Optimal Combined Purchasing Strategies for a Risk-Averse Manufacturer Under Price Uncertainty

Purpose: Our paper is to analyze optimal purchasing strategies when a manufacturer can buy raw materials from a long-term contract supplier and a spot market under spot price uncertainty. Design/methodology/approach: The procurement model is solved by using dynamic programming. First, we maximize the DM’s utility of the second period, obtaining the optimal contract quantity and spot quantity for the second period. Then, maximize the DM’s utility of both periods, obtaining the optimal purchasing strategy for the first period. We use a numerical method to compare the performance level of a pure spot sourcing strategy with that of a mixed strategy. Findings: Our results show that optimal purchasing strategies vary with the trend of contract prices. If the contract price falls, the total quantity purchased in period 1 will decrease in the degree of risk aversion. If the contract price increases, the total quantity purchased in period 1 will increase in the degree of risk aversion. In period 2, the relationship between the optimal contract quantity and the degree of risk aversion depends on whether the expected spot price or the contract price is larger. Finally, we compare the performance levels between a combined strategy and a spot sourcing strategy. It shows that a combined strategy is optimal for a riskaverse buyer.


Introduction
Prices of raw materials fluctuate a lot in the era of globalization.The fluctuation brings a great risk to the procurement process of a company.Many manufacturers fall into trouble as they may suffer huge losses.For example, a can making company lost 20 million in 2010, because this company stored 10,000 tons of steel plate and the price fell 2000 yuan one ton.
The procurement cost of raw materials accounts for 60-80% of the revenue in manufacturing.
The fluctuation of raw-material prices greatly influences the stability of profits.Therefore, many companies are adopting proper strategies to control procurement cost.For example, Hewlett-Packard (HP) has implemented an active procurement risk management (PRM) program.Different and flexible purchasing methods, such as a long-term contract, a spot market and an option contract, are used to meet demands based on a project's risk evaluation in this program (Nagali, Hwang, Sanghera, Gaskins, Pridgen, Thurston et al., 2008).
Traditionally, a manufacturer can buy raw materials from a supplier via a long-term contract.A purchasing price and a quantity are specified in a long-term contract.Recently, the spot market plays an important role in resource allocation, as many raw materials are traded in the spot market.Many online exchange markets emerge with the rapid development of information technology, such as ChemConnect for chemical products, E-Steel for steel and Converge for semiconductors, which significantly reduce the transaction cost through a spot market.
Therefore, more and more spot purchasing is used for raw materials.It is recognized that a long-term contract and a spot market can be combined to manage the procurement risk in a volatile environment.
In this paper a manufacturer need to purchase raw materials to meet the demand of two periods.Our aim is to analyze the optimal purchasing strategies when the manufacturer can buy raw materials from a long-term contract supplier and a spot market.The remainder of the paper is organized as follows.In section 2, we review the relevant research.In section 3, we describe our purchasing model in detail, and derive the optimal purchasing strategies.Section 4 compares the performance level of a combined strategy with that of a spot sourcing strategy.
We give a summary of important results in Section 5.

Literature Review
Our work is to analyze optimal purchasing strategies when a manufacturer can buy raw materials from a long-term contract supplier and a spot market.A lot of literature in operations management has addressed different aspects of procurement.The most relevant literature is the research on dual sourcing and sourcing with a spot market.
The effect of dual sourcing on optimal inventory policies is studied by some scholars.Donohue (2000) studies efficient supply contracts for fashion goods with two production mode.The buyer may order goods at a lower price at the beginning of a sales period; or make additional order at a higher price in the sales period.Chung and Flynn (2001), and Warburton and Stratton (2005) discuss a newsboy problem with two ordering opportunities.The later ordering point can be regarded as a spot purchase.
Optimal purchasing strategies are then studied in the presence of a spot market.Serel, Dada, and Moskowitz (2001) examine sourcing decisions of a firm in the presence of a spot market.
Their study shows that inclusion of a spot sourcing reduces the capacity commitments from a long-term supplier.Moreover, Serel (2007) study capacity reservation under supply uncertainty.Lee and Whang (2002) consider the impact of a secondary market, where buyers can trade their excess inventory.They show that the introduction of a secondary market will improve allocative efficiency but the welfare of the supplier may not increase.In these works, the spot price is deterministic.
Spot price uncertainty is considered for an inventory policy by Cohen and Agrawal (1999).
They evaluate the tradeoff between long-term contracts and short-term contracts.Peleg, Lee and Hausman (2002) compare procurement strategies among three arrangements, a long-term contract, online search and a combination strategy.Kleindorfer et al. (Kleindorfer & Wu, 2003, 2005;Wu, Kleindorfer, & Zhang, 2002) have studied procurement problems via integrating long-term and short-term contracts.Their research mainly focuses on capital-intensive industries, and has contributed a lot to this area.
Araman, Kleinknecht and Akella (2001) model a buyer who can procure either through a contract, a spot market, or a combination of both.A spot market is used when preserved capacity can not fulfill demand.It is demonstrated that the spot market is beneficial from the perspective of the buyer.Seifert et al. (2004) develop a model for a buyer's optimal strategy; their results show that significant profit improvement can be achieved by adopting a combined strategy.Arnold and Minner (2011) find the best mix of advance procurement, spot market procurement, and financial options to satisfy demand.A multi-period setting is not considered in their works, which is one main concern in our paper.
Recently, some authors are exploring the purchasing problem in a multi-period setting.Ganeshan, Boone and Aggarwal (2009) show managers can integrate risk management tools to mitigate risk over multiple time periods.Inderfurth and Kelle (2011) show that the combined strategy is superior over single sourcing strategy if there is large spot price variability.But risk aversion is not addressed in their papers.Kouvelis, Li and Ding (2013) study a procurement problem for a risk-averse buyer who procures a single commodity from a supplier via a long-term contract and via short-term purchases from a spot market.Multi-period optimal inventory and financial hedging policies are obtained.Our research model is similar to their model, as we study a two-period procurement problem for a risk-averse manufacturer.The main difference is that the contract price changes in our model, and it remains constant in theirs.We investigate how the trend of the contract price influences optimal purchasing strategies.In addition, we compare a combined strategy and a single sourcing strategy based on their profits and utilities.And we also analyze how the profits and utilities depend on the degree of risk aversion.

