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Demand Response Programs and Their Impacts on Production and Inventory Management

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Title: Demand Response Programs and Their Impacts on Production and Inventory Management


1
Demand Response Programs and Their Impacts on
Production and Inventory Management
  • Xiuli Chao (???)
  • North Carolina State University
  • Tsinghua University
  • Summer Workshop on Stochastic Models of Supply
    Chain and Logistics
  • Beijing, July 18, 2005

2
MS/OR Research
  • MS/OR is a very dynamic area.
  • It originates from real world applications during
    WWII.
  • Its areas of research are motivated and guided by
    applications.
  • And, it helps improve real-world applications.

3
Standard procedure to do research in OR/MS
  • Standard procedure (for scientifically solving
    problems)
  • Identifying the problem
  • Formulating the problem
  • Solving the problem
  • Analyzing the solution (for insights) and
    implementing the solution.

4
Identification of MS/OR Research Problems
  • Consulting with industry is one very important
    way of learning real world problems.
  • Through reading newspapers, magazines (such as
    Wall Street Journal, Fortune, Forbes, Business
    Week, etc)
  • Discussion with colleagues, and reading research
    articles in technical journals, e.g., in MS, OR,
    etc..

5
One Example Energy market
  • US Energy Market
  • Started in 2000, a number of urgent problems
    happened in the US energy market, e.g.,
    electricity.
  • Blackout occurred during summer time in
    California, and it contributed to the step down
    of the former governor, Wilson.

6
Demand Response Programs
  • As a result, many demand response program were
    offered by the US energy market.
  • Under these programs, the energy users, such as
    manufacturers, are offered certain incentives to
    curtail their energy consumption during peak
    periods.

7
The New York Times
  • On July 17, 2000, the following article, written
    by M.L. Ward, appeared in New York Times
  • Utilities try new ways
  • to vary energy pricing

8
Dollar for Power (DFP) Program
  • as a way for Wisconsin Electric to pay
    participating customers a market-based premium
    for voluntarily reducing their energy use.
    Customers are compensated by Wisconsin Electric
    at pre-established prices for the portion of the
    electric load they reduce during periods of
    program activation.

9
Demand Response Incentive Programs
  • Indeed, as of December 2002, more than 45 States
    (out of 50) in the US have implemented some form
    of DR programs (Source US Department of Energy,
    2002).

10
DR Incentive Programs
  • The DR incentive programs have two things in
    common
  • Because of contract, energy suppliers are obliged
    to provide steady supply of resources to the
    firms, thus participation in these programs have
    to be completely voluntary
  • The amount of reward the user receives depends on
    the amount of time it participates in the
    program.

11
Questions facing the users
  • If you are an energy user, such as manufacturer,
    how should you respond to the incentive programs?
  • If you decide to participate in the incentive
    program, how long and which portion of time
    should you participate?

12
How to proceed?
Time
  • Continuous time or discrete time
  • Any point in time can be either peak or off-peak
  • which is not controllable
  • 3. Off-peak regular time
  • 4. Peak Energy crisis state, demand response
    program
  • is available
  • 5. Either suppliers and energy users
    perspectives.

13
How to proceed in our MS/OR research
  • We focus on energy user.
  • Consider a manufacturer (the firm), who uses
    electricity (or one particular type of resource)
    for its production, and without it the production
    comes to a stop.
  • The firm faces random demand for its product, and
    its objective is to satisfy the customer demand
    while minimizing its cost.
  • We first consider periodic review system.

14
The two classical models for the firm
  • All periods are off-peak.
  • Periodic review inventory system without setup
    cost
  • Periodic review inventory system with setup cost

15
Classical model with no setup cost
  • Backorder model or lost sales model. Let us
    consider backorder model
  • Purchasing price c
  • Holding cost h
  • Shortage cost b
  • Objective is to minimize total cost over planning
    horizon
  • Main result base-stock policy is optimal.

16
Classical model with setup cost
  • In addition to the cost in previous model, there
    is a setup cost K.
  • Ordering cost for x is
  • C(x)K1xgt0cx
  • Main result is that optimal policy for each
    period is (s, S).

17
Proposed model 1 A naïve model
  • Model the problem as a periodic review production
    system.
  • Each period can be in a peak or a off-peak
    period.

Time
18
Suppliers Incentive program
  • Suppose R is the average usage rate of the firm
    in a period.
  • During off-peak period, charging rate of the
    resource is always c per period.
  • During peak period, the energy supply uses the
    following policy If the firm uses less than R in
    a peak period, charge regular rate c, but if more
    than R, then the additional amount is charged a
    higher rate c.

19
Supplys charge during peak period
C
c
Resource (energy) usage
20
Analysis
  • If the system is currently in a (off-) peak
    period, the next period will be peak with
    probability p (q), and, and it will be off-peak
    with probability 1-p (1-q).
  • For simplicity suppose one unit of energy can
    produce one unit of product.
  • The state of the system is (x, y), where y0
    represents off-peak period, and y1 represents
    peak period x is the inventory level of the
    firm.

