Page 53 - IJOCTA-15-2

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D. Balpınarlı, M. Onal / IJOCTA, Vol.15, No.2, pp.245-263 (2025)
re-manufacturing. See 74–80 for other examples of Decision variables:
successful application of TSA to lot sizing prob- D it = demand of item i in period t
lems. x it = quantity of item i produced in period t
y it = binary variable that indicates production
setup for item i in period t
3. Mathematical modelling I it = amount of item i left over at the end of
period t (after demand is satisfied)
The multiple item ELSIDD is an extension of the
and carried in inventory to period t + 1
multiple item ELS problem where the demand for
U it = amount of item i available in inventory
each item is a function of the amount of available
at the beginning of period t, right after
stock of that item. The problem is defined for N
production but before demand is satisfied
items over a discrete and finite planning horizon
Mathematical Model:
of T periods. At the beginning of each period t,
N T N T
t = 1, . . . , T, if there is production for an item, X X X X
max p it D it − (S it y it + c it x it + h it I it )
the produced amount in that period is added to
i=1 t=1 i=1 t=1
the stocks. For each item i, i = 1, . . . , N, we (1)
let x it be the production quantity of item i in
subject to
period t for t = 1, . . . , T, and let I it be the left
over items in the inventory at the end of period I i,t-1 + x it = D it + I it , i=1,...,N, t=1,...,T (2)
t (i.e., after demand is satisfied in period t), for U i,t = I i,t-1 + x it , i=1,...,N, t=1,...,T (3)
t = 0, . . . , T. The demand for each item i in a
D it = g it (U it ), i=1,...,N, t=1,...,T (4)
period t is a piecewise linear concave function of
N
U it , amount of item i in the stocks at the begin- X
x it ≤ C t , t=1,...,T (5)
ning of period t right after production. In that
i=1
regard, U it = I i,t−1 + x it , i.e., it is equal to the
x it ≤ C t y it , i=1,...,N, t=1,...,T (6)
stock quantity carried from period t − 1 plus pro-
y it ∈ {0, 1}, i=1,...,N, t=1,...,T (7)
duction quantity of that item in period t. We call
U it the available stock after production (but be- I it , U it , D it ≥ 0, i=1,...,N, t=1,...,T (8)
fore demand is realized). After the demands for I i0 = 0, i=1,...,N (9)
items are realized (as a function of U it ), leftover
products are carried over to the following period. The first summation term of the objective
function (1) stands for the total revenues, and the
There are item and period dependent unit pro-
second summation stands for total setup, produc-
duction (p it ) and holding costs (h it ), and fixed tion, and holding costs. Constraint (2) guarantees
setup costs (S it ). The total amount that can be the balance between inventory, production, and
produced in period t is limited by the production demand in each period. Constraint (3) is the def-
capacity C t . The problem is planning production inition of U it , the total amount of item i available
after production in period t. Constraint (4) states
for each item to satisfy the resulting demands
that D it , the demand of item i in period t, is a
and maximize profit. The parameters and de- function of U it , available stock of that item in pe-
cision variables are summarized below, followed riod t. Constraint (5) states the capacity restric-
by a formulation of the multiple item ELSIDD tions for production. Constraint (6) is the setup
enforcing constraint. Constraints (7) & (8) state
problem:
the domains of decision variables. Constraint (9)
states that there are no initial inventories.
Parameters: Functions g it (.), i = 1, . . . , N, t = 1, . . . , T,
C t = production capacity in period t
which are also called as Demand Functions, posses
4
S it = fixed setup cost of production of item exactly the same properties as in. To summa-
i in period t rize, they are piecewise linear and concave in U it ,
p it = unit selling price of item i in period t and after U it reaches a certain point, the func-
tions flatten out (i.e., their slopes become zero).
c it = unit production cost of item
This implies that after some point, increasing the
i in period t
stocks, i.e. U it values, does not cause any further
= unit inventory holding cost of item i
h it increase in demand, i.e., D it . A demand function
max
in period t g it (.) consists of J it segments. The j th segment
j
j-1
g it (.) = demand function for item i in period t of this function is defined in the interval [u ,u )
it it
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