation can be had when the profit is shared on the incurred cost basis (Scheme III). The use of Scheme III provides buyer 3 a profit of 7,092. This profit amount is much less compared to the loss in overall profit of 11,499. Thus, keeping buyer 3 in the supply chain helps the other partners to earn combined together an extra amount of 11,499 – 7,092 = 4,407. This also shows that leaving a weak partner may not be an advantageous strategy for the other chain partners. In case the cost
parameters of a buyer are very high, this partner has to keep its selling price very high and thus may not be in a position to have any demand from its customers. Naturally, in such a scenario, the related buyer will be out of the scene and will not practically remain to be the supply chain partner. The above numerical experiment shows that the inefficiency of one partner does affect the others. However, leaving a partner does not make sense as the overall profit drops. Instead, an effort is required to be made by the other chain partners to put in their money and experience on this inefficient partner to make it more efficient, enabling everyone then to accrue more of profit. The reverse of dropping a partner can be taken as the addition of new buyers, if possible, to increase the profit share of everyone. 6.2. Analysis of the efficiency of proposed GA and TLBO heuristics
An effort has been made to analyze the efficiency of the proposed heuristic approaches based on the framework of GA and TLBO. For this purpose, five problems were randomly chosen, with the one already used in illustrating the proposed schemes in Section 5. All these problems have one manufacturer, but with an increasing number of buyers. These problems were solved using the proposed GA and TLBO approaches incorporating the profit distribution philosophy of Scheme III. Resulting objective function values for these problems and execution time taken by Intel (R) Core (TM) i7-6700 CPU with 3.40GHz processor have been recorded, and the same are presented in Table 10. In Table 10, Gtot and Itot respectively denote the number of the generations in the case of GA and the number of the iterations in the case of TLBO. They are taken to be 25 times the number of the chain partners. For a single manufacturer and 3 buyers supply chain, these will be equal to 25 × 4 = 100. Gbest is the generation in which the best objective function value was obtained in the case
of GA. Ibest shows the iteration number in the case of TLBO for the same purpose. From Table 10, it can be noticed that CPU time required for the proposed GA is less compared to the one required by LINGO for small size problems; but it is more when the problem size increases. TLBO has taken more time in all the cases compared to that taken by GA or LINGO. Wilcoxon Signed Rank test finds this difference to be statistically significant. Thus, the proposed GA and TLBO approaches are expected to take more time as compared to that required in the use of LINGO software. GA approach was also found to take more time compared to that by optimization software by Ghasimi et al. (2014). From Table 10, it can also be noticed that the best objective function values were obtained by GA and TLBO much before all the iterations could be completed. For example 1 (Table 10), GA and TLBO respectively took 0.010 secs 0.004 secs to achieve the best objective values, while LINGO took 0.110 secs. From this perspective, the Wilcoxon test identifies TLBO to be the best and LINGO to be the poor, with GA falling in between the two. The values of the objective function obtained in various generations/iterations of the proposed GA and TLBO approaches are shown in Fig. 9 for experiment number 1 with 3 buyers (Table 10). This figure also shows early achievement of the best objective function values in the case of TLBO but of an inferior value compared to that obtained from the GA approach. Further, in all of the experiments, the proposed GA and TLBO heuristics resulted in solutions with inferior objective function values as compared to that obtained from the use of LINGO software (Table 10). Wilcoxon Signed Rank test identifies LINGO to the best, TLBO to be the poor and GA in between the two. This difference in objective function values is found to increase with the increase in the number of buyers.
The above observations are strong evidence to suggest the use of optimizing algorithms instead of the proposed heuristics, both from the perspective of solution quality and CPU time requirement. [Insert Table 10] [Insert Fig. 9]
8. Conclusion This paper presents some mathematical models and schemes for maximizing the overall supply chain profit and allocating the same among the chain partners while determining optimal production-inventory and pricing decisions for the integrated supply chain composed of a single manufacturer-multiple buyer where each buyer faces price-dependent demand. This paper assumes that the manufacturer produces the item at a finite rate. Shipment to a buyer is made in multiple shipments of equal-sized batches to save upon holding cost at the additional expense of transportation cost. Previous researches treat the holding cost for the buyers as a fixed quantity and the manufacturer’s selling price as pre-specified. In the present work, the manufacturer’s selling price is a decision variable. Further, the holding cost of the buyers is taken to depend upon the manufacturer’s selling price for being more realistic and practical. In the proposed integrated model, the supplies from the manufacturer to the buyers are coordinated to ensure no occurrence of shortage at the end of the buyers. Instead of classical minimization of overall inventory management cost, this study strategizes for the maximization of the total supply chain profit to arrive at the optimal inventory decisions as the sale price may change the demand and, thus, the cost structure. In the present research work, four different schemes have been proposed for the maximization of the overall supply chain profit and its distribution among the chain partners considering different
levels of coordination and faith among them. The first scheme is applicable when the coordination has just begun among the partners, and they minimize their maximum opportunity loss instead of insisting on the best position while managing their inventory independently. Because of a low level of understanding and a low level of alliance, the maximum overall profit achieved is found not to be as high as it could be. The second proposed scheme, based on somewhat better understanding among the partners and for being more fair and rational in profit distribution, provides for the same profit margin to all the chain partners and thus does away with the dissatisfaction that was very likely to result in from the unequal profit margin in the first scheme. However, the numerical experiment found the overall profit to be less than that in the case of the first scheme. An analysis, in the form of numerical experimentation, was carried out to visualize the effect of the increase in the markup on the cost price of the manufacturer on the overall supply chain profit. It was learnt that the earning of the profit, more than what was achieved in the first two schemes, is possible only when the manufacturer sells the item to its buyers at no markup. In this case, all the profit was with the buyers only and they were required to share their additional profit with the manufacturer in a rational way. In this paper, it has been proposed to carry out sharing of the profit in a fair manner by having the manufacturer a forward contract signed with its buyers (Cachon and Lariviere, 2005) reflecting mutual trust for each other. For this purpose, additional two schemes were proposed. The third scheme is based on the philosophy of the distribution of the profit among the partners in proportion to their incurred costs, as was carried out by Chauhan and Proth (2005) or by Jaber and Osman (2006). This scheme was found to have an inherent fallacy of promoting cost inefficiency among the chain partners as they may not work to reduce their incurred cost due to the allocation of a lower amount of profit on account of less incurred cost. The fourth proposed
scheme does not have this issue. It, in the fairest manner, distributes the overall profit among the supply chain partners in proportion to their contribution indices that are computed using Shapley function (Shapley, 1953). Further numerical experimentation with this scheme shows that the inefficiency of any buyer does not affect the profit of the other buyers, but that of the manufacturer and of that buyer itself. This observation seems to be quite practical as a buyer would not like to suffer because of the inefficiencies of other buyers. On experimenting with the third and the fourth schemes, it was found that the dropping of any buyer from the chain will cause a loss in profit to every other chain member. Therefore, it is being suggested that no buyer should be dropped from the chain simply on the basis of cost inefficiency. If a buyer is cost-inefficient, it will have less of the demand from the customers and, thus, less of the profit will be allocated to it in use of the Shapley function. If the cost becomes prohibitive, the buyer will have no demand and will be practically out of the supply chain. Since the proposed profit distribution models are mixed-integer nonlinear mathematical programming formulations, Genetic Algorithm (GA) and Teaching-learning Based Optimization (TLBO) heuristics have been proposed with a view to solving the considered problem in a computationally efficient manner. However, on numerical experimentation, GA and TLBO heuristics were found to take more of CPU time to solve the problem in comparison to that taken by optimizing software LINGO. The finding on the CPU time requirements is the same as was observed by Ghasimi et al. (2014). Proposed heuristics were also found to result in more deterioration in the objective function value as the number of buyers increases. Since GA and TLBO approaches were not found to be serving the purpose of being computationally-efficient, no effort has been made to determine the robust values of their parameters.
The scope of the present problem can be extended to analyze the effect of various possibilities of transporting the item from the manufacturer to the buyers using either owned transportation facilities or the ones roped in from by a third-party logistics service provider. In the present paper, a strict and deterministic relationship of the price with the demand was assumed. In the real world, a change in the price may not necessarily bring a typical change in the demand all the time. Therefore, the present work may be extended to consider the price-dependent stochastic demand.
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https://doi.org/10.1080/00207543.2016.1249431 Zanoni, S., Mazzoldi, L., Zavanella, L.E., Jaber, M.Y., 2014. A joint economic lot size model with price and environmentally sensitive demand. Prod. Manuf. Res. 2:1, 341–354. Appendix A
The marginal contribution of the chain partners in a particular coalition, in the use of Shapley function, depends upon the order of the partners in which they are arranged in that coalition. In a coalition as (M,B1,B2,B3), all the partners are involved and, for this combination, overall supply chain profit was found to be as 51,783 mentioned at serial number 15 of Table 7. Now going by the order and including one more partner at a time, the contribution of the partners will be determined. Recognizing the order in this coalition, first partner is observed to be the manufacturer. If it is considered alone, its contribution will be zero as was recorded at serial number 1 of Table 7. Considering the order, next partner to be added to the manufacturer (M) will be buyer 1 (B1). For the manufacturer and buyer 1 combined, the overall profit is 17,517, mentioned at serial number 5 of Table 7. Since the contribution of the manufacturer is zero, the amount of 17,517 is taken as the one generated by buyer 1. Hence the contribution of buyer 1 becomes 17,517. Next, buyer 2 (B2) is added. With the manufacturer associated with buyer 1 and 2, the overall profit was found to be 40,284 recorded at serial number 11 of Table 7. So the additional contribution on joining of buyer 2 is 40,284 ― 17,517 = 22,767. This extra contribution is naturally the contribution from buyer 2. In the last, buyer 3 (B3) is also included. Now the whole supply chain has been considered for which the overall profit was found to be 51,583. Therefore, the additional contribution on taking buyer 3 into the chain will be 51,583 ― 40,284 = 11.499. This extra profit of 11,499 is only due to addition of buyer 3 to the chain. For this reason, the contribution of buyer 3 in the coalition is considered as 11,499.
These contributions of the partners will depend upon the order in the coalition considered. For the single manufacturer and three buyers’ case, the total number of distinct coalition will be 4𝑃4 = 24. Following the methodology described above, the contributions of the partners are determined for all the 24 permutations or coalitions and the same are detailed in Table A.1.When all the partners are making the chain, consideration of the order is meaningless. To nullify the effect of the order on the contribution, Shapley, (1953) suggested to make use of average contribution. It is this profit that would be allocated to the respective partners. Relating this average contribution to the overall profit, contribution index of a partner has been determined. These contribution indices give an idea about the strength of a partner in generating the overall profit. By not charging any markup on cost price, the manufacturer has made highest contribution of 49% to the overall profit. The weakest partner is buyer 3, contributing the least (11%) to the overall profit.
Table A.1 Computation of Shapley indices
Possible order of partners M,B1,B2,B3 M,B1,B3,B2 M,B2,B1,B3 M,B2,B3,B1 M,B3,B1,B2 M,B3,B2,B1 B1,M,B2,B3 B1,M,B3,B2 B1,B2,M,B3 B1,B2,B3,M B1,B3,M,B2 B1,B3,B2,M B2,B1,M,B3 B2,B1,B3,M B2,M,B1,B3 B2,M,B3,B1 B2,B3,B1,M B2,B3,M,B1 B3,B1,B2,M B3,B1,M,B2 B3,B2,B1,M B3,B2,M,B1 B3,M,B1,B2 B3,M,B2,B1 Total Average contribution Contribution index
Marginal contribution of manufacturer 0 0 0 0 0 0 17,517 17,517 40,284 51,783 28,811 51,783 40,284 51,783 21,358 21,358 51,783 32,702 51,783 28,811 51,783 32,702 10,014 10,014 612,068
Marginal contribution of buyer 1 17,517 17,517 18,926 19,080 18,796 19,080 0 0 0 0 0 0 0 0 18,926 19,080 0 19,080 0 0 0 19,080 18,796 19,080 224,960
Marginal contribution of buyer 2 22,767 22,972 21,358 21,358 22,972 22,688 22,767 22,972 0 0 22,972 0 0 0 0 0 0 0 0 22,972 0 0 22,972 22,688 271,459
Marginal contribution of buyer 3 11,499 11,294 11,499 11,344 10,014 10,014 11,499 11,294 11,499 0 0 0 11,499 0 11,499 11,344 0 0 0 0 0 0 0 0 134,297
25,503
9,373
11,311
5,596
0.49
0.18
0.22
0.11
List of Figures Fig. 1. The supply chain considered Fig. 2. Inventory cycle of manufacturer and buyers Fig. 3. Proposed genetic algorithm Fig. 4. A chromosome Fig. 5. Functioning of 1-point crossover Fig. 6. Functioning of mutation Fig. 7. A student with subjects as variables Fig. 8. Profit of supply chain partners and maximum opportunity loss for different values of α Fig. 9. Overall profit at different number of generations of GA and iterations of TLBO
Fig. 1. The supply chain considered
Fig. 1. Inventory cycle of manufacturer and buyers
Step 1: Start with a randomly generated population. Step 2: Evaluate the fitness function value of each chromosome in the population. Step 3: Create a new population by repeating following steps of genetic operations until the current generation is completed Step 3.1: Selection - Select the chromosomes for the formation of mating pool by Roulette wheel selection method. Step 3.2: Crossover - Use multivariate to crossover the parents to form a new offspring. Step 3.3: Mutation - Use creep mutation to mutate new genes of the population. Step 4: Place new offsprings in the new population. Step 5: Test if the end condition is satisfied. If yes go to Step 7, otherwise to Step 6. Step 6: Go to Step 2. Step 7: Stop.
