The Multiplicative MMFE

We denote the ratio of successive forecasts by δi

n ≡ Xni+1/Xni for n < N and δNi ≡

XN+1/XNi for each i ∈ {s, m}. We consider only two decision makers for the sake of brevity and without loss of generality. We refer to these decision makers as the supplier (s) and the manufacturer (m) because we apply the model to a forecast sharing problem between these two decision makers in §3.3.

The classical MMFE assumes that the multiplicative forecast update for each decision maker, i.e., δi

n, is log-normally distributed for every n. Part (c) and (d) of

3_{From the tower property of conditional expectation, E[X}

N+1|Fni] = E[XN+1|Xni] =

E[E[XN+1|Fni+1]|Xni] = E[Xni+1|Xni] = Xni +E[∆in|Xni]. If ∆in is correlated to Xni, the value of

E[∆i

n|Xni] may not be 0. In this case, decision makeri’s best demand forecast at periodtwould be

Xi

Theorem 3.1 imply that δi

n is independent of Xni and has a mean value of 1. Hence, the initial forecast X₁i and the variances of log(δ_ni) fully characterize the evolution of

X_ni for each decision maker i4. However, the variance of log(δ_ni) is not sufficient to characterize the interaction between the two forecast processes Xs

n and Xnm. One may determine the correlation coefficient between log(δs

ns) and log(δ

m nm) for

every ns and nm to characterize the interaction between the two forecast sequences. However, this approach can lead to inconsistency. We provide one such example. Suppose that the correlation coefficient between log(δ₁s) and log(δ₁m) is 1 and the correlation coefficient between log(δs₂) and log(δm₁ ) is also 1. Then, by obtaining the value of δs

1, the supplier obtains the full information of δ1m, which also contains the

full information of δs

2. Hence, the property that δ1s and δ2s are independent does not

hold.

We propose a different approach to model the interaction between Xs

n and Xnm, which does not suffer from any inconsistency such as the one discussed above. Deci- sion makers update their forecasts by obtaining information about events that affect demand. Following Hausman (1969), suppose there are in total K such events and let ej be the random variable that models the impact of event j. According to the theory of proportional effect (Aitchison and Brown 1957), the change in the forecast by each event is proportional to the size of the current forecast. In other words, after obtaining the information of eventj, decision makeriupdates the forecast fromX_ni to

X_niej. Following this explanation, we first express demand byXN+1 =

QK

j=1ej. Next, we divide the set of all events into (N + 1)×(N + 1) sets by the time at which the information is obtained by each decision maker. More specifically, we define Ens,nm

as the set of events whose information is obtained by the supplier during period ns and by the manufacturer during period nm.

We define δns,nm ≡ Q

j∈Ens,nmej, which indicates the total information obtained

by the supplier at period ns and by the manufacturer at periodnm. We assume that eachδns,nm is log-normally distributed and has a mean value of 1 exceptδ0,0

5_{. When}

4_{The assumption that E[δ}i

n] = 1 for a log-normal random variable δ i

n implies E[log(δ i n)] = −V ar(log(δ_ni))/2. Therefore, the variance of log(δi_n) is sufficient to characterizeδ_ni.

5_{By taking the logarithm ofδ}

ns,nm, we get log(δns,nm) = P

j∈E_ns,nmlog(ej). When the number of

events in Ens,nm becomes large, P

Ens,nm is an empty set, i.e., when no information is obtained by the supplier at period

ns and by the manufacturer at periodnm, thenδns,nm = 1. Note that by construction

a distinct piece of information is contained in one event set, hence δns,nm forms an

independent set of random variables.

Given this construction, we can express demand as XN+1 =

QN ns=0

QN

nm=0δns,nm.

The supplier’s information set at the beginning of period n is

F_ns≡σ([δ0,0, . . . , δ0,N], . . . ,[δn−1,0, . . . , δn−1,N]).

Then, the supplier’s demand forecast is Xs

n = E[XN+1|Fns] =

Qn−1

ns=0 QN

nm=0δns,nm,

and the ratio of successive forecasts is δs n =

QN

nm=0δn,nm. Because the multipli-

cation of log-normal random variables is also a log-normal random variable, δs_n is also log-normally distributed. Therefore, from the supplier’s perspective, the fore- cast evolution is consistent with the classical MMFE. The manufacturer’s forecast can be expressed in a similar way. Figure 3.1 illustrates the information structure of the MMFE for two decision makers. During each period, the supplier obtains all information given in the row corresponding to that period, whereas the manufacturer obtains all information given in the corresponding column.

Figure 3.1: Information Structure of the MMFE

n 0 1 · · · N (s) 0 δ0,0 × δ0,1 × · · · × δ0,N X1s × × × × × 1 δ1,0 × δ1,1 × · · · × δ1,N δ1s × × × × × .. . ... × ... × . .. × ... ... × × × × × N δN,0 × δN,1 × · · · × δN,N δsN (m) Xm 1 × δ1m × . . . × δNm XN+1

From this construction, we can fully characterize the evolution of X_ns and X_nm by

limit theorem. Because both decision makers have the information δ0,0 before the beginning of

the forecast horizon, we assume that δ0,0 is a deterministic value. When E[δns,nm] 6= 1 for some (ns, nm), we can push this information toδ0,0 and normalizeδns,nm byδns,nm/E[δns,nm], hence the assumptionE[δns,nm] = 1 is without loss of generality.

determining the value of δ0,0 and the variances of log(δns,nm).

Note that the demand and the forecast revisions have the following relationship;

XN+1 = X1i

QN

k=1δ

i

k for each decision maker i. At the beginning of period n, deci- sion maker i has the information Xi

1, δ1i, . . . , δni−1, but does not have the information

δi

n, δin+1, . . . , δiN. Hence, the multiplication

QN

k=nδki represents the demand uncertainty faced by decision maker i at the beginning of periodn.