Supply and Demand Uncertainty in Multi-Echelon Supply Chains Lawrence V. Snyder1
Z.-J. Max Shen2
1 Department
of Industrial & Systems Engineering Center for Value Chain Research Lehigh University Bethlehem, PA
2 Department
of Industrial Engineering & Operations Research University of California, Berkeley Berkeley, CA
INFORMS, Pittsburgh, PA November 6, 2006 Snyder and Shen (Lehigh and Berkeley)
Supply and Demand Uncertainty
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Outline 1
Introduction
2
Cost of Unreliability
3
Order Frequency
4
Inventory Placement
5
Supply Chain Structure
6
Cost of Reliability
7
Conclusions
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Introduction
Outline 1
Introduction Motivation Literature Review Roadmap Methodology and Assumptions
2
Cost of Unreliability
3
Order Frequency
4
Inventory Placement
5
Supply Chain Structure
6
Cost of Reliability
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Introduction
Motivation
Supply vs. Demand Uncertainty
Demand uncertainty (DU) Randomness in demand quantity, timing, product mix, etc. Purview of SCM/OM for decades
Supply uncertainty (SU) Disruptions Capacity/yield uncertainty Lead-time uncertainty etc. Increased attention only recently
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Introduction
Motivation
Are DU and SU the Same? Under both DU and SU, the main issue is the same: Not enough supply to meet demand May be irrelevant whether the mismatch came from DU or SU
Mitigation strategies are similar: Safety stock Multiple sourcing Improved forecasts Demand management Excess capacity etc.
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Introduction
Motivation
Are DU and SU the Same? Under both DU and SU, the main issue is the same: Not enough supply to meet demand May be irrelevant whether the mismatch came from DU or SU
Mitigation strategies are similar: Safety stock Multiple sourcing Improved forecasts Demand management Excess capacity etc.
The good news: We know a lot about supply chains under DU
The bad news: The “conventional wisdom” from DU is often wrong under SU
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Introduction
Literature Review
Literature Review
Classical inventory models + disruptions Parlar and Berkin (1991), Berk and Arreola-Risa (1994), Parlar and Perry (1995,1996), Gupta (1996), Mohebbi (2003,2004), many others
Classical inventory models + yield uncertainty Gerchak et al. (1988), Bassok and Akella (1991), Yano and Lee (1995), Wang and Gerchak (1996), many others
Strategic questions Tomlin (2006): optimal mitigation strategy Tomlin and Snyder (2006): advanced warning Lewis, Erera, and Whilte (2005): border closures Chopra, Reinhardt, and Mohan (2005): “bundling” disruptions and yield uncertainty
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Introduction
Literature Review
Multi-Echelon Models
Very few multi-echelon models with disruptions Kim, Lu, and Kvam (2005): Yield uncertainty in 3-echelon supply chain, risk-averse objective Hopp and Yin (2006): Optimal placement and size of inventory and capacity buffers in assembly network
Must study disruptions in multi-echelon setting Disruptions are never local Cascading effect
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Introduction
Literature Review
A Newsboy-Style Result Theorem (Tomlin 2006) In a single-stage base-stock system with deterministic demand and stochastic supply disruptions, the optimal base-stock level is given by p ∗ −1 , S = d + dF p+h where d is the demand per period and F is the cdf of supply. F (x) = P(we are in a disruption lasting x periods or fewer) Cycle/safety stock interpretation Similar (but less sharp) result given by G¨ ull¨ u et al. (1997)
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Introduction
Roadmap
SU vs. DU: Roadmap
1
The cost of unreliability
2
Order frequency Inventory placement
3
Centralization vs. decentralization Upstream vs. downstream 4
Supply chain structure Hub-and-spoke vs. point-to-point Supplier redundancy Supplier flexibility
5
The cost of reliability
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Introduction
Methodology and Assumptions
Methodology
Some of our results are proved analytically Others we demonstrate using simulation BaseStockSim software Rough optimization of base-stock levels
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Introduction
Methodology and Assumptions
Supply Chain Assumptions Multi-echelon SC Each stage has processing function and output buffer:
2
1
May represent physical location, processing activity, or SKU
Backordered demand Costs h, p Processing (lead) time T Under DU, demands are N(µ, σ 2 ) Under SU, disruption process follows 2-state Markov process Disruption probability α Recovery probability β
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Cost of Unreliability
Outline 1
Introduction
2
Cost of Unreliability
3
Order Frequency
4
Inventory Placement
5
Supply Chain Structure
6
Cost of Reliability
Conclusions Introduction Motivation Snyder and Shen (Lehigh and Berkeley) 7 1
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Cost of Unreliability
The Cost of Unreliability
Two stages: supplier and retailer Supplier cannot hold inventory, is subject to disruptions
Under either DU or SU, base-stock policy is optimal at retailer Suppose firm fails to plan for either type of uncertainty
2
- 1
i.e., it sets S = µ
Key Question: Is this a bigger mistake under DU or SU?
