Adelman (2007) studies a dynamic fleet management problem on a closed logistics queueing network. The focal of this paper is to resolve the spacial (geographic) imbalance of the supply and demand from a central planner’s point of view. First, Adelman’s paper takes a centralized view for controlling the system. Second, the primary concern in his work is geographic (spacial) imbalance of supply and demand. He traces the state (location and idle time) of each box in a closed queueing network. The market segmentation (as he pointed in the end) is not taken into account in his work. In summary, his work is a centralized dynamic fleet-management problem with flavor. In terms of methodology, he uses LP and nonlinear LP to approximate the problem.

Price-Directed Control of a Closed Logistics Queueing Network.” Daniel Adelman; Operations Research, 2007, 55(6), pp. 1022-38

Technorati : LP, approximation, math programming, operations management, revenue management

This paper also investigates the effects of capacitated supply, and shows that the supply capacity limits the effectiveness of such threat advisory system as it limits the firm’s ability to take advantage of threat-dependent coverage. This is intuitive since the information is valuable only when you can act upon on it. Due to the limited capacity, the system may not attain the optimal base stock level in one period and therefore, the coverage functions as a protection not only for the possible disruption in the next period but also the possible future disruptions, during which the inventory level has yet to reach the optimal level because of the capacity. An interesting effect unique in capacitated system is that due to the delay of bringing the inventory to the optimal level, the system can be down again during its recovery, i.e., ‘recovery effect’. Another similar effect is the disruption during the transition from one up state to another up state, ‘threat transition effect’. Therefore, compared to infinite capacity system, the base stock levels in capacitated system should be higher.

The paper also examines two-suppler cases. Two tactics are possible under such situations: (1) sourcing mitigation refers to a sourcing policy in which the firm routinely sources a positive fraction $w$ of its demand from a reliable but expensive supplier, besides the main unreliable supplier; (2) contingent rerouting refers to a sourcing policy whereby the firm temporarily increases the quantity sourced from the reliable supplier during the disruption of the unreliable supplier. Rerouting is only allowed during disruption. The paper finds that sourcing mitigation is preferable for low supplier reliability and high expected disruption duration, while the inventory mitigation is preferable for high reliability and short disruption during cases.

The paper also investigates the impact of the structure of the supply process upon the disruption management strategy. Two structures of the Markov chain are investigated, completely connected and sequential connected (birth and death process). It is intuitive that the sequential connected chain yields more value due to its higher predictivity.

The authors characterize the parameter range so that the action of each type of supplier can be specified, where is the unit revenue, is the unit backup option cost. The value of supplier backup option for the manufacturer is not necessarily larger under asymmetric information. The manufacturer is willing to pay the most for information when supplier backup option is moderately expensive. This is because when the backup option either extremely cheap or expensive, the actions of suppliers have already been certain and their is no need for the additional information to infer it. The value of information may increase as supplier types become uniformly more reliable. Higher reliability need not be a substitute for better information. In other words, they can be complements to each other.

See also, Supply disruptions, asymmetric information and a dual-sourcing option, by Yang, Aydin, Babich, and Beil, working paper, 08.

Tow points are worth attention. DP is not widely adopted by the practitioners largely due to the different mind set. Traditional RM models based on the independent demand and ignorance of the network effect have severe flaws. It also impacts how the systems should be controlled. Bid price should a more promising and superior control mechanism, compared with virtual nesting. But there is no consensus among practitioners yet.

The major issue in the forecasting is the level of detail. The paper suggests that a bottom-up approach may be more accurate than a top-down one. Also, the passenger name records (PNR) is a more desirable data source than historical transaction data.

A more nature approach is to model the acceptance decision directly, say and , the amount of order accepted. This paper builds the structure of the policy into the model, and therefore, the actual amount of acceptance is implies by the policy embedded. On advantage of this approach is that, we can explicitly express each quantities and assess their costs, so the objective function is expressed in terms of the policy parameters, which are easy to handle. In particular, different quantities of reservation and backlogs can be written out explicitly.

The benchmark case is the system that has no customer differentiation, zero leadtime (due date), and no tactical inventory. NDS case is the one with leadtime 1, but no customer differentiation, therefore reservation is impossible but backlog can be done. The optimal policy allows leadtime 1, thus backlog, and customer differentiation (so reservation is possible).

The paper also discusses the value of price differentiation. When both premium and discount prices are offered, rather than a single discount or premium prices, the increased capacity flexibility (availability due leadtime 1) can outweight the lost due to the lower average price.

