Integrated ProductionInventory Supply Chain Model
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Read this article. An integrated productioninventory model is constructed to address supplier, manufacturer, and retailer uncertainties. According to the author, what are the three types of uncertainties in supply chain management?
Abstract
In this paper, an integrated productioninventory model is presented for a supplier, manufacturer, and retailer supply chain under conditionally permissible delay in payments in uncertain environments. The supplier produces the item at a certain rate, which is a decision variable, and purchases the item to the manufacturer. The manufacturer has also purchased and produced the item in a finite rate. The manufacturer sells the product to the retailer and also gives the delay in payment to the retailer. The retailer purchases the item from the manufacture to sell it to the customers. Ideal costs of supplier, manufacturer, and retailer have been taken into account. An integrated model has been developed and solved analytically in crisp and uncertain environments, and finally, corresponding individual profits are calculated numerically and graphically.
Source: Dipak Kumar Jana, Kalipada Maity, and Tapan Kumar Roy, https://juaajournal.springeropen.com/articles/10.1186/2195546816
This work is licensed under a Creative Commons Attribution 2.0 License.
Introduction
Supply chain management has taken a very important
and critical role for any company in the increasing globalization and
competition in the market. A supply chain model (SCM) is a network of
suppliers, producers, distributors, and customers which synchronizes a
series of interrelated business process in order to have (1) optimal
procurement of raw materials from nature, (2) transportation of raw
materials into a warehouse, (3) production of goods in the production
center, and (4) distribution of these finished goods to retailers for
sale to the customers. With a recent paradigm shift to the supply chain
(SC), the ultimate success of a firm may depend on its ability to link
supply chain members seamlessly.
One of the earliest efforts to
create an integrated SCM has been developed by Oliver and Webber,
Cohen and Baghanan, and Cachon and Zipkin. They developed a
production, distribution, and inventory (PDI) planning system that
integrated three supply chain segments comprising supply,
storage/location, and customer demand planning. The core of the PDI
system was a network model and diagram that increased the decision
maker's insights into supply chain connectivity. The model however was
confined to a singleperiod and singleobjective problem. Viswanathan
and Piplani were concerned an integrated inventory model through
common replenishment in the SC. Khouja was the first to consider a
threestage supply chain with one or more firms at each stage. Agarwal
et al. have developed a dynamic balancing of inventory model in
supply chain management. Rau et al. developed an integrated SCM of a
deteriorating item with shortages. Lee added a new dimension to the
single vendorsingle buyer problem by setting the number of raw
material shipments received by the vendor per cycle to be a decision
variable. BenDaya et al. have developed an integrated
productioninventory model with raw material replenishment
considerations in a threelayer supply chain. Sana has integrated a
productioninventory model of imperfect quality products in a
threelayer supply chain. Recently, Pal et al. have developed a
threelayer supply chain model with productioninventory model for
reworkable items. All of the abovementioned SCMs are considered with
constant, known demand and production rates in a crisp environment.
Different
types of uncertainty such as fuzziness, randomness, and roughness are
common factors in SCM. In many cases, it is found that some inventory
parameters involve fuzzy uncertainty. For example, inventoryrelated
costs such as holding cost and setup cost, demand, and selling price
depend on several factors such as bank interest, stock amount, and
market situation which are uncertain in a fuzzy sense. To be more
specific, inventory holding cost is sometimes represented by a fuzzy
number, and it depends on the storage amount which may be imprecise and
range within an interval due to several factors such as scarcity of
storage space, market fluctuation, human estimation, and/or thought
process. The following papers have been developed in these environments.
Wang
and Shu developed a fuzzy decision methodology that provides an
alternative framework to handle SC uncertainties and to determine SC
inventory strategies, while there is a lack of certainty in data or even
a lack of available historical data. Fuzzy set theory is used to model
SC uncertainty. A fuzzy SC model based on possibility theory is
developed to evaluate SC performances. Based on the proposed fuzzy SC
model, a genetic algorithm approach is developed to determine the
orderupto levels of stockkeeping units in the SC to minimize SC
inventory cost subject to the restriction of fulfilling the target fill
rate of the finished product. The proposed model allows decision makers
to express their risk attitudes and to analyze the tradeoff between
customer service level and inventory investment in the SC, so that
better SC inventory strategies can be made.
