Every week our industrial partner for more than 4000 products has to decide on mark-downs. Hence, a fast approach for solving the Price Optimization Problem is necessary.
In praxis there remain only two days to decide for an eventual mark-down. Our results show that therefore applying state-of-the-art solvers on the mixed-integer programming formulation of price optimization are not an alternative.
A standard approach for pricing problems is dynamic programming. We applied this idea on the Price Optimization Problem how it takes part in DISPO and extended the approach by dominance rules to a label setting algorithm. Now we can solve the Price Optimization Problem for one article averagely in less than one second. Our label setting algorithm with dominance checks against an enumeration of all mark-down strategies would reduce the runtime for deciding mark-downs for weekly4000articles from four hours to one hour. In this form our algorithm POP-DYN with dominance checks could be applied as a standard approach for deciding on mark-downs at our industrial partner.
Chapter 5
Stochastic Optimization
In this chapter we outline some basics of stochastic programming. We are mainly guided by [BL97].
In contrast to deterministic programs stochastic programs can contain random data.
Thus, stochastic programming extends the field of mathematical programming by pro-gramming under uncertainty. Uncertain input data are reproduced by random variables with known distribution.
Stochastic programs are applied when not all of the input data is known at the time of decision making. The aim is, e.g., to optimize the expected costs over all possible scenarios.
Economical problems often have to deal with uncertainties. Demand for prod-ucts, resources, etc. are not always known a priori. By treating this uncertainties in a stochastic program a more realistic problem formulation is anticipated which shall lead to better decisions at the end.
In Section5.1we will introduce so-called two-stage stochastic programs. Two-stage stochastic programs consist of a so-called first Two-stage decision, a decision which has to be made before the scenario in effect is known. The second stage – or recourse decision – responds to the realization of the scenario. If the random events follow a discrete distribution with a finite number of scenarios the two-stage stochastic program can be formulated as a so-called deterministic equivalent. In Section5.2we sketch out approaches from literature to solve the deterministic equivalent. Our focus in this chapter is on dual bounds for general two-stage stochastic programs, Section 5.3– later on we will apply them to the Integrated Size and Price Optimization Problem in the context of a customized Branch&Bound approach.
In Section5.4we sketch out the idea of multi-stage stochastic programs. Again a first-stage decision has to be made before anything about future behavior is known. But in contrast to two-stage programs – in which we only deal with one recourse decision – multi-stage programs involve sequences of recourse decisions over time depending on the realizations of the particular outcomes.
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5.1 Two-stage stochastic programs
We start with an example of a popular stochastic program, the newsvendor problem, as it is for example stated in [BL97].
Example 6(newsvendor problem). In the morning a newsvendor buysxnewspapers at a price c per paper from a publisher to sell them at the price ofq on the street.
The numberxof bought newspapers is bounded above byu. The newsvendor sells as many papers as possible for the sales priceq. At the end of the day he can return the remaining newspapers to the publisher at a price ofrwithr < c. The demand per day is varying and described by a random variableξ.
The newsvendor problem can be formulated as a two-stage stochastic linear pro-gram. Thefirst stageis the decision on how many newspapers the newsvendor should buy from the publisher. Assecond stageorrecoursethe newsvendor can compensate a wrong first-stage decision by returning overbought newspapers to the publisher.
We get to the general formulation of a two-stage stochastic program.
Definition 8(two-stage stochastic program with recourse).
minx cTx+EξQ(x, ξ) (5.1)
subject toAx=b, (5.2)
x≥0. (5.3)
It is
Q(x, ξ) = min{qξTy|Wξy=hξ−Tξx, x, y≥0}. (5.4)
The functionQ(x, ξ)is also calledrecourse function. The recourse is calledfixed if the so-calledrecourse matrixWξ does not depend on any uncertainties, then it is Wξ =W. The recourse is calledcompleteif there is a valid second-stage decision for every first-stage decision and relative completeif for every valid first-stage decision for every scenario a valid second-stage decision exists. The matrix T is also denoted as technology matrix.
In the case that the random events follow a discrete distribution with a finite number of scenarios it is possible to reduce the two-stage stochastic program to a deterministic program. Then the expected valueEξQ(x, ξ)can be computed explicitely: For every variabley of the second stage one introduces a random variable for every particular scenario and obtains an equivalent linear program – the so-calleddeterministic equiv-alent.
Definition 9(two-stage stochastic problem in its extensive form – deterministic equiv-alent).
minx cTx+X
ξ∈Ξ
pξqTξyξ (5.5)
subject toAx=b, (5.6)
Tξx+Wξyξ =hξ, ∀ξ∈Ξ, (5.7)
x≥0, y≥0. (5.8)
The setΞcontains all scenarios which follow from the discrete distribution. The prob-ability for the occurrence of scenarioξis given bypξ.
The number of the second stage variables in terms of the classical formulation from Definition8in this models multiplies with the number of scenarios. This may lead to a large deterministic program. It may be that this deterministic equivalent can not be handled with standard approaches from linear programming.