The impact of pricing trend is also investigated numerically. Let be the rate of price growth. Then when the growth is slow, then backlog is a more effective level for single class problem since it allows more capacity availability to meet the realized demand and the future will not be much better than now. If we can fulfill the realized orders now, we would rather borrow future to serve the current realized demand.

When the growth is strong, i.e., future is significantly better than the current, then reservation (rationing) is more effective because the system want to sell to more profitable future. However, the system may still backlog, since the realized demand has its value, in case there is no demand in the future at all. In this case, the demand variability plays an important role. If the future is certain, then there may be no need to backlog.

Even if part of demand may switch to discount classes when two prices are offered, it may still be profitable to do so. One reason is that, the discount class allows more capacity flexibility to capture the realized demand, which may be lost if the system only offers premium price and service (zero leadtime).
The price should be average by the demand: , not just arithmetic average.

The other paper, “POLICIES UTILIZING TACTICAL INVENTORY FOR SERVICE-DIFFERENTIATED CUSTOMERS”, is almost the same as this model.

]]> http://pureideals.wordpress.com/2008/09/16/optimal-production-and-inventory-policies-of-priority-and-price-differentiated-customers-by-duran-liu-simchi-levi-swann-iie07/feed/ 0 pureideals http://pureideals.wordpress.com/2008/09/15/capacity-allocation-to-sales-agents-in-a-decentralized-logistics-network-by-demirag-swann-nrl-07/ http://pureideals.wordpress.com/2008/09/15/capacity-allocation-to-sales-agents-in-a-decentralized-logistics-network-by-demirag-swann-nrl-07/#comments Mon, 15 Sep 2008 01:32:15 +0000 pureideals http://pureideals.wordpress.com/2008/09/15/capacity-allocation-to-sales-agents-in-a-decentralized-logistics-network-by-demirag-swann-nrl-07/ ]]> This paper considers a capacity allocation problem in sea cargo industry. The model incorporates revenue management idea for order acceptance in a network flow framework. The problem is formulated as several deterministic math programs. Except for single route problems, the optimal solution is NP hard. Therefore, several heuristics are developed: (1) marginal revenue; (2) priority OD based allocation heuristics; (3)equal allocation; (4) conservative allocation. There are not many theoretical results due to the complexity of the problem. The computational experiments show that heuristics performances well when relative capacity is high (defined as capacity/demand). The paper also shows that the average service level improves as the capacity level improves.

Note that it is possible that the residual inventory after is lower than the leadtime inventory. In this case, the is to reserve for the demands \emph{beyond} the leadtime window .

The difficulty of establish when is that we are not sure about the relation between the optimal actions and . Without a definite relation, the comparison can go either way.

If the definite relation is elusive, then we should just go with the computation.

Approximation property of supremum: assume the supremum of a real set exists (bounded above). Then for each , there exists such that , and . The first asserts the existence of points in such that they can be arbitrarily close to . The second says that there exists points of such that with arbitrary small help , they can be larger than . Another useful property says that if [, ], then .

The construction of reals start from the naturals, with the principle of math induction (well ordering principle), we can construct . Then we can define rationals by . After define the supremum, we can define a real by . The existence of is guaranteed by the Completeness Axiom of reals. .

Note that much of this part is to prove the consistence among different constructions and definitions.

The Archimedean Property: If and are two positive reals, then there exists a natural number such that .

The density property of rationals: between any two distinct real numbers there exists a rational number. That is, for any , , for some .

Existence of roots of reals: for , 1) if and $n$ is even, there exists a unique such that . 2) if and $n$ is odd, there exists a unique such that . The n-th root of is denoted or .

, , .

.

Technorati : MATH

The substitution effect is captured by the customer choice behavior. In particular, each customer has a preference mapping that maps products (flights) into their ranking. Then the choice is made base on both the preference ranking and the availability of seats, which is controlled by the system. A customer will choose a product if it is the top choice among the available alternatives. The demand is then defined as the sum over all choices made by the customers. Note that the distribution of the preferences of the population is known. Then the sales of a product is the sum over all choices made by each customers who bought this product, , where is the choice function, given the capacity and the preference of customer . The pure demand of class is defined as the total sales of class if there are infinite capacity of class , give others are , . Note the following holds . This construction takes the view that the group behavior is some aggregation of individual behaviors. Begin with modeling individuals, this model is able to model the choice behavior at both the individual and the group level.