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Cost of Unreliability
Level of Uncertainty
We need a way to compare DU and SU fairly Let level of uncertainty = % of demands backordered when S = µ LOU = 1 − fill rate A DU process and an SU proces are equivalent if they have the same LOU
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Cost of Unreliability
Level of Uncertainty, cont’d
Under DU, fill rate is 1−
σL(z) , µ
where L(z) is standard normal loss function and z = (S − µ)/σ. Therefore LOUDU =
Snyder and Shen (Lehigh and Berkeley)
σ σL(0) ≈ 0.3989 . µ µ
Supply and Demand Uncertainty
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Cost of Unreliability
Level of Uncertainty, cont’d
Under DU, fill rate is 1−
σL(z) , µ
where L(z) is standard normal loss function and z = (S − µ)/σ. Therefore LOUDU =
σ σL(0) ≈ 0.3989 . µ µ
Under SU, fill rate = % of periods in which supplier is up Therefore LOUSU = P(supplier down) =
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α . α+β
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Cost of Unreliability
Simulation Experiment
Varied LOU from 0.0 to 0.2 (by 0.01) For each LOU, find σ and α that achieve it
80 70 60 50 40 30 20 10 0 0. 2
0. 18
0. 16
0. 14
0. 1 0. 12
0. 08
0. 06
DU Cost SU Cost
0. 04
0 0. 02
Mean Cost per Period
(Keeping µ and β fixed)
Level of Uncertainty
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Cost of Unreliability
Insights
More costly to fail to plan for SU than for DU Holds under a wide range of parameters Cost difference is greater when Holding cost is smaller Stockout cost is larger Recovery probability is smaller
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Order Frequency
Outline 1
Introduction
2
Cost of Unreliability
3
Order Frequency
4
Inventory Placement
5
Supply Chain Structure
6
Cost of Reliability
Conclusions Introduction Motivation Snyder and Shen (Lehigh and Berkeley) 7 1
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Order Frequency
Order Frequency
Two-stage supply chain µ = 20, p = 100 at retailer T = 1 at supplier Under DU, σ = 5 Two possible cost structures: 1 2
2
- 1
h = 2.85 and K = 0 h = 0.1 and K = 250
Key Question: Does firm prefer #1 (one-for-one ordering) or #2 (batch ordering)?
