ScienceAsia 27 (2001)

Material balance between stages constraints
I _{i 4 ,t1 }I I I
i1,t1
_{i3,t1 } X
^{i2,t1 } X
X
i3t
i1t
I
i2t
i4t
i3t
I I
X I X
X
i1t
i2t
i4t
i4t
X X
i2t
i4t
i5t
t; i 13,14,15; k 3,
, t; k 4,
, t; k 2,(8) , t; k 1, (9)
(6) (7)

Capacity constraints
X_{ikt }
= Quantity of product i to be produced at
I_{ikt }
stage k in period t (units) = Inventory of product i at stage k at the
R_{kt }
end of period t (units) = Regular time used at stage k during
O_{kt }
period t (manhours) = Overtime used at stage k during period
Subject to

Finished product requirement constraints
I _{i 5,t1 }
X I d i t i t i t 5 5
, t; k 5, (5)

Decision variables:
N a X i k i k t 1
R k t
O k t
k , t,
(10)
Available regular and overtime constraints.
k , t, k , t,
(11) (12)
t (manhours)
R r m k t k t ( ) O o m k t k t ( )
, (4)
LP model:
M i n i m i z e Z h I c O i k i k t t T k K i N o k t t T k K 1 1 1 1 1

Inventory
constraints.
capacity
of
finished
product
I W i k t N 1
t; k 5,
(13)

Safety stock of finished product constraints.
I i k t
S S i t
, t; k 5,
(14)
ikt X 0
, k, t, , t; k 1,2,3, 4
(15)
ikt I 0
(16)

Nonnegativity conditions
277
Eq.7 represents the material balance constraint in Stage 3, which produces the body of three component products, for Products 13, 14, and 15. Constraint (13) must be included since the finished products are very bulky and require significant warehouse space that is quite limited. Workin process inventory does not require significant storage space because it can be stacked. The nonnegativity constraint (16) ensures that shortages of workin process inventory do not occur.
Input Parameters
The initial inventory of product i at stage k, I_{ik0}, is collected from real data of workinprocess or finished good inventories on the factory floor at the beginning of the planning horizon. The inventory holding cost of product i at stage k, h_{ik}, is estimated by assuming that the annual inventory holding cost is 25% of the cost per unit of the product at the respective production stage. Since the cost per unit is constant over the planning horizon, the annual inventory holding cost is timeinvariant. The factory has enough space in the warehouse to store not more than 40,000 units of finished products.
The total available regular time, (rm)_{kt}, is estimated based on the fact that the factory is normally operated 16 hours a day and six days a week, and the total available overtime, (om)_{kt}, is calculated by assuming that the overtime could not be more than six hours a day.
The overtime cost, c_{o}, is assumed to be constant throughout the planning horizon, and is estimated to be 60 Baht per manhour.
After all related parameters have been estimated and entered into the planning model, the optimal values of all decision variables are calculated using the LINGO software. The computation time takes less than one minute on a Pentium PC.
Results of the Production Planning Models with Different Levels of Safety Stock
In this section, two production planning models with different safety stock levels (as shown in Table 5) are solved to determine the total cost savings when the recommended forecasting models are applied in place of the current practice. The inventory holding, overtime, and total costs of both models are presented in Table 6.
Based on the optimal total cost of the current practice (4,078,746 Baht per year) and the optimal total cost of the recommended forecasting models (3,541,772 Baht per year), the total cost saving is 536,974 Baht per year, or 13.2 %. It can be also seen