Due to the curse of dimensionality, the problem cannot be solved optimally for practical size problems. Therefore, the paper derives several bounds and heuristics. The basic ideas are stochastic ordering and pooling, where stochastic ordering decomposes the multiple dim problem to several easily solvable 1-dim problems, and pooling reduces the problem in one single product problem. For the bounds based on stochastic ordering, the paper first construct two sequences of demand independent of actions and states such tath . Note that depends on states and actions. and is constructed from the different individual behaviors. In particular, assumes that all customers only buy their first choice, and do not buy if the first choice is not available. assumes the only choice the choices are . Then for each decomposed problem, we can solve it easily to get the optimal booking limits. Note that the original problem is now single dim problems. After they get the policy (booking limits ), they then use simulation to evaluate the policy. The other approach is to solve LP in stead of 1-dim DP to get the policy, and use simulation to evaluate the performance.

The results show that the policy from weighted value function approximation (stochastic ordering) works well better than mixed policy and LP. Note that LP can be executed on a rolling horizon manner in order to take into account the updates. Note that in approximation step what really matters is the policy, not the value function, the value function will by reevaluated via simulation using the policy from the approximation step.

The numerical study investigates the effect of demand variability, demand correlation, price gap, and a real size example.

For an infinite set , the followings are equivalent:

1) is countable;

2) there exists an onto (surjective) , where ;

3) there exists a one-to-one (injective) .

The countable union of countable sets is countable.

The trick to proof countability is to construct a map with respect to . It is also quite often we need to construct objects relative to the set of rationals .

If for cardinalities, we have , then and are equivalent.

Consider an partition by a function , where . Then . It can also be viewed as a relation . Note that defines an equivalent relation on .

A Cartesian product of a family with each nonempty is defined as . is the choice function. The Axiom of Choice says that if each , then . In other words, there exist a set such that consists of precisely one element. If , then . That is, if the image (range) sets are homogenous, then the product is simply the set of all possible functions mapping the index set into the image . In particular, power set of . That is, the family of all subsets of is equivalent to the set of indication functions of subsets of .

Well ordering principle: any nonempty subset of has a least element.

Principle of math induction: For any subset of , if

1) ;

2) implies ;

then .

Note that the Well ordering principle is equivalent to the principle of math induction.

Technorati : math

This paper formulates a multi-flight RM problem, proposes and evaluates several bounds and heuristics. The bounds based on price or capacity pooling ideas work well when the customers’ valuation and fights capacities are homogeneous. The heuristics based on separable bounds (roll out policy), however, perform well regardless the demand and capacity conditions. In particular, value approximation does better than policy approximation, where both of them are of the form , where is chosen to maximize the value function.

It is observed that infrequent change of prices can extract most of the benefits from the real-time pricing policy.

For a given policy, it is easy to evaluate its performance by simulation.

References

G Gallego, SG Kou, R Phillips – Management Science, 2008

G Gallego, R Phillips – Manufacturing & Service Operations Management, 2004

Technorati : operations management, revenue management

Using queueing models, this paper compares the performance of several policies with/without inventory rationing, shipment flexibility (leadtime or shipping mode), lost sales, backlog and their combination. Note that in this paper the rationing is a static threshold independent of the inventory level. The production follows base stock policy. Shipment flexibility allows the firm to pay additional shipping cost in order to serve more demand and avoid backlog costs. Inventory rationing provides the firm with the lever to shut down less profitable channel when the supply is tight. Note that the underlying processes of different policies are of birth-and-death type, with the birth rate equals production rate, and the death rate equals effective demand rate.

It is assumed that the callable demand arrive before the high reward demand. The callable products function as backup demands, which the system can dispose with the cost to make up the capacity for realized high profit demands. If there are sufficient capacity, then the committed callable demand is also fulfilled.

The paper models this system at an aggregate demand level with two periods. It is strong in presenting concepts, but the model is not detailed enough for daily operations.

This is a good paper with many insightful results. It also summarizes several previous results regarding rationing. With the choice model, the lead time option induces three effects: 1)demand induction (w.p. ) due to the option of longer lead time; 2) risk pooling (fulfillment flexibility) from the long leadtime customers due to the enlarged action space; 3) demand cannibalization (w.p. ) due to the introduction of the long leadtime option. Note that both demand induction (enlarge the market size) and fulfillment flexibility will improve the system performance, while demand cannibalization may decrease the performance, because of decreased class-1 customers. , becomes , where 1 and 2 indicate customer classes.

