The whole textile-apparel and retailing system is one of the domains which undergo great economical and market pressures. In such a fast changing environment, the skill of the manager at any stage of the supply chain is greatly solicited in order to make the best, or more likely, the least bad decisions.
Some of these decisions may be of strategical nature, for example dealing with long term investment on warehouse facilities, or choice of strategic partners, manufacturers or third-party logistic providers, with the objectives of unifying some common processes. Other decisions are for very short term, if not in real time, such as those taken at the workshop level: scheduling problems, production decisions etc. The last type of decisions is intermediate of the previous, at the tactical planning level and covers few months of time horizon, e.g. one year budget decisions, or products purchasing policy for the next sales season. All of these decisions are interconnected, and involve a mix of exact data or knowledge and a greater or lesser amount of uncertainties. As obtaining extra information on these uncertainties may be expensive in terms of workload and information system investment, the main objective of this work is to identify the optimization methods and the minimal required information for these decision-making problems, based on the assessment of the value of information.
This chapter, which is an extended version of (Douillet and Rabenasolo, 2005), is focused on the quality and the robustness of a purchasing policy, especially the optimal order quantity. The purchasing problem is stated as follows: assuming a given knowledge of the future demand, the decision maker has to buy now a certain quantity of products in order to maximize his revenue during a future sales period.
In the textile-apparel context, the demand is hardly identified because of many factors typical to textile items: high competition between companies, consumer volatility, unpredictable fashion trends, high degree of diversity, and short product life cycle. The short life-cycle of most of the textile-apparel products implies that there is no opportunity to correct any error that the retailer may have made on this order quantity. The catalog of products is generally organized in two or four seasons of three months sales (spring, summer, autumn and winter seasons). Because of the specialization to one of these seasons, the product life-cycle rarely spans over two consecutive sales periods, for example summer-autumns. Moreover, the procurement delay for a fabric may be about three months, while the logistic and transportation delay may range from one to three weeks or more for delocalized industries. In the case of make-to-stock systems, the purchase decision should be taken about four months before any sale occurs. Any error cannot be corrected if this sales period is of three months. The best decision is a trade-off between the risk that the retailer can miss sales opportunity if he does not buy enough products (costs due to stock-out phenomena), and the risk that he may loose some revenue if the purchased quantity is greater than necessary (costs due to excess of inventory). The main problem is then, from the actual knowledge on the future demand, to define as precisely as possible these two risks and to solve the trade-off decision problem in a realistic context, without the bias of any extra information that is impossible to prove or to verify at the time of this decision.
In order to get a higher comprehension of the phenomena, the problem will be simplified and illustrated by the well known newsboy paradigm which is still largely used, with various extensions such as multi products environment with multiple discounts (Khouja, 1995), multi period sales and ordering (Groenevelt and Rudi, 2003), capacity constraints (Voros and Szidarovszky, 2001), or random procurement delays.
While in many real life cases, the only available knowledge is an
expectation of the mean demand 
, the newsboy paradigm proposes
an advanced model where the probability distribution of the demand
is exactly known. This assumption is a strong limitation because in
reality, the knowledge we can ever have upon the probability distribution
function (pdf) of the future demand is far lower than the limited
knowledge we can have over the future demand itself. For example,
it is impossible to prove, even after the sales season, that
the future demand will exactly follow the normal Gaussian or any pdf.
Therefore the best order decision must be analyzed under various weaker
hypotheses. The main assumption of this work is that, instead of propositions
similar to 'the total demand for the next 4 weeks is Poisson distributed
with mean 52500', we suppose that the retailer can reasonably make
assertions like 'we expect to sell in average 52500 units within an
interval of 
%'.
In all the studied cases, it will be assumed that the mean demand
is identifiable with enough precision through various statistical
techniques, and that, additionally, some second-order characteristic
of the demand is also identifiable. For example, the Scarf's (1958)
founding result addresses the case where the mean and the
standard deviation 
of the future demand are known. In (Douillet and Rabenasolo, 2005),
we have investigated for another second-order characteristic namely
, using notations
of Section 1.2 that appears to be interesting.
In such a condition, various families of demand pdfs 
or 
need to be investigated. The determination
of the optimal decision is then a max-min problem, where the objective
is to optimize the gain for the worst case over a family of demand
models, in order to guarantee a lower bound for the expected performance.
The present chapter is organized as follows. In Section 1.2, we restate the hypotheses, fix the notations and recall the formulae for the optimal order quantity and the associated gain are recalled. Thereafter, in Section 1.3, we compare the results obtained when using different families of models: normal, lognormal, triangular and the "two Diracs" model.
In Section 1.4, we discuss the hypotheses that were at the basis of these results. The fact is highlighted that actual measures can hardly provide sufficient knowledge to allow a direct application of these formulae, requiring a carefull evaluation of the robustness of the purchase decision obtained from what we really know upon the future demand.
In Section 1.5, we address this question by
the usual max-min method: for each order quantity 
, the worst-case
demand probability distribution function among a given family of models
is determined, and thereafter the value of that maximize the
''gain in the worst case'' is determined. The case of the triangular
models with given and is examined, and compared
with the general Scarf's solution (that uses "two Diracs"
models).
The chapter ends by some concluding remarks and a bibliography.