Ahmed Othman El-meehy
Faculty of Engineering, Ain Shams
University, Egypt
E-mail: ahmed.othman.shawky@eng.asu.edu.eg
Amin K. El-Kharbotly
Faculty of Engineering, Ain Shams
University, Egypt
E-mail: amin_elkharbotly@eng.asu.edu.eg
Mohammed M. El-Beheiry
Faculty of Engineering, Ain Shams
University, Egypt
E-mail: mohamed.m.mohamed@eng.asu.edu.eg
Submission: 1/07/2019
Revision: 2/10/2019
Accept: 2/27/2019
ABSTRACT
The
joint lot sizing and scheduling problem can be considered as an evolvement of
the joint economic lot size problem which has drawn researchers’ interests for
decades. The objective of this paper is to find the effect of a capacitated
multi-period supply chain design parameters on joint lot sizing and scheduling
decisions for different holding and penalty costs. The supply chain deals with
two raw materials suppliers. The production facility produces two products
which are shipped to customers through distribution centers. A mathematical
model is developed to determine optimum quantities of purchased raw materials,
production schedule (MPS), delivered quantities and raw material and products
inventory for predetermined number of periods. The model is solved to maximize
total supply chain profits. Results showed that at high capacity and low
holding cost, the supply chain tends to produce only one product each period,
for limited capacity and high value of holding cost, the supply chain may
produce the two products together each period.
Keywords: Joint Lot-Sizing and Scheduling,
Supply Chain optimization, Integrated Supply Chain
1. INTRODUCTION
In
production entities, planning activities are done to allocate resources
efficiently to satisfy customer demands while balancing contradicting
objectives. The key operational planning decisions are those related to
material purchasing, production, and delivery (SAWIK, 2016) as well as the
inventory management decisions (CUNHA et al.,
2018; SENOUSSI et al.,
2016).
Taking
each of these decisions independently result in conflicting issues that might
harm supply chain performance (ZHAO; WU; YUAN, 2016). For this
reason, and especially after mathematical tools have witnessed great
improvement, many researchers are currently directed towards integrating the
optimization of various supply chain decisions (GHARAEI; JOLAI, 2018). In this
context, the problem of joint lot-sizing and scheduling attains its importance
among different integrated decisions due its effects on supply chain
performance.
The
basic Joint Economic Lot size Problem (JELP) as defined by (BEN-DAYA; DARWISH; ERTOGRAL, 2008), and then
adopted by (SADJADI; ZOKAEE; DABIRI, 2014) is to determine
the order and delivery quantities for two echelons (vendor and buyer) supply
chain. In a later review (GLOCK, 2012), JELP is defined as determining order, production and
delivery (shipment) quantities for multi echelon supply chain minimizing total
costs.
The
basic JELP with various costs modelling is tackled by a number of researchers (ERTOGRAL; DARWISH; BEN-DAYA, 2007; LEE; FU, 2014; WANG; LEE, 2013; MARCHI et al.,
2016).
Extensions of the basic JELP can be found in (VAN HOESEL et
al., 2005), were the
decision variables were taken over a number of periods. Another extension was
done by (POURAKBAR; FARAHANI; ASGARI, 2007; GHARAEI; JOLAI, 2018) as
they solved supply chain designs with multi echelons and/or multi actor per
echelon.
Another
decision integration is to decide the lot size and schedule
simultaneously. This approached is
defined as lot size and scheduling problem where production sequence is
determined integrally with the production quantities (HUANG; YAO, 2013), this
definition is similar to the Economic Lot size and Scheduling Problem (ELSP).
Another definition to the problem is introduced by (TORABI; FATEMI GHOMI; KARIMI, 2006; HARIGA et al.,
2013)
which is to determine production quantities and delivery schedules in two
echelon supply chains.
The
two previous problems are considered integrally where the joint lot size and
scheduling problem emerged, to decide on the quantities of raw material
purchased, production, delivery, …etc. These are considered taken integrally
with determining the production schedule.
