Constrained Optimization in Newsboy Problems under Uncertainty via Statistical Inference Equivalence Principle

The aim of the present paper is to show how the statistical inference equivalence principle (the idea of which belongs to the authors) may be employed in the particular case of finding the effective statistical solutions for the multi-product newsboy problems with constraints. To our knowledge, no analytical or efficient numerical method for finding the optimal policies under parameter uncertainty for the multi-product newsboy problems with constraints has been reported in the literature. Using the (equivalent) predictive distributions, this paper represents an extension of analytical results obtained for unconstrained optimization under parameter uncertainty to the case of constrained optimization. An example is given. INTRODUCTION The last decade has seen a substantial research focus on the modeling, analysis and optimization of complex stochastic service systems, motivated in large measure by applications in areas such as transport, computer and telecommunication networks. Optimization issues, which broadly focus on making the best use of limited resources, are recognized as of increasing importance. However, stochastic optimization in the context of systems and processes of any complexity is technically very difficult. Most stochastic models to solve the problems of control and optimization of system and processes are developed in the extensive literature under the assumptions that the parameter values of the underlying distributions are known with certainty. In actual practice, such is simply not the case. When these models are applied to solve real-world problems, the parameters are estimated and then treated as if they were the true values. The risk associated with using estimates rather than the true parameters is called estimation risk and is often ignored. When data are limited and (or) unreliable, estimation risk may be significant, and failure to incorporate it into the model design may lead to serious errors. Its explicit consideration is important since decision rules that are optimal in the absence of uncertainty need not even be approximately optimal in the presence of such uncertainty. In this paper, we propose a new approach to solve constrained optimization problems under parameter uncertainty. This approach is based on the statistical inference equivalence principle, the idea of which belongs to the authors. It allows one to yield an operational, optimal information-processing rule and may be employed for finding the effective statistical solutions for problems such as multi-product newsboy problem with constraints, allocation of aircraft to routes under uncertainty, airline set inventory control for multi-leg flights, etc. STATISTICAL INFERENCE EQUIVALENCE PRINCIPLE In the general formulation of decision theory, we observe a random variable X (which may be multivariate) with distribution function F(x;θ) where a parameter θ (in general, vector) is unknown, θ∈Θ, and if we choose decision d from the set of all possible decisions D, then we suffer a loss l(d;θ). A “decision rule” is a method of choosing d from D after observing x∈X, that is, a function u(x)=d. Our average loss (called risk) Eθ{l(u(X);θ)} is a function of both θ and the decision rule u(⋅), called the risk function r(u;θ), and is the criterion by which rules are compared. Thus, the expected loss (gains are negative losses) is a primary consideration in evaluating decisions. We will now define the major quantities just introduced. Definition 1. A general statistical decision problem is a triplet (Θ,D,l) and a random variable X. The random variable X (called the data) has a distribution function F(x;θ) where θ is unknown but it is known that θ∈Θ. X will denote the set of possible values of the random variable X. θ is called the state of nature, while the nonempty set Θ is called the parameter space. The nonempty set D is called the decision space or action space. Finally, l is called the loss function and to each θ∈Θ and d∈D it assigns a real number l(d;θ). Definition 2. For a statistical decision problem (Θ,D,l), X, a (nonrandomized) decision rule is a function u(⋅) which to each x∈X assigns a member d of D: u(X)=d. Definition 3. The risk function r(u;θ) of a decision rule u(X) for a statistical decision problem (Θ,D,l), X (the expected loss or average loss when θ is the state of nature and a decision is chosen by rule u(⋅)) is r(u;θ)=Eθ{l(u(X);θ)}. This paper is concerned with the implications of group theoretic structure for invariant loss functions. Our underlying structure consists of a class of probability models (X, A, P), a one-one mapping ψ taking P onto an index set Θ, a measurable space of actions (D, B), and a real-valued loss function { } ) ; ( ) ; ( X d l E d l o θ θ = (1) defined on Θ × D, where ) ; ( X d l o is a random loss function with a random variable X∈(0,∞) (or (−∞,∞)). We assume that a group G of one-one A measurable transformations acts on X and that it leaves the class of models (X, A, P ) invariant. We further assume that homomorphic images G and G~ of G act on Θ and D, respectively. ( G may be induced on Θ through ψ; G~ may be induced on D through l). We shall say that l is invariant if for every (θ, d) ∈ Θ × D ), ; ( ) ; ~ ( θ θ d l g d g l = g∈G. (2) A loss function, ) ; ( θ d l , can be transformed as follows: ), ; ( ) ; ~ ( ) ; ( # 1 ˆ 1 ˆ V η l g d g l d l = = − − θ θ θ θ (3) where V=V(θ, θ ) ) is a pivotal quantity whose distribution does not depend on unknown parameter θ; η=η(d, θ ) ) is an ancillary factor; θ ) is a maximum likelihood estimator of θ (or a sufficient statistic for θ). Then the best invariant decision rule (BIDR) is given by ), , ( 1 BIDR θ ) ∗ − = ≡ η η d u (4) where { } ) ; ( inf arg # V η η η l E = ∗ (5) and a risk function { } { } ) ; ( ) ( ) ( # BIDR BIDR V ∗ = ; = ; η l E u l E u r θ θ θ (6) does not depend on θ. Consider now a situation described by one of a family of density functions f(x;μ,σ) indexed by the vector parameter θ=(μ,σ), where μ and σ(>0) are respectively parameters of location and scale. For this family, invariant under the group of positive linear transformations: x→ax+b with a>0, we shall assume that there is obtainable from some informative experiment (a random sample of observations X=(X1, ..., Xn)) a sufficient statistic (M,S) for (μ,σ) with density function h(m,s;μ,σ) of the form ] / , / ) [( ) , ; , ( 2 σ σ μ σ σ μ s m h s m h − = • − (7)