Optimal Procurement Strategy Combining Contract Market and Spot Market
Considering a raw-material purchase for a manufacturer, this company will need two batches of raw materials in 3 months and 6 months later (marked as period 1 and period 2 below, Figure 1), one batch for each period.Raw materials are used to produce end products for customers, and the amount of materials needed is proportional to the output of end products.
The manufacturer should buy materials to meet the demand for production, and try to reduce the purchasing cost.The manufacturer can buy materials from a supplier via a long-term contract or from a spot market.At the beginning of period 1, the inventory level is x 1.The spot price p 1 is unknown.A decision maker (DM) decides the order quantity from a supplier for period 1.At the end of period 1, p 1 is realized.The quantity ordered q 1 is received and demand D 1 occurs.And the DM decides quantity Q 1 to purchase from a spot market.Insufficient materials should be purchased from the spot market in shortage, or excess materials are left for use at period 2.
At the beginning of period 2, the inventory level is x 2. The spot price p 2 is unknown.Similarly, the DM decides the order quantity for period 2. At the end of period 2, p 2 is realized.
Insufficient materials are purchased from a spot market, or excess materials are sold in the spot market.The notations used in this paper are given in Table 1.Demand for production in each period must be satisfied, that is, shortage of materials is not allowed.x 1 equals zero, and x 2 is not less than zero.The spot price of raw materials is exogenous, determined by the raw materials market.The sales price is assumed to be higher than the corresponding cost of materials consumed, that is r This procurement model can be solved by using dynamic programming.First, we maximize the DM's utility of the second period, obtaining the optimal contract quantity and spot quantity for the second period.Then, maximize the DM's utility of both periods, obtaining the optimal purchasing strategy for the first period.
The manufacturer's profit is the revenue of products minus the cost of raw materials, so the profit in period 2 is where the first item in RHS is the revenue of products, the second item is the ordering cost via a long-term contract, and the third item is the purchasing cost via a spot market.
Correspondingly, the DM's mean-variance utility function (Gan, Sethi & Yan, 2004) of period 2 is where k is bigger than 0. The larger k is, the more conservative the DM is.
At the beginning of period 2, the DM determines an ordering policy (q 2 , Q 2 ) so as to maximize his utility.That is (1) The first constraint implies that demand for production in period 2 is satisfied.
The DM determines an ordering policy (q 1 , Q 1 ) at the start of period 1, maximizing the sum of utilities for both periods. (2)

Optimal Purchasing Strategy in Period 2
Proposition 1.The optimal contract quantity and spot quantity purchased are , in period 2, if ; The optimal purchasing strategy is in period 2, if .
Proof:  From the two cases above, we can obtain that the buyer's expected profit is not necessarily higher in a mixed sourcing model than in a spot sourcing model.However, the buyer's utility of using a combined strategy is always higher.The profit when buying via a spot market has a bigger variance, resulting in a lower utility.As k increases, both of the utility levels will decrease.
The utility of using a combined strategy decreases less, and the utility of using a spot market drops more.Therefore, a combined strategy is the optimal purchasing strategy for a riskaverse DM.

Conclusion
This paper studies a two-period purchasing model for a risk-averse manufacturer who procures raw materials under spot price uncertainty.The manufacturer can procure raw materials via a long-term contract and a spot market.Optimal purchasing strategies are obtained through dynamic programming.
Our results show that optimal purchasing strategies in period 1 are different, depending on the trend of the contract price.When the contract price falls, the total quantity purchased in period 1 decreases in k.When the contract price increases, the total quantity purchased in period 1 increases in k.Then the purchasing strategy in period 2 will be determined.The relationship between the optimal contract quantity and the degree of risk aversion k depends on whether the expected spot price or the contract price is larger in period 2. The contract quantity and spot quantity in each period are also given.
Finally, the performance levels are compared while the manufacturer adopts a combined strategy and a spot sourcing strategy.A numerical analysis shows that a combined strategy is optimal for a risk-averse buyer.The buyer's risk aversion factor k has a smaller influence on the utility of a mixed strategy than on the utility of a spot sourcing strategy.
This study provides a solution that is easy to implement in practice for a buyer who is confronted with the price risk of raw materials.A long-term contract and a spot market can be combined to reduce procurement risk and help to increase operational flexibility.

Figure 1 .
Figure 1.A two-period procurement model

Figure 3 .
Figure 3. Utility comparison when the contract price falls

Figure 4 .
Figure 4. Profit comparison when the contract price increases