21
Analysis (contd)
  • The firm faces random demand D1, D2, .
  • Back-order model is considered (lost sales model
    can be similarly studied).
  • The firm faces holding cost rate h and shortage
    cost rate b, as in classical models.
  • The objective is to minimize the total discounted
    or average cost for the firm.

22
Formulation
  • We use MDP (or SDP).
  • First consider finite horizon problem, and then
    infinite horizon problem.
  • Let alpha be the discount factor.
  • Let Vn(x, y) be minimum expected discounted cost
    for an n-period problem.

23
Before proceed, we need some prerequisites
Convexity and concavity
24
Special case
  • If cc, we obtain the classical model
  • Base-stock policy is optimal

25
A preliminary result
  • If f(x) is convex, then
  • Minygtx b 1ygtxaf(y) is determined by two
    numbers, L and U.
  • L is the minimum of bxf(x), and U is the minimum
    of f(x).
  • If xltL-a, optimal yL, if L-altxltU-a, optimal
    yxa, if U-altxltU, optimal yU, and if xgtU, yx.

26
Remark
  • This result can be extended to piece-wise linear
    convex ordering cost.

27
Analysis
  • Optimality equation
  • Claim V(x, y) is convex in x for given y.
  • Based on this property and the previous result we
    can obtain the following result.

28
Result 1.
  • The optimal strategy in an off-peak period is
    base-stock level y.
  • The optimal strategy for a peak period is
    determined by two numbers L and U, such that
  • If xltL-R, yR
  • If L-RltxltU-R, yxR
  • If U-RltxltU, yU
  • If xgtR, yx.

29
x -gt x
x -gt U
U
U-R
x -gt xR
L
L-R
x -gt L
30
Some further pre-requisite
  • K-convexity
  • Graphical definition of K-convexity using
    visibility
  • Some properties of K-convexity
  • (i), (ii), (iii), and (iv)
  • Implications of K-convexity
  • Existence of s, S, and other things
  • Optimality of (s, S) policy
  • Classical model with setup cost

31
Proposed Model 2
  • Consider the case where at each period, the firm
    can either produce, or not produce.
  • If not produce, then there is a reward K.
  • If produce, the reward is lost, regardless of how
    much it produces.
  • This is the model of Chen, Sethi and Zhang
    (2004).

32
Analysis
  • Optimality equation
  • The argument is reduced to classical models
  • Value function V(x, y) is K-convex for given y.
  • Main result
  • Policy for off-peak period
  • Policy for peak period.

33
Proposed Model 3
  • The main disadvantage of the first model is that
    the firm charges higher rate for using more than
    R during peak period.
  • The main disadvantage for the second model is
    that the firm is discouraged to not any energy at
    all, but in reality, we do not want them to use
    too much.
  • Revised incentive program If the firm uses more
    than R in one period, charge the regular rate c,
    but if the firm uses less than R, it is charged
    at rate c but is awarded an amount K as an
    incentive.

34
Model 3 (contd)
  • Let C(z) be the charging rate of the supplier
    when the firms usage level is z, then
  • C(z) cz1zgtR(cz-K)1zlt R
  • cz K 1zgtR-K
  • The cost K can be ignored (why?).

35
Model 3 (contd)
  • The ordering cost function is
  • C(z)cz K 1zgtR
  • This is a natural extension of the classical
    inventory model with setup cost, which is
  • C(z)cz K 1zgt0

36
C(z)czK1zgtR
K
z
R
37
Analysis
  • Optimality equation
  • Lemma 1 If f(x) is K-convex, so is
  • g(x)minxltyltxRf(y)
  • Define V(x) as value function
  • Rewrite the optimality equation.
  • The following result can be obtained.

38
Result 2
  • The optimal strategy for the firm is the
    following
  • At the beginning of an off-peak period, if the
    inventor level is less than y, then produce up
    to y. If more than y, then produce up to y(x),
    which is an non-decreasing function.
  • At the beginning of a peak period, the optimal
    policy is determined by four numbers, s, u,
  • S-R, and S.

39
Result 2 (contd)
  • If inventory level x is less than s, produce more
    than R to replenish to S, if x between s and u,
    produce exactly R to replenish to less than S, if
    x between u and S-R, produce less than R to
    replenish to less than S, if initial inventory
    level is between S-R and S, produce less than R
    to replenish to exact S.

40
Produce gt R to S
Produce R to lt S
Produce lt R to lt S
Produce lt R to S
s
u
S-R
S
Strategy for peak periods
41
Remark
  • If R0, then the model is reduced to classical
    inventory model with setup cost.
  • The policy then reduces to two points, (s,S)
    policy, where us, and S-RS.