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Order Frequency
Order Frequency: DU Option 1: h = 2.85, K = 0 Base-stock policy is optimal, with ∗
S = µ + σΦ
−1
p p+h
≈ 30
E[cost] ≈ 32.8
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Order Frequency
Order Frequency: DU Option 1: h = 2.85, K = 0 Base-stock policy is optimal, with ∗
S = µ + σΦ
−1
p p+h
≈ 30
E[cost] ≈ 32.8
Option 2: h = 0.1, K = 250 (s, S) policy is optimal with s ∗ ≈ 31,
S ∗ ≈ 349
E[cost] ≈ 32.8
So the firm is indifferent between the two options under DU
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Order Frequency
Order Frequency: SU Option 1: h = 2.85, K = 0 Base-stock policy is optimal (Tomlin 2006), with p S ∗ = µ + µF −1 ≈ 60 p+h E[cost] ≈ 497.7
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Order Frequency
Order Frequency: SU Option 1: h = 2.85, K = 0 Base-stock policy is optimal (Tomlin 2006), with p S ∗ = µ + µF −1 ≈ 60 p+h E[cost] ≈ 497.7
Option 2: h = 0.1, K = 250 Optimal policy not known (deterministic demand, stochastic disruptions, fixed cost) Lemma s ∗ and S ∗ are integer multiples of µ. s ∗ ≈ 40, S ∗ ≈ 340, E[cost] ≈ 391.1
So the batch ordering policy is preferred Snyder and Shen (Lehigh and Berkeley)
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Order Frequency
Insights
Why is batch policy preferred? If an order is disrupted, the impact is the same under either policy But the likelihood of a disruption affecting an order is smaller under batch policy
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Order Frequency
Simulation Experiment 600
(s,S) Cost
500 400 300 200 100 0 0
100
200
300
400
500
600
Base-Stock Cost
Batch policy is usually—though not always—preferred s and S may not be optimal
Instances are generated so that batch and base-stock policies are equivalent under DU Snyder and Shen (Lehigh and Berkeley)
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Inventory Placement
Outline 1
Introduction
2
Cost of Unreliability
3
Order Frequency
4
Inventory Placement Centralization vs. Decentralization Upstream vs. Downstream
5
Supply Chain Structure
6
Cost of Reliability Conclusions
7 and Shen (Lehigh and Berkeley) Snyder
Supply and Demand Uncertainty
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Inventory Placement
Centralization vs. Decentralization
Centralization vs. Decentralization
One warehouse, multi-retailer (OWMR) system Cost of holding inventory is equal at the two echelons
1 - 2
4 @
@ R 3 @
Lead times are negligible Key Question: Should we hold inventory at the warehouse or at the retailers?
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Inventory Placement
Centralization vs. Decentralization
OWMR under DU
Let CD , CC be cost under decentralized and centralized systems, resp. Theorem (Eppen 1979) Under DU, E [CD ] ∝ N √ E [CC ] ∝ N Therefore, centralization is optimal The risk-pooling effect
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Inventory Placement
Centralization vs. Decentralization
OWMR under SU Under SU: Disruptions affect inventory sites In decentralized system, a disruption affects one retailer In centralized system, a disruption affects the whole supply chain
Theorem Under SU, (a) E [CD ] = E [CC ] (b) V [CD ] ∝ N V [CC ] ∝ N 2 Therefore decentralization is preferable We call this the risk-diversification effect Snyder and Shen (Lehigh and Berkeley)
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Inventory Placement
Centralization vs. Decentralization
Implication for Facility Location
Joint location–inventory model by Daskin et al. (2002) and Shen et al. (2003) Considers DU via concave inventory costs in location model Optimal # of facilities decreases because of risk-pooling effect (and inventory economies of scale)
Reliability model by Snyder and Daskin (2005) Considers SU in the form of facility failures Optimal # of facilities increases—related to risk-diversification effect
Model by Jeon et al. (working paper, 2006) balances these competing tendencies
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Inventory Placement
Upstream vs. Downstream
Upstream vs. Downstream
Serial supply chain Cost of holding inventory is non-increasing as we move downstream
3
- 2
- 1
Lead times are negligible Key Question: Should we hold inventory upstream or downstream?