Note that the incentive to use onhand inventory to meet class-2 demand (long leadtime) is to reduce both the holding cost and purchasing cost (clear the backlog of class-2), while the motivation to backlog class-2 is to reserve onhand inventory for potential future class-1 demand in the current cycle.

The numerical study provides the following insights: when price gap is small , demand induction along with fulfillment flexibility improves the profitability. When the price gap is large , the choice model describes that substantial class-1 customers switch to class-2, therefore decrease system performance. The study also shows that when holding cost is low, cycle time is long, dynamic policy (with the time-dependent threshold) significantly outperforms the static policy ( with a constant threshold for the entire cycle?). This is quite intuitive given that the static policy is some kind of average of the dynamic policy, as the decision horizon increases, the variation of relative to increases, and so is the performance gap.

Technorati : MDP, operations management, revenue management

Dong’s paper is a good paper, offering many useful insights. I haven’t gone through technical details. But it is a must read for dynamic pricing with consumer choice behavior. One point is that the marginal profit of each variate should be set to the same for the optimal policy.

The inventory decision in this paper is somehow superficial. It is simply a newsventor problem on top of the pricing problem.

The substitution is somehow similar to the inventory pooling.

The idea for the heuristic is that instead of multiple products, we can define an ‘average product’ that is equivalent in the sense that has the same no-purchasing probability, and the purchasing cost is the average weighted by their attractiveness.

This is an excellent paper that one should fully understand.

The low-orthant property is proved through concavity and submodularity, neat!

The switch-over policy disregards the inventory information and only depends on the class information. If we assume that all arrivals are of unit one, then the bid price can be transformed to the critical time epochs thresholds beyond which the coming classes can be accepted.

Note that the main idea of this paper is that, if we specify a class of policies, then it is easier to analyze it, especially we can write out the value functions explicitly.

Technorati : MDP, Operations research, Poisson Process, knapsack, operations management, revenue management, stochastic processes

The backward pass computation of the gradients can significantly improve the performance of the algorithm.

ADP performs better than rolling horizon approach.

Overall, this is a good tutorial paper deserving a second read.

Technorati : 9/10 Approximate Dynamic Programming for Large-Scale Resource Allocation Problems, Topaloglu and Powell, Tutorials 05

This paper summarizes several bounds for network RM. In particular, it establishes the orders of these bounds in a unified way. In particular, , where is the Lagrangian Relaxation, and is strong Affine approximation. However, this is still an ongoing research. More definite results may take a while.

The paper also points out that in terms of computation, (or RLP) is the most practical one. Also, Adleman and Zhang’s choice model paper, Kunnumkal and Topaloglu’s choice paper, are the extensions of and methods to network RM under the choice model.

Technorati : Kalyan Talluri, WK08, on bounds for network revenue management

The key idea of time decomposition is that using the simple control policy, such as LP, before the switch time, and after the switch time uses MDP (or the optimal). The rationale is that, only at the end of the horizon, or only when the scarcity is prominent the rationing policy makes difference. When there are ample capacities, one can simply follow FCFS rules, no control policy is necessary.

The numerical results, though many, are not very insightful. Many of them are intuitive or obvious.

This is an interesting and well written paper, but the analysis is weak. I like two aspects of this paper:

1. The paper models project selection problem at a tactical level with sufficient realistic details using stochastic dynamic programming. It is interesting to see the application of SDP in this problem.

2. The details of the model include several interesting notation, such as capacity profile (similar to distribution function), rush order, etc.

However, this detail and explicit approach is also a curse to the problem. With so many irregularities, especially the combinatorial nature of the capacity profit evolution, the paper cannot go further than presenting the model and solve it numerically. In fact, the numerical examples are more like solutions for homework than real cases for the journal, too simple and too obvious. This leads to the following problems:

1. The paper states many obvious observations, no matter they are termed as lemma or theorem, but either lack of rigorous proof, or too trivial to prove.

2. Theorem 1 requires the assumption equivalent to infinite backlog without penalty. But this is the case, along with the perishable nature of capacity, of course there is no incentive to use future capacity rather than use the current capacity.

3. The discussion of rush orders are also so obvious and lack of rigorous proof. So are the discussions of reinvestment revenues.
4. The issue of non-regular capacity can be modeled with a convex cost function without getting into the cumbersome details of two type of capacity and so that the state space can be simplified.

