A threelayer supply chain integrated productioninventory model under permissible delay in payments in uncertain environments
 Dipak Kumar Jana^{1}Email author,
 Kalipada Maity^{2} and
 Tapan Kumar Roy^{3}
DOI: 10.1186/2195546816
© Jana et al.; licensee Springer. 2013
Received: 11 April 2013
Accepted: 26 July 2013
Published: 13 August 2013
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.
Keywords
Production inventory system Uncertain variables Threelayer supplier chain Delay in paymentIntroduction
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 [1], Cohen and Baghanan [2], and Cachon and Zipkin [3]. 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 [4] were concerned an integrated inventory model through common replenishment in the SC. Khouja [5] was the first to consider a threestage supply chain with one or more firms at each stage. Agarwal et al. [6] have developed a dynamic balancing of inventory model in supply chain management. Rau et al. [7] developed an integrated SCM of a deteriorating item with shortages. Lee [8] 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. [9] have developed an integrated productioninventory model with raw material replenishment considerations in a threelayer supply chain. Sana [10] has integrated a productioninventory model of imperfect quality products in a threelayer supply chain. Recently, Pal et al. [11] 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 [12] 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. [13] 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. [14] developed a multicriteria fuzzy optimization for locating warehouses and distribution centers in a supply chain network.
Peidro et al. [15] 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 [16] 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. [17] 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. [18] 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 [19], 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 [20] proved some convergence theorems of uncertain sequences. Liu [21] has defined uncertain process and Liu [22] 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 [23] 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 [24] have developed an EOQ model for a deteriorating item with timedependent quadratic demand under permissible delay in payment. Also, Maihami and Abadi [25] 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 [19] 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 Γ be a nonempty set and L be a σ algebra over Γ. Each element Λ ∈ L is called an event. A set function $\mathcal{M}\{\mathrm{\Lambda}\}$ is called an uncertain measure if it satisfies the following four axioms of Liu [19]:
Axiom 1
(Normality) $\mathcal{M}\{\mathrm{\Lambda}\}=1$
Axiom 2
(Monotonicity) $\mathcal{M}\{\mathrm{\Lambda}\}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}\mathcal{M}\{{\mathrm{\Lambda}}^{C}\}=1$, for any event Λ
Axiom 3
Definition 2
Definition 3
provided that at least one of the two integrals is finite.
Theorem 1
(Liu [26]) Let ξ be an uncertain variable with uncertainty distribution Φ. If the expected value exists, then $E[\xi ]={\int}_{0}^{1}{\mathrm{\Phi}}^{1}(\alpha )d\alpha $.
Lemma 1
Theorem 2
Theorem 3
Assumptions and notations
Assumptions
 (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:

p _{ s } = production rate for the suppliers, which is a decision variable.

p _{ m } = demand rate or production rate for the manufacturer.

D _{ r } = constant demand rate for the retailer.

D _{ c } = constant demand rate of customer.

C _{ s } = purchase cost of unit item for suppliers.

C _{ m } = selling price of unit item for suppliers which is also the purchase cost for manufacturer.

C _{ r } = selling price of unit item for manufacturer which is also the purchase cost for retailers.

${C}_{{r}_{1}}$ = selling price for retailers.

t _{ s } = production time for supplers.

T _{ s } = cycle length for the suppliers.

T _{ r } = time duration where order is supplied by the manufacturer, by retailer’s cycle length.

T^{′} = last cycle length of the retailer.

T = total time for the integrated model.

h _{ s } = holding cost per unit per unit time for suppliers.

h _{ m } = holding cost per unit per unit time for manufacturer.

h _{ r } = holding cost per unit per unit time for retailers.

A _{ s } = ordering cost for suppliers.

A _{ m } = ordering cost for manufacturer.

h _{ r } = ordering cost for retailers.

id _{ s }, $\stackrel{~}{{\text{id}}_{s}}$ = idle cost per unit time for suppliers in crisp and uncertain environments, respectively.

id _{ m }, $\stackrel{~}{{\text{id}}_{m}}$ = idle cost per unit time for manufacturer in crisp and uncertain environments, respectively.

id _{ r }, $\stackrel{~}{{\text{id}}_{r}}$ = idle cost per unit time for retailers in crisp and uncertain environments, respectively.

n = number of cycle for retailers.

r = number of cycles where manufacturer stops production.