The production schedule may be to determine the sequence of production
of various products as done by (MUNGAN; YU; SARKER, 2010; ZHAO; WU; YUAN, 2016; JIA et al.,
2016), or
to determine which products to be produced each period i.e. determine the
Master Production Schedule (MPS) as tackled by (SARIN; SHERALI; LIAO, 2014; CUNHA et al.,
2018; SENOUSSI et al.,
2016).
In
this paper, the optimization of joint lot-sizing and scheduling problem is
considered were the materials purchasing, production and delivered quantities
are to be determined integrally with the MPS over a number of periods in
multi-echelon supply chains. The
optimization in present joint lot sizing and scheduling problem is made for
different production capacities while investigating the change in optimum
decisions at each production capacity level.
The
rest of the paper is organized as follows: in section 2 the literature review
of the joint lot size and scheduling problem, how the problem is developed and
what solution methodologies are used, section 3 describes the definition of the
problem, while in section 4 the proposed mathematical model of maximizing the
supply chain profits is presented. Numerical Experiments and results with
discussions are given in section 5 and 6 respectively while conclusions and
recommendation of future work is given in section 7.
2. LITERATURE REVIEW
The
scheduling decisions can be operational or tactical. In operational decisions,
products’ sequencing is made, while in tactical decisions, the MPS is developed
(MUNGAN; YU; SARKER, 2010), studied the
joint lot-sizing and scheduling problem in a two stages supply chain whose
products suffer from continuous price reduction during its life cycle.
They
found optimal lot-sizes for procurement and production, and delivery schedules
that minimize total costs of raw materials (ordering and purchasing), and
finished products (setup, production and holding). The results showed that on
adopting the policy of smaller and more frequent deliveries, the considered
costs are lower (JIA et al.,
2016),
addressed the problem of unconstrained delivery consolidation for a
manufacturer and multiple buyers supply chain.
They
optimized lot-sizes at the manufacturer and hence their production starting
times, replenishment lots for different buyers and suitable delivery schedules
that minimize total costs per unit time including ordering, setup, and holding
at the manufacturer and buyers. Through numerical experiments, it is
demonstrated that adopting delivery consolidation in multi-buyer supply chains
with SPT scheduling and capacity utilization approaches improves the supply
chain costs (SAĞLAM; BANERJEE, 2018), formulated a
mathematical model that integrates batch production schedules and shipment
scheduling decisions to minimize setup costs, transportation costs and
inventory holding costs per unit time for a two-echelon supply chain with
multiple products.
In a
common cycle approach, they determined the amounts produced, carried in
inventory and shipped to the customers as well as production cycle length,
shipment interval and number of shipments considering Shipment capacity.
Results showed that when variable transportation costs are used, the optimal
shipment schedule is lot-for-lot according to the demand.
A
larger supply chain is considered by (ZHAO; WU; YUAN, 2016), as they
considered an integrated supply chain composed of four echelon that delivers
finished goods to customers having time-varying demand of a single product over
a finite planning horizon. They determined optimally the batch size of finished
goods, number of production cycles, setup time in each cycle, and raw material
order times that minimize total operational costs. They proved the problem can
be solved optimally for a time varying demand product.
Other
researchers integrated the MPS decisions with lot sizes decisions such as: (SARIN; SHERALI; LIAO, 2014) discussed the
problem of integrating lot-sizing and scheduling for different product families
in the primary manufacturing phase in a pharmaceutical supply chain. Different
pharmaceutical ingredients are to be scheduled on parallel capacitated bays for
production in batches.
Changeovers
between different production families necessitate setup times and costs. The
objective is to minimize inventory holding and setup costs. Results showed the
effectiveness of this modeling and the column-generation solution approach in
remarkable reductions in computational times (CUNHA et al.,
2018),
addressed the problem of integrating lot sizing of purchased raw materials with
production scheduling of final products to fully meet customer demands in a
chemical industry.
Purchased
materials are brought from different suppliers whose discount rates are
different and depend on the purchased quantities. The production scheduling
considers batch production of multi-stage production structure. The objective
is minimizing total costs incorporating raw material purchasing, ordering,
holding of both raw materials and final products, setup and production costs.