42
Model 4 Continuous time model
  • Suppose now time is continuous.
  • Demands for the firms products follow a batch
    Poisson process
  • Peak period arrives according to a Poisson
    process, whenever it occurs, it lasts for a
    random amount of time.
  • First, assume that the peak period is
    exponentially distributed.
  • Productions are in batches of M units and the
    production time is exponentially distributed.

43
Continuous time model (contd)
  • Production cost is c per unit of time.
  • Holding cost and shortage cost are the same as
    before, h and b per unit of time per item.
  • During peak period, the firm is offered the
    incentive program that, each unit of time it
    shuts down, it receives a reward r.
  • The objective is to find the optimal production
    and shut down strategy so that the total
    discounted cost is minimized.

44
Prerequisite Continuous time MDP model
  • If all events have exponential clock times, then
    all events can be generated using a single
    Poisson process with rate being sum of all rates.
  • This is the so-called uniformization technique.
    It allows a continuous time MDP to be transformed
    to a discrete time MDP.

45
Prerequisite Continuous time MDP
  • Since time unit can be anything, a common trick
    to assume that the rate of the Poisson process is
    1. This is equivalent to that the sum of rate of
    all exponential clocks, including the discount
    rate which is considered as killing process, is
    1.
  • For the problem under consideration, we have

46
Analysis
  • Optimality equation
  • The optimal strategy is determined by the
    structure of the value function.
  • The structure of the value function is common
    obtained from that of finite horizon problem
    first consider the problem with n Poisson arrival
    epochs.

47
Analysis (contd)
  • The value function V(x, y) can be proved to be
    convex in x for y0, 1.
  • As a result, it can be shown the following result
    for the first continuous time model.

48
Result 3
  • The optimal production strategy is determined by
    two numbers, AltB.
  • During off-peak period, produce if and only if
    the inventory level is less than B.
  • During peak period, produce if and only if
    inventory level is less than A.
  • A is decreasing in the reward rate r, and B is
    increasing in the reward rate r.
  • If r0, then AB.

49
Optimal strategy
B
Up too B if off-peak
A
Up to A if peak
50
More on continuous time model
  • What if the duration of peak period is not
    exponentially distributed?
  • For example, suppose, the peak period is always
    a, a fixed number.
  • More generally, suppose that the duration of peak
    time is not known in advance. It becomes known,
    however, when the peak period arrives.
  • This is equivalent to, the announced length of
    the peak period is drawn from a distribution
    function, say F.

51
More on continuous time models
  • That is, before the peak period arrives, no one
    knows how long the next peak period will last.
  • However, as the peak period arrives, the energy
    supplier announces the length of the incentive
    program.
  • The firm needs to determine the optimal
    production and shutdown strategy during peak as
    well as off-peak periods.

52
Result 4
  • The optimal production and shut down strategy is
    determined by a number B and a function A(t),
    A(t)ltB, such that during off-peak period, the
    firm produces if and only if its inventory level
    is less than B.
  • During peak period, if there are t units of time
    left before the current peak period expires, the
    firm produces if and only if the inventory level
    is less than A(t), t gt0.
  • A(t) is decreasing in incentive rate r and B is
    increasing in r.

53
Optimal production and shutdown strategy
B
Up to B during off-peak period
A(t)
Up to A(t) if t units of time left before peak
ends
54
Some possible extensions
  • More general arrival times of peak periods.
  • Seasonal effects.
  • And others
  • Some results can be obtained but as more features
    are included, the model and results become much
    more complicated (it is usually state-dependent
    optimal policy)

55
Energy suppliers problem
  • The previous analysis focuses on the energy user.
  • How about the energy supplier?
  • Different incentive programs yield different
    customer responses.
  • This gives rise to a Stackelberg game problem,
    with energy user being leader, and energy users
    being followers.
  • The suppliers objective should not only be its
    profit, but also social effects.

56
Summary
  • The field of MS/OR came from application, its
    research problems are motivated by applications,
    and it serves real world applications.
  • The field of OR/MS is dynamic and it evolves
    rapidly as the world is becoming a global
    economy. This is precisely why supply chain
    management and logistics are hot research topics
    in recent years.
  • The emerging new economy presents opportunities
    as well as challenges to MS/OR researchers.

57
Summary (contd)
  • We expect Chinese MS/OR researchers will make
    important contribution to the field, apply the
    MS/OR studies to Chinas rapidly growing economy,
    and make contribution to the economic development
    in China.

58
Summery (contd)
  • This presentation is based on papers
  • X. Chao and F.Y. Chen, Optimal production and
    shutdown strategy when the supplier offers an
    incentive program. Manufacturing and Service
    Operations Management, Vol 7, No. 2, 130-143,
    2005.
  • X. Chao and P. H. Zipkin, Optimal inventory
    strategy for periodic review system with
    transportation contract. Second revision under
    review in Operations Research.
  • F.Y. Chen, Sethi, S. and Zhang, H. A production
    inventory problem under energy buy-back program.
    Working paper.

59
Thank you ... for your attention
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