M = retailer’s trade credit period offered by the manufacturer to the retailers in years, which is the fraction of the years.

I _{ p } = interest payable to the manufacturer by the retailers.

${I}_{e},\stackrel{~}{{I}_{e}}$ = interest earned by the retailers in crisp and uncertain environments, respectively.

ATP = average total profit for the integrated models.

${P}_{m}^{\ast}$ = optimum value of P _{ m } for integrated models.

ATP^{∗} = optimum value of average total profit for the integrated models.
Model description and diagrammatic representation
Mathematical formulation of the model
Formulation of suppliers’ individual average profit
Formulation of manufacturer individual average profit
The total idle cost $={\text{id}}_{m}\left[\frac{{pm}_{Tm}{\mathit{\text{nD}}}_{R}}{{D}_{c}}\right]$, purchase cost = c _{ m } p _{ m } T _{ s }, selling price = c _{ r } p _{ m } T _{ s }, and ordering cost = A _{ m }.
Case 1
Case 2
Formulation of retailer individual average profit
The total idle cost = id_{ r } T _{ R }, purchase cost = c _{ r } p _{ m } T _{ s }, selling price = c _{ r 1} p _{ m } T _{ s }, and ordering cost = A _{ r }.
Case 1
(When M ≤ T^{′} ≤ T _{ R })
Case 2
(When T^{′} ≤ M ≤ T _{ R })
Crisp environment
Case 1
Case 2
Proposed inventory model in uncertain environment
Let us consider ${\stackrel{~}{\mathit{\text{id}}}}_{s}$, ${\stackrel{~}{\mathit{\text{id}}}}_{r}$, ${\stackrel{~}{\mathit{\text{id}}}}_{m}$, and ${\stackrel{~}{I}}_{\mathit{\text{re}}}$ as zigzag uncertain variables where $\stackrel{~}{{\text{id}}_{s}}=L({\text{id}}_{{s}_{1}},{\text{id}}_{{s}_{2}},{\text{id}}_{{s}_{3}})$, $\stackrel{~}{{\text{id}}_{r}}=L({\text{id}}_{{r}_{1}},{\text{id}}_{{m}_{2}},{\text{id}}_{{r}_{3}})$, $\stackrel{~}{{\text{id}}_{m}}=L({\text{id}}_{{m}_{1}},{\text{id}}_{{m}_{2}},{\text{id}}_{{m}_{3}})$, and $\stackrel{~}{{I}_{\mathit{\text{re}}}}=L({I}_{{\mathit{\text{re}}}_{1}},{I}_{{\mathit{\text{re}}}_{2}},{I}_{{\mathit{\text{re}}}_{3}})$. Then, the objective is reduce to the following:

For Case 1 (M ≤ T^{′} ≤ T _{ R })$\begin{array}{ll}\phantom{\rule{6.5pt}{0ex}}\mathrm{A}\stackrel{~}{\mathrm{T}}{\mathrm{P}}_{1}& =\frac{{D}_{c}}{{p}_{m}{T}_{s}+{D}_{R}}\left[0({c}_{m}{c}_{s}){p}_{m}{T}_{s}{h}_{s}\left(\frac{{p}_{s}{t}_{s}^{2}}{{p}_{m}}{p}_{s}{t}_{s}^{2}\right){\stackrel{~}{\mathit{\text{id}}}}_{s}\left({T}_{R}+{p}_{s}{t}_{s}\left(\frac{1}{{D}_{c}}\frac{1}{{p}_{m}}\right)\right)\right.\\ \phantom{\rule{1em}{0ex}}{A}_{s}+({c}_{r}{c}_{m}){p}_{m}{T}_{s}{h}_{m}\left({\mathit{\text{np}}}_{m}{T}_{s}{T}_{R}\frac{{n}^{2}+n2r2}{2}{T}_{R}{D}_{R}\frac{{p}_{s}^{2}{t}_{s}^{2}}{2{p}_{m}}\right)\\ \phantom{\rule{1em}{0ex}}{A}_{m}+({c}_{{r}_{1}}{c}_{r}){p}_{m}{T}_{s}\frac{{h}_{r}}{2}\left(\frac{{p}_{m}^{2}{T}_{s}^{2}}{{D}_{c}}2{\mathit{\text{np}}}_{m}{T}_{s}{T}_{R}(2n+1){T}_{R}{D}_{R}\right)\\ \phantom{\rule{1em}{0ex}}\left(\right)close="]">\phantom{\rule{0.3em}{0ex}}{\stackrel{~}{\mathit{\text{id}}}}_{m}\left(\frac{{p}_{m}{T}_{m}{\mathit{\text{nD}}}_{R}}{{D}_{c}}\right)+\frac{(n+1){c}_{{r}_{1}}\stackrel{~}{{I}_{\mathit{\text{re}}}}{D}_{c}{M}^{2}}{2}{\stackrel{~}{\mathit{\text{id}}}}_{r}{T}_{R}{A}_{r}& .\end{array}$(24)