To
highlight the importance of integrated scheduling and purchasing decisions, the
authors solved the problem once on an integrated approach and compared the
results to the independent (disintegrated) approach. Results have shown that
the integrated approach outperforms the disintegrated one in all instances of
their experimentation (SENOUSSI et al.,
2016),
introduced the integration of vehicle routing to the joint lot-sizing problem
in a supply chain composed of a single supplier production facility and
multiple retailers performing under Vendor Managed Inventory (VMI) policy.
The
authors considered vehicle capacity limitations, production capacities, and
retailers' inventory capacities. The objective is obtaining optimal production,
inventory and delivered quantities along with scheduling of production,
vehicles and receiving retailers each period that minimize total supply chain
costs. Numerical results show that the valid inequalities used improved the
quality of the formulations. Also, the parameters influencing computational
times are analyzed.
From
the aforementioned review, it is obvious that the Joint Lot sizing and
scheduling problem is gaining attention in the last few years, yet it is seldomly tackled with the effect of supply chain design
parameters such as capacity and location.
Most of the work reviewed solved the 2 echelons supply chain problem. (ZHAO; WU; YUAN, 2016) solved the
problem for a larger supply chain, while in real life the integration of more
members in the supply chain is increasing.
Thus,
the objective of this paper is to study the effect of the production facility
capacity on the Joint Lot Sizing and Scheduling decisions for three echelons
supply chain containing two suppliers. In the following section a detailed
problem definition is illustrated.
3. PROBLEM DEFINITION
In
the supply chain under consideration, non-identical products are produced. Each
product uses the same two different raw materials in its production yet with
different ratios. The processing time of each product is different from one to
another. The considered supply chain is a three-echelon-supply-chain as shown
in figure 1. The production facility has two suppliers from which raw materials
are acquired; the first supplier can supply both types of raw materials, while
the second supplier can only supply one type of the raw materials.
The
quality of materials received from both suppliers is consistent. Both suppliers
dedicate part of their capacities for the production facility, and therefore,
the facility is obliged to purchase a minimum quantity from each supplier for
the whole planning period. The
production facility can produce one or more of a batch of each product during
the same period. If only one product is
produced at any period, no changeover will take place.
If
more than one product is produced during the same period, no changeover will be
needed for the first product and changeovers will be done for the production of
the next product. The production facility
ships its production to a distribution center at which the products are either
sent directly to customer(s) or kept as inventory to meet future demand of next
periods.
The
customers' demands are all confirmed orders of different products per period.
Since no transportation costs are considered, all customers are assumed to be
only one customer and its demand is the total demand from each product.
A
Mixed Integer Non- Linear programming model is developed to maximize the supply
chain profits with fixed selling price of each product at different periods.
The costs considered are: material(s) purchasing cost, inventory cost for raw
materials at the production facility and finished products at the DC,
changeover cost at production facility, processing cost and penalty cost
incurred for undelivered quantities to the customer.
It is
required to determine the joint lot size (purchased quantities from each
supplier, production quantities and delivery quantities) for the two products
and the Master Production Schedule (MPS) that maximize the supply chain profit
over number of periods composing a planning horizon.
Figure 1:
Supply Chain Structure
4. RESEARCH METHODOLOGY
4.1.
Mathematical
model
A
mathematical model is developed to maximize the supply chain profits when
optimum purchase, production and delivery quantities are determined jointly, in
addition to determining the MPS.
4.1.1. Nomenclature
Indices
t:
Periods (t = 1, 2, ..., T)
i:
Items (raw materials) (i = 1, 2, ..., I)
j:
Suppliers (j = 1, 2, ..., J)
n:
Products (n = 1, 2, ..., N)
Parameters
Sn: Selling
price per finished product 'n'
Cmij: Cost
of one item of raw material 'i' from supplier 'j'
Cpn: Cost
of processing of product 'n' per unit time
Chmi: Inventory
holding cost of one item of raw material 'i' for one period
Chpn: Inventory
holding cost of one product 'n' for one period
Co: Changeover
cost for each product except that at the start of the period
Cs: Penalty
cost paid for each undelivered unit
Tpn: Production
time per product 'n' in the facility
Tmij: Production
time of a single item of raw material 'i' at supplier 'j'
tc: Changeover
time at the facility
Dtn: Demand
of product 'n' at any period 't'
vtj: Max.