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Inventory Placement
Upstream vs. Downstream
Upstream vs. Downstream, cont’d
Under DU, conventional wisdom says hold inventory upstream Holding costs increase as we move downstream
But under SU, downstream inventory may be preferable Protects against stockouts anywhere in the system Depends on relative holding costs
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Inventory Placement
Upstream vs. Downstream
Savings Increases as Disruption Probability Increases
% Savings from Holding Downstream
300% 250% 200% 150% 100% 50% 0% 0
0.05
0.1
0.15
0.2
0.25
Failure Prob at Stage 2
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Supply Chain Structure
Outline 1
Introduction
2
Cost of Unreliability
3
Order Frequency
4
Inventory Placement
5
Supply Chain Structure Hub-and-Spoke vs. Point-to-Point Supplier Redundancy Supplier Flexibility
6
Cost of Reliability
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Supply Chain Structure
Hub-and-Spoke vs. Point-to-Point
Hub-and-Spoke vs. Point-to-Point Systems Hub-and-Spoke:
Point-to-Point: 1
7
1
- 2
@
@ @ 3 R
9 J 4 J J ^ 8 J - 5 @ @ R 6 @
2 * : X 9 HXX @H XXX z @HHH j H @ @ @ R @
3 4 5 6
Key Question: Which type of network is preferred?
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Supply Chain Structure
Hub-and-Spoke vs. Point-to-Point
Hub-and-Spoke vs. Point-to-Point Systems, cont’d
Under DU, hub-and-spoke systems are optimal Due to risk-pooling effect: fewer stocking locations =⇒ smaller inventory requirement
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Supply Chain Structure
Hub-and-Spoke vs. Point-to-Point
Hub-and-Spoke vs. Point-to-Point Systems, cont’d
Under DU, hub-and-spoke systems are optimal Due to risk-pooling effect: fewer stocking locations =⇒ smaller inventory requirement
Under SU, point-to-point systems are optimal Related to risk-diversification effect: more stocking locations =⇒ reduced severity of disruptions
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Supply Chain Structure
Hub-and-Spoke vs. Point-to-Point
Mean Cost of H-S Network
Simulation Results
4000 3500 3000 2500 2000 1500 1000 500 0 0
200
400
600
800
1000
1200
Mean Cost of P-P Network
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Supply Chain Structure
Supplier Redundancy
Supplier Redundancy
2 Single retailer with one or more suppliers Suppliers are identical in terms of cost, capacity, reliability
@ @ R 1 @
Key Question: What is the value of having backup suppliers?
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Supply Chain Structure
Supplier Redundancy
Supplier Redundancy
2 Single retailer with one or more suppliers Suppliers are identical in terms of cost, capacity, reliability
@
3
@ R 1 @ -
Key Question: What is the value of having backup suppliers?
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Supply Chain Structure
Supplier Redundancy
Supplier Redundancy
2 Single retailer with one or more suppliers Suppliers are identical in terms of cost, capacity, reliability
@
3
@ R 1 @
4
Key Question: What is the value of having backup suppliers?
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Supply Chain Structure
Supplier Redundancy
Supplier Redundancy under DU
Under DU, second supplier provides value if capacities are tight e.g., if capacity = µ + σ But value decreases quickly as capacity increases Third, etc. suppliers provide little value
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Supply Chain Structure
Supplier Redundancy
Avg % Savings
Value of Backup Suppliers: DU
8.0% 7.0% 6.0% 5.0% 4.0% 3.0% 2.0% 1.0% 0.0% -1.0%
25
30
35
40
Supplier Capacity Second Supplier
Third Supplier
µ = 20, σ = 5
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Supply Chain Structure
Supplier Redundancy
Supplier Redundancy under SU
Under SU, second supplier provides great benefit Fills in when primary supplier is disrupted Also helps ramp back up after disruption Even third+ supplier provides some benefit
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Supply Chain Structure
Supplier Redundancy
Value of Backup Suppliers: SU
Avg % Savings from Second Supplier
120% 100% (0.001, 0.1) (0.01, 0.3) (0.05, 0.5) (0.1, 0.7) (0.2, 0.9) Overall
80% 60% 40% 20% 0% 25
30
35
40
Supplier Capacity
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Supply Chain Structure
Supplier Flexibility
Supplier Flexibility
Related concept: supplier flexibility Multiple suppliers, multiple retailers
5
- 1
6
- 2
7
- 3
8
- 4
How many suppliers per retailer?
Closely related to process flexibility (Jordan and Graves 1995) Bipartite network of jobs and workers How much cross-training is required? i.e., how dense should network be?
Results are similar
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Supply Chain Structure
Supplier Flexibility
Supplier Flexibility
Related concept: supplier flexibility Multiple suppliers, multiple retailers How many suppliers per retailer?
Closely related to process flexibility (Jordan and Graves 1995) Bipartite network of jobs and workers How much cross-training is required? i.e., how dense should network be?
5
@ R @ 6 @ @ R @ 7 @ @ R @ 8 @
1 2 3 4
Results are similar
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Supply Chain Structure
Supplier Flexibility
Supplier Flexibility
Related concept: supplier flexibility Multiple suppliers, multiple retailers How many suppliers per retailer?
Closely related to process flexibility (Jordan and Graves 1995) Bipartite network of jobs and workers How much cross-training is required? i.e., how dense should network be?
5
A@ R @ 6 A@ A@A AAU R @ 7 A@ @ A AAU R @ 8 @
1 2 3 4
Results are similar
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Supply and Demand Uncertainty
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Supply Chain Structure
Supplier Flexibility
Supplier Flexibility
Related concept: supplier flexibility Multiple suppliers, multiple retailers How many suppliers per retailer?
Closely related to process flexibility (Jordan and Graves 1995) Bipartite network of jobs and workers How much cross-training is required? i.e., how dense should network be?
5
BA@ B A @ R @ 6 BA A@ B A A @ R @ 7 ABAU @ AAUBBN R @ 8 @
1 2 3 4
Results are similar
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Cost of Reliability
Outline 1
Introduction
2
Cost of Unreliability
3
Order Frequency
4
Inventory Placement
5
Supply Chain Structure
6
Cost of Reliability
Conclusions Introduction Motivation Snyder and Shen (Lehigh and Berkeley) 7 1
Supply and Demand Uncertainty
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Cost of Reliability
The Cost of Reliability
Firms are accustomed to planning for DU Often reluctant to plan for SU if it requires large investment Key Question How much DU cost must be sacrificed to achieve a given level of reliability?
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Cost of Reliability
The Cost of Reliability
Firms are accustomed to planning for DU Often reluctant to plan for SU if it requires large investment Key Question How much DU cost must be sacrificed to achieve a given level of reliability? The short answer: Not much
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Cost of Reliability
Tradeoff Curve Each point represents a solution (set of base-stock levels) for serial system Left-most point is “optimal” solution considering DU only Second point: 21% fewer stockouts, 2% more expensive
“Steep” left-hand side of tradeoff curve is fairly typical Especially for combinatorial problems 0.14 Backorder Rate
0.12 0.1 0.08 0.06 0.04 0.02 0 0
20
40
60
80
100
120
140
DU Cost
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Conclusions
Outline 1
Introduction
2
Cost of Unreliability
3
Order Frequency
4
Inventory Placement
5
Supply Chain Structure
6
Cost of Reliability
Conclusions Introduction Motivation Snyder and Shen (Lehigh and Berkeley) 7 1
Supply and Demand Uncertainty
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Conclusions
Conclusions
Planning for SU is critical Optimal strategy under SU is often exact opposite from that under DU That’s not to say firms are doing everything wrong But SU should be accounted for more than it is Strategy chosen should account for both
Many of these results are related to risk-diversification effect Disruptions are less severe when eggs aren’t all in one basket
Tradeoff between cost and reliability is often steep Large improvements in reliability with small increases in cost
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Conclusions
Notes and Acknowledgments
Supported by National Science Foundation grant #DMI-0522725 Thanks to Jae-Bum Kim (Lehigh) for assistance with simulation study Working paper available at www.lehigh.edu/∼lvs2/research.html BaseStockSim software available at www.lehigh.edu/∼lvs2/software.html
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Conclusions
Questions?
[email protected] [email protected]
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