5. On page 20, paragraph 2, ” The overall lesson to learnt is that optimal acceptance and planning decisions depend on (1) the immediate reward as well as (2) the expected future rewards.” But wouldn’t this be so obvious that even a graduate student after the first week dynamic programming class would know this?

6. Most equations presented in the paper should only show up in a student’s homework, not a paper, since they are simply the computation of the DP problem. In fact, the paper would be more readable if these irrelevant equations can be omitted.

7. The tardiness penalties discussion in section 5.7 again can be seen from the convex cost function perspective.

In summary, this is a well written (before the analysis part) and interesting paper, including good literature review on project selection from different research domain. But from an OR perspective, the paper lacks of necessary sophistication to substantiate the insights.

Technorati : Operations Management, Project Selection, RM, Reading

This is a very solid paper. Actually it is from Song’s thesis. It summarizes many inventory control results up to that point. {\bf In particular, the ordered vectors and differences survive value iteration deserve further investigation.}

{\bf References:} Denardo, 1976, contraction mappings in the theory underlying dynamic programming. SIAM Rev.

Lovejoy, 1987, Ordered solutions for DP. Math Opns. Res.

Topkis, 1978, Minimizing a submodular function on a lattice. Opns. Res.

Technorati : LIFE, LOG

DONE @2.48AM, MARCH 28, 2008

Technorati : LOG, PROJECTS

Instead, we result to the exhausted searching method. That is, we compute all the possible constrained policy set and pick the best one.

{\bf Note:} when the effects of information are not significant, we should change the MC to more extreme case, such as $q_2=.01$ and $q_3=.9$.

Technorati : CODING, LOG, MDP

For the optimal policy and the value function , it holds that

Policy Iteration Method:

1) Policy evaluation: given , find by DE ;

2) Policy improvement: given , find by OE , where is the optimal operator.

Value Iteration Method:

When action space is large and state space is relatively small, use PI; otherwise, use VI.

Simulation is good to evaluate a give policy, but not efficient to solve for the optimal policy.

Technorati : DYNAMIC PROGRAMMING, MATH, MDP

We can view the whole construction from another point of view. First, if we think continuously, then iid r.v. is actually the differences or derivatives of the renewal sequences. The renewal process is the inverse function of renewal sequences.

Given ,

define ,

and .

Therefore, .

Note that the alternating renewal process is a different concept. It actually defined by two r.v. . The process is the state of the process, not the number of events.

We don’t have to advance systems universally or evenly with real time clock. As in continuous time Markov chains, we can proceed systems by their event epoch, with the inter-event times not necessarily equal.

Technorati : probability, stochastic processes

In any case, all systems can be simulated, even without mathematical model description. Then why don’t you build a simulation model first?

Technorati : SIMULATION, THOUGHTS

The interesting part of this part is its two approximating schemes. (1) for its convex objective functions, the authors use \emph{piecewise linear functions} to approximate; (2) for the expected future reward function, they use Monte Carlo simulation to estimate.

The major insight from numerical part is that the near-term forecasts contain more significant information than long-term ones, which I hardly read from their numerical results.

For a queue, the major conclusions include ,
, where is the optimal profit, and is the mean lead time, is the optimal quoting lead time.

For finite case, it argues that
“\emph{In general, if firms compete on both price and lead time, it is possible, though by no means guaranteed, that the market equilibrium could result in all firms offering the same price and lead time. This appears to be approximately the case in the overnight mail and pizza delivery markets.}”
The customer choice function is $ latex p(a) = \chi(a\le \hat{a})$ under the assumption of standard industry lead times. The main theory is that for any , there exists a time such that a new customer is accepted if the first job has been in process for longer than time units and rejected otherwise. If the service time distribution has finite support or is exponential, there exists a and such that a new order will be accepted iff either (a) there are less than customers in the system, or (2) there are exactly customers in the system and the first one has been in process for longer than time units. The optimal solution has increasing in .

The \emph{idea for bounds} in this case is by sandwich concept that , where and are the solutions to the systems with 0 and customers. By proving that both and are increasing in , it is sensible to conjecture that is also increasing in , the duration of the first order that has already been in service.

The case allowing scheduling consideration is somehow less interesting to me. The proof of the ‘earlist due date (EDD)’ protocol is largely borrowing from Duenyas’s paper. One point is worth noting that the assumption that the penalty is proportion (or convex) to tardiness is necessary in the proof, whereas replacing it with a fixed cost, the result does not hold.