time capacity at supplier 'j' dedicated to the production facility
during period 't'
utj: Min.
capacity at supplier 'j' dedicated to the production facility during
period 't'
Wt: Max.
capacity available at production facility during period 't'
ain: Amount
of raw material 'i' required for production of one product 'n'
Decision variables
Qmtij: Quantity
of material 'i' purchased by production facility from supplier 'j'
during period 't'
Qptn: Quantity
of product 'n' processed at facility during period 't'
Qdtn: Quantity
of product 'n' delivered from DC to customer during period 't'
: Inventory
level of product 'n' at end of period 't' at the DC
: Inventory
level of raw material 'i' at end of period 't' at the production
facility.
Ltn: Binary Matrix where ltn
= 0 if the product 'n' is not listed in the MPS in period 't'
otherwise equal 1.
4.1.2. The
Developed Model
The
objective function is to maximize total supply chain profits which is given by
total supply chain revenues minus total costs of purchasing, production,
inventory holding for final products and raw materials, penalty and changeover.
Profit Model
The
supply chain profit given in equation (1) is modelled as supply chain revenues
from selling products to customers from the DC minus the costs incurred by the
production facility and the DC. The
revenue is modelled as the selling price 'Sn' multiplied by
the delivered quantity 'Qdtn' to
the customer each period from each product.
The first cost element is the cost of purchasing material 'i' from supplier 'j' and it is modelled by
multiplying the material cost per unit 'Cmij'
by the purchased quantity 'Qmtij'. The processing cost of product 'n' is
calculated as the processing time 'tpn'
multiplied by the cost of processing 'Cpn'
multiplied by the sum of quantities manufactured during the planning horizon 'T'.
Products
and materials inventory costs are calculated as the inventory holding cost per
unit ('Chpn' for products and 'Chmi' for materials) multiplied by the sum
of end of period inventory level ('Iptn'
for products and 'Imti' for
materials) during the planning horizon. The penalty cost is formulated as the
difference between the required demand 'Dtn'
and actual delivered quantities 'Qdtn',
multiplied by the penalty paid for each undelivered unit. The total supply
chain changeover cost for all periods along the planning horizon is modeled as
the cost per changeover 'Co' multiplied by the sum of the number of
products processed each period minus one to exclude the first product.
|
(1) |
Model
Constraints
Total
quantity of items purchased from supplier 'j' at any period 't'
lies between the minimum and maximum capacity limits determined by the
suppliers. This is ensured by
constraints (2) and (3) where the time needed by supplier 'j' to produce
quantity 'Qmtij' for all materials 'I'
is greater than the minimum capacity in time units 'utj'
and smaller than maximum time capacity 'vtj'
dedicated to the production facility.
|
(2) |
|
|
(3) |
|
, " t є T |
(4) |
Constraint
(4) ensures that the production capacity is not violated, the sum of processing
and changeover times at any period 't' cannot exceed the time capacity
at the production facility 'wt'.
Total processing time of any product 'n' at any period 't' is the
multiplication of the quantity produced from this product 'Qptn' by the processing time for one item
of this product. Total changeover time is the time of a single changeover 'tc' multiplied by the number of products manufactured
during the period 't' excluding the first product.
|
(5) |
This
constraint ensures that total purchased quantity of material 'i' from all suppliers at any period 't' is
greater than or equal to the required quantity from this raw material to
produce 'Qptn' products in the same
period. This is given in constraint (5) by having the purchased quantity of raw
material 'Qmtij' from all suppliers
is greater than the amount of materials required for one product 'ain' multiplied by the produced quantity of
the same product 'Qptn'.
, " t є T, n є N |
(6) |
Delivered
quantity of product 'n' at any period 't'; 'Qdtn'
is less than or equal to the customer's demand 'Dtn'.
This constraint ensures that the delivered quantities may not exceed the
customer demand.
