The Operational-level Planning Model

The variable costs are estimated from the output (decision variables) of the strategic-level sub-model, customer demand requirements, minimum required service and flexibility levels, cost and lead-time data, and bill of material data.

Additionally, variable costs are estimated under uncertainty. Also, various operational variables are calculated by optimizing inventory variables including lot sizes, reorder points and safety stock. A multi-objective function is developed incorporating all tradeoffs in cost, customer service level (fill rate), and flexibility (delivery). Four sub-models are addressed at the operational level: (1) GC-module control, (2) bare-bone assembly control, (3) bare-bone stockpile control, and (4) full-set configuration control. The GC-module control and full-set configuration control sub-models are solved with analytical techniques, while the bare-bone assembly control and bare-bone configuration control sub-models are simultaneously optimized using non-linear programming. A single solution for the operational-level planning model is derived by a heuristic approach, as described in the following subsections. Table 6.2 presents the notations utilized in the operational sub-model.

A. GC-module Control sub-model

This model assumes continuous review of the inventory position for each GC-module r involved in producing bare-bone set F at plant k, using an (s, Q) _rk inventory control policy. A fixed quantity (Q ) is ordered whenever the inventory _rk position drops to the exact reorder point s. The demand requirement for GC-module r is calculated from the assembly requirement of bare-bone i at plant k (X ), which _ik is determined at the strategic level, and the unit usage rate of r in i (τ_ri) is specified in the BOM data. The GC-module shortages are assumed to be back-ordered. The GC-module control analytical sub-model is formulated as the following equations.

To simplify the computations, a normal lead-time demand distribution is also assumed. Using standard terms, as in Silver and Peterson (1985), Eq. (17) indicates the total cost of controlling GC-module inventory at assembly plant k, which involves setup, holding, and backorder (delay) costs. Eq. (18) calculated the on-hand inventory level (average inventory level plus safety stock) is given by. The safety factor n is selected to control the safety stock associated with a specified _rk customer service level.

(17)

(18)

The approximate expression for n is given as Silver and Peterson (1985) and _rk Johnson et al. (1996). The required reorder point s can be calculated directly _rk using Eq. (19), where L indicates the expected demand over a replenishment _rk lead-time, and Θ indicates the average total replenishment lead-time of r at k. _rk

Θ is calculated as the sum of the GC-module lead-time and delay time, rk

considering all suppliers.

(19)

Table 6.2 Notations for operational-level planning model

Inputs Definitions – GC-module control sub-model

θrk Order setup cost of replenishing r at k ($) u_rvk Expected lead-time of r from v to k (period)

Hrk Unit holding cost of r at k ($/period/unit) ρ_rv Expected delay time of r at v (period)

πrk Unit backorder penalty cost for shortage r at k

($/unit) p^rv Module availability (fill rate) for r at v

Outputs Definitions

∗

Qrk Optimal batch size of module r at plant k (units) P_rk^∗ Optimal fill rate for r at k

Irk Inventory holding level of r at k (units/period) n_rk Safety stock factor of r at k

σrk Standard deviation of replenishment lead-time

demand of r at k S^rk Reorder point for r at k (units)

Lrk Expected demand of r over a replenishment

lead-time at k u^rk Unit cost involved in controlling r required at k ($/unit)

Θrk Average total replenishment lead-time for r at k u_ik^G Unit cost involved in controlling all r required at k for i ($/unit)

Inputs Definitions -- Bare-bone assembly control and Bare-bone stockpile control sub-models

θik Production setup cost of i at k ($) H_ik Unit holding cost of i at k ($/period/unit)

Γik Unit processing cost of i at k ($/unit) x_ikm Unit holding cost for i en-route from k to m ($/period/unit)

Ωik Unit work-in-process holding cost for i at k

($/period/unit) N^ikm Normal transportation lead-time for i from k to

m (period)

gik Production setup time for i at k (period) E_ikm Expedited transportation lead-time for i from k to m (period)

lik Waiting time at the work station for i at k

(period) e^ik Cost of initiating an expedited production order

for i at k

η Customer service performance index T '_ikm Standard delivery time at k when i is out of stock at m (period)

