Dynamic Pricing and Inventory Management under Fluctuating Procurement Costs

Transcription

1 1 Dynamic Pricing and Inventory Management under Fluctuating Procurement Costs Philip (Renyu) Zhang (Joint work with Guang Xiao and Nan Yang) Olin Business School Washington University in St. Louis June 20, 2014

2 2 Motivation Inventory management: To mitigate the demand uncertainty risk.

3 2 Motivation Inventory management: To mitigate the demand uncertainty risk. Current global market: Prices of many commodities are now fluctuating as much in a single day as they did in a year in the early 1990s (Wiggins and Blas 2008).

4 2 Motivation Inventory management: To mitigate the demand uncertainty risk. Current global market: Prices of many commodities are now fluctuating as much in a single day as they did in a year in the early 1990s (Wiggins and Blas 2008).

5 3 Motivation (Cont d) Dynamic Pricing: Dynamically adjust the sales price in each period. 1. Demand control effect. 2. Risk-pooling effect.

6 3 Motivation (Cont d) Dynamic Pricing: Dynamically adjust the sales price in each period. 1. Demand control effect. 2. Risk-pooling effect. Dual-Sourcing: spot purchasing + forward-buying. 1. Cost-responsiveness tradeoff: differentiated costs and leadtimes. 2. Portfolio effect among different sourcing channels.

7 3 Motivation (Cont d) Dynamic Pricing: Dynamically adjust the sales price in each period. 1. Demand control effect. 2. Risk-pooling effect. Dual-Sourcing: spot purchasing + forward-buying. 1. Cost-responsiveness tradeoff: differentiated costs and leadtimes. 2. Portfolio effect among different sourcing channels. Goal of our paper: To understand how to coordinate the dynamic pricing and dual-sourcing strategies to hedge against demand uncertainty and procurement cost fluctuation risks.

8 4 Research Questions 1. What is the impact of procurement cost volatility?

9 4 Research Questions 1. What is the impact of procurement cost volatility? 2. How to optimally respond to the cost fluctuation?

10 4 Research Questions 1. What is the impact of procurement cost volatility? 2. How to optimally respond to the cost fluctuation? 3. How does the dual-sourcing flexibility affect the optimal policy?

11 4 Research Questions 1. What is the impact of procurement cost volatility? 2. How to optimally respond to the cost fluctuation? 3. How does the dual-sourcing flexibility affect the optimal policy? 4. What is the relationship between dynamic pricing and dual-sourcing?

12 5 Outline Related Literature Model Impact of Cost Volatility Impact of Dual-Sourcing Conclusion: Takeaway Insights

13 6 Literature Review

14 6 Literature Review Inventory management under fluctuating costs: Kalymon (1971), Berling and Martínez-de-Albéniz (2011), Chen et al. (2013).

15 6 Literature Review Inventory management under fluctuating costs: Kalymon (1971), Berling and Martínez-de-Albéniz (2011), Chen et al. (2013). Joint price&inventory control: Federgruen and Heching (1999), Zhou and Chao (2014).

16 6 Literature Review Inventory management under fluctuating costs: Kalymon (1971), Berling and Martínez-de-Albéniz (2011), Chen et al. (2013). Joint price&inventory control: Federgruen and Heching (1999), Zhou and Chao (2014). Our paper: Joint pricing & inventory management under demand uncertainty, cost fluctuation, and dual-sourcing.

17 7 Model Formulation: Basics A risk-neutral firm modeled as a T period stochastic inventory system, labeled backwards, with discount factor α (0, 1). Maximize the total expected profit over the planning horizon.

18 7 Model Formulation: Basics A risk-neutral firm modeled as a T period stochastic inventory system, labeled backwards, with discount factor α (0, 1). Maximize the total expected profit over the planning horizon. Endogenous pricing.