, ("
t=2, 3, 4, …, t-1) |
(7) |
Constraint
(7) is a balance constraint, ensures that the inventory of final products 'Iptn' at any period 't'; is equal to
inventory level at the end of the previous period 'Ip(t-1)n'
plus remaining from production quantity '' that is not
delivered in the same period 't'.
, ("
t = 2, 3, 4, …, t) |
(8) |
Constraint
(8) shows that the inventory level of material 'i'
at any period 't'; 'Imti'
equals its inventory level at the end of the previous period 'Im(t-1)i' plus
amount of purchased quantity at period 't'; 'Qmtij'
minus the amount required for the production of this period 'ain*Qptn'.
The capacity of raw materials storage at the production facility is unlimited.
|
(9) |
Constraint
(9) implies that Supplier (2) cannot produce the second raw material
|
(10) |
Constraint
(10) shows that is to prevent not producing any quantity 'Qptn' from product 'n', yet it is in
the MPS at the same period i.e. ltn
= 1.
|
(11) |
Constraint
(11) prevents having zero production quantity while the product is listed in the
MPS
5. NUMERICAL EXPERIMENTATION
The
main objective is to study how the joint lot sizing and scheduling decision may
change with the change in production capacity and inventory holding costs. In order to achieve this objective three
different production levels are assumed; low in which the available capacity
can only produce the demand from one product or slightly more, moderate
capacity in which the capacity is sufficient to produce the demand of both
products with slight shortages, and high capacity where the capacity is enough
to produce both products with setup each period. Table 1 illustrates the values of the
capacities and input parameters considered.
Table 1: Input Parameters for Numerical
Experimentation
Sn |
1500, 1500 |
Cs |
100 &
1000 |
vtj |
500, 250 |
Cmij |
200, 200,
200 |
Tpn |
1, 1.2 |
utj |
50, 25 |
Cpn |
150, 150 |
Tmij |
0.2, 0.2,
0.15 |
Wt |
150, 250
& 350 |
Chmi |
20 |
tc |
10 |
ain |
1, 1, 2, 1 |
Chpn |
25 & 100 |
Dt1 |
150 |
|
|
Co |
5000 |
Dt2 |
100 |
|
|
The
mathematical model is coded using LINGO 17.0 software which yielded the global optimum
of the problem. LINGO is run using a
workstation with Intel Xeon E3-1246 v3 (3.50 GHz) processor and 16 GB RAM, the
run time varied drastically from few seconds to more than 100 hours for some
instances.
6. RESULTS AND DISCUSSION
It is
clear from table 2 that as the production capacity increases the supply chain
profits increases due to the decrease in penalties paid and the increased
delivered quantities. In cases of low and moderate capacities, the increase in
penalty costs decreased the profits than high capacity case by an average 77.7%
and 20.2% respectively, as at these two capacities shortage occur.
While
at lower holding cost the profits are higher by an average 2.2% for moderate
and high capacities, while it has no effect as no inventory is kept at low
capacity. Even at high capacity where
the capacity is enough to produce total demand of both products each period and
there is no need to keep inventory, yet inventory is kept from products as this
will be illustrated using figures 2-7.
Table 2: Supply Chain Profits at different
Capacities, Holding Costs and Penalty Costs
w
(hours per period) |
Chp
(Unit cost per period) |
Cs
= 100 (unit cost per undelivered unit) |
Cs
= 1000 (unit cost per undelivered unit) |
150 |
25 |
2460000 |
300000 |
100 |
2460000 |
300000 |
|
250 |
25 |
4306430 |
3867265 |
100 |
4176000 |
3690560 |
|
350 |
25 |
4817100 |
4817100 |
100 |
4788000 |
4788000 |
6.1.
