Procurement Decisions in Multi-period Supply Chain

. Pricing and ordering decision in multi-period supply chain environments is not explored comprehensively. We consider three pragmatic procurement scenarios where the retailer can procure products (i) by maintaining strategic inventory, (ii) in bulk in ﬁrst-period and distribute them in forthcoming selling period, and (iii) without maintaining any inventory. The results suggest that conventional single period planning exhibit sub-optimal characteristics. Build-up strategic inventory is not always proﬁtable for the retailer. The retailer can also earn more proﬁts by employing a bulk procurement strategy.


Introduction
Efficient inventory management is one of the key issues in retailing.Retailers maintain inventory to reduce transportation cost, take advantage of quantity discounts, ensure continuity of selling activities, evade variations in wholesale price and demand etc. ( [4], [8], [9]).However, Anand et al. [1] reported that retailer's decision to maintain inventory in multi-period supply chain interactions under manufacturer-stackelberg game can reduce the degree of double marginalization.They found that the retailer can force the manufacturer to reduce the wholesale price of forthcoming periods by maintaining surplus order quantities as strategic inventory.Arya and Mittendorf [3] proved that the manufacturer can curtail advantage of the retailer in building strategic inventory by introducing consumer rebate.Consumer rebate prevents the retailer to maintain high amounts of SI.Arya et al. [2] extended this enticing stream of research and compare the effect of SI in the presence of multiple retail outlets.Hartwig et al. [5] conducted empirical investment to explore the effect of SI and found that the retailer can immensely induce differentiated wholesale pricing behaviour by building up SI.Mantin and Jiang [6] explored the impact of the product quality deterioration in the presence of SI.Moon et al. [7] analyzed the impact of SI in perspective of supply chain coordination.They found that the optimal supply chain profit cannot be achieved by implementing quadratic quantity discount contract mechanism.All the above cited contributions consider multi-period interaction among supply chain member to explore the consequences of SI.In the existing literature on supply chain models, it is assumed that the retailer procures products to satisfy demand in each selling period.However, in practice, the retailer maintains SI to satisfy future demand.But, to the best of the authors' knowledge, the advantage of SI is not fully explored in current state.We consider three procurement decision for the retailer and explore the pricing and ordering behaviour under five consecutive selling period.It is found that the pricing behaviour is correlated with procurement decision.The singleperiod procurement decision always leads to suboptimal solution.The supply chain members can receive higher profit if the retailer maintains SI or procures in bulk.

Problem description
We explore the interaction in a serial supply chain with one retailer and one supplier under price-sensitive demand in a fifth-period game.The retailer in the supply chain has a downstream retail monopoly and rely solely on the upstream supplier for the retailed good.Three procurement strategies are considered.In first procurement strategy (WSI), the retailer may maintain SI in between twoconsecutive selling period.In Second procurement strategy (BP), the retailer procures in bulk for the first selling period and distribute those in forthcoming periods.Third procurement strategy (BM) is similar to the conventional literature, where the retailer procures products to satisfy demand for each period.We consider linear price sensitive demand and derive optimal solution.For feasibility of the optimal solution, it is assumed that the retail (p t ) and wholesale prices (w t ) at each period satisfy the following relations p t > w t > 0, ∀t = 1, ...., 5.The unit holding cost for the retailer is h.All the parameters related to market demand are common knowledge between supply chain members [5].

