Large inventory-service optimization in configure-to-order systems

US 6,970,841 B1
Filed: 04/17/2000
Issued: 11/29/2005
Est. Priority Date: 04/17/2000
Status: Expired due to Fees

First Claim

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1. A computer implemented process of managing manufacturing logistics of configure-to-order end products comprising the steps of:

a) initializing a process of managing manufacturing logistics of configure-to-order end products by setting x_i;

=0 for each iε

S, setting r_mi;

=P(X_mi>

0), setting β

_m;

=0 for each mε

M, and setting β

;

=0, where S is a set of components indexed by i, M is a set of end products indexed by m, x_iis a probability of no-stockout of a component of index i, r_miis a probability that a positive number of units of component i is used in the assembly of an end product indexed by m, β

_mis a probability of stockout of an end product of index m, and β

is an upper limit on the stockout probability over all end products;

b) setting a set of active components to A;

={ };

c) considering each iε

S, followed by considering each end product m that uses component i in its bill-of-material;

d) setting β

_m;

=β

_m+r_miΔ

, for all m such that iε

S_mwhere Δ

is a unit step size;

e) computing the a difference δ

_i;

=max_m{β

_m}−

β

;

f) determining if δ

_i≦

0, and if so, then adding component index i to the set of active components, A;

=A+{i};

g) determining if the set of active components is non-empty, and if so, then setting B;

=A, otherwise setting B;

=S where B is a set of component indexes;

h) finding i*;

=arg max_iε

B{−

c_iσ

_i/r_mig′

(x_i+Δ

/2)}, where −

g′

(●

) follows the equation $- g^{'} (x) = - Φ ({\overline{Φ}}^{- 1} (x)) \cdot \frac{- 1}{ϕ ({\overline{Φ}}^{- 1} (x))} = \frac{1 - x}{ϕ ({\overline{Φ}}^{- 1} (x))},$ where Φ

(.) is a probability distribution function of the standard normal variate, and φ

(.) is a probability density function of the standard normal variate;

i) setting x_i*;

=x_i*+Δ

to update the probability of no-stockout of component i*;

j) computing β

;

=max_mε

Mβ

_m, and checking whether inequality $\sum_{i \in S}^{} c_{i} σ_{i} g (x_{i}) \leq β,$ where B is the budget limit on the expected overall inventory cost, is satisfied and if so, stop and replenish components identified by said set B from suppliers following a base-stock policy that minimizes a total cost of inventory of said components i,wherein said cost c_iof at least one component differs from said cost c_iof at least one other component;

k) otherwise, updating β

_mand for each mε

M_i*, set β

_m;

=β

_m+r_miΔ

, and going to step b).

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Abstract

A manufacturing process is migrated from an existing operation to a configure-to-order (CTO) system. As the CTO operation will eliminate the “machine-type model” (MTM) inventory of the existing operation, the emphasis is shifted to the components, or “building blocks”, which will still follow the build-to-stock scheme, due to their long leadtimes, and hence still require inventory. The solution involves an inventory-service trade-off of the new CTO system, resulting in performance gains, in terms of reduced inventory cost and increased service level. Other benefits include better forecast accuracy through parts commonality and risk-pooling, and increased customer demand, as orders will no longer be confined within a restricted set of pre-configured MTMs.

Citations

9 Claims

1. A computer implemented process of managing manufacturing logistics of configure-to-order end products comprising the steps of:
- a) initializing a process of managing manufacturing logistics of configure-to-order end products by setting x_i;
  
  =0 for each iε
  
  S, setting r_mi;
  
  =P(X_mi>
  
  0), setting β
  
  _m;
  
  =0 for each mε
  
  M, and setting β
  
  ;
  
  =0, where S is a set of components indexed by i, M is a set of end products indexed by m, x_iis a probability of no-stockout of a component of index i, r_miis a probability that a positive number of units of component i is used in the assembly of an end product indexed by m, β
  
  _mis a probability of stockout of an end product of index m, and β
  
  is an upper limit on the stockout probability over all end products;
  
  b) setting a set of active components to A;
  
  ={ };
  
  c) considering each iε
  
  S, followed by considering each end product m that uses component i in its bill-of-material;
  
  d) setting β
  
  _m;
  
  =β
  
  _m+r_miΔ
  
  , for all m such that iε
  
  S_mwhere Δ
  
  is a unit step size;
  
  e) computing the a difference δ
  
  _i;
  
  =max_m{β
  
  _m}−
  
  β
  
  ;
  
  f) determining if δ
  
  _i≦
  
  0, and if so, then adding component index i to the set of active components, A;
  
  =A+{i};
  
  g) determining if the set of active components is non-empty, and if so, then setting B;
  
  =A, otherwise setting B;
  
  =S where B is a set of component indexes;
  
  h) finding i*;
  
  =arg max_iε
  
  B{−
  
  c_iσ
  
  _i/r_mig′
  
  (x_i+Δ
  
  /2)}, where −
  
  g′
  
  (●
  
  ) follows the equation $- g^{'} (x) = - Φ ({\overline{Φ}}^{- 1} (x)) \cdot \frac{- 1}{ϕ ({\overline{Φ}}^{- 1} (x))} = \frac{1 - x}{ϕ ({\overline{Φ}}^{- 1} (x))},$ where Φ
  
  (.) is a probability distribution function of the standard normal variate, and φ
  
  (.) is a probability density function of the standard normal variate;
  
  i) setting x_i*;
  
  =x_i*+Δ
  
  to update the probability of no-stockout of component i*;
  
  j) computing β
  
  ;
  
  =max_mε
  
  Mβ
  
  _m, and checking whether inequality $\sum_{i \in S}^{} c_{i} σ_{i} g (x_{i}) \leq β,$ where B is the budget limit on the expected overall inventory cost, is satisfied and if so, stop and replenish components identified by said set B from suppliers following a base-stock policy that minimizes a total cost of inventory of said components i,wherein said cost c_iof at least one component differs from said cost c_iof at least one other component;
  
  k) otherwise, updating β
  
  _mand for each mε
  
  M_i*, set β
  
  _m;
  
  =β
  
  _m+r_miΔ
  
  , and going to step b).

2. A method that translates end-product demand forecast in an assemble-to-order (ATO) environment into a forecast for components, taking into account outbound leadtime comprising the steps of:
- defining in an assemble-to-order (ATO) environment an end product demand D_m(t) of type m in period t, each unit of type m demand requiring a subset of components, denoted S_m⊂
  
  S, as $D_{i} (t) = \sum_{m \in M_{i}}^{} D_{m} (t + L_{m}^{out}); [and]$ deriving mean and variance for component demand D_i(t) as $\begin{matrix} E [D_{i} (t)] = \sum_{m \in M_{i}}^{} \sum_{l}^{} E [D_{m} (t + l)] P [L_{m}^{out} = l], and \\ Var [D_{i} (t)] = \sum_{m \in M_{i}}^{} \sum_{l}^{} E [D_{m}^{2} (t + l)] P [L_{m}^{out} = l] - \\ \sum_{m \in M_{i}}^{} {(\sum_{l}^{} E [D_{m} (t + l)] P [L_{m}^{out} = l])}^{2}, respectively; and \end{matrix}$ replenishing said components from suppliers following a base stock policy that minimizes a total cost of inventory of said components, each said component having a cost,wherein said cost of at least one component differs from said cost of at least one other component, and wherein said difference determines the result of said replenishing step.
- View Dependent Claims (3, 4, 5, 6)
- - 3. The method recited in claim 2, wherein the ATO environment is extended to a configure-to-order (CTO) environment for stationary demand, taking into account batch sizes comprising the steps of:
    - translating end-product demand into demand for each component i (per period) as $D_{i} = \sum_{m \in M_{i}}^{} \sum_{k = 1}^{D_{m}} X_{m i} (k) .$ where X_mi(k), for k=1, 2, . . . , are independent, identically distributed (i.i.d.) copies of X_mi;
      
      deriving marginal distributions, and then the mean and the variance of X_mias $\begin{matrix} E [D_{i}] = \sum_{m \in M_{i}}^{} E [X_{m i}] E [D_{m}], and \\ Var [D_{i}] = \sum_{m \in M_{i}}^{} (E [D_{m}] Var [X_{m i}] + Var [D_{m}] E^{2} [X_{m i}]) \\ = \sum_{m \in M_{i}}^{} (E^{2} [X_{m i}] E [D_{m}^{2}] + Var [X_{m i}] E [D_{m}] - \\ E^{2} [X_{m i}] E^{2} [D_{m}]), \end{matrix}$ respectively.
  - 4. The method recited in claim 3, extended to non-stationary demand, wherein the mean and the variance of X_miare generalized as $E$
    - [Di⁡
      