υ Delivery flexibility performance index Output Definitions

p

TCik Total cost of assembling i at k TC_ik^S Total cost of stockpile for i at k

∗

Qik Optimal production batch size for i at k (units) P_ik^∗ Optimal fill rate of i at k p

uik Unit production cost for i at k ($/unit) u_ik^S Unit cost of stockpile for i at k

tik Total production lead-time for i at k (period) s_ik Reorder point of i at k (units)

hik Production processing time for i at k (period) L_ik Expected demand of i over production lead-time at k

Θik Module supply delay time for i at k (period) T_ikm Expected replenishment lead-time for i from k to m

TCik Total expected cost of production and stockpile

of i at k n^ik Safety stock factor of i at k

PSik Customer service (fill rate) availability of i at k σ_ik Standard deviation of replenishment lead-time demand of i at k

PDikm Delivery flexibility availability of i from k to m Inputs Definitions – Full-set configuration control sub-model

Hjm Unit holding cost of j at m ($/period/unit) I_jm Unit final assembly cost of j at m ($/unit)

θjm Order setup cost of j at m ($) Ω_sj Utilization rate for each KC-module s per unit of j

πjm Unit backorder penalty cost for shortage j at m ($/unit)

Outputs Definitions F

TCjm Total cost of full-set configuration of j at m S_jm Order-up-to level for j at m (units)

∗

Qjm Optimal final assembly batch size of j at m

(units) L^jm Expected demand of j over a replenishment

lead-time at m

Ujm Unit cost of throughput for j at m ($/unit) n_jm Safety stock factor of j at m

sjm Reorder point for j at m (units) σ_jm Standard deviation of replenishment lead-time demand of j at m

The variance of Θ is calculated as Eq. (20). The variance of _rk L is _rk calculated as Eq. (21). The optimal lot size (Q ) is then considered by Eq. (22), by _rk^∗

finding the first derivative of the total cost with respect to Q , and setting it equal _rk to zero.

(20)

(21)

(22)

The optimal service level (P ) for GC-module r at assembly plant k is _rk^∗ calculated as Eq. (23). The unit cost associated with GC-module r control at plant k is given by Eq. (24). The unit cost associated with controlling all GC-module required for bare-bone i at assembly plant k is given by Eq. (25).

(23)

(24)

(25)

B. Bare-bone Assembly Control and Bare-bone Stockpile Control sub-models

Eq. (26) indicates the cost function to be minimized of controlling bare-bone assembly system, which involves setup costs, processing costs and work-in-process carrying costs. More specifically, the total costs for the production of bare-bone i at

( )

(26)

The total production lead-time (t ) is given by Eq. (27), which determines the _ik sum of the setup time (g ), the waiting time at the workstations (_ik l ), the processing _ik time (h ) and the GC-module delay time (_ik Θ ). _ik

(27)

The processing time of a batch of bare-bone i at plant k can be calculated as Eq.

(28), where r denotes the average work rate for the processing of bare-bone i at _ik plant j. As long as r >1, then bare-bone i utilizes more than one GC-module type, and if the manufacturer cannot begin production until all GC-modules have been received, then the lead-time or delay-time in the model is given by the maximum average realized lead-time or delay-time from suppliers. The material delay time can be then determined from Eq. (28). Additionally, the unit cost of producing bare-bone i at plant k is given by Eq. (29).

(28)

(29)

An (s, Q) inventory control policy is adopted to operate the bare-bone stockpile control system. The distribution demand shortages are assumed to be met by expedited shipment. Using standard terms, as in Cohen and Lee (1988), the total costs related to the stockpile for bare-bone i at plant k per period are given by Eq.

(30), where the total cost is the sum of the stockpile holding cost, transportation holding cost from plant k to the configuration hubs and the expedited order setup cost.

The relative parameters for the bare-bone stockpile are calculated similarly to those in the GC-module control sub-model. These parameters, including reorder point (s ), variance of expected demand over production lead-time (_ik var

( )

Lik ), the expected replenishment lead-time or bare-bone i from plant k to configuration hub m (T_ikm) and the unit bare-bone stockpile cost (u ) are given by Eq. (31)－(35). _ik^S

(31)

(32)

(33)

(34)

(35)

A multiple objective function that addresses cost, customer service level (fill rate), and delivery flexibility tradeoffs is proposed to find the optimalQ_ik,P_ik,T_ikm. The first objective function applies cost as a performance measure, and is given by Eq. (36).

(36)

The second objective function represents service levels (fill rates) for replenishing the configuration hubs from the bare-bone stockpile at plant k, and is given by Eq.