Das et al. have
presented a joint performance of an SC with two warehouse facilities in a
fuzzy environment. A realistic twowarehouse and
multicollectionproductioninventory model with
constant/stockdependent demand, defective production system, and fuzzy
budget constraint has been formulated and solved in an SC context. Later
Chen et al. developed a multicriteria fuzzy optimization for
locating warehouses and distribution centers in a supply chain network.
Peidro
et al. developed a fuzzy linear programming model for tactical
supply chain planning in a multiechelon, multiproduct, multilevel,
multiperiod supply chain network in a fuzzy environment. In this
approach, the demand, process, and supply uncertainties are jointly
considered. The aim is to centralize multinode decisions simultaneously
to achieve the best use of the available resources along the time
horizon so that customer demands are met at a minimum cost. This
proposal is tested using data from a real automobile SC. The fuzzy model
provides the decision maker with alternative decision plans with
different degrees of satisfaction.
Chu developed the supply
chain flexibility that has become increasingly important. This study
thus builds a group decisionmaking structure model of flexibility in
supply chain management development. Recently, Jana et al. have
developed a fuzzy simulation via contractive mapping genetic algorithm
approach to an imprecise productioninventory model under volume
flexibility. This study presents a framework to evaluate the supply
chain flexibility that comprises two parts: (1) an evaluation hierarchy
with flexibility dimensions and related metrics and (2) an evaluation
scheme that uses a threestage process to evaluate the supply chain
flexibility. This study then proposes an algorithm to determine the
degree of supply chain flexibility using a fuzzy linguistic approach.
Evaluations of the degree of supply chain flexibility can identify the
need to improve supply chain flexibility and identify specific
dimensions of supply chain flexibility as the best directions for
improvement. The results of this study are more objective and unbiased
for two reasons. First, the results are generated by group
decisionmaking with interactive consensus analysis. Second, the fuzzy
linguistic approach used in this study has more advantages to preserve
no loss of information than other methods. Additionally, this study
presents an example using a case study to illustrate the availability of
the proposed methods and compare it with other methods.
Kristianto
et al. developed an adaptive fuzzy control application to support a
vendormanaged inventory (VMI). This paper also guides the management
in allocating inventory by coordinating with suppliers and buyers to
ensure minimum inventory levels across a supply chain. Adaptive fuzzy
VMI control is the main contribution of this paper.
However, the
uncertainty theory was developed by Liu, and it can be used to
handle subjective imprecise quantity. Much research work has been done
on the development of the uncertainty theory and related theoretical
work. You proved some convergence theorems of uncertain sequences.
Liu has defined uncertain process and Liu has discussed
uncertain theory. In this paper, we developed for the first time a
threelayer supply chain model under delay in payment in an uncertain
environment.
In the traditional economic order quantity (EOQ)
model, it often assumed that the retailer must pay off as soon as the
items are received. In fact, the supplier offers the retailer a delay
period, known as trade credit period, in paying for the purchasing cost,
which is a very common business practice. Suppliers often offer trade
credit as a marketing strategy to increase sales and reduce onhand
stock level. Once a trade credit has been offered, the amount of the
tied up retailer's capital in stock is reduced, and that leads to a
reduction in the retailer's holding cost of finance. In addition, during
the trade credit period, the retailer can accumulate revenues by
selling items and by earning interests. As a matter of fact, retailers,
especially of small businesses which tend to have a limited number of
financing opportunities, rely on trade credit as a source of shortterm
funds. In this research field, Goyal was the first to establish an
EOQ model with a constant demand rate under the condition of permissible
delay in payments. Khanra, Ghosh, and Chaudhuri have developed an
EOQ model for a deteriorating item with timedependent quadratic demand
under permissible delay in payment. Also, Maihami and Abadi have
established joint control of inventory and its pricing for
noninstantaneously deteriorating items under permissible delay in
payments and partial backlogging.
