An Available-to-Promise Production-Inventory System with Pseudo Orders joint work with Susan Xu University of Dayton Penn State University MSOM Conference, June 6, 2008
Outline Introduction 1 Introduction Motivation Research Questions 2 Pseudo Order Model Order Promising Model 3
Outline Introduction Motivation Research Questions 1 Introduction Motivation Research Questions 2 Pseudo Order Model Order Promising Model 3
What is a pseudo order? Motivation Research Questions Pseudo Orders Intended purchase orders, short-term forecasts, (imperfect) advance demand information in a B2B environment Subject to change and cannot be enforced Contain information of the likelihood of becoming actual orders, due date, requested quantity, etc. Trade-off Maintained and revised by sales personnel Lumpy, nonstationary, volatile, highly uncertain The presence of pseudo orders makes confirmed orders less likely to be accepted because it is more desirable to reserve limited resources for future higher value pseudo orders.
A Pseudo Order Example Motivation Research Questions Siebel.com, (Forbes, January 21, 2002) Acting upon the information of the sudden cancellation of hundreds of potential deals in February 2000, Thomas Siebel (CEO) anticipated the recession months ahead of rivals and economists. He realigned his sales force, readjusted resource allocation decisions, and avoided the worst in 2001. However, Pseudo order information is not well-integrated into business planning and control systems. The cost of ignoring such information can be very high.
Available-To-Promise System Motivation Research Questions What is an ATP system? A business function that matches incoming customer orders to planned resources Different from traditional planning, scheduling and inventory management processes Operate within a short-term operational environment Most resources are considered fixed because of procurement leadtime limitations Deal with multiple customer classes
Motivation Research Questions Production Capacity Comitted Orders Pseudo Orders Component Availability accept/process orders ATP System reserve comp avail & prod cap supplier order lead time modify supplier orders Inventory Mgmt System
Motivation Research Questions ATP Examples: Dell and Toshiba Dell two-stage order promising practice Customer Differentiation: Home, Small Business, Medium & Large Business, Government, etc. Provide initial soft confirmation via email Generate hard confirmation after checking resource availability, based on batch ATP Toshiba electronic product ATP system Orders for several thousand models are collected and processed by a single central order processing system ATP execution every 1/4 1/2 hour Book pseudo orders up to 10 weeks in advance of delivery
Research Questions Introduction Motivation Research Questions Research Questions How to model the lumpy, non-stationary and volatile natures of pseudo order information? What is the optimal order promising policy in an ATP system with pseudo order information? How robust is the optimal policy? What are the costs of using suboptimal policies? What is the value of the pseudo order information?
Outline Introduction Pseudo Order Model Order Promising Model 1 Introduction Motivation Research Questions 2 Pseudo Order Model Order Promising Model 3
Pseudo Order Model Introduction Pseudo Order Model Order Promising Model Three major characteristics of pseudo orders Lumpiness: non-negligible probability of cancellation Non-stationarity: demands are not identically distributed Volatility: attributes change before either confirmed or cancelled. For each future pseudo order, Random demand distribution: Y k t (e k ) F k e k, where e k E k is a distribution state, evolving according to a Markov chain, q k t (e k e k). Random confirmation date: s k evolves according to h k t (s k s k).
Pseudo Order Model Order Promising Model An Example: Zero-Inflated Poisson Distribution Two dist. states: E = { 0, 1 }, cancellation or PP(λ) Time-homogeneous transition probabilities of distribution states [ ] [q k 1 0 ( )] = π k (1 π k ) However, if cancellation information is unknown, the demand distribution is the mix of mass 0 and PP(λ), resulting in Zero-Inflated Poisson (ZIP) distribution { P(Y k π k + (1 π k )e λ, if j = 0, = j) = (1 π k )e λ λ j /j!, if j > 0. Conclusion Information updates can remove one source of uncertainties.
