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Significance of stochastic programming in addressing production planning under uncertain demand...

not encountered in previous models have been included, E it demand for product i in month t
adding diversity to the literature. Finally, by examining F j daily working time capacity of machine j
the model deterministically and stochastically and making y i unit cost of product i
comparisons, clarity on the efficiency of the application was h i unit cost of product i that is an unmet amount
achieved. D i inventory holding cost for the product i
W i number of units of product i per box
Q warehouse capacity
G j number of machines of type j
p s probability of scenario s
3. Method
B i safety stock quantity for the product i
R 1is initial stock of product i for scenario s
3.1. Definition of problem
Decision Variables:
Although significant advancements have been made in the
production field due to technological developments, con-
trolling external factors that influence production is not U i production quantity of product i for the 1st
feasible. Political, social, economic, and societal conditions month
production quantity of product i in month t
U it
substantially impact companies’ operations. The most
X its quantity of product i produced in month t ac-
measurable outcome of these influences is the uncertainty cording to scenario s
in demand. Accepting uncertainty and incorporating it inventory quantity of product i in month t
R it
into problem-solving is possible with stochastic modeling. inventory quantity of product i in month t ac-
R its
This study develops a stochastic model for production cording to scenario s
planning in a factory that produces gas clamps, aiming L it quantity of product i that is unmet in month t
to anticipate demand uncertainty and prepare accordingly. L its quantity of product i that is unmet in month t
Based on the annual sales volumes of this clamp-producing according to scenario s
factory, a production plan for the following year is created. N it sales quantity of product i in month t
N its sales quantity of product i in month t according
to scenario s
The production area contains seven machines: two presses, box quantity of product i in month t
K it
three guides, and two welders. Goods are processed in box quantity of product i in month t according
K its
related machines respectively to complete the production to scenario s
cycle. While product 1 goes through welder machines after
press, product 2 goes through guide machine after press.
All products are processed into a press machine first. Each 3.2. Mathematical model
machine operates for a total of eight hours. The processing
The model aims to minimize costs according to different
times of products on these machines have been calculated.
scenarios per the factory’s request. The problem is solved
The business starts the year with an initial inventory. The
total capacity of the warehouse is 30.000 boxes. The di- by establishing two models: deterministic and stochastic.
mensions of the boxes are the same for all products, but
the number of units per box varies. Inventory costs per 3.2.1. Deterministic model
product have been determined accordingly. Using all the
obtained data, it is assumed that demand for the coming The classical deterministic mathematical modeling ap-
year is uncertain, and an ideal production plan is to be proach determines the optimal solution based on the ex-
created using various scenarios. pected value without integrating the uncertain demand
into the model. So, the effect of stochastic or random de-
mand is not considered. The problem of cost minimization,
In the model, the decision variables include the quantities considering deterministic parameters, is given by equation
of products produced, quantities left unmet under different 1. The operating capacity of the business must not be ex-
scenarios, quantities of products sold, quantities of prod- ceeded, and each product has a specific processing time on
ucts in the shipping boxes used for product transportation, the machine. The capacity constraint in equation 2 is to
and inventories. Notations and parameters are provided ensure the total working hours are not exceeded. The in-
below: ventory remaining at the end of the month must equal the
production quantity within the month plus the previous
Notations: month’s inventory minus the quantity sold. Equation 3 is
added to ensure the inventory balance. The demand con-
straint is expressed by equation 4, where the sales quantity
i : product type, where i ∈ I
within the month must equal the total of the sales quantity
j : machine type, where j ∈ J
and the unmet quantity. The business wishes to maintain
t : month, where t ∈ T
a safety stock for products, and the inventory remaining
s : scenario, where s ∈ S
at the end of the month must not be less than the safety
stock. This constraint is expressed by equation 5. The
storage capacity is 30.000 packages. Accordingly, equa-
tion 6 calculates the number of packages required based
Parameters:
on the remaining inventory at the end of the month. In
contrast, equation 7 ensures that the package quantity for
a ij the time that product i spend on machine j
each product does not exceed the total storage capacity.
c i unit selling price of product i
Before production, the business has an initial inventory,
E its demand for product i in month t according to
and the inventory quantity for the first month must equal
scenario s
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