Title: Demand Response Programs and Their Impacts on Production and Inventory Management
1Demand 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
2MS/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.
3Standard 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.
4Identification 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..
5One 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.
6Demand 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.
7The 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
8Dollar 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.
9Demand 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).
10DR 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.
11Questions 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?
12How 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.
13How 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.
14The 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
15Classical 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.
16Classical 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).
17Proposed 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
18Suppliers 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.
19Supplys charge during peak period
C
c
Resource (energy) usage
20Analysis
- 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.
21Analysis (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.
22Formulation
- 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.
23Before proceed, we need some prerequisites
Convexity and concavity
24Special case
- If cc, we obtain the classical model
- Base-stock policy is optimal
25A 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.
26Remark
- This result can be extended to piece-wise linear
convex ordering cost.
27Analysis
- 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.
28Result 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.
29x -gt x
x -gt U
U
U-R
x -gt xR
L
L-R
x -gt L
30Some 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
31Proposed 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).
32Analysis
- 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.
33Proposed 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.
34Model 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?).
-
35Model 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
36C(z)czK1zgtR
K
z
R
37Analysis
- 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.
38Result 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.
39Result 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.
40Produce 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
41Remark
- 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.
42Model 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.
43Continuous 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.
44Prerequisite 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.
45Prerequisite 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
46Analysis
- 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.
47Analysis (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.
48Result 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.
49Optimal strategy
B
Up too B if off-peak
A
Up to A if peak
50More 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.
51More 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.
52Result 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.
53Optimal 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
54Some 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)
55Energy 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.
56Summary
- 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.
57Summary (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.
58Summery (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.
59Thank you ... for your attention