Collaborative Forecasting, Delayed Information and Informa-

formation Sharing

Consider the collaborative forecasting process discussed in Aviv (2001, 2002, and 2007). When the two decision makers collaborate to forecast demand, they share all available information. Based on Definition 3.1, we define the collaborative information set as Fcf

n ≡ Fns∪ Fnm. Because the union of two σ-fields is also a σ-field, Fncf is a well-defined information set. Then, the collaborative forecast (CF) of the two decision makers is Xcf

n ≡ E[XN+1|Fncf]. Next we derive the most important property of the CF.

Theorem 3.2. The CF has a smaller mean-squared-error than the forecast of a single decision maker, i.e., E[(XN+1−Xncf)2]≤E[(XN+1−Xni)2] for every i∈ {s, m}. This result states that two decision makers who collaborate can predict demand more accurately.

For the case of multiplicative MMFE, the collaborative information set Fcf n in- cludes all δns,nm such that ns ≤ n or nm ≤ n. Hence, the initial forecast is X

cf 0 = δ0,0 QN ns=1δns,0 QN

nm=1δ0,nm, and the ratio of successive forecasts are

δ_ncf =δn,n N Y ns=n+1 δns,n N Y nm=n+1 δn,nm.

Figure 3.2(a) illustrates the information structure available to decision makers under a collaborative forecasting scheme. Because eachδcf

n is a log-normal random variable with the mean value of 1, the collaborative forecast is also a multiplicative MMFE.

Figure 3.2: Information Structure of the multiplicative MMFE

(a) Collaborative Forecast (b) Asymmetric Forecast Evolution

Aviv (2001) also uses the MMFE to describe the forecast sequences of the two decision makers. He first models the forecast sequence of each decision maker as an MMFE with the initial forecast ofXi

1 =µδi0 and the multiplicative forecast revisions

of δi

n. Then, he assumes that V ar(log(δni)) = (ηiσn)2 for every n = 0, . . . , N −1, and V ar(log(δi_N)) =σ2−PN−1

n=0(η

i_σ

n)2. The value ofσ represents the degree of total demand uncertainty, and the value of ηi represents the forecasting power of decision maker i. He models the interaction between the two forecast sequences by assuming that the correlation coefficient between log(δs

ns) and log(δ

m

nm) is ρ for ns = nm, and

0 for ns 6= nm. In other words, the forecast revisions of two decision makers are correlated, but not inter-temporally6_.

In the MMFE for multiple decision makers, we construct a single demand model, which automatically constructs the forecast sequence of each decision maker. By construction, our model does not suffer from inconsistency. In contrast, Aviv (2001) constructs the forecast sequence of each decision maker separately and then models the interaction between them. This approach may lead to inconsistency. For example, whenηs _=ηm _{= 1 andσ}

N−1 =σ, both decision makers obtain all demand information

during period N −1, hence the correlation coefficient ρ should be exactly 1 for the demand models to be consistent. For this reason, Aviv (2001) provides a sufficient

6_{Iida and Zipkin (2009) also use the MMFE to describe the forecast sequences of two decision}

condition on (ρ, ηs_{, η}m_{) that guarantees consistency.}

As mentioned above, Aviv (2001) assumes the inter-temporal independence be- tween the two decision makers’ forecast revisions. However, some important forecast- ing scenarios do not follow this assumption. We discuss two such examples (delayed information and asymmetric forecast evolution) shortly. The MMFE for multiple decision makers covers the general case including that of Aviv (2001). Here, we de- scribe his model with our framework. The inter-temporal independence means that

δns,nm = 1 unlessns=nm orni =N fori∈ {s, m}. Hence, the information obtained

by the two decision makers at period n consists of three parts, δn,n, δn,N and δN,n. They correspond to the three corners of a single block in Figure 3.2(a). The correlated part of δs

n and δmn corresponds to δn,n, hence we can set V ar(log(δn,n)) = ρηsηmσ2n. Similarly, the uncorrelated parts ofδs

n and δnm correspond toδn,N and δN,n, hence we can setV ar(log(δn,N)) = (ηsσn)2−ρηsηmσ2nandV ar(log(δN,n)) = (ηmσn)2−ρηsηmσ2n. Next consider the delayed information scenario discussed by Chen (1999). In this case, the manufacturer observes demand of a product at each period and makes a replenishment order to the supplier. The supplier of the product receives the man- ufacturer’s order with the delay of l periods. The decision makers use the demand history to update forecasts. Therefore, the information sets of the two decision mak- ers are identical withl periods of delay. In other words, we haveFs

n+l =Fnm for every

n. Then, the supplier and the manufacturer have the same sequence of forecasts with a delay of l periods. In the multiplicative MMFE, the delayed information can be represented by δn,k = 1 for every k+l 6=n.