Low
Production Capacity
For
the low production capacity, the decisions are the same for various holding and
penalty costs, as there is enough capacity to produce the demand of product 1
which has higher demand and lower processing time. Figure 2 shows various
decisions made in case of low production capacity at holding cost equal 100
units cost per unit per period and penalty cost 100 units cost penalty per each
undelivered, it is clear that the production quantities are from one product
(product 1) which consumes less capacity and has higher demand to minimize the
penalty cost and hence maximize the profit.
|
|
|
(a) Purchased
Materials Quantities |
(b) Production Quantities |
|
|
|
|
(c) Products Inventory Levels |
(d) Delivered Quantities |
|
Figure 2: Various Decision Variables at Low
Capacity and Chp = 100 and Cs = 100
6.2.
Moderate
Production Capacity
In
the case of moderate capacity and having equal holding and penalty costs as
shown in figure 3, there are three main production schedules altering in this
solution. The first is producing full demand of product1 (150 units) and using
the rest of the capacity to produce product2 (75 units with a shortage of 25
units) as in periods (3, 5, 7, 8, 9, 11, 13, 15, 17, 19, 20 and 21).
The
second solution is producing the full demand of product2 (100 units) and using
the rest of the production capacity to produce product1 (120 units with a
shortage of 30 units) as in periods (1, 2, 4, 6, 10, 12, 14, 18, 20, 22, 24).
The third solution found in period (16) only is a solution in which shortages
is shared between the two final products; producing 132 units from product1
with a shortage of 18 units and 90 units of product2 with 10 shortage units.
This
schedule is not a unique optimal one, as when any of the three schedules is
fixed for the all periods each yielded the same supply chain profit. Material purchased will be exactly equal the
same amounts needed for production and the delivered quantities are the same as
the production quantities.
|
|
(a) Purchased
Materials Quantities |
(b) Production Quantities |
|
|
(c) Products Inventory Levels |
(d) Delivered Quantities |
Figure 3: Various
Decision Variables at Moderate Capacity
and Chp = 100 and Cs = 100
Similar
decisions are observed on having the penalty cost higher than the holding cost
as given in figures 4 & 5, the production decision favors the product that
consumes less capacity (having less processing time). When the difference is
low as in figure 4 the reached inventory levels is lower and shifting from one
product to the other is more frequent.
While on having high difference between the holding cost and penalty
cost as in figure 5, the fill rate of product 1 is 100% and residual capacity
is used to produce product 2.
Furthermore,
a building of inventory is made from product 1 on the expense of not delivering
product 2 in periods 3, 4 & 9 and having high shortage in period 8, the
service level of product 2 in this period is only 23%. This enabled the supply chain to build
inventory from both products as the capacity was used to produce only one
product in 9 periods, which is used when the production quantity is less than
demand. This production schedule enables
the supply chain to have 100% service level in both products in 15 periods of
the planning horizon (62.5% of the periods) which in return minimize the
penalty cost.
The
materials purchased quantities were exactly equal to the amounts needed for the
production quantities. In figure 6 as the holding cost increase the same
pattern of decision is made yet the 100% service level of both products was
reached in 12 periods only and the favoring of product 1 is less as there 3
periods its service level was 93.3% and twice it was 97.3% (periods 5, 9 and
13). This is due to the fact that
building inventory is becoming more expensive, so a tradeoff is made choosing
to build less inventory from product 1.
|
|
(a) Purchased
Materials Quantities |
(b)
Production Quantities |
|
|
(c) Products Inventory Levels |
(d) Delivered Quantities |
Figure 4: Various
Decision Variables at Moderate Capacity
and Chp = 25 and Cs = 100
|
|
(a) Purchased
Materials Quantities |
(b) Production Quantities |
|
|
(c) Products Inventory Levels |
(d) Delivered Quantities |
Figure 5:
Various Decision Variables at Moderate Capacity and Chp = 25 and Cs = 1000
|
|
(a) Purchased
Materials Quantities |
(b) Production
Quantities |
|
|
(c) Products Inventory Levels |
(d) Delivered Quantities |
Figure 6:
Various Decision Variables at Moderate Capacity and Chp = 100 and Cs = 1000
6.3.
High
Production Capacity
In
the case of high capacity and low inventory holding costs as shown in figure 7,
although it is possible to produce exactly the demand of both products each
period, yet the production decision makes use of this low holding costs by
producing one product in most periods. For 21 periods (87.5% of the periods)
one product is produced and inventory quantity is kept for next period(s) where
the other product is produced. During
the remaining 3 periods both products are produced.