The proposed model considers a
threelayer supply chain involving the supplier, manufacturer, and
retailer who are responsible in transforming the raw materials into
finished product and making them available to satisfy the customer's
demand time. Inventory and production decisions are made at the
supplier, manufacturer, and retailer levels in uncertain environments.
The problem is to coordinate production and inventory decisions across
the supply chain so that the total profit of the chain is maximized.
Necessary knowledge about uncertain variables
To better describe subjective imprecise quantity, Liu in proposed an uncertain measure and further developed an uncertainty theory which is an axiomatic system of normality, monotonicity, selfduality, countable subadditivity and product measure.
Definition 1
Let be a nonempty set and be a algebra over . Each element is called an event. A set function is called an uncertain measure if it satisfies the following four axioms of Liu:
Axiom 1
(Normality)
Axiom 2
(Monotonicity) , for any event
Axiom 3
(Countable subadditivity) For every countable sequence of events we have .
Definition 2
The uncertainty distribution of an uncertain variable is defined by .
Definition 3
Let be an uncertain variable. Then the expected value of is defined by , provided that at least one of the two integrals is finite.
Theorem 1
Let be an uncertain variable with uncertainty distribution . If the expected value exists, then .
Lemma 1
Let \ be a zigzag uncertain variable. Then its inverse uncertainty distribution , and it can be expressed as .
(1)
Theorem 2
Let be independent uncertain variables with
uncertainty distributions , respectively. If f is a
strictly increasing function, then is an uncertain
variable with inverse uncertainty distribution
Theorem 3
Let and be independent uncertain variables with finite
expected values. Then for any real numbers a 1 and a 2, we have
Assumptions and notations
Assumptions
The following assumptions are considered to develop the model:
(i) Models are developed for single item product.
(ii) Lead time is negligible.
(iii) Joint effect of supplier, manufacturer, retailer is consider in a supply chain management.
(iv) Supplier produced the item with constant rate unit per unit time, which is a decision variable.
(v) Total production rate of manufacturer is equal to the demand rate of manufacturer.
(vi) The manufacturer give the opportunity to the retailer conditionally
permissible delay in payment.
(vii) Idle cost of suppliers, manufacturer and retailer are also assumed.
Notations
The following notations are considered to develop the model:
 = production rate for the suppliers, which is a decision variable.
 = demand rate or production rate for the manufacturer.
 = constant demand rate for the retailer.
 = constant demand rate of customer.
 = purchase cost of unit item for suppliers.
 = selling price of unit item for suppliers which is also the purchase cost for manufacturer.
 = selling price of unit item for manufacturer which is also the purchase cost for retailers.
 = selling price for retailers.
 = production time for supplers.
 = cycle length for the suppliers.
 = time duration where order is supplied by the manufacturer, by retailer's cycle length.
 = last cycle length of the retailer.
 = total time for the integrated model.
 = holding cost per unit per unit time for suppliers.
 = holding cost per unit per unit time for manufacturer.
 = holding cost per unit per unit time for retailers.
 = ordering cost for suppliers.
 = ordering cost for manufacturer.
 = ordering cost for retailers.
 = idle cost per unit time for suppliers in crisp and uncertain environments, respectively.
 = idle cost per unit time for manufacturer in crisp and uncertain environments, respectively.
 = idle cost per unit time for retailers in crisp and uncertain environments, respectively.
 = number of cycle for retailers.
 = number of cycles where manufacturer stops production.
 = retailer's trade credit period offered by the manufacturer to the retailers in years, which is the fraction of the years.
 = interest payable to the manufacturer by the retailers.
 = interest earned by the retailers in crisp and uncertain environments, respectively.
 = average total profit for the integrated models.
 = optimum value of P m for integrated models.
 = optimum value of average total profit for the integrated models.