Order Aggregation Scheme Pseudo Order Model Order Promising Model Less volatile, aid ATP decision making, increase operations and computation efficiency Aggregate demands with order confirmation dates s: X t,s = Yt k (e k ) k Fe k k { k:s k =s } Aggregated demands are temporally dependent, governed by P { E t 1 E t } = k K t h k t (s k s k ) q k t (e k e k )
Pseudo Order Model Order Promising Model Order Aggregation Scheme: Poisson Example Two dist. states E = {0, 1}, cancellation or PP(λ) System state can be simplified to the total number of uncancelled orders E t = (n t,1,..., n t,t 1 ). Aggregated demand X t,s follows PP(n t,s λ). Transition probability of P t { E t 1 E t } is given by P t ((n t 1,1, n t 1,2,..., n t 1,t 1 ) (n t,1, n t,2,..., n t,t 1 )) = t 1 n t,s! s=1 n(0 s)!(n t,s n(0 s))! πn(0 s) (1 π) nt,s n(0 s) n ( t 1 (nt,s n(0 s))! ) t 1 s=1 n(1 s)! n(t 1 s)! s =1 (h t(s s)) n(s s). Conclusion Our Markov chain model completely describes the evolution of pseudo orders at both the individual and the aggregate level.
Pseudo Order Model Order Promising Model Assumptions for the Order Promising Model T-period ATP system with MTO manufacturing strategy Multiple classes of orders bring in revenue r 1 > > r I Each order consumes one unit of production capacity, takes a single production period Pseudo order forecast E t is updated by P(E t 1 E t ) Newly confirmed orders N t Accepted orders x t = {x i t} must be fulfilled within L periods Production: first-accepted, first-served
Pseudo Order Model Order Promising Model Sequence of Events in Each Period t N t orders confirmed Future pseudo orders updated to E t Observe net capacity Q t Planned capacity K t becomes available Decision: accept orders x t = {x i t : i I} Objective Maximize the expected total profit over the ATP execution horizon T
Pseudo Order Model Order Promising Model A Markov Decision Process Formulation V t (Q t, N t, E t ) = max x t A t The action space A t is defined by Nonlinear system dynamics r x t p(q t + K t x t ) + + E t 1 N t 1 p(n t 1 E t )P(E t 1 E t ) V t 1 (Q t 1, N t 1, E t 1 ),(1) 0 x t N t, (2) x t Q t + [K] t t L. (3) Q t 1 = [Q t + K t x t ] 0. (4)
Pseudo Order Model Order Promising Model Characterization of the Optimal Policy Optimal order acceptance policy Accept in an increasing order of the index Reject class i if class i 1 are not fully accepted Accept class i until 1 all N i t are accepted (demand) 2 cumulative leadtime capacity for i is exhausted (supply) 3 the net capacity rationing level is reached (rationing) Formally, for class i I, the optimal acceptance is N t, i ˆx t i [ = min Qt + [K] t t L [N] i 1 ] +, 1 [ Qt + K t [N] i 1 1 ηt 1 i (E t) ]. (5) +
Pseudo Order Model Order Promising Model Contributions of the Characterization Explicitly reveal the dependence on demand quantity, lead time capacity, and capacity rationing level in a simple form Result in (Q t, N t )-state independent threshold η i t 1 (E t), depending on forecast only Ease the Curse of Dimensionality for such multi-dim MDP O(Q E I) v.s. O(Q E N I ) For example, if N = 100, I = 3, save 0.3 10 6 times in computation efforts!
Outline Introduction 1 Introduction Motivation Research Questions 2 Pseudo Order Model Order Promising Model 3
Introduction When are rationing and pseudo order information necessary? What are the costs of using suboptimal policies? Is it beneficial to use short term volatile forecast, or just use long term forecast? How robust is the optimal policy?