For Case 2 (T^{′} ≤ M ≤ T _{ R })$\begin{array}{ll}\phantom{\rule{6.5pt}{0ex}}\mathrm{A}\stackrel{~}{\mathrm{T}}{\mathrm{P}}_{2}& =\frac{{D}_{c}}{{p}_{m}{T}_{s}+{D}_{R}}\left[({c}_{m}{c}_{s}){p}_{m}{T}_{s}{h}_{s}\left(\frac{{p}_{s}{t}_{s}^{2}}{{p}_{m}}{p}_{s}{t}_{s}^{2}\right){\stackrel{~}{\mathit{\text{id}}}}_{s}\left({T}_{R}+{p}_{s}{t}_{s}\frac{1}{{D}_{c}}\frac{1}{{p}_{m}}\right){A}_{s}\right.\\ \phantom{\rule{1em}{0ex}}+({c}_{r}{c}_{m}){p}_{m}{T}_{s}{h}_{m}\left({\mathit{\text{np}}}_{m}{T}_{s}{T}_{R}\frac{{n}^{2}+n2r2}{2}{T}_{R}{D}_{R}\frac{{p}_{s}^{2}{t}_{s}^{2}}{2{p}_{m}}\right)\\ \phantom{\rule{1em}{0ex}}{\stackrel{~}{\mathit{\text{id}}}}_{m}\left(\frac{{p}_{m}{T}_{m}{\mathit{\text{nD}}}_{R}}{{D}_{c}}\right)\\ \phantom{\rule{1em}{0ex}}{A}_{m}+({c}_{r1}{c}_{r}){p}_{m}{T}_{s}\frac{{h}_{r}}{2}\left(\frac{{p}_{m}^{2}{T}_{s}^{2}}{{D}_{c}}2{\mathit{\text{np}}}_{m}{T}_{s}{T}_{R}(2n+1){T}_{R}{D}_{R}\right)\\ \phantom{\rule{1em}{0ex}}+\left(\right)close="]">\left\{\frac{{\mathit{\text{nc}}}_{{r}_{1}}{D}_{c}{M}^{2}}{2}+\frac{{c}_{{r}_{1}}}{2}({p}_{m}{T}_{s}{\mathit{\text{nD}}}_{R})(2M{T}^{\prime})\right\}\stackrel{~}{{I}_{\mathit{\text{re}}}}{\stackrel{~}{\mathit{\text{id}}}}_{r}{T}_{R}{A}_{r}& .\end{array}$(25)
The equivalent crisp model
Using Lemma 1 and applying Theorem 2, the expected total average profit is given by the following:

For Case 1 (M ≤ T^{′} ≤ T _{ R })$\begin{array}{ll}\phantom{\rule{6.5pt}{0ex}}E[\mathrm{A}\stackrel{~}{\mathrm{T}}{\mathrm{P}}_{1}]& =\frac{{D}_{c}}{{p}_{m}{T}_{s}+{D}_{R}}\left[0({c}_{m}{c}_{s}){p}_{m}{T}_{s}{h}_{s}\left(\frac{{p}_{s}{t}_{s}^{2}}{{p}_{m}}{p}_{s}{t}_{s}^{2}\right)\right.\\ \phantom{\rule{1em}{0ex}}E[{\stackrel{~}{\mathit{\text{id}}}}_{s}]\left({T}_{R}+{p}_{s}{t}_{s}\left(\frac{1}{{D}_{c}}\frac{1}{{p}_{m}}\right)\right)\\ \phantom{\rule{1em}{0ex}}{A}_{s}+({c}_{r}{c}_{m}){p}_{m}{T}_{s}{h}_{m}\left({\mathit{\text{np}}}_{m}{T}_{s}{T}_{R}\frac{{n}^{2}+n2r2}{2}{T}_{R}{D}_{R}\frac{{p}_{s}^{2}{t}_{s}^{2}}{2{p}_{m}}\right)\\ \phantom{\rule{1em}{0ex}}{A}_{m}+({c}_{{r}_{1}}{c}_{r}){p}_{m}{T}_{s}\frac{{h}_{r}}{2}\left(\frac{{p}_{m}^{2}{T}_{s}^{2}}{{D}_{c}}2{\mathit{\text{np}}}_{m}{T}_{s}{T}_{R}(2n+1){T}_{R}{D}_{R}\right)\\ \phantom{\rule{1em}{0ex}}\left(\right)close="]">E[{\stackrel{~}{\mathit{\text{id}}}}_{m}]\left(\frac{{p}_{m}{T}_{m}{\mathit{\text{nD}}}_{R}}{{D}_{c}}\right)+\frac{(n+1){c}_{{r}_{1}}E[\stackrel{~}{{I}_{\mathit{\text{re}}}}]{D}_{c}{M}^{2}}{2}E[{\stackrel{~}{\mathit{\text{id}}}}_{r}]{T}_{R}{A}_{r}& .\end{array}$(26)

For Case 2 (T^{′} ≤ M ≤ T _{ R })$\begin{array}{ll}\phantom{\rule{6.5pt}{0ex}}E[\mathrm{A}\stackrel{~}{\mathrm{T}}{\mathrm{P}}_{2}]& =\frac{{D}_{c}}{{p}_{m}{T}_{s}+{D}_{R}}\left[({c}_{m}{c}_{s}){p}_{m}{T}_{s}{h}_{s}\left(\frac{{p}_{s}{t}_{s}^{2}}{{p}_{m}}{p}_{s}{t}_{s}^{2}\right)\phantom{\rule{0.3em}{0ex}}E[{\stackrel{~}{\mathit{\text{id}}}}_{s}]\phantom{\rule{0.3em}{0ex}}\left({T}_{R}+{p}_{s}{t}_{s}\frac{1}{{D}_{c}}\frac{1}{{p}_{m}}\right)\right.\phantom{\rule{0.3em}{0ex}}{A}_{s}\\ \phantom{\rule{1em}{0ex}}+({c}_{r}{c}_{m}){p}_{m}{T}_{s}{h}_{m}\left({\mathit{\text{np}}}_{m}{T}_{s}{T}_{R}\frac{{n}^{2}+n2r2}{2}{T}_{R}{D}_{R}\frac{{p}_{s}^{2}{t}_{s}^{2}}{2{p}_{m}}\right)\\ \phantom{\rule{1em}{0ex}}E[{\stackrel{~}{\mathit{\text{id}}}}_{m}]\left(\frac{{p}_{m}{T}_{m}{\mathit{\text{nD}}}_{R}}{{D}_{c}}\right){A}_{m}+({c}_{r1}{c}_{r}){p}_{m}{T}_{s}\\ \phantom{\rule{1em}{0ex}}\frac{{h}_{r}}{2}\left(\frac{{p}_{m}^{2}{T}_{s}^{2}}{{D}_{c}}2{\mathit{\text{np}}}_{m}{T}_{s}{T}_{R}(2n+1){T}_{R}{D}_{R}\right)\\ \phantom{\rule{1em}{0ex}}\left(\right)close="]">+\left\{\frac{{\mathit{\text{nc}}}_{{r}_{1}}{D}_{c}{M}^{2}}{2}+\frac{{c}_{{r}_{1}}}{2}({p}_{m}{T}_{s}{\mathit{\text{nD}}}_{R})(2M{T}^{\prime})\right\}E[\stackrel{~}{{I}_{\mathit{\text{re}}}}]E[{\stackrel{~}{\mathit{\text{id}}}}_{r}]{T}_{R}{A}_{r}& .\end{array}$(27)
Numerical example
Crisp environment
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 
Optimum results for objective functions and other parameters
Parameter  Case 1  Case 2 

ATP^{∗}  1,039.45  1,108.93 
${p}_{s}^{\ast}$  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
Optimum value changes due to parametric changes
Parameter name  Parametric value  T ^{∗}  ${\mathit{P}}_{\mathit{s}}^{\mathbf{\ast}}$  ATP ^{∗}  

Case 1  Case 2  Case 1  Case 2  Case 1  Case 2  
c _{ s }  1.5  1.92  1.86  79.18  78.31  10.89  1,529.43 
2.5  1.61  1.76  46.76  78.31  10.14  1,526.25  
3.5  1.12  1.45  18.18  78.31  10.06  1,522.47  
c _{ m }  1.5  1.82  1.82  56.14  78.31  10.56  1,528.36 
2.5  1.75  1.65  69.76  78.31  10.46  1,525.64  
3.5  1.10  1.33  64.15  78.31  10.15  1,520.25  
c _{ r }  1.5  1.75  1.96  65.18  78.31  10.56  1,528.19 
2.5  1.52  1.69  62.17  78.31  10.56  1,522.58  
3.5  1.06  1.14  58.14  78.31  10.56  1,520.43  
h _{ s }  1.5  1.86  1.76  69.18  78.49  10.56  1,536.45 
2.5  1.64  1.54  64.76  78.31  10.14  1,533.25  
3.5  1.25  1.19  60.19  78.04  10.56  1,520.43  
h _{ m }  1.5  1.09  1.76  88.76  78.74  10.56  1,530.52 
2.5  1.21  1.52  76.78  78.54  10.18  1,525.19  
3.5  1.13  1.01  55.45  78.17  10.56  1,523.57  
h _{ r }  1.5  1.75  1.89  69.76  78.31  10.19  1,520.43 
2.5  1.25  1.15  54.76  78.01  10.56  1,518.29  
3.5  1.12  1.06  48.16  78.00  10.04  1,518.21 
Uncertain environment
Input data of different zigzag parameters for Case 1 and Case 2
Parameters in uncertain environments  ${\stackrel{\mathbf{~}}{\mathbf{\text{id}}}}_{\mathit{s}}$  ${\stackrel{\mathbf{~}}{\mathbf{\text{id}}}}_{\mathit{m}}$  ${\stackrel{\mathbf{~}}{\mathbf{\text{id}}}}_{\mathit{r}}$  ${\stackrel{\mathbf{~}}{\mathit{I}}}_{\mathit{e}}$ 

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) 
Optimal values of objective and decision variables
Parameter  Case 1  Case 2 

ATP^{∗}  1,056.43  1,520.43 
${p}_{s}^{\ast}$  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.
Declarations
Authors’ Affiliations
References
 Oliver RK, Webber MD: Supplychain management: logistics catches up with strategy (reprint from Outlook). 1982.Google Scholar
 Cohen MA, Baghanan MP: The stabilizing effect of inventory in supply chains. Oper. Res 1998, 46: 72–83. 10.1287/opre.46.3.S72View ArticleGoogle Scholar
 Cachon GP, Zipkin PH: Competitive and cooperative inventory policies in a twostage supply chain. Manag. Sci 1999, 45: 936–953. 10.1287/mnsc.45.7.936View ArticleGoogle Scholar
 Viswanathan S, Piplani R: Coordinating supply chain inventories through common replenishment epochs. Eur. J. Oper. Res 2001, 129: 277–286. 10.1016/S03772217(00)002253View ArticleGoogle Scholar
 Khouja M: Optimizing inventory decisions in a multistage multi customer supply chain. Transportation Research 2003, 39: 193–208. 10.1016/S13665545(02)000364View ArticleGoogle Scholar
 Agarwal V, Chao X, Seshadri S: Dynamic balancing of inventory in supply chains. Eur. J. Oper. Res 2004, 159: 296–317. 10.1016/j.ejor.2003.08.017View ArticleGoogle Scholar
 Rau H, Wu MY, Wee HM: Deteriorating item inventory model with shortage due to supplier in an integrated supply chain. Int. J. Syst. Sci 2004, 35: 293–303. 10.1080/00207720410001714833MathSciNetView ArticleGoogle Scholar
 Lee W: A joint economic lot size model for raw material ordering, manufacturing setup,and finished goods delivering. Int. J. Manag. Sci 2005, 33: 163–174.Google Scholar
 BenDaya M, Seliaman M: An integrated production inventory model with raw material replenishment considerations in a three layer supply chain. Int. J. Prod. Econ 2010, 28: 11–21.Google Scholar
 Sana S: A productioninventory model of imperfect quality products in a threelayer supply chain. Decision Support Syst 2011, 50: 539–547. 10.1016/j.dss.2010.11.012View ArticleGoogle Scholar
 Pal B, Sana SS, Chaudhuri K: Threelayer supply chain  a productioninventory model for reworkable items. Appl. Math. Comput 2012, 219: 530–543. 10.1016/j.amc.2012.06.038MathSciNetView ArticleGoogle Scholar
 Wang J, Shu YF: Fuzzy decision modelling for supply chain management. Fuzzy Sets Syst 2005, 150: 107–127. 10.1016/j.fss.2004.07.005MathSciNetView ArticleGoogle Scholar
 Das B, Maity K, Maiti M: A two warehouse supplychain model under possibility /necessity /credibility measures. Math. Comput. Model 2007, 46: 398–409. 10.1016/j.mcm.2006.11.017View ArticleGoogle Scholar
 Chen C, Yuan T, Lee W: Multicriteria fuzzy optimization for locating warehouses and distribution centers in a supply chain network. J. Chin. Inst. Chem. Eng 2007, 38: 393–407. 10.1016/j.jcice.2007.08.001View ArticleGoogle Scholar
 Peidro D, Mula J, Jimenez M, Botella MM: A fuzzy linear programming based approach for tactical supply chain planning in an uncertainty environment. Eur. J. Oper. Res 2010, 205: 65–80. 10.1016/j.ejor.2009.11.031View ArticleGoogle Scholar
 Chu SJ: Interactive group decisionmaking using a fuzzy linguistic approach for evaluating the flexibility in a supply chain. Eur. J. Oper. Res 2011, 213: 279–289. 10.1016/j.ejor.2011.03.022View ArticleGoogle Scholar
 Jana DK, Maity K, Das B, Roy TK: A fuzzy simulation via contractive mapping genetic algorithm approach to an imprecise production inventory model under volume flexibility. J. Simulation 2013,7(2):90–100. 10.1057/jos.2012.23View ArticleGoogle Scholar
 Kristianto Y, Helo P, Jiao J, Sandhu M: Adaptive fuzzy vendor managed inventory control for mitigating the Bullwhip effect in supply chains. Eur. J. Oper. Res 2012, 216: 346–355. 10.1016/j.ejor.2011.07.051MathSciNetView ArticleGoogle Scholar
 Liu B: Uncertainty Theory. Berlin: Springer; 2007.View ArticleGoogle Scholar
 You C: Some convergence theorems of uncertain sequences. Math. Comput. Model 2009, 49: 482–487. 10.1016/j.mcm.2008.07.007View ArticleGoogle Scholar
 Liu B: Fuzzy process, hybrid process and uncertain process. J. Uncertain Syst 2008, 2: 3–16.Google Scholar
 Liu B: Some research problems in uncertainty theory. J. Uncertain Syst 2009, 3: 3–10.Google Scholar
 Goyal SK: Economic order quantity under conditions of permissible delay in payments. J. Oper. Res. Soc. 1985, 36: 35–38.View ArticleGoogle Scholar
 Khanra S, Ghosh SK, Chaudhuri KS: An EOQ model for a deteriorating item with time dependent quadratic demand under permissible delay in payment. Appl. Math. Comput 2011, 218: 1–9. 10.1016/j.amc.2011.04.062MathSciNetView ArticleGoogle Scholar
 Maihami R, Abadi INK: Joint control of inventory and its pricing for noninstantaneously deteriorating items under permissible delay in payments and partial backlogging. Math. Comput. Model 2012, 55: 1722–1733. 10.1016/j.mcm.2011.11.017MathSciNetView ArticleGoogle Scholar
 Liu B: Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty. Berlin: Springer; 2010.View ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.