Model description and diagrammatic representation
The integrated inventory model (Figure 1) starts when and stock is zero. At that time, the suppliers start their production with the rate unit per unit time and purchase at the rate unit per unit time to the manufacturer. When , suppliers stop their production, and at , the inventory level of suppliers become zero. The total time of the integrated model is , so the idle time for suppliers is . Similarly, the manufacturers start their production at the same time with the production rate unit per unit time and purchase this production unit to the retailer in the time gap , which is the bulk pattern. At time , manufacturers stop their production, and at , the stock of manufacturer is zero. Thus, idle period for the manufacturer is . Retailers start selling this product to the customers at time and end selling at . The idle period for retailers is .
Figure 1 Inventory level for the integrated model.
Mathematical formulation of the model
Formulation of suppliers' individual average profitDifferential equation for the supplier in Figure 2 in [0,T] is given by
(3)
Formulation of manufacturer individual average profit
Inventory level of manufacturer in Figure 3 in [0,T] is given byFormulation of retailer individual average profit
Inventory level of retailer in Figure 4 in [0,T] is given byCase 2
(19)
where A and B are given in (16) and (17), respectively and
Numerical example
Crisp environment
The input data of different parameters for Case 1 and Case 2 are shown in Table 1, and the expected optimum value of the total profit is given in Table 2.
Table 1 Input data of different parameters for Case 1 and Case 2
Parameters  c _{ s }  c _{ m }  c _{ r }  c _{ r 1}  h _{ s }  h _{ m }  T _{ s }  p _{ m }  n  r  ρ  id _{ s }  id _{ m }  id _{ r }  I _{ e }  A _{ s }  A _{ m }  A _{ r }  m  i _{ p }  D _{ C }  D _{ R } 

Case 1  7  14  25  35  0.15  0.8  10  16  4  4  0.4  1  2  3  1  25  40  45  20  16  55  100 
Case 2  10  12  26  25  0.17  0.9  12  25  17  5  0.3  1.5  2.5  3.5  1  30  28  52  24  45  120  130 
Table 2 Optimum results for objective functions and other parameters
Parameter  Case 1  Case 2 

ATP^{∗}  1,039.45  1,108.93 
70.49 
79.23 

T  2.85  1.59 
APS^{∗}  246.51  253.56 
APM^{∗}  468.25  463.18 
APR^{∗}  321.43  390.47 
Sensitivity analysis
The major contribution of the supply chain is mainly in the inclusion of the manufacturer. We consider product reworking of defective items which are reworked just after regular production, with a different holding cost for good and defective items in the threelayer supply chain. An integrated productioninventory model is presented for the supplier, manufacturer, and retailer supply chain under conditionally permissible delay in payments which has developed in both crisp and uncertain environments (Table 3).Uncertain environment
Parameters in uncertain environments  

Case 1  L(0.8,1.2,1.4)  L(1.5,2.0,2.5)  L(1.4,2,2.3)  L(0.04,0.06,0.08) 
Case 2  L(1.4,1.8,2.4)  L(2,2.3,2.9)  L(1.4,2.1,2.5)  L(0.06,0.08,0.1) 
Parameter  Case 1  Case 2 

ATP^{∗}  1,056.43 
1,520.43 
69.76 
78.31 

T  1.75  1.76 
APS^{∗}  248.71  276.76 
APM^{∗}  463.43  467.35 
APR^{∗}  324.43  393.73 
Achievements and conclusion of the model
In this model, we
developed a threelayer productioninventory supply chain model in an
uncertain environment. Here, the suppliers are also the manufacturers;
they collect the raw material (ore) and produce the raw material of the
actual manufacturer. For example, in the petroleum industry, suppliers
collect the ore and produce the naphthalene, which is the raw material
of the manufacturer. Then manufacturer produces the usable product to
sell to the retailer. In this paper, we have developed a
productioninventory supply chain model under an uncertain environment.
The paper can be extended to imperfect productioninventory system.
Deterioration can be allowed for produced items of the retailer and
manufacturer. In the case of the retailer, it might be interesting to
consider the effect that only a percent of imperfect quality products
could be reworked by manufacturers and that other scrap items must be
eliminated immediately. In order to show the uncertainties, the present
model could be extended, applying stochastic demand and production rate
in each member of the supply chain. These are some topics of ongoing and
future research, among others.