Two
out of these three periods are the first two periods, in which the buildup of
inventory is taking place using the excess available capacity. In period 13
both products are produced as there are not enough inventory to cover the
demand of both products. This production schedule allows the reduction of the
number of changeovers, and consequently reduce the changeover costs while
maintaining 100% service levels for each product.
As in
previous cases the material purchase follows the production schedule and no
materials inventory is kept.
|
|
(a) Purchased
Material Quantities |
(b)
Production Quantities |
|
|
(c) Products
Inventory Levels |
(d) Delivered
quantities |
Figure
7: Different Decisions variables at each period at w=350, Chp=25, Cs=100
|
|
(a) Purchased Material Quantities |
(b) Production Quantities |
|
|
(c) Products Inventory Levels |
(d) Delivered quantities |
Figure
8: Different Decisions variables at each period at w=350, Chp=100, Cs=100
The
high production capacity and high inventory holding cost case shown in figure 8
both products are produced in quantities equal to the demand each period and
hence all produced quantities are delivered, and no inventory is kept from
products. The materials purchased each
period equal to the amounts needed to produce the demanded quantities.
7. CONCLUSIONS
The
results showed that the Joint Lot Sizing and Scheduling problem is modelled and
solved optimally for a large number of planning periods (24 periods). The MPS depends hugely on the available
capacity at the manufacturing facility while, the purchased material is done as
a lot-for-lot to fulfill production needs each period. In case of low capacity, only one product is
produced, whatever the costs were, as there isn’t enough capacity to produce
both products.
The
produced product is the one with higher demand to reduce the penalty costs. In
moderate and high capacities, the holding cost has a great impact on the
decision; as the holding cost decreases the tendency to produce only one
product each period, minimizing changeover costs, and keep inventory to satisfy
demand in future periods increases.
Another
factor affected the decisions which is the ending inventory at the last period,
since its optimum value is zero, the solution resulted in steady production of
both products in the last periods to assure that no ending inventory is kept at
the end of the last period. The effect of having variable demand, different
suppliers' quality and lead times and more real bill of materials for the
product family can be researched in the future.
REFERENCES
BEN-DAYA,
M.; DARWISH, M.; ERTOGRAL, K. (2008) The joint economic lot sizing problem:
Review and extensions, European Journal of Operational Research,
v. 185, n. 2, p. 726–742. doi: 10.1016/j.ejor.2006.12.026.
CUNHA, A. L. et al. (2018) An integrated approach for
production lot sizing and raw material purchasing, European Journal of
Operational Research. Elsevier B.V., v. 269, n. 3, p. 923–938. doi:
10.1016/j.ejor.2018.02.042.
ERTOGRAL, K.; DARWISH, M.; BEN-DAYA, M. (2007) Production and
shipment lot sizing in a vendor-buyer supply chain with transportation cost, European
Journal of Operational Research, v. 176, n. 3, p. 1592–1606. doi:
10.1016/j.ejor.2005.10.036.
GHARAEI, A.; JOLAI, F. (2018) A multi-agent approach to the
integrated production scheduling and distribution problem in multi-factory
supply chain, Applied Soft Computing Journal. Elsevier B.V., n. 65, p.
577–589. doi: 10.1016/j.asoc.2018.02.002.
GLOCK, C. H. (2012) The joint economic lot size problem: A review,
International
Journal of Production Economics. Elsevier, v. 135, n. 2, p. 671–686.
doi: 10.1016/j.ijpe.2011.10.026.
HARIGA, M. et al. (2013) Scheduling and lot sizing
models for the single-vendor multi-buyer problem under consignment stock
partnership, Journal of the Operational Research Society. Nature
Publishing Group, v. 64, n. 7, p. 995–1009. doi: 10.1057/jors.2012.101.
HUANG, J. Y.; YAO, M. J. (2013) On the optimal lot-sizing and
scheduling problem in serial-type supply chain system using a time-varying
lot-sizing policy, International Journal of Production Research,
v. 51, n. 3, p. 735–750. doi: 10.1080/00207543.2012.662604.