19 7 Model Formulation: Basics A risk-neutral firm modeled as a T period stochastic inventory system, labeled backwards, with discount factor α (0, 1). Maximize the total expected profit over the planning horizon. Endogenous pricing. Dual-sourcing: Spot market: immediate delivery, fluctuating cost ct. Forward-buying contract: postponed delivery, with unit cost Ft (c t ).

20 8 Sequence of Events The firm reviews inventory I t and spot market price c t.

21 8 Sequence of Events The firm reviews inventory I t and spot market price c t. The firm makes the following decisions: xt I t 0: spot-purchasing, delivered immediately; qt 0: forward-buying, delivered at the beginning of the next period; pt [p, p]: sales price in the customer market.

22 8 Sequence of Events The firm reviews inventory I t and spot market price c t. The firm makes the following decisions: xt I t 0: spot-purchasing, delivered immediately; qt 0: forward-buying, delivered at the beginning of the next period; pt [p, p]: sales price in the customer market. Demand D t (p t ) realized, revenue collected.

23 8 Sequence of Events The firm reviews inventory I t and spot market price c t. The firm makes the following decisions: xt I t 0: spot-purchasing, delivered immediately; qt 0: forward-buying, delivered at the beginning of the next period; pt [p, p]: sales price in the customer market. Demand D t (p t ) realized, revenue collected. Net inventory fully carried over to the next period: Excess inventory fully carried over with unit cost h; Unsatisfied demand fully backlogged with unit cost b.

24 9 Demand Model D t (p t ) = d(p t ) + ϵ t. ϵ t : independent continuous random variables, with E{ϵ t } = 0. d( ): strictly decreasing function of p t, with a strictly decreasing inverse p( ) in the expected demand, d t.

25 9 Demand Model D t (p t ) = d(p t ) + ϵ t. ϵ t : independent continuous random variables, with E{ϵ t } = 0. d( ): strictly decreasing function of p t, with a strictly decreasing inverse p( ) in the expected demand, d t. We use d t = d(p t ) [d, d] as the decision variable.

26 9 Demand Model D t (p t ) = d(p t ) + ϵ t. ϵ t : independent continuous random variables, with E{ϵ t } = 0. d( ): strictly decreasing function of p t, with a strictly decreasing inverse p( ) in the expected demand, d t. We use d t = d(p t ) [d, d] as the decision variable. Assumption 1 R(d t ) := p(d t )d t is continuously differentiable and strictly concave.

27 10 Spot-Market Price Fluctuation c t 1 = s t (c t, ξ t ). ξ t : The random perturbation in the cost dynamics.

28 10 Spot-Market Price Fluctuation c t 1 = s t (c t, ξ t ). ξ t : The random perturbation in the cost dynamics. s t (, ) > 0 a.s., and s t (ĉ t, ξ t ) s.d. s t (c t, ξ t ) for any ĉ t > c t.

29 10 Spot-Market Price Fluctuation c t 1 = s t (c t, ξ t ). ξ t : The random perturbation in the cost dynamics. s t (, ) > 0 a.s., and s t (ĉ t, ξ t ) s.d. s t (c t, ξ t ) for any ĉ t > c t. µ t (c t ) := E{s t (c t, ξ t )} < + is increasing in c t. Perfect market: µt (c t ) = c t /α.

30 10 Spot-Market Price Fluctuation c t 1 = s t (c t, ξ t ). ξ t : The random perturbation in the cost dynamics. s t (, ) > 0 a.s., and s t (ĉ t, ξ t ) s.d. s t (c t, ξ t ) for any ĉ t > c t. µ t (c t ) := E{s t (c t, ξ t )} < + is increasing in c t. Perfect market: µt (c t ) = c t /α. Examples: GBMs, mean-reverting processes.

31 10 Spot-Market Price Fluctuation c t 1 = s t (c t, ξ t ). ξ t : The random perturbation in the cost dynamics. s t (, ) > 0 a.s., and s t (ĉ t, ξ t ) s.d. s t (c t, ξ t ) for any ĉ t > c t. µ t (c t ) := E{s t (c t, ξ t )} < + is increasing in c t. Perfect market: µt (c t ) = c t /α. Examples: GBMs, mean-reverting processes. Inventory resale is not allowed: no room for arbitrage.

32 11 Forward-Buying Contract To mitigate cost volatility at the expense of responsiveness.

33 11 Forward-Buying Contract To mitigate cost volatility at the expense of responsiveness. Forward-buying contract: (f t, q t ): The firm pays ftq t to the supplier in period t e ; The supplier delivers qt to the firm in period t e ; For technical tractability, t e = t 1.

34 11 Forward-Buying Contract To mitigate cost volatility at the expense of responsiveness. Forward-buying contract: (f t, q t ): The firm pays ftq t to the supplier in period t e ; The supplier delivers qt to the firm in period t e ; For technical tractability, t e = t 1. f t = γc t /α. Effective unit cost: γct. Perfect market: γ = 1. In general, ft is determined through bilateral negotiations. Most results hold for general ft = F t (c t ).

35 11 Forward-Buying Contract To mitigate cost volatility at the expense of responsiveness. Forward-buying contract: (f t, q t ): The firm pays ftq t to the supplier in period t e ; The supplier delivers qt to the firm in period t e ; For technical tractability, t e = t 1. f t = γc t /α. Effective unit cost: γct. Perfect market: γ = 1. In general, ft is determined through bilateral negotiations. Most results hold for general ft = F t (c t ). Focus on the operational effect of forward-buying. The contract cannot be traded in the derivatives market.

36 12 Bellman Equation V t (I t c t ) =the maximal expected discounted profit in periods t, t 1,, 1 with starting inventory level I t and cost c t in period t. Terminal condition: V 0 (I 0 c 0 ) = 0.

37 12 Bellman Equation V t (I t c t ) =the maximal expected discounted profit in periods t, t 1,, 1 with starting inventory level I t and cost c t in period t. Terminal condition: V 0 (I 0 c 0 ) = 0. Bellman equation: V t (I t c t ) =c t I t + max x t I t,q t 0,d t [d, d] J t (x t, q t, d t c t ), where J t (x t, q t, d t c t ) = c t I t + E{p(d t )D t c t (x t I t ) γc t q t h(x t D t ) + b(x t D t ) + αv t 1 (x t + q t D t s t (c t, ξ t ))} =R(d t ) c t x t γc t q t + Λ(x t d t ) + Ψ t (x t + q t d t c t ) with Λ(y) =E{ h(y ϵ t ) + b(y ϵ t ) }, and Ψ t (y c t ) =E{V t 1 (y ϵ t s t (c t, ξ t )) c t }.

38 13 Optimal Policy (x t (I t, c t ), q t (I t, c t ), d t (I t, c t )): the optimal decisions in period t. t (I t, c t ) := x t (I t, c t ) d t (I t, c t ): the optimal safety stock.

39 13 Optimal Policy (x t (I t, c t ), q t (I t, c t ), d t (I t, c t )): the optimal decisions in period t. t (I t, c t ) := x t (I t, c t ) d t (I t, c t ): the optimal safety stock. The cost-dependent order-up-to/pre-order-up-to list-price policy. If I t x t (c t ), order from both channels and charge a list price. If I t [x t (c t ), I t (c t )], order via the forward-buying contract only and charge a discounted price. If I t I t (c t ), order nothing and charge a discounted price.

40 14 Impact of Cost Volatility Higher demand variability lower profit.

41 14 Impact of Cost Volatility Higher demand variability lower profit. Surprisingly, the prediction is reversed for cost volatility. Theorem 1 For two procurement cost processes {c t } 1 t=t and {ĉ t} 1 t=t, assume that for every t = T, T 1,, 1, s t (c t, ξ t ) and ŝ t (c t, ξ t ) are concavely increasing in c t for any realization of ξ t. The following statements hold: (a) V t (I t c t ) is convexly decreasing in c t, for any I t. (b) If {c t } 1 t=t and {ĉ t} 1 t=t are identical except that ŝ τ (c τ, ξ τ ) cx s τ (c τ, ξ τ ) for some c τ and τ, ˆV t (I t c t ) V t (I t c t ) for each (I t, c t ) and t, where cx refers to larger in convex order, and { ˆV t (I t c t )} 1 t=t are the value functions associated with {ĉ t} 1 t=t.

42 15 Impact of Cost Volatility (Cont d) Higher cost volatility higher profit.

43 15 Impact of Cost Volatility (Cont d) Higher cost volatility higher profit. The fundamental difference between demand and cost risks: Demand risk: decisions made prior to demand realization. Betting on demand uncertainty.

44 15 Impact of Cost Volatility (Cont d) Higher cost volatility higher profit. The fundamental difference between demand and cost risks: Demand risk: decisions made prior to demand realization. Betting on demand uncertainty. Cost risk: decisions made posterior to cost realization. Responding to cost volatility.

45 15 Impact of Cost Volatility (Cont d) Higher cost volatility higher profit. The fundamental difference between demand and cost risks: Demand risk: decisions made prior to demand realization. Betting on demand uncertainty. Cost risk: decisions made posterior to cost realization. Responding to cost volatility. The impact of decision timing with respect to uncertainty realization in capacity management and newsvendor network models with responsive/postponed pricing: Van Mieghem and Dada (1999), Chod and Rudi (2005) and Bish et al. (2012).

46 16 Impact of Cost Volatility: Assumptions Risk neutrality is necessary for Theorem 1 to hold.

47 16 Impact of Cost Volatility: Assumptions Risk neutrality is necessary for Theorem 1 to hold. The concavity of s t (c t, ξ t ) generally can be satisfied (e.g., GBMs, mean-reverting processes).

48 16 Impact of Cost Volatility: Assumptions Risk neutrality is necessary for Theorem 1 to hold. The concavity of s t (c t, ξ t ) generally can be satisfied (e.g., GBMs, mean-reverting processes). When s t (c t, ξ t ) is not concave in c t, the result holds for the majority of numerical cases (except when the initial cost is low), in particular when the initial cost follows the stationary distribution.

49 17 Optimal Response to Cost Volatility Optimal sales price: d t (I t, c t ) c t, i.e., p t (I t, c t ) c t. The firm passes (part of) the cost risk to customers.

50 17 Optimal Response to Cost Volatility Optimal sales price: d t (I t, c t ) c t, i.e., p t (I t, c t ) c t. The firm passes (part of) the cost risk to customers. J t (x t, q t, d t c t ) =[R(d t ) c t d t ] + [Λ( t ) (1 γ)c t t ] + [Ψ t ( t + q t c t ) γc t ( t + q t )]. Three objectives: (a) generating revenue, (b) hedging against demand uncertainty, and (c) speculating on future costs.

51 17 Optimal Response to Cost Volatility Optimal sales price: d t (I t, c t ) c t, i.e., p t (I t, c t ) c t. The firm passes (part of) the cost risk to customers. J t (x t, q t, d t c t ) =[R(d t ) c t d t ] + [Λ( t ) (1 γ)c t t ] + [Ψ t ( t + q t c t ) γc t ( t + q t )]. Three objectives: (a) generating revenue, (b) hedging against demand uncertainty, and (c) speculating on future costs. Optimal safety-stock and spot-purchasing: t (c t ), x t (c t ) c t, if γ 1; t (c t ) c t, if γ > 1.

52 17 Optimal Response to Cost Volatility Optimal sales price: d t (I t, c t ) c t, i.e., p t (I t, c t ) c t. The firm passes (part of) the cost risk to customers. J t (x t, q t, d t c t ) =[R(d t ) c t d t ] + [Λ( t ) (1 γ)c t t ] + [Ψ t ( t + q t c t ) γc t ( t + q t )]. Three objectives: (a) generating revenue, (b) hedging against demand uncertainty, and (c) speculating on future costs. Optimal safety-stock and spot-purchasing: t (c t ), x t (c t ) c t, if γ 1; t (c t ) c t, if γ > 1. Optimal forward-buying quantity: generally not monotone in c t. Higher future cost trend d t (I t, c t ), x t (I t, c t ), t (I t, c t ), and q t (I t, c t ).

53 18 Impact of Dual-Sourcing Flexibility γ: the cost ratio between forward-buying and spot-purchasing. Lower γ implies higher dual-sourcing flexibility.

54 18 Impact of Dual-Sourcing Flexibility γ: the cost ratio between forward-buying and spot-purchasing. Lower γ implies higher dual-sourcing flexibility. Intuition suggests q t (I t, c t ) γ.

55 18 Impact of Dual-Sourcing Flexibility γ: the cost ratio between forward-buying and spot-purchasing. Lower γ implies higher dual-sourcing flexibility. Intuition suggests q t (I t, c t ) γ.due to procurement cost fluctuation, this may not be true in general: γ d t (I t, c t ), x t (I t, c t ), t (I t, c t ). q t (I t, c t ) may not be monotone in γ, because lower γ also decreases the marginal value of inventory in future periods.

56 18 Impact of Dual-Sourcing Flexibility γ: the cost ratio between forward-buying and spot-purchasing. Lower γ implies higher dual-sourcing flexibility. Intuition suggests q t (I t, c t ) γ.due to procurement cost fluctuation, this may not be true in general: γ d t (I t, c t ), x t (I t, c t ), t (I t, c t ). q t (I t, c t ) may not be monotone in γ, because lower γ also decreases the marginal value of inventory in future periods. When γ is big enough (γ sup{ αµ t(c t ) c t }), the model is reduced to one with sole-sourcing from spot market alone. Dual-sourcing d t (I t, c t ), xt (I t, c t ), t (I t, c t ) (Zhou and Chao, 2014).

57 19 Value of Dynamic Pricing and Dual-sourcing Dynamic pricing and dual-sourcing are strategic complements, i.e., the application of one strategy increases the value of the other.

58 19 Value of Dynamic Pricing and Dual-sourcing Dynamic pricing and dual-sourcing are strategic complements, i.e., the application of one strategy increases the value of the other. When sourcing from the less responsive forward-buying channel, the flexibility to control demand via pricing becomes more valuable. In Zhou and Chao (2014), they are strategic substitutes.

59 19 Value of Dynamic Pricing and Dual-sourcing Dynamic pricing and dual-sourcing are strategic complements, i.e., the application of one strategy increases the value of the other. When sourcing from the less responsive forward-buying channel, the flexibility to control demand via pricing becomes more valuable. In Zhou and Chao (2014), they are strategic substitutes. Compared with Zhou and Chao (2014), cost volatility renders the value of dynamic pricing and dual-sourcing significantly higher.

60 20 Conclusion: Takeaway Insights

61 20 Conclusion: Takeaway Insights A risk-neutral firm benefits from the procurement cost volatility.

62 20 Conclusion: Takeaway Insights A risk-neutral firm benefits from the procurement cost volatility. Timing of decision making and uncertainty realization.

63 20 Conclusion: Takeaway Insights A risk-neutral firm benefits from the procurement cost volatility. Timing of decision making and uncertainty realization. Dynamic pricing and dual-sourcing are strategic complements.

64 20 Conclusion: Takeaway Insights A risk-neutral firm benefits from the procurement cost volatility. Timing of decision making and uncertainty realization. Dynamic pricing and dual-sourcing are strategic complements. Dynamic pricing dampens both demand and cost risks, while dual-sourcing mitigates the cost risk but intensifies the demand risk.

65 21 Q&A Thank you! Questions?