Experiment Design Introduction Two-class inventory ATP system, horizon T = 10 4 demand settings: { SL, SH, NL, NH } 3 resource availability ρ levels: scarce, intermediate, ample ρ = S/[EX 1 t + EX 2 t ] 3 profit ratio γ = r 1 /r 2 levels, r 1 + r 2 = 10 Holding cost: h = 0.5 2 lead time levels: L = { 0, 2 } 72 scenarios, each generates 100 instances, total 7200 instances
Introduction OPT: rationing with complete pseudo order information MVE: rationing with mean demand, ignore stochasticity PRO: priority rule only, ignore pseudo order information FS: fair share or first-come first-served, ignore both prioritization and pseudo order information Performance Gap: percentage difference of total profits, e.g., V M = [V V M ]/V 100%
What are the costs of using suboptimal policies? V FS V PRO V MVE V Prioritization is effective regardless of the capacity level Rationing with mean value is necessary when the capacity level is low to intermediate Stochasticity of pseudo orders cannot be ignored when capacity is at intermediate level
2 2 Introduction Policy selection: partitioning of the parameter space 5.0 4.5 4.0 Profit Ratio: γ 3.5 3.0 2.5 MVE OPT 2 2 2.0 1.5 1.0 2 2 2 2 PRO / MVE 0.2 0.4 0.6 0.8 1.0 1.2 Resource Availability: ρ 2 FS / PRO / MVE
Are later due dates always beneficial? Figure: Impact of Lead Time L OPT, MVE and FS benefit from increased lead time resource availability PRO may suffer from later due date! Customer cannibalization: larger percentage of class-2 acceptance due to increased resource availability
Value of Pseudo Order Information Updates Volatility and dynamic information availability Question: Given the volatile nature of pseudo orders, is it beneficial to use the short term forecast, or just use long term forecast? The percentage difference of the systems with and without updating quantifies the value of pseudo order updating
2 4 2 4 6 6 4 2 Introduction Value of Pseudo Order Information Updates Updating is always beneficial Significant region: V 2%, scarce capacity, heterogenous customers Profit Ratio γ 3.0 2.8 2.6 2.4 2.2 2.0 1.8 6 4 2 In this region, updating can further strengthen the effectiveness of rationing by 2% 7% 1.6 1.4 1.2 significant region: V * S > 2% 2 4 2 1.0 0 0.2 0.4 0.6 0.8 1 Resource Availability ρ
How robust is the optimal policy? Robustness What if the the forecast is inaccurate? What if the underlying distributions changed? Is OPT still better than others, especially forecast independent policies?
Robustness Comparison Table: Robustness for Forecast Errors over 4200 Instances Forecast Errors Dominance over suboptimals Type (ε 2, ε 3) µ 1 % cv 1 % P MVE % P PRO % P FS % I (+3, +3) 25.42 2.32 64.74 71.85 85.45 II ( 3, +3) 5.08 26.67 76.73 94.42 100.00 III ( 3, 3) 25.42 0.97 87.35 100.00 100.00 IV (+3, 3) 5.08 3.61 83.08 93.76 98.85 Overall (±3, ±3) ±15.25 ±8.58 77.98 90.01 96.08 OPT is robust for small to moderate forecast errors. OPT should be implemented with forecast updating mechanisms.
Conclusions Introduction We quantify lumpy volatile pseudo order information, and characterize the optimal order acceptance policy. Commonly used policies may suffer severe losses due to ignoring pseudo order information and rationing. Prioritization without rationing may reduce the profitability with extended due dates! OPT is fairly robust and should be implemented with pseudo order updating mechanisms.
Future Research Introduction Multiple components, class-specific lead time Impact of pseudo order information on strategic or tactical resources planning Random supply and production processes Other Applications: Hub Group, Inc. intermodal shipping load acceptance
Characterization of the Value Function: Proof Sketch of the Proof Difficulty: nonlinear dynamics of capacity Induction for three cases: both positive, both negative, and one each Observe that: r 1 V t 1 (Q t 1 E t ) p Use complementary property of max{x, 0} and min{x, 0}: at least one of them is 0 Regarding N t, there is no lost sale penalty and the action space is convex
Literature Review: Rationing Continuous time rationing models Ha. (1997) Benjaafar & ElHafsi (2006) Discrete time rationing models Topkis. (1968) Only one nonperishable resource, available at the beginning, no pseudo orders Wang and Gupta. (2007) Two classes, single resource, no pseudo orders Our model deals with multi-period, both perishable and nonperishable resources, incorporating pseudo orders
Characterization of the Optimal Value Function Lemma (i) V t (Q t, N t, E t ) is concave in net capacity Q t. (ii) V t (Q t, N t, E t ) is increasing concave in realized demand N t. Managerial Insights Marginal value of unit capacity diminishes when capacity increases. Carefully plan and allocate capacity over time [T, 1], using pseudo order information. Marketing activities on demand management, such as order expedition and postpone, need to be coordinated with the planned resources.