Fig. 2. Proposed genetic algorithm
𝐷1
𝐷2
𝐷3
𝑡
𝛼
𝑛
400
250
705
0.25
0.20
3
Fig. 3. A chromosome
400
250
705
0.25
0.20
3
750
150
486
0.06
0.50
5
(a) Two parent chromosomes
400
250
486
0.25
0.20
3
750
150
705
0.06
0.50
5
(b) Two children chromosomes after the crossover with interchange in only one gene
400
250
486
0.06
0.50
5
750
150
705
0.25
0.20
3
(c) Two children chromosomes after the crossover with changes in second substrings Fig. 4. Functioning of 1-point crossover
400
250
705
0.25
0.20
3
0.50
5
(a) A chromosome
400
150
486
0.06
(b) A chromosome after mutation Fig. 5. Functioning
of mutation
D1
D2
D3
t
𝛼
n
400
250
705
0.25
0.20
4
Fig. 7. A student with subjects as variables
Fig.8. Profit of supply chain partners and maximum opportunity loss for different values of α
Fig. 9. Overall profit at different number of generations of GA and iterations of TLBO
List of Tables Table 1 Buyers’ related data for the example problem Table 2 Profits of supply chain partners for different values of markup (α) charged by the manufacturer Table 3 Demand and sales price of buyers for different values of markup (α) charged by the manufacturer Table 4 Solution from Scheme I Table 1 Solution from Scheme II Table 6 Solution from Scheme III Table 7 Coalitions and corresponding supply chain profit Table 8 Solution from Scheme IV Table 9 Profit share and percentage profit margin at different levels of cost parameters of buyer 3 Table 10 Comparison of the results obtained from LINGO and with the use of GA and TLBO
Table 1 Buyers’ related data for the example problem
𝑩𝒖𝒚𝒆𝒓(𝒊) 1
𝒂𝒊 1,500
𝒃𝒊 10
𝑨𝒊 25
𝑻𝒊 30
𝒓𝒊 0.20
2
1,800
12
23
25
0.20
3
1,400
11
22
24
0.20
Table 2 Profits of supply chain partners for different values of markup (α) charged by the manufacturer Manufacturer
Buyer 1
Buyer 2
Buyer 3 𝑻
𝜶
𝑷
𝑶𝒎
𝑷𝟏
𝑶𝟏
𝑶𝟐
𝑷𝟐
𝑶𝟑
𝑷𝟑
𝒁
0%
0
49,698
18,491
0
22,284
0
11,008
0
51,783
49,698
10%
8,319
41,380
15,696
2,795
18,947
3,337
8,727
2,281
51,688
41,380
20%
16,631
33,068
12,909
5,582
15,604
6,680
6,452
4,556
51,596
33,068
35%
29,110
20,588
8,722
9,769
10,596
11,688
3,037
7,971
51,465
20,588
40%
33,263
16,436
7,332
11,159
8,929
13,355
1,903
9,105
51,426
16,436
41%
34,093
15,605
7,054
11,437
8,596
13,688
1,676
9,332
51,418
15,605
42%
34,923
14,775
6,776
11,715
8,262
14,022
1,449
9,559
51,411
14,775
43%
35,754
13,945
6,498
11,993
7,929
14,355
1,222
9,786
51,403
14,355
44%
36,585
13,114
6,220
12,271
7,596
14,688
995
10,013
51,395
14,688
45%
37,415
12,283
5,942
12,549
7,262
15,022
768
10,240
51,388
15,022
50%
41,218
8,481
4,540
13,951
5,580
16,704
0
11,008
51,337
16,704
60%
46,843
2,855
1,658
16,833
2,118
20,166
0
11,008
50,619
20,166
66%
49,698
0
0
18,491
0
22,284
0
11,008
49,699
22,284
70%
48,671
1,028
0
18,491
0
22,284
0
11,008
48,671
22,284
80%
43,331
6,367
0
18,491
0
22,284
0
11,008
43,331
22,284
Table 3 Demand and sales price of buyers for different values of markup (α) charged by the manufacturer Manufacturer
Buyer 1
Buyer 3
Buyer 2
𝒑𝟑
𝜶
𝑫𝟏
𝒑𝟏
𝑫𝟐
𝒑𝟐
𝑫𝟑
0% 5% 10% 20% 35% 40% 41% 42% 43% 44% 45% 50% 60% 66% 70% 80%
441 441 441 441 440 440 440 440
106 106 106 106 106 106 106 106
529 529 529 529 528 528 528 528
106 106 106 106 106 106 106 106
360 360 360 360 359 359 359 359
95 95 95 95 95 95 95 95
440 440 440 440 440 440 412 338
106 106 106 106 106 106 109 116
528 528 528 528 528 528 497 409
106 106 106 106 106 106 109 116
359 359 359 347 274 229 197 109
95 95 95 96 102 106 109 117
Table 4 Solution from Scheme I (D1 + D2+ D3 = D = 1,327)
Variable
Value
Variable
Value
Variable
Value
𝑇
51,403
𝛼
0.43
𝑛
6
𝑃
35,754
𝑤
89
𝑚
1
𝑃1
6,498
𝐷1
440
𝑝1
106
𝑃2
7,929
𝐷2
528
𝑝2
106
𝑃3
1,222
𝐷3
359
𝑝3
95
Table 5 Solution from Scheme II (D1 + D2+ D3 = D = 1,333)
Variable
Value
Variable
Value
Variable
Value
𝑇
50,506
𝛼
0.26
𝑛
5
𝑃
22,069
𝑤
79
𝑚
1
𝑃1
10,242
𝐷1
480
𝑝1
102
𝑃2
12,324
𝐷2
579
𝑝2
102
𝑃3
5,871
𝐷3
274
𝑝3
102
Table 6 Solution from Scheme III (D1 + D2+ D3 = D = 1,330)
Variable
Value
Variable
Value
Variable
Value
Variable
Value
𝑇
51,783
𝛼
0.0
𝑛
4
𝑚
1
𝐶
83,127
𝑃
25,609
𝑤
62
𝑝1
106
𝐶1
28,198
𝑃1
8,687
𝐷1
441
𝑝2
106
𝐶2
33,742
𝑃2
10,395
𝐷2
529
𝑝3
95
𝐶3
23,020
𝑃3
7,092
𝐷3
360
Table 7 Coalitions and corresponding supply chain profit
Sr. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Coalition Buyer 1 ✔ ✔ ✔ ✔ ✔ ✔ ✔
Manufacturer ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔
Buyer 2 ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔
Buyer 3 ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔
Total profit 0 0 0 0 17,517 21,358 10,014 0 0 0 40,284 32,702 28,811 0 51,783
Table 8 Solution from Scheme IV (D1 + D2+ D3 = D = 1,330)
Variable
Value
Variable
Value
Variable
Value
Variable
Value
𝑇
51,783
𝛼
0.0
𝑛
4
𝑚
1
0.49
𝑃
25,503
𝑤
62
𝑝1
106
0.18
𝑃1
9,373
𝐷1
441
𝑝2
106
0.22
𝑃2
11,311
𝐷2
529
𝑝3
95
0.11
𝑃3
5,596
𝐷3
360
Contribution index of manufacturer Contribution index of buyer 1 Contribution index of buyer 2 Contribution index of buyer 3
Table 9 Profit share and percentage profit margin at different levels of cost parameters of buyer 3
Profit to chain partner
Model
No change
Chain profit
Manufacturer Value
margin
Buyer 1
Buyer 2
Value margin
Value
margin
Buyer 3 Value margin
III
51,783
25,609
30.80
8,687
30.80
10,395
30.80
7,092
30.80
IV
51,783
25,503
30.68
9,373
33.24
11,311
33.52
5,596
24.31
III
51,277
25,293
30.47
8,599
30.47
10,288
30.47
7,099
30.47
IV
51,277
25,251
30.42
9,371
33.21
11,310
33.52
5,344
22.94
III
50,775
24,971
30.17
8,515
30.17
10,184
30.17
7,105
30.17
IV
50,775
25,001
30.21
9,370
33.20
11,310
33.51
5,094
21.63
Doubled
Tripled
Table 10 Comparison of the results obtained from LINGO and with the use of GA and TLBO
LINGO
In use of GA
In use of T
CPU Time (secs) Ex. No.
1 2 3 4 5
Number
of Buyers
3 5 7 8 10
Overall Profit (A)
51,783 94,194 125,190 148,543 179,692
CPU Time (secs)
0.110 0.110 0.120 0.140 0.150
Overall Profit (B)
51,061 91,943 121,820 144,226 173,110
Gbest
44 87 104 167 173
Up to
Up to
Gbest
Gtot
0.010 0.030 0.040 0.090 0.110
0.044 0.075 0.126 0.146 0.193
CPU Time (se (1-B/A)×100
1.39% 2.39% 2.69% 2.91% 3.66%
Overall Profit (C)
Ibest
50,933 91,406 121,282 142,736 170,152
3 4 4 4 6
Up to Ibest
0.004 0.005 0.014 0.020 0.020
Up t
0. 0. 0. 0. 0.
Author contributions statement Prof. Anil K Agrawal and Susheel Yadav conceived the idea of the project. Prof. Anil K Agrawal and Susheel Yadav planned the algorithms and numerical studies. Susheel Yadav performed the numerical computations. Prof. Anil K Agrawal and Susheel Yadav both contributed towards the final manuscript. Prof. Anil K Agrawal supervised the project.
S0360-8352(19)30677-1 https://doi.org/10.1016/j.cie.2019.106208 CAIE 106208
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
5 June 2018 20 November 2019 27 November 2019
Please cite this article as: Agrawal, A.K., Yadav, S., Price and profit structuring for single manufacturer multibuyer integrated inventory supply chain under price-sensitive demand condition, Computers & Industrial Engineering (2019), doi: https://doi.org/10.1016/j.cie.2019.106208
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Price and profit structuring for single manufacturer multi-buyer integrated inventory supply chain under price-sensitive demand condition Anil Kumar Agrawal Department of Mechanical Engineering, Indian Institute of Technology (Banaras Hindu University), Varanasi, India- 221005
Email: [email protected]
Susheel Yadav* Department of Mechanical Engineering, Indian Institute of Technology (Banaras Hindu University), Varanasi, India- 221005 Email: [email protected]. Phone No.: +91-8409969147
*Corresponding Author
Highlights An integrated production-inventory model for a single manufacturer-multiple buyers.
Issue of profit division among the partners of a supply chain is discussed.
Four different models of profit sharing have been proposed.
Profit sharing based on Shapley function values turns to be fair and supportive.
GA and TLBO meta-heuristics have also been proposed.
Price and profit structuring for single manufacturer multi-buyer integrated inventory supply chain under price-sensitive demand condition
Abstract This paper undertakes the study of an integrated production-inventory and pricing decision problem for a single manufacturer-multiple buyers supply chain where each buyer faces pricedependent demand. The item manufactured at a finite production rate is shipped to the buyers in multiple equal-sized shipments. Earlier researchers have not studied this problem for maximizing the overall supply chain profit and its allocation among the chain partners. For the allocation of the maximized profit, four different schemes are being proposed, each for a different level of understanding and faith among the partners. Demand, taken as price-sensitive, affects the costs to the manufacturer as the same is dependent upon the sales volume. Since the manufacturer fixes its sales price depending upon the sales volume affected cost, the volume will affect the buyer in terms of its unit purchase cost, and the holding cost which is charged as a percentage of the unit cost. This reality is attempted in this paper in addition to maximization of the chain profit and its allocation among the chain partners for integrated inventory management. Associated problems, formulated as mixed-integer non-linear programming problems, are computationally very hard, and thus evolutionary heuristics as Genetic Algorithm and Teaching-Learning Based Optimization are proposed. The strength of various proposed schemes for profit distributions has been analyzed using numerical experimentation. It is found that a forward contract between manufacturer and buyers will help to generate the maximum profit. It has also been shown that the distribution of profit based on Shapley function values is most fair and logical.