JIA, T. et al. (2016) Optimal production-delivery
policy for a vendor-buyers integrated system considering postponed simultaneous
delivery, Computers and Industrial Engineering. Elsevier Ltd, n. 99,
p. 1–15. doi: 10.1016/j.cie.2016.07.002.
LEE, S. D.; FU, Y. C. (2014) Joint production and delivery
lot sizing for a make-to-order producer-buyer supply chain with transportation
cos’, Transportation Research Part E: Logistics and Transportation Review.
Elsevier Ltd, n. 66, p. 23–35. doi: 10.1016/j.tre.2014.03.002.
MARCHI, B. et al. (2016) A joint economic lot size
model with financial collaboration and uncertain investment opportunity, International
Journal of Production Economics. Elsevier, n. 176, p. 170–182. doi:
10.1016/j.ijpe.2016.02.021.
MUNGAN, D.; YU, J.; SARKER, B. R. (2010) Manufacturing
lot-sizing, procurement and delivery schedules over a finite planning horizon, International
Journal of Production Research, v. 48, n. 12, p. 3619–3636. doi:
10.1080/00207540902878228.
POURAKBAR, M.; FARAHANI, R. Z.; ASGARI, N. (2007) A joint
economic lot-size model for an integrated supply network using genetic
algorithm, Applied Mathematics and Computation, v. 189, n. 1, p.
583–596. doi: 10.1016/j.amc.2006.11.116.
SADJADI, S. J.; ZOKAEE, S.; DABIRI, N. (2014) A single-vendor
single-buyer joint economic lot size model subject to budget constraints, International
Journal of Advanced Manufacturing Technology, v. 70, n. 9–12, p.
1699–1707. doi: 10.1007/s00170-013-5382-2.
SAĞLAM, Ü.; BANERJEE, A. (2018) Integrated multiproduct batch
production and truck shipment scheduling under different shipping policies, Omega
(United Kingdom), n. 74, p. 70–81. doi: 10.1016/j.omega.2017.01.007.
SARIN, S. C.; SHERALI, H. D.; LIAO, L. (2014) Primary
pharmaceutical manufacturing scheduling problem, IIE Transactions (Institute of
Industrial Engineers), v. 46, n. 12, p. 1298–1314. doi:
10.1080/0740817X.2014.882529.
SAWIK, T. (2016) Integrated supply, production and
distribution scheduling under disruption risks, Omega (United Kingdom).
Elsevier, n. 62, p. 131–144. doi: 10.1016/j.omega.2015.09.005.
SENOUSSI, A. et al. (2016) Modeling and solving a
one-supplier multi-vehicle production-inventory-distribution problem with
clustered retailers, International Journal of Advanced
Manufacturing Technology, v. 85, n. 5–8, p. 971–989. doi:
10.1007/s00170-015-7966-5.
TORABI, S. A.; FATEMI GHOMI, S. M. T.; KARIMI, B. (2006) A
hybrid genetic algorithm for the finite horizon economic lot and delivery
scheduling in supply chains, European Journal of Operational Research,
v. 173, n. 1, p. 173–189. doi: 10.1016/j.ejor.2004.11.012.
VAN HOESEL, S. et al. (2005) Integrated Lot Sizing in
Serial Supply Chains with Production Capacities, Management Science,
v. 51, n. 11, p. 1706–1719. doi: 10.1287/mnsc.1050.0378.
WANG, S. P.; LEE, W. (2013) A joint economic lot-sizing model
for the hospital’s supplier with capacitated warehouse constraint, Journal
of Industrial and Production Engineering, v. 30, n. 3, p. 202–210.
doi: 10.1080/21681015.2013.805700.
ZHAO, S. T.; WU, K.; YUAN, X. M. (2016) Optimal
production-inventory policy for an integrated multi-stage supply chain with
time-varying demand, European Journal of Operational Research.
Elsevier B.V., v. 255, n. 2, p. 364–379. doi: 10.1016/j.ejor.2016.04.027.