      (t)]=∑
      
      m∈
      
      Mi⁢
      
      E⁡
      
      [Xm⁢
      
      ⁢
      
      i]⁢
      
      ∑
      
      l⁢
      
      E⁡
      
      [Dm⁡
      
      (t+l)]⁢
      
      P⁡
      
      [Lmout=l],and $Var [D_{i} (t)] = \sum_{m \in M_{i}} E^{2} (X_{m i}) \sum_{l} E [D_{m}^{2} (t + l)] P [L_{m}^{out} = l] + \sum_{m \in M_{i}} Var (X_{m i}) \sum_{l} E [D_{m} (t + l)] P [L_{m}^{out} = l] - \sum_{m \in M_{i}} E^{2} (X_{m i}) {(\sum_{l} E [D_{m} (t + l)] P [L_{m}^{out} = l])}^{2}, respectively .$
  - 5. The method recited in claim 3, further comprising the steps of:
    - defining R_i(t) as a reorder point (or, base-stock level) in period t as
      R_i(t);
      
      =μ
      
      _i(t)+k_i(t)σ
      
      _i(t),where k_i(t) is a desired safety factor, while μ
      
      _i(t) and σ
      
      _i(t) can be derived (via queuing analysis) as $μ_{i} (t) = \underset{s = t}{\sum^{t + l_{i}^{i n} - 1}} E [D_{i} (s)], and$ $σ_{i}^{2} (t) = \underset{s = t}{\sum^{t + l_{i}^{i n} - 1}} Var [D_{i} (s)], respectively,$ where $l_{i}^{i n};= E [L_{i}^{i n}]$ is expected in-bound leadtime; and
      
      translating R_i(t) into “
      
      days of supply”
      
      (DOS), where the μ
      
      _i(t) part of R_i(t) translates into periods of demand and the k_i(t)σ
      
      _i(t) part of R_i(t) is turned into $\frac{k_{i} (t) σ_{i} (t)}{\frac{μ_{i} (t)}{l_{i}^{i n}}}$ periods of demand so that R_i(t) is expressed in terms of periods of DOS as ${DOS}_{i} (t) = l_{i}^{i n} [1 + k_{i} (t) \frac{σ_{i} (t)}{μ_{i} (t)}] .$
  - 6. The method recited in claim 5, wherein demand is stationary in which for each demand class m, D_m(t) is invariant in distribution over time, so that the mean and the variance of demand per period for each component i reduce to
    - μ
      
      i=lii⁢
      
      ⁢
      
      n⁢
      
      E⁡
      
      [Di],and⁢
      
      ⁢
      
      σ
      
      i2=lii⁢
      
      ⁢
      
      n⁢
      
      Var⁡
      
      [Di],respectively,⁢
      
      and $R_{i} = l_{i}^{i n} E [D_{i}] = k_{i} \sqrt{l_{i}^{i n}} sd [D_{i}], and hence, {DOS}_{i} = \frac{R_{i}}{E [D_{i}]} = l_{i}^{i n} + k_{i} θ_{i} \sqrt{l_{i}^{i n}} = l_{i}^{i n} [1 + k_{i} \frac{θ_{i}}{\sqrt{l_{i}^{i n}}}],$ where θ
      
      _i;
      
      =sd[D_i]/E[D_i] is the coefficient of variation of the demand per period for component i, and hence $θ_{i} / \sqrt{l_{i}^{i n}}$ is the coefficient of variation of the demand over the leadtime l_iⁱⁿ.