Keywords: price-sensitive demand; integrated inventory; single manufacturer multiple buyers; profit-sharing; pricing; genetic algorithm ______________________________________________________________________________ 1. Introduction In supply chain management (SCM), efforts are made to provide a win-win strategy for all the chain partners, right from buyers to supplier's vendors (Lummus and Vokurka, 1997). It would require a lot of coordination among the chain partners (Power, 2005; Simatupang et al., 2002). Out of the several important factors for the coordination, Thomas and Griffin, (1996) identified the following important issues: the selection of batch size, choice of transportation mode, and choice of production quantity. These issues are suggested to be resolved by having operational coordination — Buyer-Vendor coordination, Production-Distribution coordination, and InventoryDistribution coordination. Arshinder et al. (2011) also emphasized on coordination in the supply chain. They urged that the members of the supply chain should behave as parts of a unified system and coordinate with each other to improve the overall performance of the supply chain. This coordination can be at an operational level in the form of synchronized cycle times of all the buyers (Lu, 1995; Hoque, 2008; Chen and Sarker, 2017) and also at the financial level through a revenuesharing contract (Cachon, 2003; Govindan et al., 2013; Arani et al., 2016; Hu and Feng, 2017). In the absence of proper coordination, the supply chain has to face several operational and tactical issues. If a partner considers its inventory management problem independently of the other chain partners, it is found to bring in cost inefficiency and coordination issues (Bagchi et al., 2005). In order to avoid such situations, the focus is on adopting the Joint Economic Lot Sizing (JELS) framework right from buyers to the suppliers' suppliers. JELS models deal with the problems where the total relevant cost for the inventory is considered across the whole supply chain and not
separately and independently for individual partners. Working on this principle helps in resulting in a net saving in the total cost of the supply chain (Ben-Daya and Hariga, 2004; Hill and Omar, 2006; Shyu et al., 2006; Chung, 2008; Ben-Daya et al., 2008). Pando et al. (2012) asserted that, in real life, the main objective of inventory management is to maximize profit. Accordingly, the revenue from the sale of items should be considered in inventory models along with ordering, setup and holding costs. Since supply chain members are primarily concerned with their profits, it is natural to look into the joint inventory problem from the total profit perspective rather than simply and only considering the total cost. This has prompted some researchers to analyze the problem from the total supply chain profit perspective (Giri and Roy, 2015; Chen and Sarker, 2017). In the early years, integrated production-inventory management researches focused on the manufacturer's production capacity assumption and delivery methods. A review of the researches related to joint economic lot-sizing problems can be had from the work of Glock (2012). The concept of JELS for the first time came from Goyal (1976). He considered the integrated inventory management problem consisting of a single vendor and a single buyer where the vendor has an infinite production rate capability and meets the demand of the purchaser using a lot-for-lot shipment policy. Later, Banerjee (1986) relaxed the unrealistic assumption on the vendor's infinite production rate capability and instead considered a finite production rate for the vendor. The idea of the lot-for-lot shipments was scrapped by Goyal (1988) in his further work. The profit potential of a supply chain was found to depend upon the supply policy: a single batch against an order or multiple ones. Instead of meeting the whole order from one supply, he considered the production lot to be shipped in equal-sized sub-batches, but only when the production for the entire lot was over. Agrawal and Raju (1996) took this concept of sub-batching in which a lot can be divided
into many equal-sized sub-batches. However, deliveries of these sub-batches were to be made even during the production of the lot. Models with different shipment policies consisting of unequal sub-batch and combination of equal and unequal sub-batches have also been studied by several other researchers (Goyal and Gupta, 1989; Hill, 1997; Viswanathan, 1998; Hill, 1999; Goyal, 2000; Goyal and Nebebe, 2000). A single manufacturer supplying a product to a single buyer (SMSB) is hard to find in today’s business environment (Sarmah et al., 2008), but SMSB will always be the basic building block for such problems. In today's business world, manufacturers usually sell the product through several retailers rather than relying on a single retailer. On realizing this fact, Joglekar and Tharthare (1990) proposed another JELS model considering multiple homogeneous buyers who can also be viewed as retailers. However, these retailers need not be identical and may have different operating characteristics (Banerjee and Burton, 1994). Incorporation of such considerations into JELS models will make it more realistic. A Common Replacement Epoch policy was considered by Viswanathan and Piplani (2001) for a case where a vendor supplies to its buyers at the same time. It naturally reduces the flexibility available to the buyers and increases their cost as well. To compensate for this increased cost, the vendor offers a price discount to the buyers. Their proposed model takes into consideration these factors and accordingly determines the optimal common replacement period and the price discount. A case of single vendor multiple buyer supply-chain, where all the partners are interested in creating a strategic alliance for higher profits, was studied by Woo et al. (2001). The effect of investments was evaluated in terms of ordering cost reduction on the total cost reduction through electronic data interchange (EDI). Siajadi et al. (2006) studied a single vendor multiple buyer (SVMB) supply chain model and proposed a shipment policy where the buyers can have a varying number of equal-sized shipments. In their model, the hierarchy of
the buyers affects the number of shipments and saving in the total cost. Hoque (2008) proposed three policies for shipments in SVMB scenario with integrated inventory consideration: (i) equalsized shipments are to be made as soon as the manufacturer finishes the production lot, (ii) equalsized batches are to be dispatched such that they are available by the time the buyers consume the whole inventory and (iii) unequal-sized batches are to be supplied with the next shipment size increased in the ratio of the production rate to the sum of the demand rates of all the buyers. Optimal solutions are found analytically for all the policies. Sarmah et al. (2008) also studied an SVMB supply chain with two dominance conditions: (i) the dominating manufacturer ships large size lot to the buyers at a common replacement time through common carrier and compensates their increased cost by giving them a uniform credit policy, and (ii) the dominating buyers’ association offer a common replacement time proposal to the manufacturer and acceptance of the offer depends on negotiation between the association and the manufacturer. Jha and Shanker (2013) investigated the effect of lead time reduction on total supply chain cost with a service level constraint in an SVMB supply chain. Giri and Roy (2015) also studied the controllable lead time in an SVMB supply chain, where all the buyers are homogeneous, with their demand being dependent on the price of the product. All these research papers have shown that supplying the product in multiple sub-batches to the buyers helps in reducing the overall cost by saving on holding and setup costs more than the additional expenses to be made on transportation of the items from the manufacturer to the buyers. While going through the literature, it was observed that inventory management decisions are largely governed by the cost parameters involved. The holding cost constitutes an essential component of well-established inventory management models such as Economic Order Quantity (Harris, 1913; Wilson, 1934) and JELS models of (Lu, 1995; Kosadat, 2000; Mahata et al., 2005;
Yang et al., 2008; Zanoni et al., 2014). Berling (2008), in his research, has mentioned that little effort had been put in accurately determining the cost parameters. A good number of research papers related to work on JELS (Lu, 1995; Kosadat, 2000; Mahata et al., 2005; Yang et al., 2008; Zanoni et al., 2014) considered the holding cost rate as a fixed cost per unit per year. This is different from the thought of Berling (2008) as well as of Singhal and Raturi (1990), who expressed the holding cost rate as an annual rate, and the annual carrying cost was determined by relating it to the item’s value. It is only the latter case where the unit cost of the item will have a bearing on the determination of the optimal order quantity. In the case of price-sensitive demand, the unit cost will have a major role in deciding or fixing the sales price and thus will dictate the annual demand and, finally, the optimal order quantity. If the demand is not price-sensitive, the cost minimization approach will yield the same optimal inventory management policies as for profit-maximization since the revenue received from the customers is going to be fixed due to constant values of the sales price and the demand. When the demand is not fixed and is price-sensitive, it is likely to affect the costs to the manufacturer and the buyers, most significantly their holding cost. In this environment, the unit purchase cost for the buyers may also vary if the manufacturer fixes its sales price depending upon its cost incurred in manufacturing the items. This reality has not been addressed in the available research literature that focuses on the maximization of the total supply chain profit and its judicious allocation among the chain partners. The problem considered in this paper assumes holding cost rate to depend upon the unit cost of the item and the end demand to be price-sensitive. In the present paper, the focus is on maximizing the overall profit for a single vendor and multiple buyer supply-chain and, to that end, also on determining optimal values of retailers' selling prices, number and size of multiple shipments to
fulfill an order. Even though coordination and integration will lead to more of overall supply-chain profit, but the challenge will still be there in distributing the profit among the chain partners, particularly when the markup on the cost price of the upper echelon partners is not fixed. Profit allocation is an essential issue in supply chain management on which the success of the supply chain depends. Yue and You (2014) reported that there are a number of supply chain optimization models available in the literature that consider the supply chain as a whole, without taking into account the conflicting interests of its partners. This type of optimization results in a high degree of dissatisfaction among the partners due to uneven profit distribution as these models do not incorporate any fair profit division schemes. The problem of profit allocation has been addressed in several ways. Zahran et al. (2017) considered a three-level supply chain model where each partner charges a predefined profit margin from its downstream partner. The demands of the end customers were taken to be different and deterministic. The problem was solved for four different combinations of coordination and nocoordination scenarios of consignment stock policies for its partners. Jaber and Osman (2006) proposed to share the overall profit among the partners of a two-level supply chain in proportion to their incurred costs. Their model does not maximize the profit but minimizes the supply chain costs. The two objectives are equivalent as they took the demand to be pre-specified and constant. However, the demand is not necessarily pre-specified, particularly in the case of price-dependent demand. For such a demand, the minimization of the supply-chain cost will target zero demand resulting in overall supply chain cost as zero. Naturally, the maximization of overall profit will serve as the right objective in the case of price-dependent demand. For this reason, Chauhan and Proth (2005) aim to maximize the supply chain profit for the case of price-dependent demand. They also divided the profit among the partners in proportion to their investments. Revenue
sharing is a relatively new coordination contract mechanism. Under this contract, the retailer pays the supplier a wholesale price for each unit purchased and a percentage of the revenue that the retailer generates (Yao et al., 2008). Sharing of revenue can be taken as sharing of profit as the retailers will never share its revenue with the vendor beyond a particular limit so as to incur a loss. Cachon and Lariviere (2005) studied the revenue sharing contract in coordinating the supply chain of a video cassette rental industry. They showed that revenue sharing induces the retailer to choose supply-chain actions for determining the optimal quantity and selling price. In the present paper, besides the profit-sharing mechanism due to Chauhan and Proth (2005), three more schemes have also been proposed for the case of price-dependent demand while taking the holding cost of the buyers as the function of their unit purchase cost. One of the schemes minimizes the maximum opportunity loss to a supply chain partner. This scheme will be useful in situations where the level of understanding among the chain partners has not matured enough. The remaining two schemes will be more applicable for those supply chains where the understanding among the chain partners is reasonably good, and the perfect coordination is being sought by the partners in order to maximize their overall profit. For the first two schemes, the proposed mathematical programming formulations directly allocate the profit while deciding sales price at each of the supply chain partners. For the remaining two schemes, one more mathematical model has been presented to determine optimal inventory policies resulting in maximum overall supply chain profit. This optimal profit of the chain is later distributed among the chain partners using two other different schemes proposed in this paper. The organization of the paper is as follows. Section 2 describes the problem as well as assumptions and notations involved. It also presents the expressions for various elements of the cost to the manufacturer and the buyers. The four different schemes of profit-sharing and the related
mathematical models are presented in Section 3. The solution procedures are described in Section 4 in the form of two heuristics. The relevance of the four schemes is discussed in Section 5 from the perspective of their effectiveness in different scenarios of supply chain coordination. It also presents the strengths and weaknesses of each scheme by conducting experiments using an illustrative numerical example. Section 6 discusses the merits and demerits of these schemes and provides some important managerial insights. It also presents results on the comparative performance of the two proposed heuristics. Section 7 concludes the research work presented in this paper and also suggests some future research directions. 2. The problem The supply chain considered in the present paper is shown in Fig. 1. It has a single manufacturer dealing in a single item with multiple buyers. [Insert Fig. 1] Further characteristics of the problem can be best understood from the assumptions and notations described below. 2.1. Assumptions and notations
Some of the important characteristics of the problem, considered in this paper, are explained in terms of the following assumptions: 1. The buyers have equal replenishment cycle times. 2. No shortages and backordering are allowed. 3. Replenishment is instantaneous. 4. The planning horizon is considered to be infinite. 5. The manufacturer and all the buyers have enough storage capacity to accommodate the required inventory and, therefore, there is no space constraint for storing the stock of items. 6. The production capacity of the manufacturer is considered to be finite. 7. Lead times are constant. It need not be the same for all the buyers. Notations used in the description and mathematical formulation of the problem are as follows.