(37). Finally, the delivery flexibility objective function is given by Eq. (38).

(37)

(38)

Using theε -constraint method, the multi-objective is formulated as Eq. (39)-Eq.

(41). The values of η, are specified to ensure the desired minimum levels of fill ν ( )ik

ik ik ik

ik ik ik ik

ik L

p t P

X n

L

s var

ln 1 2 2

1 ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ + −

= +

= σ π

( ) ( )Lik Xik var( )tik

var = ²

(_{ik ikm})( _ik)

ik ikm

ikm N P t E p

T = + + 1−

( ) [ ikm ik ikm ik ]

ikm

ikm x N P E P

c = + 1−

ik S S ik

ik X

u =TC

S ik P ik

ik TC TC

TC = +

ik ik

ik P P

PS = − '

ikm ikm

ikm T T

PD = ' −

rate and delivery flexibility.

(39) (40) (41)

C. Full-set Configuration Control sub-model

The full-set configuration control sub-model is formulated as the following equations. A continuous-review (s, S) inventory control policy is assumed, in which a replenishment quantity is made whenever the inventory position drops exactly to the reorder point s. The replenishment quantity is large enough to increase the inventory position to the order-up-to level S.

The simple sequential determination algorithm is adopted to determine the order-up-to level S. Demand is periodic, stochastic, and independently distributed among customer zones and over time. Additionally, the lead-time demand at each configuration hub is assumed to be normally distributed. Further, customer demand shortages are assumed to be backordered. The total cost of the full-set configuration system, which consists of holding cost, reorder, backorder cost, and configuration cost for full-set j at configuration hub m per period, is given by Eq. (42).

(42)

Relevant parameters, including expected replenishment lead-time for full-set j at configuration hub m (t_jm), expected demand of j over a replenishment lead-time at

m (L ), reorder point (_jm s_jm), and order-up-to level (S_jm), are also calculated similarly to those in the previous sub-model, and are given by Eq. (43)-Eq. (47).

(43)

(46)

(47)

To calculate the optimal batch size for full-set j at configuration hub m, the total cost equation is differentiated with respect toQ_jm, and set equal to zero (Eq. (48)).

Additionally, the optimal service level for full-set j at configuration hub m is calculated by setting the derivative (with respect toP ) of the total cost equation _jm (Eq. (49)). Unit cost of throughput for full-set j at configuration hub m is calculated by Eq. (50).

(48)

(49)

(50)

Figure 6.1 summarizes the interactive relationships between each control sub-model, according to these descriptions of operational-level planning model. The GC-module control sub-model calculates the optimal values ofQ ,_rk^∗ P_rk^∗ , and

calculates the relative parameters (u_ik^G,I_rk,s_rk,L_rk,Θ_rk) by an analytical process.

∗

p and rk Θ indicate inputs in the bare-bone assembly control sub-model. In the _rk bare-bone assembly and stockpile control sub-models, the optimal values of

ikm ik

ik p T

Q^∗, ^∗, are calculated from the cost, fill rate, and delivery flexibility tradeoffs.

The relative parameters (u ,_ik^p t_ik) can then be calculated, and t is input in the _ik

bare-bone stockpile control sub-model. Parameters (u_ik^S,s_ik,L_ik,c_ikm) are calculated

( )

jm jm

jm jm

jm jm jm

jm L

p L p

n L

s var

ln 1 2 2 1

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ + −

= +

= σ π

jm jm

jm s Q

S = +

jm z

mz jz jm

jm F jm

jm H

y D Q

Q TC

⎟⎠

⎜ ⎞

⎝

⎛

∂ =

=∂ ∑

∗

θ 2

jm jm jm

F jm jm

H p

P TC

−π

∂ =

=∂

∗ 1

=∑

k jkm F jm

jm B

U TC

configuration control sub-model. Additionally, the optimal values of Q ,_jm P_jmare

calculated, from which the relative parameters (U_jm,s_jm,S_jm,L_jm) can be calculated.

Now, we can summarize the actual unit variable costs (U_ik =u_ik^G+u_ik^P +u_ik^S,U_jm,c_ikm), which are adopted as inputs to the strategic-level planning model.

Figure 6.1 Interactive Relationships in Operational Parameters

在文檔中 筆記型電腦產業營運模式分析與供應鏈設計 (頁 54-62)