7. A method that relates service requirements to base-stock levels of components in an assemble-to-order (ATO) environment comprising the steps of:
- defining in an assemble-to-order (ATO) environment each order of type m as requiring exactly one unit of component iε
  
  S_m, α
  
  as a required service level, referred to as off-shelf availability of all the components required to configure a unit of type m product, for any m, and E_ias an event that component i is out of stock;
  
  determining a probability P for each end product mε
  
  M,
  P[⊂
  
  _iε
  
  S_mE_i]≦
  
  1−
  
  α
  
  , and $P [⋃_{i \in S_{m}} E_{i}] = \sum_{i} P (E_{i}) - \sum_{i < j} P (E_{i} ⋂ E_{j}) + \sum_{i < j < k} P (E_{i} ⋂ E_{j} ⋂ E_{k}) - \dots, and P [⋃_{i \in S_{m}}] ≅ \sum_{i \in S_{m}} P (E_{i}) = \sum_{i \in S_{m}} \overline{Φ} (k_{i}) \leq 1 - α; and$ establishing base stock levels for each component i that minimize a total cost of inventory of said components, each said component having a cost,wherein said cost of at least one component differs from said cost of at least one other component, and wherein said difference determines the result of said step of establishing base stock levels.

8. The method recited in 7, wherein the method is extended to a configure-to-order (CTO) environment taking into account batch sizes, further comprising the steps of:
- defining A⊂
  
  S_mas a certain configuration, which occurs in a demand stream with probability P(A);
  
  weighting a no-stockout probability, Π
  
  _iε
  
  AΦ
  
  (k_i), by P(A);
  
  changing the service requirement to $\begin{matrix} α \leq \sum_{A \subseteq S_{m}} P (A) \prod_{i \in A} Φ (k_{i}) \\ \approx \sum_{A \subseteq S_{m}} P (A) [1 - \sum_{i \in A} \overline{Φ} (k_{i})] \\ = 1 - \sum_{A \subseteq S_{m}} P (A) \sum_{i \in A} \overline{Φ} (k_{i}) \\ = 1 - \sum_{i \in S_{m}} (\sum_{i \in A} P (A)) \overline{Φ} (k_{i}); and \end{matrix}$ extending the CTO environment the service requirement to $\sum_{i \in S_{m}} r_{m i} \overline{Φ} (k_{i}) \leq 1 - α$ where r_miis the probability that a positive number of units of component i is used in the assembly of an end product indexed by m.

9. A method that translates service requirements in terms of leadtimes into requirements for off-shelf availability of components comprising the steps of:
- relating an off-shelf availability requirement to standard customer service requirements expressed in terms of leadtimes, W_m, where a required service level of type m demand is
  P[W_m≦
  
  w_m]≧
  
  α
  
  ,mε
  
  M where w_m'"'"'s are given data and P is probability;
  
  when there is no stockout at any store iε
  
  S_m, denoting the associated probability as π
  
  _0m(t), a delay being L_i^out, the out-bound leadtime;
  
  when there is a stockout at one or several stores in the subset s⊂
  
  S_m, denoting the associated probability as π
  
  s_m(t), so that the delay becomes L_i^out+τ
  
  _s, where τ
  
  _sis the additional delay before the missing components in s become available;
  
  determining $P [W_{m} \leq w_{m}] = π_{0 m} (t) P [L_{m}^{out} \leq w_{m}] + \sum_{s \in S_{m}} π_{sm} (t) P [L_{m}^{out} + τ_{s} \leq w_{m}];$ assuming that $L_{m}^{out} \leq w_{m} and L_{m}^{out} + τ_{s} > w_{m}$ both hold almost surely, so that when the (nominal) outbound leadtime is nearly deterministic and shorter than what customers require, whereas the replenish leadtime for any component is substantially longer; and
  
  replenishing said components from suppliers following a base stock policy that minimizes a total cost of inventory of said components, each said component having a cost,wherein said cost of at least one component differs from said cost of at least one other component, and wherein said difference determines the result of said replenishing step.

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Current Assignee
International Business Machines Corporation
Original Assignee
International Business Machines Corporation
Inventors
Yao, David Da-Wel, Ettl, Markus, Cheng, Feng, Lin, Grace Yuh-Jiun
Primary Examiner(s)
Jaketic, Bryan
Assistant Examiner(s)
Laneau, Ronald

Application Number

US09/551,118
Time in Patent Office

2,052 Days
Field of Search

705/28, 705/29, 705/10, 705/22, 705/26, 705/27
US Class Current

705/28
CPC Class Codes

G05B 19/4097   characterised by using desi...

G06Q 10/0631   Resource planning, allocati...

G06Q 10/087   Inventory or stock manageme...

G06Q 10/0875   Itemisation or classificati...

G06Q 20/203   Inventory monitoring

Large inventory-service optimization in configure-to-order systems

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Large inventory-service optimization in configure-to-order systems

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