𝑐
Profit margin charged by the manufacturer on the unit cost (fraction of the unit cost price) Cost of one unit of the item after it has been manufactured
𝑖
Index for a buyer
𝑚
Number of sub-batches during the production cycle of the manufacturer
𝑛
Number of equal-sized sub-batches per order
𝑡
Common replenishment cycle time for all the buyers in year
𝑣
Overall unit cost of the manufacturer
𝑤
Unit purchase price to buyers
𝑥
Ratio of annual production rate to annual demand rate
𝐷
Annual demand for the manufacturer being equal to cumulative demand of all the buyers
𝑁
Total number of buyers
𝑃
Annual production rate of the manufacturer
𝑄
Production lot size of the manufacturer
𝑇
Total annual supply chain profit
𝑍
Maximum of the opportunity loss to a supply chain partner
𝐶
Total annual cost of the manufacturer
𝑃
Total annual profit to the manufacturer
𝑃
Total annual profit of all the buyers
𝑎𝑖
Maximum demand experienced by buyer 𝑖 in units
𝑏𝑖
Demand-price elasticity coefficient for buyer 𝑖
𝑝𝑖
Unit selling price of 𝑖𝑡ℎ buyer
𝑟𝑖
Annual inventory holding cost rate for 𝑖𝑡ℎ buyer (charged as a fraction on per unit monetary value of the inventory stock)
𝑟𝑚
Annual inventory holding rate for the manufacturer (charged as a fraction on per unit monetary value of the inventory stock)
𝐴𝑖
Ordering cost per order of 𝑖𝑡ℎ buyer
𝐴𝑚
Cost per setup of the manufacturer
𝐵𝑖
Best profit for buyer i
𝐵𝑚
Best profit level of the manufacturer
𝛼
𝐶𝑖
Total annual cost of buyer i
𝐶𝑚
Area of one inventory cycle of the manufacturer
𝐷𝑖
Annual demand of 𝑖𝑡ℎ buyer as a function of its selling price
𝐻𝑖
Annual inventory holding cost to 𝑖𝑡ℎ buyer
𝐻𝑚
Annual holding cost of the manufacturer
𝐼𝑚
Average inventory of the manufacturer
𝑀𝑖
Annual ordering cost to 𝑖𝑡ℎ buyer
𝑂𝑖
Opportunity cost for buyer 𝑖
𝑂𝑚
Opportunity cost for the manufacturer
𝑃𝑖
Annual purchase cost to 𝑖𝑡ℎ buyer
𝑃𝑖
Total annual profit of 𝑖𝑡ℎ buyer
𝑃𝑚
Annual cost of manufactured units of the item
𝑅𝑖
Annual revenue of 𝑖𝑡ℎ buyer
𝑆𝑚
Annual setup cost to the manufacturer
𝑇𝑖
Transportation cost per shipment of 𝑖𝑡ℎ buyer
𝑇𝑖
Annual transportation cost to 𝑖𝑡ℎ buyer
2.2. Expressions for cost elements
The buyers face demand from customers, and the same depends upon their sales price for the item. The relationship considered between price and demand (Chen and Sarker, 2017) is: 𝐷𝑖 = 𝑎𝑖 ― 𝑏𝑖𝑝𝑖
∀ 𝑖.
(1)
Fig. 2 depicts the integration of demand from the buyers and the supply from the manufacturer. In this figure, on-hand inventory is not shown on the same scale for the members of the supply chain. The manufacturer produces the item at a constant and finite production rate and completes a lot of size 𝑄 units in one setup. These units are sent in 𝑛 number of equal sub-batches. From a sub-batch
of size 𝑞, 𝑖𝑡ℎ buyer gets 𝑞𝑖 units in proportion of the ratio of its annual demand (𝐷𝑖) to the total 𝐷𝑖
𝑁
annual demand (𝐷 = ∑𝑖 = 1𝐷𝑖) of all the buyers. Thus 𝑖𝑡ℎ buyer gets a quantity 𝑞𝑖 = 𝐷 𝑞 (with 𝑞 = 𝑁
∑𝑖 = 1𝑞𝑖 ) and consumes it in time 𝑡. Hence
𝑞1
𝑞2
𝑞𝑁
𝐷1 = 𝐷2 = … = 𝐷𝑁 = 𝑡. From Fig. 2, it can be observed
that 𝑄
(𝑄 ― 𝑛 ) 𝐷
= (𝑛 ― 1)𝑡.
(2)
From the above, the following relationship will emerge. 𝑄
𝑡 = 𝑛𝐷
(3)
[Insert Fig. 2] Using the defined notations and the distribution framework provided in Fig. 2, the mathematical expressions for the various costs are determined. They are presented in subsequent sub-sections. 2.2.1 Manufacturer’s total cost
The total cost for the manufacturer will be the amount of money that the manufacturer will spend on procuring the items, in processing the same, and also in stocking the finished items. Annual cost on manufactured products without holding and setup costs
Per unit cost 𝑐 to the manufacturer is assumed to include all the related costs such as unit raw material purchase cost, unit raw material holding cost, unit processing cost, etc. In order to satisfy the cumulative annual demand 𝐷 of all the buyers, the manufacturer will bring the units of the item to finished form by incurring the total annual expense, as given below. 𝑃𝑚 = 𝑐𝐷 Annual inventory holding cost
(4)
This cost is taken in proportion to the average inventory that can be determined by dividing the manufacturer’s inventory cycle area by the cycle length. From Fig. 2, the area under the inventory cycle curve of the manufacturer (𝐶𝑚) can be determined as 𝐶𝑚
𝑃 𝑄 ― (𝑚 ― 1)𝑞 ― 𝑞 𝑞𝑞 𝑞𝑃 𝑞 𝑃 𝑞 𝑞𝑃 𝑞 𝑃 𝑞 𝐷 + +𝑞 ―1 + 𝑞 ― 1 (𝑚 ― 1) + = +…+ 2 2𝑃 2𝐷𝐷 𝐷 𝐷 2𝐷𝐷 𝐷 𝐷
{
( )}
{
}
( )
(
)
𝑃 𝑃 𝑄 ― (𝑚 ― 1)𝑞 ― 𝑞 𝑄 ― (𝑚 ― 1)𝑞 ― 𝑞 𝐷 𝐷 𝑃 𝑞 + 𝑄 ― (𝑚 ― 1)𝑞 ― 𝑞 ― 𝑃 𝑃 𝐷 𝐷
(
){
}
[
{
}
( )]
𝑞 + {1 + 2 + 3 + … + (𝑛 ― 𝑚 ― 1)} 𝑞 . 𝐷
The average inventory of the manufacturer (𝐼𝑚) can be calculated by dividing the area under the inventory cycle curve of the manufacturer (𝐶𝑚) by the cycle time (𝑄 𝐷). Hence, the average inventory of the manufacturer will be 𝐼𝑚 =
[
𝑞𝑞 2𝑃
+
(
𝑞𝑃 𝑞 2𝐷𝐷
(
+𝑞
𝑃 𝐷
) )+…+(
―1
𝑞𝑃 𝑞 2𝐷𝐷
𝑞 𝐷
(
+𝑞
𝑃 𝐷
)
― 1 (𝑚 ― 𝐷 𝑄
𝑞 1)𝐷
)+(
.
𝑃
)(
𝑃
) + {𝑄 ― (𝑚 ― 1)𝑞
𝑄 ― (𝑚 ― 1)𝑞𝐷 ― 𝑞
𝑄 ― (𝑚 ― 1)𝑞𝐷 ― 𝑞
2
𝑃
𝑃 𝐷
(5)
After simplification, 𝑄
[
𝐼𝑚 = 2𝑛2
1
𝑃 𝐷 + (𝑛 ― 𝑚 ― 1)(𝑛 ― 𝑚) + 𝑚(2𝑛 ― 𝑚 ― 1) ―
].
(𝑛 ― 1)2 𝑃𝐷
(6)
Therefore, the annual inventory holding cost of the manufacturer (Hm) can be calculated by multiplying the average inventory by inventory holding cost rate (rm) and per unit cost (c). Thus,
𝐻𝑚 = 𝑟𝑚𝑐𝐼𝑚
}{
―𝑞
𝑞 𝐷
―
𝑄 ― (𝑚 ― 1)𝑞 𝑃
After putting the value of 𝐼𝑚 from equation (6) in the above equation, we would get, 𝑄
[
𝐻𝑚 = 2𝑛2
1
𝑃 𝐷 + (𝑛 ― 𝑚 ― 1)(𝑛 ― 𝑚) + 𝑚(2𝑛 ― 𝑚 ― 1) ―
]𝑟 𝑐.
(𝑛 ― 1)2
(7)
𝑚
𝑃𝐷
Annual setup cost
Besides incurring the cost for bringing the item to the finished level and also on holding them in stock, the other important cost to be borne by the manufacturer is the setup cost. The annual setup cost of the manufacturer can be calculated by multiplying the number of setups per year with the cost incurred in making one setup. 𝐷
Thus,
(8)
𝑆𝑚 = 𝑄𝐴𝑚.
So, the total annual cost of the manufacturer (𝐶) will be the sum of the annual cost of manufactured products (Pm), the annual inventory holding cost (Hm) and the annual setup cost (Sm). Thus,
𝐶 = 𝑃𝑚+ 𝐻𝑚+ 𝑆𝑚.
Substituting the values of 𝑃𝑚, 𝐻𝑚 and 𝑆𝑚 into the above equation from their respective equations (4), (7) and (8), we get 𝑄
[
𝐶 = 𝑐𝐷 + 2𝑛2
1 𝑃𝐷
+ (𝑛 ― 𝑚 ― 1)(𝑛 ― 𝑚) + 𝑚(2𝑛 ― 𝑚 ― 1) ―
]𝑐 𝑐 +
(𝑛 ― 1)2
𝑚
𝑃𝐷
𝐷
𝑄 𝐴𝑚.
Per unit overall cost of the manufacturer (𝑣) can be found by dividing the total annual cost by the total annual demand. Therefore, 1
[
𝑄
{
𝑣 = 𝐷 𝑐𝐷 + 2𝑛2
2.2.2. Buyers’ total cost
1 𝑃𝐷
+ (𝑛 ― 𝑚 ― 1)(𝑛 ― 𝑚) + 𝑚(2𝑛 ― 𝑚 ― 1) ―
} 𝑐 𝑐 + 𝐴 ].
(𝑛 ― 1)2 𝑃𝐷
𝑚
𝐷 𝑄 𝑚
(9)
Each of the buyers will spend money on purchasing the item from the manufacturer, placing the order to the manufacturer and also for holding the item in stock. Annual purchase cost to 𝒕𝒉𝒆 𝒊𝒕𝒉 buyer
Since the manufacturer may have a markup on its unit overall cost price (𝑣), the unit purchase cost to the buyers (𝑤) need not be equal to the unit overall cost of the manufacturer (𝑣). The annual purchase cost for 𝑖𝑡ℎ buyer will be 𝑃𝑖 = 𝐷𝑖𝑤.
(10)
Annual inventory holding cost
Buyer 𝑖 incurs a cost to hold the item in its stock, and it depends upon the lot size (𝑞𝑖) and the inventory holding cost rate (ri). Thus, the annual inventory holding cost to 𝑡ℎ𝑒 𝑖𝑡ℎ buyer (𝐻𝑖) will be 𝐻𝑖 =
𝑞𝑖
( 𝑟 𝑤) = 2 𝑖
𝑄
2𝑛𝐷𝐷𝑖𝑟𝑖𝑤.
(11)
Annual ordering cost
Annual ordering cost to the 𝑖𝑡ℎ buyer can be determined by multiplying the ordering cost per order (𝐴𝑖) by the number of orders per year Thus,
(𝐷𝑄). 𝐷
𝑀𝑖 = 𝑄 𝐴 𝑖 .
(12)
Annual transportation cost
Annual transportation cost to the 𝑖𝑡ℎ buyer will be 𝐷
𝑇𝑖 = 𝑄𝑛𝑇𝑖 .
(13)
So, the total annual cost of the buyer i, (𝐶𝑖) will be the sum of the annual purchase cost (Pi), the annual inventory holding cost (Hi), the annual ordering cost (𝑀𝑖) and the annual transportation cost (𝑇𝑖). Thus,
𝐶𝑖 = 𝑃𝑖 + 𝐻𝑖+ 𝑀𝑖 + Ti.
(14)
3. Profit-sharing considerations Four different proposed schemes for profit sharing among the chain partners proposed are detailed and discussed below. 3.1. Scheme I: minimization of the maximum opportunity cost
In this case, it is assumed that the various supply chain partners may not have perfect and true collaborative thinking for overall profit maximization. This kind of scenario is not uncommon when the various partners, managing their inventory problem independently so far, begin to initiate to coordinate and integrate themselves to form the supply chain. The traditional mindset of working for the best for themselves is not seen to change so drastically to what is really required in profit maximization as a whole. The best policy for an individual is not necessarily the best policy for the other partners. A deviation from their best position will naturally be viewed by themselves as an opportunity loss. Therefore, while in negotiation with each other, every chain partner will try to keep such an opportunity loss as low as may be possible. These buyers will earn the maximum profit when the manufacturer sells these products at its cost price to them, i.e. when 𝛼 = 0. The moment the manufacturer will go with a markup on its cost price, the unit purchase cost of the buyers will increase and push their total cost upwards. So they will not be in a position to make their best profit. Since the manufacturer is also in business for profit-making and not for only buyers to make the profit, it will try to push up its selling price. In this process, the opportunity
loss to the manufacturer will reduce, but it will increase the same for the buyers. Going beyond a particular limit, the opportunity loss to the manufacturer will become less as compared to some buyers. Such buyers are quite likely not to accept such an increase in the sales price of the manufacturer, which will make them suffer a loss more than that for the manufacturer. In such a situation, ultimate negotiation would converge on the philosophy of minimization of opportunity loss to every partner. A relevant, pragmatic and logical strategy in such an ecosystem will have to minimize the maximum opportunity loss experienced by any of the supply chain partners. This philosophy can be further understood from the following elaborations. If the manufacturer charges a profit margin (𝛼) on its incurred overall unit cost (𝑣), then unit selling price to the buyers (𝑤) would be 𝑤 = (1 + 𝛼)𝑣 .
(15)
Overall profit to the manufacturer (𝑃) will be 𝑃 = 𝛼𝑣𝐷.
(16)
Annual revenue of 𝑖𝑡ℎ buyer will be 𝑅𝑖 = 𝐷𝑖𝑝𝑖. The above equation, after substituting for 𝐷𝑖 from equation (1), will become 𝑅𝑖 = (𝑎𝑖 ― 𝑏𝑖𝑝𝑖)𝑝𝑖.
(17)
Annual profit of 𝑖𝑡ℎ buyer will be its annual revenue minus its annual expenses. Thus, 𝑃𝑖 = 𝑅𝑖 ― (𝑃𝑖 + 𝐻𝑖 + 𝑀𝑖 + 𝑇𝑖).
(18)
After substituting the expressions for variables from equations (11), (12), (13) and (14), the above equation can be rewritten as
(
𝑃𝑖 = (𝑎𝑖 ― 𝑏𝑖𝑝𝑖)𝑝𝑖 ― 𝐷𝑖(1 + 𝛼)𝑣 +
)
𝑄 𝐷 𝐷 𝐷𝑖𝑟𝑖(1 + 𝛼)𝑣 + 𝐴𝑖 + 𝑛𝑇𝑖 . 2𝑛𝐷 𝑄 𝑄
It is obvious that the maximum value of 𝑃𝑖, denoted as 𝐵𝑖, will occur only when 𝛼 = 0. It has been mentioned earlier that the increase in the value of 𝛼 may bring more profit to the manufacturer at a higher cost to the buyer and, ultimately, a lower level of demand. With the increase in the value of 𝛼, the profit to the manufacturer can be thus seen to increase initially to reach the peak and then to reduce thereafter. The opportunity costs to the chain partners can be mathematically expressed as 𝑂𝑚 = 𝐵𝑚 ― 𝑃, and
(19) ∀ 𝑖.
𝑂 𝑖 = 𝐵 𝑖 ― 𝑃𝑖
(20)
The profit maximization philosophy discussed herein can be summarized in the form of the following mathematical model.
{𝑂 }, 𝑂 ) (maximum 𝑖
minimize 𝑍 = maximum
𝑖
𝑚
subject to equations (1), (3), (9), (10), (11), (12), (13), (15), (16), (17), (18), (19), (20) and 𝑁
(21)
𝐷 = ∑𝑖 = 1𝐷𝑖 𝑎𝑖
𝑝𝑖 ≤ 𝑏𝑖 𝛼,𝑡,𝑄,𝐷,𝑣,𝑤,𝑃 ≥ 0
𝑛,𝑚 ≥ 0 𝑎𝑛𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
∀𝑖
(22) (23) (24)
𝐷𝑖, 𝐻𝑖, 𝑂𝑖, 𝑝𝑖, 𝑃𝑖, 𝑅𝑖, 𝑃𝑖, 𝑇𝑖 ≥ 0
(25)
∀𝑖
3.2. Scheme II: all partners have the same markup on their unit overall cost
Scheme I, of the profit division among the chain partners, will be more suitable when the partners have less amount of faith for each other and have not developed a reasonable level of confidence in the sustenance of coordination among themselves. At the next level of understanding, the partners may not be so self-centered and may perceive a win-win situation for themselves in terms of having earned the same profit margin. Thus, as compared to Scheme I, Scheme II will represent the comparatively less conservative position of the partners as they agree to have the same markup on their unit overall cost. In view of this, the selling price (𝑤) of the manufacturer will be the same as given by equation (15); but the selling price of the 𝑖𝑡ℎ buyer (𝑝𝑖) will change and will be
𝑝𝑖 =
or,
𝑝𝑖 =
(1 + 𝛼) (𝑃𝑖 + 𝐻𝑖 + 𝑂𝑖 + 𝑇𝑖) 𝐷𝑖
(𝑤 +
(1 + 𝛼) 𝐷𝑖
𝑄
2𝑛𝐷𝐷𝑖𝑟𝑖𝑤
𝐷
𝐷
)
+ 𝑄( 𝐴𝑖) + 𝑄𝑛𝑇𝑖 .
(26)
Now, the problem of profit distribution can be expressed by the following mathematical model. 𝑁
maximize T = 𝑃 + ∑𝑖 = 1 𝑃𝑖 subject to equations (1), (3), (9), (10), (11), (12), (13), (15), (16), (17), (18), (21), (22), (23), (24), (25) and (26). In this case, the chain partners enjoy the same profit margin. The manufacturer supplies the units to the retailers treating this profit margin as its markup on its overall unit cost price. Since the unit
purchase cost to the buyers is going to increase, their selling price is expected to increase while causing a fall in their annual demand. It is going to reduce the overall chain profit and, thus, it will not be in the interest of the chain naturally. Having understood the reason for a lower level of profit earning due to a low level of coordination, the buyers will become interested in going in with a forward contract with the manufacturer to share their profits with the manufacturer while receiving the item from the manufacturer at the cost price. The following two schemes are based on this concept. However, profit distribution mechanisms are different. In the two schemes, the manufacturer will supply the item to the buyers without any markup and will share the profits of the buyers under a forward contract. In such a case, the thought of Ingene and Parry (2005) can be employed for having a forward contract by a manufacturer with its buyers. The forward contract has to ensure that the buyers share their profit with the manufacturer in a logical and rational manner. 3.3. Scheme III: profit distribution in the proportion of incurred costs
This scheme aims to distribute the resulting maximum overall profit among the chain partners in proportion to their incurred costs. In order to find the maximum overall supply chain profit, the following mathematical programming model is being presented. maximize
𝑁
𝑇 = 𝑃 + ∑ 𝑖 = 1 𝑃𝑖
(27)
subject to equations (1), (3), (9), (10), (11), (12), (13), (15), (16), (17), (18), (21), (22), (23), (24) and (25) The distribution of the supply chain profit among the partners is to take place in proportion to their incurred costs. That is,
𝐶 𝑁
𝐶 + ∑𝑖 = 1𝐶𝑖
=
𝐶1 𝑁
𝐶 + ∑𝑖 = 1𝐶𝑖
=
𝐶2 𝑁
𝐶 + ∑𝑖 = 1𝐶𝑖
=… =
𝐶𝑁
. 𝑁 𝐶 + ∑𝑖 = 1𝐶𝑖
(28)
The above mathematical programming model will not allow any amount of profit to the manufacturer requiring it to supply all the units to the buyers at its cost price. Because of the reduction in the cost price, the holding cost of the buyers will also come down as it is charged in proportion to the unit purchase cost of the buyers. As a result, the overall cost to a buyer will reduce. The retailers can now think of reducing the sales price to target more of demand, even keeping the profit margin from each sold item unchanged. Thus the overall profit is expected to increase. 3.4. Scheme IV: profit distribution in proportion to contribution indices
Here, for the distribution of the profit, a methodology due to Shapley (1953) is employed. In his proposed approach, the contribution of each chain partner has to be evaluated. Based on the input as the contribution made by a chain partner, its contribution index is worked out. The overall supply chain profit is shared by the chain partners in proportion to these contribution indices. The use of the Shapley function for distribution of profit would require solving of mathematical programming formulation of Scheme III for several times, with the manufacturer in common and the buyers in various combinations. Since there are ‘N’ buyers, the buyers are to be associated in 𝑁
𝐶1 + 𝑁𝐶2+ 𝑁𝐶3+ … +𝑁𝐶𝑁 combinations. In case N is large, the mathematical programming
formulation of Scheme III has to be used for a very large number of combinations before profit distribution can be worked out. Total combinations will be 7 for the number of buyers as 3, will be 15 for 4, 31 for 5 and will increase exponentially with the number of the buyers. In other cases, where either the manufacturer alone is taken or only the buyers in any combination, the overall contribution is taken to be zero. Even though this process of profit distribution may be time-
consuming, but it is worth trying as it will have more of practicality in terms of its acceptance by the chain partners. The utility of the schemes, proposed above, is also elaborated later by taking a numerical example. 4. Proposed heuristic approaches 4.1. Genetic Algorithm (GA) as a solution methodology
For the integrated multi-echelon supply chains, GAs have been found to yield solutions close to optimal ones by Nachiappan and Jawahar (2007), and also by Jawahar and Balaji (2009). With the motivation from their findings, Genetic Algorithm (GA) is being proposed to solve the mathematical model stated in this paper mainly due to non-linearity in the model with integer value requirement on some of the variables. GAs have been reported to lead to the best solution if the fitness functions are properly defined (Yang et al., 2018; Prakash et al., 2011). Holland (1975) introduced the Genetic Algorithms (GAs). These are population-based adaptive heuristic search algorithms. GAs are based on the Darwinism, the theory of biological evolution developed by Charles Darwin in his book "On the Origin of Species" in 1859. GAs have been theoretically and empirically proven to be a robust search technique (Goldberg, 1989; Behzadi, et al. 2008). The available research, over the last few decades, has demonstrated the enormous success of GAs in terms of their ability to handle complex, nonlinear optimization and real-world problems (Ali et al., 2016). Large number of researchers have used GAs to solve the problems from different fields such as, inventory management (Dania, 2010; Priya and Iyakutti, 2011; Maiti and Maiti, 2008), supply chain management (Han and Damrongwongsiri, 2005), group technology (Moon et al., 2006), machine layout (Kovačič et al.,
2013), advertising display optimization (Miralles-Pechuán, et al., 2018), airspace configuration (Sergeeva et al., 2017), optimal land uses (Palogos et al., 2017), etc. GA begins with the initial population, which is a set of arbitrarily initialized solutions that evolves toward better and better regions of the search space by means of randomized processes of selection, cross-over (recombination) and mutation (Bäck and Schwefel, 1993). The selection operator chooses the “best” parents in the population for mating, the crossover operator picks two parents and generates offsprings by interchanging the segments of the parent genes, and the mutation operator alters some of the genes of a chromosome to create a new chromosome. The crossover operator has to maintain a proper balance between exploitation and exploration that leads towards the success or failure of the search process (Gallard and Esquivel, 2001). The exploitation strategy focuses on local search in a narrow optimistic region consisting of solutions that have already been achieved. Whereas, the exploration strategy is used in carrying out a global search over a wider region in the hope of a better solution. The proposed GA has been used for finding the maximum total supply chain profit and corresponding profits of individual partners for the mathematical formulation given in Section 3.3. Various steps of the algorithm are as described in Fig. 3. [Insert Fig. 3] Real-coded chromosomes are taken to define the population of the solutions with the genes showing the real values of the decision variables. The variables included in the problem are: demands of the buyers (𝐷𝑖), replenishment time(𝑡), the profit margin of manufacturer (𝛼) and number of the shipments (𝑛). Since demands and the sales prices of the buyers are interrelated,
the demands are taken as the variables with the prices are calculated using the price-demand relationship. The solution procedure consists of the following steps. (a) Representation of individual chromosome (solution): Each individual chromosome consists of genes (variables) that are real values. For a single manufacturer and three buyers problem, the total number of genes is 6 (3 for each buyer’s demand; and 1 each for replenishment time (𝑡), the profit margin of the manufacturer (α) and number of the shipments (𝑛)). A solution chromosome is generated by randomly choosing values of the variables, where the choice on the values depends on some realistic criteria. For example, the demand values (𝐷𝑖) are generated randomly anywhere between minimum demand (0) to maximum demand (𝑎𝑖), and replenishment time value(𝑡) between
1 365
to 1 year. Similarly, for the manufacturer's profit
margin (𝛼) and the number of shipments (𝑛), respective values between 0.00 to 0.80 and from 1 to 10 are generated randomly. A sample chromosome is shown in Fig. 4 for a problem with a single manufacturer and three buyers. [Insert Fig. 4] Further steps of the proposed GA are clarified using this single manufacturer and three buyers chain. (b) Generation of the initial random population: An initial population of solutions (chromosomes) is generated by randomly choosing the variable values. In the implementation of GA for numerical experimentation, the population size is taken as 20. (c) Evaluation of fitness function: A specific function, using which the strength of each generated solution is determined, is known as the fitness function. This fitness function can be identical to the objective function (Fahimnia et al., 2018).
The objective function for the profit, defined in Scheme III and represented by expression (27) that determines the value of total supply chain profit (𝑇), serves as the fitness function. (d) Genetic operations: Once the fitness function value for each of the population solutions is determined, then control is handed to genetic operators: selection, crossover and mutation. Selection: Selection is the first operation to be performed. It selects the chromosomes from the present generation for further operations of the GA. A number of selection operations are available in the GA literature, viz., Tournament Selection, Roulette Wheel Selection, Proportionate Selection, Rank Selection, etc. The roulette wheel selection process is applied to the selection process in the present work. Roulette wheel selection is a stochastic selection process where fittest individuals have a higher chance of survival than weaker ones (Pasala et al., 2013). Crossover: In the traditional crossover, a cross-site will be used to decide a string of continuous genes to be interchanged over the two parent chromosomes. In the other case, only the genes at the cross-site are interchanged. A crossover rate of 20% is applied in the present work. The strategy chosen for the crossover is 1-point crossover, in which case a single cross-site is randomly chosen. Another random number would be used to decide whether conventional crossover will be followed or not. This random number for the present work is kept as 0.50. It means that only one gene will be interchanged in 50% of the descendants, and it can be even more than this in the remaining 50% cases. This philosophy strikes a balance between the solutions with modified values of one to many parameters to visualize their effect in a more rational manner. The working of the proposed crossover is explained in Fig. 5.
Fig. 5(b) depicts the case of crossover of one gene and Fig. 5(c) for the crossover of multiple genes. [Insert Fig. 5] Mutation: After the crossover, the mutation operation is carried out. The mutation prevents the algorithm from being trapped in a local minimum. It plays the role of recovering the lost genetic materials as well as for randomly disturbing genetic information. It is an insurance policy against the irreversible loss of genetic material (Abdoun et al., 2012). A creep mutation with 5% mutation rate is carried out in the present work. Creep mutation selects a gene randomly and changes the value with a random value lying between lower and upper bounds (Sivanandam and Deepa, 2007). The process of mutation has been explained in Fig. 6, where the second gene value after mutation becomes 150 from its original value of 250.
[Insert Fig. 6]
(e) Stopping criteria: After the mutation operation, one generation of the GA is complete. The best value, obtained so far, is recorded. The process iterates until some fixed number of generations evolve or some fitter chromosomes are not obtained. In the present work, the number of generations as 25 times the number of supply-chain partners is selected as a stopping criterion. 4.2. Teaching-Learning Based Optimization (TLBO) as a solution methodology
In solving the proposed mathematical models, Teaching-Learning Based Optimization (TLBO) meta-heuristic developed by Rao et al. (2011) is also being proposed. Among several metaheuristic techniques, TLBO is a relatively new technique with an edge over the other techniques. Since it does not require any algorithm parameters to be tuned, the implementation of TLBO is simpler (Rao et al., 2012). In this algorithm, a random population of the solutions is generated. A population of 20 learners is generated randomly in the implementation of the heuristic in the present paper. The population size represents the number of the students in a class and each design variable to represent a subject taught. In encoding the problem, a learner, with D1, D2, D3, t, α and n as its subjects with the marks in the respective subjects as 400, 250, 705, 0.25, 0.20 and 4, is shown in Fig. 7, with the subject values being generated randomly within the range as described below. [Insert Fig. 7]
The choice for the values will depend upon some realistic criteria. For example, the demand values
(𝐷𝑖) are generated randomly anywhere between the minimum demand value (0) to the maximum demand value (𝑎𝑖), replenishment time value (𝑡) between
1 365
year to 1 year, the manufacturer’s
profit margin value (𝛼) between 0.00 to 0.80, and the number of shipments value (𝑛) from 1 to 10. The algorithm is divided into two major steps: (i) Teacher’s phase and (ii) Students’ phase.
Teacher’s phase: In this phase, each of the randomly generated solutions will lead to an objective function value that is analogous to the result or the overall grade of an individual student. The solution (student) with the highest objective function value (the highest grade) is declared as a
teacher. The teacher is considered to be a highly learned person who tries to improve the knowledge of the students. The mean result of the students in each of the subjects is calculated. A factor namely “difference_ meani” is calculated for subject i using the following expression difference_ meani = ri(Xi_teacher – TFXi_learner) where ri is a uniformly distributed random number between 0 and 1, Xi__teacher is marks in the 𝑖𝑡ℎ subject of the teacher, TF is taken to denote “Teaching Factor” with its value being either equal to 0 or 1, and Xi_learner is the mean value of the marks of the students in 𝑖𝑡ℎ subject. At the subsequent stages of learning, the students are supposed to improve their learning by having the subject knowledge from the teachers. The change brings the students to a different level. The same is expressed as Xi_learner = Xi_learner + difference_ meani A student’s new learning level is accepted only when it results in a better value of the objective function compared to the old one. In this way, the whole population is updated. The updated population serves as input to the students’ phase.
Students’ phase: This phase mimics the learning of students through discussion among them. In this phase, two students are selected randomly. The one, with the higher knowledge, is assumed to improve the knowledge of the other. Let X1 and X2 (being vectors) represent marks in different subjects of some two students and f(Xi) to represent objective function value (grade) for the ith student. Since the amount of learning is going to be random, the same is controlled by a uniformly distributed random number rn taking values in the range zero to one in the following manner:
and
X1 =X1 + rn (X2 - X1)
if f(X2 ) > f(X1),
X2 =X2+rn (X1 - X2)
if f(X1) > f(X2 ).
If the new solution is better compared to the corresponding old solution, then it is updated; otherwise, the old one is retained. In the present work, the number of unique pairs of students to be identified for learning under this phase is taken to be equal to the number of the students, i.e. the size of the class or the population. After the two phases are over, the current iteration gets completed. This procedure is being iterated for 25 times the number of chain partners in the present paper. The proposed GA and TLBO have been implemented on the platform of MATLAB (R2015a) and are run on a computer system with Intel (R) Core (TM) i7-6700 CPU with 3.40 GHz processor. 5. Numerical experimentation The proposed profit distribution schemes are further described using an illustrative example. The supply chain considered consists of a single manufacturer and three retailers (𝑁 = 3). This problem takes most of the data from the numerical example of Jokar and Sajadieh (2009) and Jha and Shanker (2013). The manufacturer related data are: P = 12,000, c = 60, 𝐴𝑚 = 1,000 and 𝑟𝑚= 0.10. The buyer’s related data are given in Table 1. [Insert Table 1] The solutions to the example problem from the proposed schemes have been obtained using a commercially available software LINGO. The solutions, along with some insights, are been presented below. 5.1. On the application of Scheme I
The example problem is solved for different values of 𝛼 and the results are presented in Tables 2 and 3. Fig. 8 shows the variation in the profit to chain partners and maximum opportunity loss with
change in the value of α. A careful observation of these tables and the figure will provide the following relevant information. i)
As the profit margin (𝛼) increases, the total supply chain profit (T) decreases. It happens because of the increase in the unit purchasing cost to the buyers with the increase in the value of 𝛼. As a result, the sales price would increase in order to maximize the profit and thus to cause demand value to go down (Table 3).
ii)
Profit to the buyers is the maximum for 𝛼 = 0 or when the manufacturer sells the item with no markup on its unit overall cost. On the other hand, when the profit to the manufacturer is the maximum (𝛼 = 66%), the buyers make no profit. These two situations are extreme ones. The optimal solution has to naturally lie in between these two extremes (Table 2).
iii)
The least possible profit for any of the supply chain partners is zero because the scheme denies the possibility of a partner being in the chain even when it makes a loss.
iv)
Increasing 𝛼 value beyond 66%,
the buyers are not making any profit and the
manufacturer’s profit moves downwards instead of increasing. This clearly indicates that the manufacturer cannot afford to charge an exorbitantly high profit-margin in order to earn a very high profit (Table 2). v)
From Table 3, it can be noticed that, with the increase in the value of α upto 66%, the change in the sales price and the demand is either not there or is much less compared to when α is increased beyond 66%.
From Table 2 and Figure 8, it can also be noticed that when the manufacturer charges a profit margin of 0% (or no profit for itself), the buyers earn the respective maximum profit as 𝐵1 = 18,491, 𝐵2 = 22,284 and 𝐵3 = 11,008. Any deviation from this position is seen as a loss of
opportunity (𝑂𝑖 for buyer i). For example, 𝑂1 = 2,795 for 𝛼 = 10%. It is obtained by subtracting the resulted profit to buyer 1 as 15,696 from its maximum profit value of 18,491. The manufacturer, to make itself better-off, will go for a higher value of 𝛼. As the value of 𝛼 increases, the profit to manufacturer also increases, but to a certain limit (α = 66%). Increasing 𝛼 beyond 66% causes the profit to the manufacturer to decrease. This value of 𝛼 as 66% corresponds to the best profit to the manufacturer, thus 𝐵𝑚 = 49,698. Any deviation from this profit level will be seen as a loss of opportunity by the manufacturer for earning the maximum possible profit. The value of 𝑂𝑚 for the manufacturer is calculated by subtracting the current profit from this maximum possible profit. For example, for 𝛼 = 10%, 𝑂𝑚 = 49,698 ― 8,319 = 41,380. Opportunity loss from the best position for each supply chain partner is calculated, and the maximum of the opportunity loss to a supply chain partner is recorded in the last column of Table 2. The minimum of this maximum opportunity loss is found to be 14,355 for the corresponding value of 𝛼 as 43% with the values of some of the decision variables given in Table 4. Thus, the profit distribution philosophy of this scheme yields an overall supply chain profit as 51,403 for 𝛼 = 0.43. Out of this supply chain profit, the manufacturer makes a profit of 35,753 while the buyers 1, 2 and 3 make respective profits of 6,498, 7,929 and 1,222. While the manufacturer receives a markup of 43% on its incurred cost, the buyers 1, 2 and 3 earn respective profit margins as 16.18%, 16.49% and 3.73% on their incurred costs. Since the profit percentage on the incurred cost of buyer 3 is very small as compared to the other two buyers and significantly smaller compared to that for the manufacturer. Hence this scheme of profit distribution is quite likely not to be appreciated by buyer 3 because of the high level of disparity involved. In fact, every chain partner would like to have the same profit margin. With this very reason, profit distribution using the philosophy built into Scheme II will make a lot of sense.
[Insert Table 2] [Insert Table 3] [Insert Table 4] [Insert Fig. 8]
5.2. On the application of Scheme II
Scheme II of profit-sharing is based on the idea that every chain partner shall have the same profit margin, i.e., ratios of profits earned to their investment or incurred costs are going to be equal. The optimal solution to the problem is shown in Table 5, with the optimal value of 𝛼 as 26.44%. In this scheme, the manufacturer makes a profit of 22,069 while the buyers 1, 2 and 3 make profits of 10,242, 12,324 and 5,871, respectively (Table 5). [Insert Table 5] From Tables 4 and 5, the following can be observed: i) For a high value of α as 0.43, the unit selling price to the buyers (w) is 89. The same is 79 for α as 0.26. Because of the fall in the value of α, the profit to the manufacturer can be observed to decrease and that for the buyers to increase. ii) There is not much change in the cumulative demand faced by the buyers. It changes from 1,327 (Table 4) to 1,333 (Table 5). However, the demand for buyers changes. Demands to the buyers 1 and 2 increase while that to the buyer 3 decreases.
The use of rationality in profit distribution helps buyer 3 to lift profit for itself, but overall supply chain profit drops from 51,403 to 50,506. Rational distribution of profit without compromising on the total supply chain profit is obtained from the application of the philosophy of Schemes III and IV. 5.3. On the application of Scheme III
Since the manufacturer is not having any markup on its cost price to earn profit for itself, it has to sign a forward contract with its buyers for proper profit distribution. Since Scheme III believes in earning maximum overall supply chain profit, the application of the respective mathematical model, presented in Section 3.3, yields maximum profit as 51,783 for 𝛼 = 0 (Table 2). The solution for the example problem under this philosophy of profit distribution is given in Table 6. From this table and Tables 4 and 5, it can be noticed that the overall demand has changed by a very small amount. The unit purchase cost price to the buyers experiences a lot of change. From a high value of 89, it has come down to 62. In Table 6, profit to the manufacturer and to the buyers are in the proportion of their incurred costs and are obtained by distributing the overall profit as 51,783 using the relationship given by equation (28). The profit to manufacturer as 25,609 is received under a forward contract with the buyers due to which buyer 1will share an amount of 9,804 from its profit of 18,491 (Table 2) to finally have profit amount as 8,687, buyer 2 will share 11,889 and buyer 3 as 3,916 to finally retain the respective profits as 10,395 and 7,092. The distribution in the proportion of the incurred costs has inherent fallacy. More costs one incurs, more of the profit is earned by it. Thus, this philosophy has a weakness of bringing inefficiencies if some partner chooses to incur more costs in order to just earn more profit for itself. This point is further explained in Section 6.1. [Insert Table 6]
5.4. On the application of Scheme IV
As mentioned earlier, the overall chain profit obtained in Scheme III will be distributed among the chain partners using the Shapley function. The application of the Shapley function would require the determination of maximum profit when chain partners are used in different combinations, as shown in Table 7. In this table, the overall profit is zero whenever either the manufacturer is taken alone or only the buyers in any combination without it. In rest other combinations, the manufacturer is there with some of the buyers. For each of these combinations, the mathematical model of Scheme III (presented in Section 3.3) is used to determine the overall profit that is recorded in the last column of Table 7. From Table 7, the average contribution for each chain partner has been worked out. The same is detailed in Appendix-A. It is the contribution that is likely to be highly appreciated by every chain partner as their contribution to overall profit. Naturally, the profit shared by a chain partner would be equal to its average contribution. Thus the manufacturer will get a profit of 25,503; whereas buyers 1, 2 and 3 will respectively receive the profit of 9,373, 11,311 and 5,596. [Insert Table 7] [Insert Table 8] 6. Discussions 6.1 On the fairness of profit distribution
In Scheme III for the profit distribution, it was mentioned earlier that it has a weakness in giving extra profit to the inefficient partner. From this perspective, fairness in the distribution of the overall profit using Shapley function vis-à-vis philosophy of Scheme III is being analyzed by modifying the cost data of buyer 3 by taking the same as a multiple by 2 and 3 of its original cost
values given in Table 1. The overall profit and its distribution among the partners under the two schemes (Scheme III and IV) are presented in Table 9. The following observations can be made with respect to the increase in the values of the cost parameters of buyer 3. i)
The overall profit keeps decreasing. The profit margin to the chain partners also decreases under both the schemes.
ii)
Except for buyer 3, profits to the chain partners also decrease. When the cost parameters as holding cost rate, ordering cost and transportation cost are doubled of the earlier respective values (Table 1), the overall profit decreases to 51,277 from 51,783 causing the manufacturer’s share to drop down from 25,609 to 25,293, the profit of 8,687 to 8,599 for buyer 1 and 10,395 to 10,285 for buyer 2. Buyer 3, who has become costinefficient, now receives an increased profit of 7,099 instead of an earlier amount of 7,092. This substantiates that Scheme III has a weakness of favoring cost-inefficient supply chain partners and thus the philosophy of the distribution of the overall profit in proportion to the incurred cost of the chain partners (Chauhan and Proth, 2005; Jaber and Osman, 2006) is not a fair policy.
iii)
In the solutions from Scheme IV, the profit decreases for every partner. Buyer 3 is not found to be favored as opposed to that under Scheme III. Buyers 2 and 3 lose the profit by a little amount, but the manufacturer and the buyer 3 by a significant amount. This is in line with the common belief that the selection of a wrong buyer (retailer) by a manufacturer (an organization owning the item) should not affect the other buyers. Naturally, such kind of inefficiencies with the buyer has to be borne by the
manufacturer and the concerned buyer only. This shows the strength of Scheme IV in terms of distributing the profit in the fairest manner. [Insert Table 9] From Table 4, it can be noticed that Scheme I has favored the manufacturer over the buyers, with buyer 3 suffering the most. Thus buyer 3 can be said to be the weakest partner in the supply chain from the profit earning perspective. Schemes II and III (Tables 5 and 6) take a somewhat better and balanced approach. But the most balanced, democratic and fair process seems to be of Scheme IV (Table 8). It was observed that the manufacturer had to suffer because of the inefficiencies of buyer 3. Even then, buyer 3 was favored and was taken as one of the supply-chain partners. This may lead to the thought of removing such a partner from the chain so that the remaining partners are well awarded. The worthiness of such consideration is being analyzed below. When buyer 3 is taken out of the supply chain, the profit available will be 40,284 (figure available against serial number 11 in Table 7) and thus less by 11,499 from a total of 51,783 when buyer 3 was also a supply chain partner (Table 6). In Scheme IV, with all the three buyers, the share of buyer 3 was 5,596 (Table 8). Therefore, even sharing profit amount of 5,596 with buyer 3, the other supply chain partners (the manufacturer and buyers 2 and 3) receive put together an extra amount of 5,903 out of the extra profit of 11,499. This means that buyer 3 benefits the supply chain at least by an amount of 5,903 when profit is shared using Shapley function. A similar observation can be had when the profit is shared on the incurred cost basis (Scheme III). The use of Scheme III provides buyer 3 a profit of 7,092. This profit amount is much less compared to the loss in overall profit of 11,499. Thus, keeping buyer 3 in the supply chain helps the other partners to earn combined together an extra amount of 11,499 – 7,092 = 4,407. This also shows that leaving a weak partner may not be an advantageous strategy for the other chain partners. In case the cost
parameters of a buyer are very high, this partner has to keep its selling price very high and thus may not be in a position to have any demand from its customers. Naturally, in such a scenario, the related buyer will be out of the scene and will not practically remain to be the supply chain partner. The above numerical experiment shows that the inefficiency of one partner does affect the others. However, leaving a partner does not make sense as the overall profit drops. Instead, an effort is required to be made by the other chain partners to put in their money and experience on this inefficient partner to make it more efficient, enabling everyone then to accrue more of profit. The reverse of dropping a partner can be taken as the addition of new buyers, if possible, to increase the profit share of everyone. 6.2. Analysis of the efficiency of proposed GA and TLBO heuristics
An effort has been made to analyze the efficiency of the proposed heuristic approaches based on the framework of GA and TLBO. For this purpose, five problems were randomly chosen, with the one already used in illustrating the proposed schemes in Section 5. All these problems have one manufacturer, but with an increasing number of buyers. These problems were solved using the proposed GA and TLBO approaches incorporating the profit distribution philosophy of Scheme III. Resulting objective function values for these problems and execution time taken by Intel (R) Core (TM) i7-6700 CPU with 3.40GHz processor have been recorded, and the same are presented in Table 10. In Table 10, Gtot and Itot respectively denote the number of the generations in the case of GA and the number of the iterations in the case of TLBO. They are taken to be 25 times the number of the chain partners. For a single manufacturer and 3 buyers supply chain, these will be equal to 25 × 4 = 100. Gbest is the generation in which the best objective function value was obtained in the case
of GA. Ibest shows the iteration number in the case of TLBO for the same purpose. From Table 10, it can be noticed that CPU time required for the proposed GA is less compared to the one required by LINGO for small size problems; but it is more when the problem size increases. TLBO has taken more time in all the cases compared to that taken by GA or LINGO. Wilcoxon Signed Rank test finds this difference to be statistically significant. Thus, the proposed GA and TLBO approaches are expected to take more time as compared to that required in the use of LINGO software. GA approach was also found to take more time compared to that by optimization software by Ghasimi et al. (2014). From Table 10, it can also be noticed that the best objective function values were obtained by GA and TLBO much before all the iterations could be completed. For example 1 (Table 10), GA and TLBO respectively took 0.010 secs 0.004 secs to achieve the best objective values, while LINGO took 0.110 secs. From this perspective, the Wilcoxon test identifies TLBO to be the best and LINGO to be the poor, with GA falling in between the two. The values of the objective function obtained in various generations/iterations of the proposed GA and TLBO approaches are shown in Fig. 9 for experiment number 1 with 3 buyers (Table 10). This figure also shows early achievement of the best objective function values in the case of TLBO but of an inferior value compared to that obtained from the GA approach. Further, in all of the experiments, the proposed GA and TLBO heuristics resulted in solutions with inferior objective function values as compared to that obtained from the use of LINGO software (Table 10). Wilcoxon Signed Rank test identifies LINGO to the best, TLBO to be the poor and GA in between the two. This difference in objective function values is found to increase with the increase in the number of buyers.
The above observations are strong evidence to suggest the use of optimizing algorithms instead of the proposed heuristics, both from the perspective of solution quality and CPU time requirement. [Insert Table 10] [Insert Fig. 9]
8. Conclusion This paper presents some mathematical models and schemes for maximizing the overall supply chain profit and allocating the same among the chain partners while determining optimal production-inventory and pricing decisions for the integrated supply chain composed of a single manufacturer-multiple buyer where each buyer faces price-dependent demand. This paper assumes that the manufacturer produces the item at a finite rate. Shipment to a buyer is made in multiple shipments of equal-sized batches to save upon holding cost at the additional expense of transportation cost. Previous researches treat the holding cost for the buyers as a fixed quantity and the manufacturer’s selling price as pre-specified. In the present work, the manufacturer’s selling price is a decision variable. Further, the holding cost of the buyers is taken to depend upon the manufacturer’s selling price for being more realistic and practical. In the proposed integrated model, the supplies from the manufacturer to the buyers are coordinated to ensure no occurrence of shortage at the end of the buyers. Instead of classical minimization of overall inventory management cost, this study strategizes for the maximization of the total supply chain profit to arrive at the optimal inventory decisions as the sale price may change the demand and, thus, the cost structure. In the present research work, four different schemes have been proposed for the maximization of the overall supply chain profit and its distribution among the chain partners considering different
levels of coordination and faith among them. The first scheme is applicable when the coordination has just begun among the partners, and they minimize their maximum opportunity loss instead of insisting on the best position while managing their inventory independently. Because of a low level of understanding and a low level of alliance, the maximum overall profit achieved is found not to be as high as it could be. The second proposed scheme, based on somewhat better understanding among the partners and for being more fair and rational in profit distribution, provides for the same profit margin to all the chain partners and thus does away with the dissatisfaction that was very likely to result in from the unequal profit margin in the first scheme. However, the numerical experiment found the overall profit to be less than that in the case of the first scheme. An analysis, in the form of numerical experimentation, was carried out to visualize the effect of the increase in the markup on the cost price of the manufacturer on the overall supply chain profit. It was learnt that the earning of the profit, more than what was achieved in the first two schemes, is possible only when the manufacturer sells the item to its buyers at no markup. In this case, all the profit was with the buyers only and they were required to share their additional profit with the manufacturer in a rational way. In this paper, it has been proposed to carry out sharing of the profit in a fair manner by having the manufacturer a forward contract signed with its buyers (Cachon and Lariviere, 2005) reflecting mutual trust for each other. For this purpose, additional two schemes were proposed. The third scheme is based on the philosophy of the distribution of the profit among the partners in proportion to their incurred costs, as was carried out by Chauhan and Proth (2005) or by Jaber and Osman (2006). This scheme was found to have an inherent fallacy of promoting cost inefficiency among the chain partners as they may not work to reduce their incurred cost due to the allocation of a lower amount of profit on account of less incurred cost. The fourth proposed
scheme does not have this issue. It, in the fairest manner, distributes the overall profit among the supply chain partners in proportion to their contribution indices that are computed using Shapley function (Shapley, 1953). Further numerical experimentation with this scheme shows that the inefficiency of any buyer does not affect the profit of the other buyers, but that of the manufacturer and of that buyer itself. This observation seems to be quite practical as a buyer would not like to suffer because of the inefficiencies of other buyers. On experimenting with the third and the fourth schemes, it was found that the dropping of any buyer from the chain will cause a loss in profit to every other chain member. Therefore, it is being suggested that no buyer should be dropped from the chain simply on the basis of cost inefficiency. If a buyer is cost-inefficient, it will have less of the demand from the customers and, thus, less of the profit will be allocated to it in use of the Shapley function. If the cost becomes prohibitive, the buyer will have no demand and will be practically out of the supply chain. Since the proposed profit distribution models are mixed-integer nonlinear mathematical programming formulations, Genetic Algorithm (GA) and Teaching-learning Based Optimization (TLBO) heuristics have been proposed with a view to solving the considered problem in a computationally efficient manner. However, on numerical experimentation, GA and TLBO heuristics were found to take more of CPU time to solve the problem in comparison to that taken by optimizing software LINGO. The finding on the CPU time requirements is the same as was observed by Ghasimi et al. (2014). Proposed heuristics were also found to result in more deterioration in the objective function value as the number of buyers increases. Since GA and TLBO approaches were not found to be serving the purpose of being computationally-efficient, no effort has been made to determine the robust values of their parameters.
The scope of the present problem can be extended to analyze the effect of various possibilities of transporting the item from the manufacturer to the buyers using either owned transportation facilities or the ones roped in from by a third-party logistics service provider. In the present paper, a strict and deterministic relationship of the price with the demand was assumed. In the real world, a change in the price may not necessarily bring a typical change in the demand all the time. Therefore, the present work may be extended to consider the price-dependent stochastic demand.
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https://doi.org/10.1080/00207543.2016.1249431 Zanoni, S., Mazzoldi, L., Zavanella, L.E., Jaber, M.Y., 2014. A joint economic lot size model with price and environmentally sensitive demand. Prod. Manuf. Res. 2:1, 341–354. Appendix A
The marginal contribution of the chain partners in a particular coalition, in the use of Shapley function, depends upon the order of the partners in which they are arranged in that coalition. In a coalition as (M,B1,B2,B3), all the partners are involved and, for this combination, overall supply chain profit was found to be as 51,783 mentioned at serial number 15 of Table 7. Now going by the order and including one more partner at a time, the contribution of the partners will be determined. Recognizing the order in this coalition, first partner is observed to be the manufacturer. If it is considered alone, its contribution will be zero as was recorded at serial number 1 of Table 7. Considering the order, next partner to be added to the manufacturer (M) will be buyer 1 (B1). For the manufacturer and buyer 1 combined, the overall profit is 17,517, mentioned at serial number 5 of Table 7. Since the contribution of the manufacturer is zero, the amount of 17,517 is taken as the one generated by buyer 1. Hence the contribution of buyer 1 becomes 17,517. Next, buyer 2 (B2) is added. With the manufacturer associated with buyer 1 and 2, the overall profit was found to be 40,284 recorded at serial number 11 of Table 7. So the additional contribution on joining of buyer 2 is 40,284 ― 17,517 = 22,767. This extra contribution is naturally the contribution from buyer 2. In the last, buyer 3 (B3) is also included. Now the whole supply chain has been considered for which the overall profit was found to be 51,583. Therefore, the additional contribution on taking buyer 3 into the chain will be 51,583 ― 40,284 = 11.499. This extra profit of 11,499 is only due to addition of buyer 3 to the chain. For this reason, the contribution of buyer 3 in the coalition is considered as 11,499.
These contributions of the partners will depend upon the order in the coalition considered. For the single manufacturer and three buyers’ case, the total number of distinct coalition will be 4𝑃4 = 24. Following the methodology described above, the contributions of the partners are determined for all the 24 permutations or coalitions and the same are detailed in Table A.1.When all the partners are making the chain, consideration of the order is meaningless. To nullify the effect of the order on the contribution, Shapley, (1953) suggested to make use of average contribution. It is this profit that would be allocated to the respective partners. Relating this average contribution to the overall profit, contribution index of a partner has been determined. These contribution indices give an idea about the strength of a partner in generating the overall profit. By not charging any markup on cost price, the manufacturer has made highest contribution of 49% to the overall profit. The weakest partner is buyer 3, contributing the least (11%) to the overall profit.
Table A.1 Computation of Shapley indices
Possible order of partners M,B1,B2,B3 M,B1,B3,B2 M,B2,B1,B3 M,B2,B3,B1 M,B3,B1,B2 M,B3,B2,B1 B1,M,B2,B3 B1,M,B3,B2 B1,B2,M,B3 B1,B2,B3,M B1,B3,M,B2 B1,B3,B2,M B2,B1,M,B3 B2,B1,B3,M B2,M,B1,B3 B2,M,B3,B1 B2,B3,B1,M B2,B3,M,B1 B3,B1,B2,M B3,B1,M,B2 B3,B2,B1,M B3,B2,M,B1 B3,M,B1,B2 B3,M,B2,B1 Total Average contribution Contribution index
Marginal contribution of manufacturer 0 0 0 0 0 0 17,517 17,517 40,284 51,783 28,811 51,783 40,284 51,783 21,358 21,358 51,783 32,702 51,783 28,811 51,783 32,702 10,014 10,014 612,068
Marginal contribution of buyer 1 17,517 17,517 18,926 19,080 18,796 19,080 0 0 0 0 0 0 0 0 18,926 19,080 0 19,080 0 0 0 19,080 18,796 19,080 224,960
Marginal contribution of buyer 2 22,767 22,972 21,358 21,358 22,972 22,688 22,767 22,972 0 0 22,972 0 0 0 0 0 0 0 0 22,972 0 0 22,972 22,688 271,459
Marginal contribution of buyer 3 11,499 11,294 11,499 11,344 10,014 10,014 11,499 11,294 11,499 0 0 0 11,499 0 11,499 11,344 0 0 0 0 0 0 0 0 134,297
25,503
9,373
11,311
5,596
0.49
0.18
0.22
0.11
List of Figures Fig. 1. The supply chain considered Fig. 2. Inventory cycle of manufacturer and buyers Fig. 3. Proposed genetic algorithm Fig. 4. A chromosome Fig. 5. Functioning of 1-point crossover Fig. 6. Functioning of mutation Fig. 7. A student with subjects as variables Fig. 8. Profit of supply chain partners and maximum opportunity loss for different values of α Fig. 9. Overall profit at different number of generations of GA and iterations of TLBO
Fig. 1. The supply chain considered
Fig. 1. Inventory cycle of manufacturer and buyers
Step 1: Start with a randomly generated population. Step 2: Evaluate the fitness function value of each chromosome in the population. Step 3: Create a new population by repeating following steps of genetic operations until the current generation is completed Step 3.1: Selection - Select the chromosomes for the formation of mating pool by Roulette wheel selection method. Step 3.2: Crossover - Use multivariate to crossover the parents to form a new offspring. Step 3.3: Mutation - Use creep mutation to mutate new genes of the population. Step 4: Place new offsprings in the new population. Step 5: Test if the end condition is satisfied. If yes go to Step 7, otherwise to Step 6. Step 6: Go to Step 2. Step 7: Stop.
Fig. 2. Proposed genetic algorithm
𝐷1
𝐷2
𝐷3
𝑡
𝛼
𝑛
400
250
705
0.25
0.20
3
Fig. 3. A chromosome
400
250
705
0.25
0.20
3
750
150
486
0.06
0.50
5
(a) Two parent chromosomes
400
250
486
0.25
0.20
3
750
150
705
0.06
0.50
5
(b) Two children chromosomes after the crossover with interchange in only one gene
400
250
486
0.06
0.50
5
750
150
705
0.25
0.20
3
(c) Two children chromosomes after the crossover with changes in second substrings Fig. 4. Functioning of 1-point crossover
400
250
705
0.25
0.20
3
0.50
5
(a) A chromosome
400
150
486
0.06
(b) A chromosome after mutation Fig. 5. Functioning
of mutation
D1
D2
D3
t
𝛼
n
400
250
705
0.25
0.20
4
Fig. 7. A student with subjects as variables
Fig.8. Profit of supply chain partners and maximum opportunity loss for different values of α
Fig. 9. Overall profit at different number of generations of GA and iterations of TLBO
List of Tables Table 1 Buyers’ related data for the example problem Table 2 Profits of supply chain partners for different values of markup (α) charged by the manufacturer Table 3 Demand and sales price of buyers for different values of markup (α) charged by the manufacturer Table 4 Solution from Scheme I Table 1 Solution from Scheme II Table 6 Solution from Scheme III Table 7 Coalitions and corresponding supply chain profit Table 8 Solution from Scheme IV Table 9 Profit share and percentage profit margin at different levels of cost parameters of buyer 3 Table 10 Comparison of the results obtained from LINGO and with the use of GA and TLBO
Table 1 Buyers’ related data for the example problem
𝑩𝒖𝒚𝒆𝒓(𝒊) 1
𝒂𝒊 1,500
𝒃𝒊 10
𝑨𝒊 25
𝑻𝒊 30
𝒓𝒊 0.20
2
1,800
12
23
25
0.20
3
1,400
11
22
24
0.20
Table 2 Profits of supply chain partners for different values of markup (α) charged by the manufacturer Manufacturer
Buyer 1
Buyer 2
Buyer 3 𝑻
𝜶
𝑷
𝑶𝒎
𝑷𝟏
𝑶𝟏
𝑶𝟐
𝑷𝟐
𝑶𝟑
𝑷𝟑
𝒁
0%
0
49,698
18,491
0
22,284
0
11,008
0
51,783
49,698
10%
8,319
41,380
15,696
2,795
18,947
3,337
8,727
2,281
51,688
41,380
20%
16,631
33,068
12,909
5,582
15,604
6,680
6,452
4,556
51,596
33,068
35%
29,110
20,588
8,722
9,769
10,596
11,688
3,037
7,971
51,465
20,588
40%
33,263
16,436
7,332
11,159
8,929
13,355
1,903
9,105
51,426
16,436
41%
34,093
15,605
7,054
11,437
8,596
13,688
1,676
9,332
51,418
15,605
42%
34,923
14,775
6,776
11,715
8,262
14,022
1,449
9,559
51,411
14,775
43%
35,754
13,945
6,498
11,993
7,929
14,355
1,222
9,786
51,403
14,355
44%
36,585
13,114
6,220
12,271
7,596
14,688
995
10,013
51,395
14,688
45%
37,415
12,283
5,942
12,549
7,262
15,022
768
10,240
51,388
15,022
50%
41,218
8,481
4,540
13,951
5,580
16,704
0
11,008
51,337
16,704
60%
46,843
2,855
1,658
16,833
2,118
20,166
0
11,008
50,619
20,166
66%
49,698
0
0
18,491
0
22,284
0
11,008
49,699
22,284
70%
48,671
1,028
0
18,491
0
22,284
0
11,008
48,671
22,284
80%
43,331
6,367
0
18,491
0
22,284
0
11,008
43,331
22,284
Table 3 Demand and sales price of buyers for different values of markup (α) charged by the manufacturer Manufacturer
Buyer 1
Buyer 3
Buyer 2
𝒑𝟑
𝜶
𝑫𝟏
𝒑𝟏
𝑫𝟐
𝒑𝟐
𝑫𝟑
0% 5% 10% 20% 35% 40% 41% 42% 43% 44% 45% 50% 60% 66% 70% 80%
441 441 441 441 440 440 440 440
106 106 106 106 106 106 106 106
529 529 529 529 528 528 528 528
106 106 106 106 106 106 106 106
360 360 360 360 359 359 359 359
95 95 95 95 95 95 95 95
440 440 440 440 440 440 412 338
106 106 106 106 106 106 109 116
528 528 528 528 528 528 497 409
106 106 106 106 106 106 109 116
359 359 359 347 274 229 197 109
95 95 95 96 102 106 109 117
Table 4 Solution from Scheme I (D1 + D2+ D3 = D = 1,327)
Variable
Value
Variable
Value
Variable
Value
𝑇
51,403
𝛼
0.43
𝑛
6
𝑃
35,754
𝑤
89
𝑚
1
𝑃1
6,498
𝐷1
440
𝑝1
106
𝑃2
7,929
𝐷2
528
𝑝2
106
𝑃3
1,222
𝐷3
359
𝑝3
95
Table 5 Solution from Scheme II (D1 + D2+ D3 = D = 1,333)
Variable
Value
Variable
Value
Variable
Value
𝑇
50,506
𝛼
0.26
𝑛
5
𝑃
22,069
𝑤
79
𝑚
1
𝑃1
10,242
𝐷1
480
𝑝1
102
𝑃2
12,324
𝐷2
579
𝑝2
102
𝑃3
5,871
𝐷3
274
𝑝3
102
Table 6 Solution from Scheme III (D1 + D2+ D3 = D = 1,330)
Variable
Value
Variable
Value
Variable
Value
Variable
Value
𝑇
51,783
𝛼
0.0
𝑛
4
𝑚
1
𝐶
83,127
𝑃
25,609
𝑤
62
𝑝1
106
𝐶1
28,198
𝑃1
8,687
𝐷1
441
𝑝2
106
𝐶2
33,742
𝑃2
10,395
𝐷2
529
𝑝3
95
𝐶3
23,020
𝑃3
7,092
𝐷3
360
Table 7 Coalitions and corresponding supply chain profit
Sr. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Coalition Buyer 1 ✔ ✔ ✔ ✔ ✔ ✔ ✔
Manufacturer ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔
Buyer 2 ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔
Buyer 3 ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔
Total profit 0 0 0 0 17,517 21,358 10,014 0 0 0 40,284 32,702 28,811 0 51,783
Table 8 Solution from Scheme IV (D1 + D2+ D3 = D = 1,330)
Variable
Value
Variable
Value
Variable
Value
Variable
Value
𝑇
51,783
𝛼
0.0
𝑛
4
𝑚
1
0.49
𝑃
25,503
𝑤
62
𝑝1
106
0.18
𝑃1
9,373
𝐷1
441
𝑝2
106
0.22
𝑃2
11,311
𝐷2
529
𝑝3
95
0.11
𝑃3
5,596
𝐷3
360
Contribution index of manufacturer Contribution index of buyer 1 Contribution index of buyer 2 Contribution index of buyer 3
Table 9 Profit share and percentage profit margin at different levels of cost parameters of buyer 3
Profit to chain partner
Model
No change
Chain profit
Manufacturer Value
margin
Buyer 1
Buyer 2
Value margin
Value
margin
Buyer 3 Value margin
III
51,783
25,609
30.80
8,687
30.80
10,395
30.80
7,092
30.80
IV
51,783
25,503
30.68
9,373
33.24
11,311
33.52
5,596
24.31
III
51,277
25,293
30.47
8,599
30.47
10,288
30.47
7,099
30.47
IV
51,277
25,251
30.42
9,371
33.21
11,310
33.52
5,344
22.94
III
50,775
24,971
30.17
8,515
30.17
10,184
30.17
7,105
30.17
IV
50,775
25,001
30.21
9,370
33.20
11,310
33.51
5,094
21.63
Doubled
Tripled
Table 10 Comparison of the results obtained from LINGO and with the use of GA and TLBO
LINGO
In use of GA
In use of T
CPU Time (secs) Ex. No.
1 2 3 4 5
Number
of Buyers
3 5 7 8 10
Overall Profit (A)
51,783 94,194 125,190 148,543 179,692
CPU Time (secs)
0.110 0.110 0.120 0.140 0.150
Overall Profit (B)
51,061 91,943 121,820 144,226 173,110
Gbest
44 87 104 167 173
Up to
Up to
Gbest
Gtot
0.010 0.030 0.040 0.090 0.110
0.044 0.075 0.126 0.146 0.193
CPU Time (se (1-B/A)×100
1.39% 2.39% 2.69% 2.91% 3.66%
Overall Profit (C)
Ibest
50,933 91,406 121,282 142,736 170,152
3 4 4 4 6
Up to Ibest
0.004 0.005 0.014 0.020 0.020
Up t
0. 0. 0. 0. 0.
Author contributions statement Prof. Anil K Agrawal and Susheel Yadav conceived the idea of the project. Prof. Anil K Agrawal and Susheel Yadav planned the algorithms and numerical studies. Susheel Yadav performed the numerical computations. Prof. Anil K Agrawal and Susheel Yadav both contributed towards the final manuscript. Prof. Anil K Agrawal supervised the project.