Optimal decision in the presence of SI
At the beginning of each period (t = 1, • • • , 5), the supplier determines a wholesale price (w wsi t ).The retailer then procures (Q wsi t ) amounts of product and sets retail price (p wsi t ) to satisfy market demand (q wsi t = a − bp wsi t ).If the procured quantity at each period is larger than the quantity sold in the that period (i.e., if Q wsi t > q wsi t ), then the retailer builds up SI (I wsi t = Q wsi t − q wsi t ) to be sold in the immediate period and invests hI wsi The optimal solution for the retailer fifth-period optimization problem presented in the first equation is obtained by solving )) 6 . Substituting optimal response, the profit function for the supplier is obtained as 2 )

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. After solving first order condition, the wholesale price for the fourth period is obtained as w wsi , respectively.Substituting optimal response for the retailer, profit function for the supplier in third-period is obtained as follows: , respectively.Substituting optimal response, the profit function for the supplier in second-period is obtained as Substituting the optimal response for the re-tailer, the profit function for the supplier in first-period is obtained as follows: and corresponding wholesale price is w wsi 1 = 0.566697a−0.548333bhb .Note that the first-period optimization problem for the retailer and supplier are concave as

Optimal decisions in Scenario BP
At the beginning of first period, the supplier determines a wholesale price (w bp 1 ) and then the retailer procures a − bp bp 1 + 4 t=1 I bp t unit of products and sets the retail price (p bp 1 ).In next four selling period, the supplier determines wholesale price (w bp t ) and then the retailer procures (q bp t = a − bp bp t − I bp t−1 )(t = 2, • • • , 5) units of product and sets retail price (p bp t ) to satisfy market demand.The profit functions of the supplier and retailer for five consecutive selling periods are obtained as follows: The optimal solution for the retailer fifth-period optimization problem is obtained by solving  .We compute the following Hessian matrix to check concavity: The values of principal minors are ∆ 1 = −2b < 0; ∆ 2 = 3 > 0; ∆ 3 = − 9 2b < 0; ∆ 4 = 27 4b 2 > 0 and ∆ 5 = − 81 8b 3 < 0, i.e. profit function for the retailer is concave.Substituting the optimal response for the retailer, the profit function for the supplier in first-period is obtained as 2 ) 18 and the corresponding wholesale price is w bp 1 = 5(9a−8bh) 82b . By using back substitution, one can obtain the following optimal solutions:

Benchmark model
In Scenario BM, the retailer does not maintain SI or procure products in bulk.
The profit functions for the retailer and supplier in each selling period are π bm r = (p bm − w bm )(a − bp bm ) and π bm m = w bm (a − bp bm ), respectively.One may obtain the optimal response function of the retailer by solving first order condition of optimization as p(w bm ) = a+bw bm The graphical representation of the profit functions of the retailer and supplier are shown in Figures 1a and 1b.
Figures 1a and 1b demonstrate the profits of the supply chain members if the retailer makes procurement planning for five consecutive cycle.It is found that Scenario BM is always outperformed by both scenarios BP and SI.It is found that the profit functions of the retailer does not demonstrate a cumulatively pattern.Due to additional procurement in the first selling period, the profit functions demonstrate that nature.However, one can not conclude with regards to the optimality of the procurement planning of the retailer.
Price elasticity and product holding cost are two extremely important factors affecting procurement decision and overall profitability.Price-elasticity is a critical factor ([10], [11]) influencing the demand.Therefore, more analytical investigations are required to obtain concrete conclusion.

Conclusion
The pricing and procurement decisions in a supplier-retailer five-period supply chain is explored in this study.Under price sensitive demand, impact of three procurement decisions are analyzed and corresponding Stackelberg equilibriums are compared.The comparison among equilibrium outcomes in perspective of profits of each supply chain members demonstrate how the procurement decision is influencing the overall preference of the supply chain members.In contrast to

dπ bp r5 dp bp 5 = 0 .= a+bw bp 5 2b.bp m5 ∂w bp 5 = 0 .a−2I bp 4 2b. 2 π si r5 dp bp 5 2 = −2b < 0 and d 2 π bp m5 dw bp 5 2=
On simplification, we have p bp 5 The optimal solution for the supplier fifth-period optimization problem is obtained by solving ∂π On simplification, one can obtain w bp 5 = The profit function for the retailer and supplier in fifth-period are concave because d −b < 0, respectively.Similar to previous subsection, the profit function for the retailer in first-period is obtained as follows: