a multi-echelon production–inventory system with supply disruption
TRANSCRIPT
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Journal of Manufacturing Systems 33 (2014) 262–276
Contents lists available at ScienceDirect
Journal of Manufacturing Systems
j ourna l ho me pa g e: www.elsev ier .com/ locate / jmansys
multi-echelon production–inventory system with supply disruption
rojeswar Pala, Shib Sankar Sanab,∗, Kripasindhu Chaudhuria
Department of Mathematics, Jadavpur University, Kolkata 700032, IndiaDepartment of Mathematics, Bhangar Mahavidyalaya, University of Calcutta, Bhangar, 24PGS (South), Kolkata 743502, India
r t i c l e i n f o
rticle history:eceived 10 November 2012eceived in revised form1 September 2013ccepted 31 December 2013vailable online 8 February 2014
eywords:ulti-echelon supply chainachine breakdown
upply disruptionafety stockorrective maintenancehortage
a b s t r a c t
The article investigates an integrated multi-layer supply chain model consisting of supplier, manufacturerand retailer while supply disruption, machine breakdown, safety stock, maintenance breakdown occursimultaneously. At beginning of the production, manufacturer keeps some raw materials in stock receivedfrom second supplier at high price, as safety stock due to supply disruption of first supplier. Correctivemaintenance is done immediately to restore its normal stage when machine breakdown occurs. Stockout situations at manufacturer and retailer are considered due to disruption of production for machinebreakdown. The integrated expected costs of the chain in centralized (collaborating) and decentralized(Stakelberg approach) system are compared. A numerical example and its sensitivity analysis are providedto test feasibility of the model.
© 2014 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
. Introduction
Supply chain management plays an important and critical role in any business organization in increasing competition and globalization.n last two decades, multi-stage supply chain management has received a very visible and influential research topics in the field of operationsesearch. In the real world, a chain faces a lot of real problems like supply disruption, production system breakdown, shortages of product,tc. Manufacturing process is one of the most important part to the companies for their business strategy. Careful production planning isecessary to ensure good deliveries and productive efficiencies. The key factors of the production system are planning, organizing, directingnd controlling of production activities. Machine breakdown, supply disruptions, shortages of products are common but vital factors forhe companies. The researchers and practitioners are facing tough challenging condition to find out the best strategies on productionisruption and supply disruption. The maintenance scheduling of systems, safety stock of the products, involving secondary members, etc.ay be implemented to rescue from supply disruption and production disruption. Generally speaking, textile, footwear, electronics, etc.
ndustries face such type of bottlenecks. In these industries, raw materials are transported from far side. Consequently, supply disruptionsue to transportation problem, labor strike, natural problem, etc. are common factors. In such situations, manufacturers keep stock theaw materials for continuing production process. Moreover, production system disruption is common for all industries due to uncertainachine breakdown. As a whole, the entire channel faces stock out situations. Now, our aim is to study a supply chain model considering
he important factors supply disruption, machine breakdown, corrective maintenance, safety stock, backlogging, multi-echelon supplyhain.
. Literature review
Inderfurth [14] studied a procedure for determining the optimal size and distribution of safety stocks in a general serial or divergentroduction or distribution process ruled by a base-stock control policy. Groenvelt et al. [11] introduced first a production inventory modelith machine breakdown where the effect of machine breakdown and corrective maintenance on the economic lot sizing decisions were
∗ Corresponding author. Tel.: +91 8926007373.E-mail addresses: brojo [email protected] (B. Pal), sana [email protected], shib [email protected] (S.S. Sana), [email protected] (K. Chaudhuri).
278-6125/$ – see front matter © 2014 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.ttp://dx.doi.org/10.1016/j.jmsy.2013.12.010
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B. Pal et al. / Journal of Manufacturing Systems 33 (2014) 262–276 263
onsidered. They also showed that the optimal lot size of their model would always be larger than that in a deterministic EPQ case. Groenveltt al. [12] extended their previous model by including general random repair times and a safety stock policy. Minner [21] studied a multi-tage safety stock policy where each stock point of the inventory system follows a periodically reviewed order-up-to policy. Moinzadehnd Aggarwal [22] studied an unreliable bottleneck production/inventory system with a constant production and demand rate that isubject to random disruptions. They considered restoration time is constant and the time between breakdowns is exponential. Makis andung [20] presented an EMQ model with inspections and random machine failure and obtained the formula for the long-run expectedverage cost per unit time under the assumption of generally distributed failure and replacement times, constant inspection times and zeroestoration times. Graves and Lee [9] introduced a single machine scheduling problems where the machine maintenance must be performedithin certain intervals and hence the machine is not available during the maintenance periods. Their main purpose is to schedule theaintenance and jobs to minimize some performance measures. Khouja [15] developed a three-stage, multi-customer, non-serial supply
hain model where a firm can supply two or more customers. He analyzed the model with three inventory coordination mechanismsetween chain members and solve a cost minimization model for each. Lin and Gong [18] introduced an EPQ model for a deterioratingroduct where manufacturing system faced a random machine breakdown and also considered a fixed corrective maintenance period. Chiut al. [4,8] developed an economic production quantity model with scrap, the reworking of random defective items, and stochastic machinereakdowns under no-resumption (NR) inventory control policy. Cardenas-Barron [3] introduced an n-stage-multi-customer supply chain
nventory model where retailer can supply products to several customers and he obtained the optimal equal cycle time and the optimalotal annual cost. Leung [16] extended the work of Khouja [15] considering the integer multipliers mechanism and next individuallyerive the optimal solution to the three-stage and four-stage model using the perfect squares method. Leung [17] extended the previousaper [16] by considering with/without lot streaming and with/without complete backorders under the integer multiplier coordinationechanism, and then individually derive the optimal solution to the three- and four-stage model. Ben-Daya and Al-Nassar [2] studiedith inventory and production co-ordination in a three-layer supply chain where lot produced at each stage be sent in equal shipments
o the downstream customers. Sana [26] developed an integrated production–inventory model of perfect and imperfect quality productsn a three-layer supply chain. Ayed [1] studied a production inventory model where they derived a joint optimization of maintenance androduction policies considering random demand and variable production rate. Chiu et al. [5] demonstrated the optimal replenishmentun time for a production system with stochastic machine breakdown and failure in rework. Lin and Chiu [19] presented a direct proof ofonvexity of long-run average cost function of the paper of Chiu et al. [5]. Pal et al. [23] investigated an imperfect production inventoryodel with stochastic demand where the regular maintenance is performed after each production run-time and free minimal repairarranty in a just-in-time production process is also provided to the customers. There are several interesting and relevant papers related
o multi-stage inventory model such as [6,7,10,13,24,25,27–30], etc. In this article, we develop a multi-echelon supply chain model whereaw material supplier may face supply disruption for different natural and mechanical reasons. The supplier supplies an ordering lot sizef raw material to the manufacturer’s warehouse but it may be disrupted by unnecessary conditions. At beginning, warehouse stores aafety stock of raw material from the another supplier with a high price to avoid the effect of supply disruption. The production may beisrupted by machine breakdown of manufacturing system after a random time. It may be taken a random time to return the productionystem at the initial condition by corrective maintenance. The inventory level of manufacturer may fall into shortages for the breakdownf production system and inventory of finished products at retailer may also fall into shortages. Finally, individual expected cost functionsf suppliers, manufacturer and retailer are formulated by trading off inventory and set up costs at each state, cost of raw materials,abour/energy and tool/die costs for production, maintenance cost and shortage costs. The above costs functions are analyzed in the lightf collaborating and Stakelberg approach. The rest of the paper is organized as follows: Section 3 illustrates fundamental assumptionsnd notations. Mathematical formulation and analysis of the model is discussed in Section 4. Section 5 demonstrates integrated inventoryodel. Stakelberg approach is delivered in Section 6. Section 7 analyzes numerical analysis. Sensitivity analysis and managerial insights
re discussed in Section 8 and finally conclusion of the paper is provided in Section 9.
. Fundamental assumptions and notation
.1. Assumptions
The following assumptions are made to develop the model:
(i) The chain is developed for single item product.(ii) Order lot size of supply chain is a decision variable.iii) Supplier may face supply disruption after a random time.iv) Production rate of manufacturer is greater than the demand rate of retailer.(v) During regular production run-time, the production system may be disrupted and take a random time to restore the system to the
working condition.vi) During the maintenance time of production system, the manufacturer’s finished product inventory may fall into shortages.
vii) Demand rate of the supply chain is deterministic and constant.iii) Safety stock of raw-materials which are stored in the manufacturer’s warehouse, are supplied by other supplier with more price.ix) Retailer inventory level may fall into shortage for the breakdown of production system.
.2. Notation
The following notation are used through out the model:
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64 B. Pal et al. / Journal of Manufacturing Systems 33 (2014) 262–276
Symbol Denotation
R ordering lot-size for supplierCr cost per unit raw material of supplierp unit selling price� non-negative random variable denoting the time to supplier disruptionf(�) probability density function of �t1 production run-time without effect from supply disruption and machine break downChs inventory holding cost per unit per unit time for supplierAs set up cost for supplier� non-negative random variable denoting the time to machine breakdowng(�) probability density function of �ı random variable denoting the time to perform corrective maintenanceh(·) probability density function of ıS(�) safety stock of manufacturer warehouseC ′
r cost per unit raw material of warehouseChw inventory holding cost per unit per unit time of warehousewm selling price per unit item for manufacturerDr demand rate for finished items for retailerT manufacturer run-timeChm inventory holding cost per unit per unit time for manufacturerSm shortage cost per unit per unit time for manufacturerCp per unit production costL fixed cost like labour, technology, energy, etc for manufacturing items per cycle� variation constant of tool/die costsCm corrective maintenance cost per cycleAm set up cost for manufacturerDc demand rate for finished items for customersT1 cycle time of the supply chainChr inventory holding cost per unit per unit for retailerSr shortage cost per unit per unit time for retailerAm set up cost for retailer
. Mathematical formulation and analysis of the model
In this article, we develop a multi-layer supply chain model with single item and deterministic demand rate where supplier, manufacturearehouse, manufacture and retailer are the members. The supplier delivers the raw materials to the manufacturer but the supplier may
all in disruption. The manufacturer rents a warehouse to store safety stock of raw material to prevent shutting down production throughupply disruption of supplier. The production system of manufacturer may fall into machine break down after a random time and takes
random time to recover the production system. In the recovering time of production system, production be stopped and shortages mayappened for this effect. Also, shortages of product for retailer may take place for the break down of production system.
.1. The supplier’s inventory model
The supplier takes a order of raw materials and supplies it to the manufacturer. After a random time �, supplier may face supply
isruption for different reasons. The raw material supply system is totally stopped when the supply disruption of the raw materials isappened (see Fig. 1). Here, the following two cases arise to formulate the model.Fig. 1. Logistic diagram of the model of supplier.
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B. Pal et al. / Journal of Manufacturing Systems 33 (2014) 262–276 265
.1.1. Case I: when 0 ≤ � ≤ t1In this case, supply disruption occurs in the actual regular production run-time t1. As a result, supply of raw material be totally stopped
fter the time �. The governing differential equation of inventory level of this case is
Is(t)dt
= −P, with Is(0) = R, 0 ≤ t ≤ � (s.1.1)
sing the boundary conditions, we have from Eq. (s.1.1)
Is(t) = R − Pt, 0 ≤ t ≤ � (s.1.2)
he inventory cost is:
ICs = Chs
∫ �
0
Is(t)dt = R� − 12
P�2 (s.1.3)
.1.2. Case II: when � ≥ t1In this case, supply disruption occurs after the actual regular production run-time t1. So, the model does not effected by the disruption.
he governing differential equation of inventory level of this case is
Is(t)dt
= −P, with Is(0) = R and Is(t1) = 0, 0 ≤ t ≤ t1 (s.2.1)
sing the boundary conditions, we have from Eq. (s.2.1)
Is(t) = R − Pt, 0 ≤ t ≤ t1 (s.2.2)
he inventory cost is:
ICs = Chs
∫ t1
0
Is(t)dt = Rt1 − 12
Pt21 (s.2.3)
ombining Cases I and II, we have the expected average cost per item of supplier is
ESC(R) = 1R
[cost of raw material + expected inventory cost + set up cast] = 1R
[CrR +
∫ t1
0
ICsf (�)d� +∫ ∞
t1
ICsf (�)d� + As
](s.1)
.2. The manufacturer’s warehouse inventory level
At the beginning, manufacturer’s warehouse stores a safety stock inventory S(�) to avoiding the discontinuation of production for supplyisruption. We consider the safety stock inventory level as
S(�) ={
P(t1 − �), if 0 ≤ � ≤ t1
0, if �≥t1
he warehouse wants to store minimum raw materials such that production system does not affected for raw materials but the cost of thearehouse is least. As, the supplying system of raw materials is disrupted after a random time �, then shortages of the raw materials are
(t1 − �). So, we assume the safety stock level as above.This safety stock uses for production if supply disruption occurs in the actual production run-time t1. In the model, production system
f manufacture may face break down after a random time �. So, production be stopped through random recovery time ı. Warehouse holdaw materials through the production run-time. We draw a rough diagram (Fig. 2) highlighting different following cases inventory leveln the warehouse. With the help of following cases, we find the total expected inventory cost of raw material for warehouse.
.2.1. Case I: 0 ≤ � ≤ t1 and 0 ≤ � ≤ �In this case, supply disruption of raw material is occurred before the machine breakdown of the production system. The machine
reakdown is also happened in regular production run-time. So, Some raw materials from safety stock are used between the time fromupply disruption to machine breakdown. Rest of raw materials from safety stock are used after the recovery of machine. So, the inventoryost of raw material of this case is
ICw1 = Chw
[S(�)� +
∫ �
�
(t1 − t)Pdt + (t1 − �)Pı +∫ t1+ı
�+ı
(t1 + ı − t)Pdt
]
.2.2. Case II: 0 ≤ � ≤ t1, � ≤ � ≤ t1 and � + ı ≤ �Here, the machine breakdown is taken place before the supply disruption of raw materials in the production run-time. Also, the machine
ecovers before the supply disruption. So, the raw materials from the supplier are stored between the time from machine breakdown toachine recover time. The total raw materials from warehouse are used after the supply disruption. Inventory cost of raw material of this
ase is
ICw2 = Chw
[S(�)� +
∫ �+ı
�
(t1 − t)P dt + (t1 + ı − �)P(� − � − ı) +∫ t1+ı
�
(t1 + ı − t)P dt
]
266 B. Pal et al. / Journal of Manufacturing Systems 33 (2014) 262–276
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Fig. 2. Logistic diagram of the model of warehouse.
.2.3. Case III: 0 ≤ � ≤ t1, � ≤ � ≤ t1 and � + ı ≥ �Here, the machine breakdown is taken place before the supply disruption of raw materials in the production run-time. The supply
isruption of raw materials is happened before the machine recovery. So, the raw materials from the supplier are stored in warehouseetween the time from machine breakdown to supply disruption time. The total raw materials from warehouse are used after the supplyisruption. Hence, the inventory cost of raw material of this case is
ICw3 = Chw
[S(�)� +
∫ �
�
(t1 − � − � + t)P dt + (t1 − �)P(� + ı − �) +∫ t1+ı
�+ı
(t1 + ı − t)P dt
]
.2.4. Case IV: 0 ≤ � ≤ t1, t1≤ � ≤ ∞ and � + ı ≤ t1Here, the machine breakdown is taken place before the supply disruption of raw materials in the production run-time and the production
ystem does not affected by the supply disruption of raw materials. So, the raw materials from the supplier are stored in warehouse betweenhe time from machine breakdown to machine recovery time. The raw materials of the warehouse are used for production after the time1. So, the inventory cost of raw material of this case is
ICw4 = Chw
[∫ �+ı
�
Pt dt + Pı (t1 − � − ı) +∫ t1+ı
t1
(t1 + ı − t)P dt
]
.2.5. Case V: 0 ≤ � ≤ t1, t1≤ � ≤ ∞ and � + ı ≥ t1In this case, the machine breakdown is taken place before the supply disruption of raw materials in the production run-time and the
roduction system does not affected by the supply disruption of raw materials. Here, the raw materials from the supplier are stored inarehouse between the time from machine breakdown to t1. After the machine recovery, the raw material of the warehouse are used forroduction. Hence, the inventory cost of raw material of this case is
ICw5 = Chw
[∫ t1
�
Pt dt + P(t1 − �) (� + ı − t1) +∫ t1+ı
�+ı
(t1 + ı − t)P dt
]
.2.6. Case VI: � ≥ t1 and � ≤ t1In this case, the production system is not affected by the machine breakdown. The supply disruption is occurred in the production
un-time. So, after the occurring of supply disruption, the production uses the raw materials from safety stock of the warehouse. So, thenventory cost of raw material of this case is
ICw6 = Chw
[S(�) � +
∫ t1
�
(t1 − t)P dt
]
.2.7. Case VII: � ≥ t1 and � ≥ t1In this case, the production system is not affected by the machine breakdown and supply disruption of raw material. Hence, the inventory
ost of raw material of this case is zero.Combining the above cases, the average expected cost per unit items is
B. Pal et al. / Journal of Manufacturing Systems 33 (2014) 262–276 267
4
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4
a
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Fig. 3. Logistic diagram of the model of manufacturer.
EWC = 1R
[expected cost of safety stock raw material + expected inventory cost]
= 1R
[C ′
rR +∫ t1
0
{∫ �
0
(∫ ∞
0
ICw1 h(ı)dı
)f (�)d� +
∫ t1
�
(∫ �−�
0
ICw2 h(ı)dı +∫ ∞
�−�
ICw3 h(ı)dı
)f (�)d�
+∫ ∞
t1
(∫ t1−�
0
ICw4 h[ı]dı +∫ ∞
t1−�
ICw5 h(ı)dı
)f (�)d�
}g(�)d� +
∫ ∞
0
(∫ t1
0
ICw1 f (�)d�
)g(�)d�
](w.1)
.3. The manufacturer’s inventory model
In the supply chain, the manufacturer produces products using the raw materials. The machine break down of the production systemay take place after a random time �. If machine breakdown is happened in the production run-time, the production system is again run
fter a random time ı. In this time, the production is totally stopped. As a result, finished products’ inventory for manufacturer may fall inhe shortage (see Fig. 3). We formulate this model in the following two cases
.3.1. Case I: when 0 ≤ � ≤ t1Here, the machine break down of production system takes place in the actual production run-time t1. So, the production system is
ffected by breakdown. The governing differential equation of the inventory level is
Im1 (t)dt
= P − Dr, with Im1 (0) = 0, 0 ≤ t ≤ � (m.1.1)
Im2 (t)dt
= −Dr, with Im2 (�) = Im1 (�), � ≤ t ≤ � + ı (m.1.2)
Im3 (t)dt
= P − Dr, with Im3 (� + ı) = Im2 (� + ı), � + ı ≤ t ≤ t1 + ı (m.1.3)
Im4 (t)dt
= −Dr, with Im4 (t1 + ı) = Im3 (t1 + ı), t1 + ı ≤ t ≤ T (m.1.4)
sing the boundary conditions, we have from the Eqs. (m.1.1)–(m.1.4)
Im1 (t) = (P − Dr)t, 0 ≤ t ≤ � (m.1.5)
Im1 (t) = P� − Drt, � ≤ t ≤ � + ı (m.1.6)
Im1 (t) = (P − Dr)t − Pı, � + ı ≤ t ≤ t1 + ı (m.1.7)
Im1 (t) = Pt1 − Drt, t1 + ı ≤ t ≤ T (m.1.8)
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68 B. Pal et al. / Journal of Manufacturing Systems 33 (2014) 262–276
fter the happening of the machine breakdown, the inventory of finished products of manufacturer can satisfy the demand of the retailerhrough the time tm = (Im1 (�))/(Dr). If the inventory level of the manufacturer falls in shortages, then he takes time to recover that shortages
m1 = Im2 (�+ı)P−Dr
.
.3.1.1. Subcase I: when ı ≤ tm, i.e. shortage does not occur. The inventory holding cost of produced items is
ICm1 = Chm
[∫ �
0
Im1 (t)dt +∫ �+ı
�
Im2 (t)dt +∫ t1+ı
�+ı
Im3 (t)dt +∫ T
t1+ı
Im4 (t)dt
](m.1.9)
.3.1.2. Subcase II: when ı ≥ tm, i.e. shortage does occur. The inventory holding cost of produced items is
ICm2 = Chm
[∫ �
0
Im1 (t)dt +∫ �+tm
�
Im2 (t)dt +∫ t1+ı
�+ı+tm1
Im3 (t)dt +∫ T+ı−tm
t1+ı
Im4 (t)dt
](m.1.10)
nd shortage cost is
SCm = Sm
[∫ �+ı
�+tm
Im2 (t)dt +∫ �+ı+tm1
�+ı
Im3 (t)dt
](m.1.11)
.3.2. Case II: when � ≥ t1Here, the machine break down of production system occurs after the actual production run-time t1. So, the production system is not
ffected by the machine breakdown. The governing differential equation of inventory level in this case is
Im1 (t)dt
= P − Dr, with Im1 (0) = 0, 0 ≤ t ≤ t1 (m.2.1)
Im2 (t)dt
= −Dr, with Im2 (t1) = Im1 (t1), t1 ≤ t ≤ T (m.2.2)
sing the boundary conditions, we have from the Eqs. (m.2.1) and (m.2.2)
Im1 (t) = (P − Dr)t, 0 ≤ t ≤ t1 (m.2.3)
Im1 (t) = Pt1 − Drt, t1 ≤ t ≤ T (m.2.4)
nventory holding cost of produced items is
ICm3 = Chm
[∫ t1
0
Im1 (t)dt +∫ T
t1
Im2 (t)dt
](m.2.5)
orrective maintenance cost of the model dependent on maintenance time. Let us take as Cm. Production cost per unit items Cp.Combining the above cases, the average expected cost per unit items is
EMC = Per unit production cost + 1R
[corrective maintenance cost + expected inventory cost + expected shortage cost + set up cost]
= Cp + 1R
[Cm +∫ t1
0
{∫ tm
0
ICm1 h(ı)dı +∫ ∞
tm
ICm2 h(ı)dı}g(�)d� +∫ ∞
t1
ICm3 g[�]d� +∫ t1
0
∫ ∞
tm
SCmh(ı)g(�)dıd� + Am]
(m.1)
.4. The retailer’s inventory model
Here, the retailer sells the finished products with constant demand rate to the customers. He collects the products from the manufacturerith constant rate which is greater than the customers’ demand rate. When manufacturing system faces disruption for machine breakdown,
he retailer inventory may also be affected by that situation. As a result, retailer inventory may meet the shortages (see Fig. 4). We discusshis model by the following two cases.
.4.1. Case I: when 0 ≤ � ≤ t1Here, the manufacturing production system is affected by the machine breakdown. As a result, the retailer may faces the problem of
he shortages. The governing differential equation of inventory level in this case is
Ir1 (t)dt
= Dr − Dc, with Ir1 (0) = 0, 0 ≤ t ≤ P�
Dr(r.1.1)
B. Pal et al. / Journal of Manufacturing Systems 33 (2014) 262–276 269
U
Iil
4
4
Fig. 4. Logistic diagram of the model of retailer.
Ir2 (t)dt
= −Dc, with Ir2
(P�
Dr
)= Ir1
(P�
Dr
),
P�
Dr≤ t ≤ � + ı (r.1.2)
Ir3 (t)dt
= Dr − Dc, with Ir3 (� + ı) = Ir2 (� + ı), � + ı ≤ t ≤ T (r.1.3)
Ir4 (t)dt
= −Dr, with Ir4 (T) = Ir3 (T), T ≤ t ≤ T1 (r.1.4)
sing the boundary conditions, we have from the Eqs. (r.1.1)–(r.1.4)
Ir1 (t) = (Dr − Dc)t, 0 ≤ t ≤ P�
Dr(r.1.5)
Ir2 (t) = P� − Dct,P�
Dr≤ t ≤ � + ı (r.1.6)
Ir3 (t) = (Dr − Dc)t + P� − Dr(� + ı), � + ı ≤ t ≤ T (r.1.7)
Ir4 (t) = DrT + P� − Dr(� + ı) − Dct, T ≤ t ≤ T1 (r.1.8)
f the manufacturer’s delivery flow of products to the retailer is stopped through a time for the reason of the machine breakdown, thenventory of the products of the retailer can satisfy the demand of the customer through the time tr = (Ir1 (P�/Dr))/(Dc). If the inventoryevel of the retailer falls in shortages, then he takes time to recover that shortages tr1 = (Ir2 (� + ı))/(Dr − Dc).
.4.1.1. Subcase I: when P�Dc
≥� + ı, i.e. shortage does not occur. The inventory holding cost of selling items is
ICr1 = Chr
[∫ P�Dr
0
Ir1 (t)dt +∫ �+ı
P�Dr
Ir2 (t)dt +∫ T
�+ı
Ir3 (t)dt +∫ T1
T
Ir4 (t)dt
](r.1.9)
.4.1.2. Subcase II: when P�Dc
≤ � + ı, i.e. shortage does occur. The inventory holding cost of selling items is
ICr2 = Chr
[∫ P�Dr
0
Ir1 (t)dt +∫ P�
Dc
P�Dr
Ir2 (t)dt +∫ T
�+ı+tr1
Ir3 (t)dt +∫ T1
T
Ir4 (t)dt
](r.1.10)
2
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4
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I
C
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Nf
70 B. Pal et al. / Journal of Manufacturing Systems 33 (2014) 262–276
hortage cost is
SCr = Sr
[∫ �+ı
P�Dc
Ir2 (t)dt +∫ �+ı+tr1
�+ı
Ir3 (t)dt
](r.1.11)
.4.2. Case II: when � ≥ t1In this case, the machine break down of production system happens after the actual production run-time t1. So, the retailer does not
ace any disruption from the manufacturer. The governing differential equation of inventory level in this case is
Ir1 (t)dt
= Dr − Dc, with Ir1 (0) = 0, 0 ≤ t ≤ T (r.2.1)
Ir2 (t)dt
= −Dc, with Ir2 (T) = Ir1 (T), T ≤ t ≤ T1 (r.2.2)
sing the boundary conditions, we have from the Eqs. (r.2.1) and (r.2.2)
Im1 (t) = (Dr − Dc)t, 0 ≤ t ≤ T (r.2.3)
Im1 (t) = DrT − Dct, T ≤ t ≤ T1 (r.2.4)
nventory holding cost of produced items is
ICr3 = Chm
[∫ T
0
Ir1 (t)dt +∫ T1
T
Ir2 (t)dt
](r.2.5)
ombining the above cases and using Eqs. (r.1.9)–(r.1.11) and Eq. (r.2.5), the average expected cost per unit items is
ERC = 1R
[Finished product cost + Expected inventory cost + Expected shortage cost + set up cost]
= 1R
[wm +∫ t1
0
{∫ P�
Dc−�
0
ICr1 h(ı) dı +∫ ∞
P�Dc
−�
ICr2 h(ı) dı}g(�) d� +∫ ∞
t1
ICr3 g[�] d� +∫ t1
0
∫ ∞
P�Dc
−�
SCrh(ı)g(�)dıd� + Ar] (r.1)
. The integrated inventory model
The integrated total expected cost for supplier, manufacturer’s warehouse, manufacturer and retailer is
TEIC = ESC + EWC + EMC + ERC (i.1)
.1. When the random variables (supplier disruption (�), time to machine breakdown (�) and time to perform corrective maintenance (ı))ollow uniform distribution
Let us consider the probability density function of random variables
f (�) =
⎧⎨⎩1b
, 0 ≤ � ≤ b
0, elsewhere
g(�) =
⎧⎨⎩1d
, 0 ≤ � ≤ d
0, elsewhere
h(ı) =
⎧⎨⎩1a
, 0 ≤ ı ≤ a
0, elsewhere
ow, with the help of Eqs. (s.1), (w.1), (m.1) and (r.1), the integrated total expected cost for supplier, manufacturer’s warehouse, manu-
acturer and retailer isTEIC(R) = ESC + EWC + EMC + ERC = U0 + U1R + U2R2 + U3R3 + U4R4 + U51R
(u.1)
w
P
P
T
PR
P
6
rom
B. Pal et al. / Journal of Manufacturing Systems 33 (2014) 262–276 271
here
U0 = Cr + C ′r + c2Dr(4Dr − P)Sm
6d(P − Dr)P+ C2(Chr(4D2
c − D2r ) + SrDc(Dr − 4Dc))
6d(Dc − Dr)P− C2ChmD2
r
2dP(P − Dr)− 1
24 bcDcD3r P4(Dr − Dc)
U1 = 14dDc(Dr − Dc)DrP2
[R[CDr{Chr(4D2c (P − Dc) − 2D2
r + DcDr(Dr + P)) + Dc(Chm(Dr − Dc)(3Dr − P) + 4D2c Sr + Dr(P(1 + Sm + Sr)
−4DrSm) − Dc(Dr(Sr − 4Sm) + P(1 + Sm + 4Sr)))} + 2d(Dr − Dc)P{ChsDcDr + ChmDc(P − Dr) + P(Chr(Dr − Dc) + 2DcDr(ws + �))}]]
U2 = 1
6bdDc(Dr − Dc)D2r P3
[b{ChrDr(D2c P(8Dr + P) − 4D2
c Dr + D2r P(4P − 3Dr) + DcDr(D2
r + DrP − 8P2)) + Dc(3Chm(Dr − Dc)D2r (P − Dr)
+D2r (4D2
r Sm + DrP(3Chs − 1 − 5Sm − 2Sr) + P2(Sm − Sr)) + D2c (P2 − 4D2
r )Sr + DcDr(D2r (Sr − 4Sm)) − P2(Sm + 3Sr)
+DrP(1 − 3Chs + 5Sm + 8Sr))} + dDc(Dr − Dc)D2r P{2 − Chs + 3P(w′
s − ws)}]
U3 = 1
24cdDc(Dr − Dc)D3r P4
[cDc(Dr − Dc)D3r (1 + (4Chs − 5)P) + b{Chr(4Dc − Dr)(P − Dr)(3D2
r P2 − 3DcDrP(P + Dr)
+D2c (D2
r + DrP + P2)) + Dc(3Chm(Dr − Dc)D2r (P − Dr)2 − D2
c P(12D2r + 12DrP + P2)Sr − D2
r (4D3r Sm − D2
r P(9Sm − 1)
+P3(6Sr − Sm) + 6DrP2(Sm + Sr)) + DcDr(4D3r Sm + P3(4Sr − Sm) + D2
r P(1 − 3Sr − 9Sm) + 3DrP2(2Sm + 9Sr)))}]
U4 = 120bcdP4
, i.e. U4 > 0
U5 = Am + Ar + As + Cm + L + wm, i.e. U5 > 0.
Now, we check the optimality of per unit expected integrated cost with respect to ordering lot size.Differentiating Eq. (u.1) with respect to R, we have
d TEIC(R)dR
= U1 + 2U2R + 3U3R2 + 4U4R3 − U51R2
(u.2)
roposition 5.1. The expected integrated per unit cost function of the supply chain is convex if 2U2R3 + 6U3R4 + 12U4R5 + 2U5 ≥ 0
roof. Differentiating Eq. (u.2) with respect to R, we have
d2TEIC(R)
dR2= 2U2 + 6U3R + 12U4R2 + 2U5
1R3
(u.3)
herefore, expected integrated per unit cost function of the supply chain is convex if d2TEIC(R)dR2 ≥0 i.e. if 2U2R3 + 6U3R4 + 12U4R5 + 2U5 ≥ 0.�
roposition 5.2. The (u.2) have unique solution in [0, B] if 2U2R3 + 6U3R4 + 12U4R5 + 2U5 ≥ 0 and U1B2 + 2U2B3 + 3U3B4 + 4U4B5≥U5 for ∈ [0, B].
roof. From the Eq. (u.3), we have (d2 TEIC(R))/(dR2) ≥ 0 within [0, B] if 2U2R3 + 6U3R4 + 12U4R5 + 2U5 ≥ 0 within [0, B].Hence, (d TEIC(R))/(dR) is monotonically increasing within [0, B].Again, R→ 0 ⇒ (U1R2 + 2U2R3 + 3U3R4 + 4U4R5 − U5) → negative number as U5 > 0 .Also, R → B ⇒ (U1R2 + 2U2R3 + 3U3R4 + 4U4R5 − U5) → U1B2 + 2U2B3 + 3U3B4 + 4U4B5 − U5.Therefore, Eq. (u.2) have unique solution within [0, B] if U1B2 + 2U2B3 + 3U3B4 + 4U4B5≥U5.�
. Stakelberg approach
Now, we study the model in such a way that manufacturer is the leader and other members of the model (supplier, warehouse andetailer) are the follower of manufacturer i.e. other member of the model obey the decision of manufacturer. The per unit expected costf the manufacturer where the random variables (supplier disruption (�), time to machine breakdown (�) and time to perform correctiveaintenance (ı)) follow uniform distribution which mention in above Section 5.1, is
EMC(R) = Um0 + Um1R + Um2R2 + Um3R3 + Um41R
(u.m.1)
2
w
N
P
P
T
P
P
7
7
pCutcccewEi
7
qopo
siSr
72 B. Pal et al. / Journal of Manufacturing Systems 33 (2014) 262–276
here
Um0 = c2Dr(4Dr − P)Sm(P − Dr)
6d(P − Dr)2− c2ChmD2
r
2d(P − Dr)P+ ws
Um1 = Chm(P − Dr)2DrP
− cChm(3D2r − 4DrP + P2)
4d(P − Dr)P2− c(P − Dr)2(4Dr − P)Sm
4d(P − Dr)2+ w′
s
2bP− ws
2bP
Um2 = −(
Chm(P − Dr)2dP3
+ (P − Dr)3(4Dr − P)Sm
6dDr(P − Dr)P3
)≤ 0 if P ≤ 4Dr
Um3 =(
Chm(P − Dr)2
8cdDrP4− (P − Dr)2(4Dr − P)Sm
24cdD2r (P − Dr)2P4
)Um4 = Am + Cm + L, i.e. Um4 > 0.
ow, we check the optimality of expected per unit cost of manufacturer with respect to ordering lot size.Differentiating Eq. (u.m.1) with respect to R, we have
d EMC(R)dR
= Um1 + 2Um2R + 3Um3R2 − Um41R2
(u.m.2)
roposition 6.1. The expected cost function of the leader (manufacturer) is convex if 2Um2R3 + 6Um3R4 + 2Um4 ≥ 0.
roof. Differentiating Eq. (u.m.2) with respect to R, we have
d2EMC(R)
dR2= 2Um2 + 6Um3R + 2Um4
1R3
(u.m.3)
herefore, expected per unit cost function of manufacturer is convex if (d2 EMC(R))/(dR2) ≥ 0 i.e. if 2Um2R3 + 6Um3R4 + 2Um4 ≥ 0.�
roposition 6.2. The (u.m.2) have unique solution in [0, B] if 2Um2R3 + 6Um3R4 + 2Um4 ≥ 0 and Um1B2 + 2Um2B3 + 3Um3B4≥Um4 for R ∈ [0, B].
roof. From the Eq. (u.m.3), we have d2EMC(R)dR2 ≥0 within [0, B] if 2Um2R3 + 6Um3R4 + 2Um4 ≥ 0 within [0, B].
Hence, dEMC(R)dR is monotonically increasing within [0, B].
Again, R→ 0 ⇒ (Um1R2 + 2Um2R3 + 3Um3R4 − Um4) → negative number as Um4 > 0 .Also, R → B ⇒ (Um1R2 + 2Um2R3 + 3Um3R4 − Um4) → Um1B2 + 2Um2B3 + 3Um3B4 − Um4.Therefore, Eq. (u.m.2) have unique solution within [0, B] if Um1B2 + 2Um2B3 + 3Um3B4≥Um4.�
We provide the following flow chart for the proposed method of our model (Chart 1.).
. Numerical analysis
.1. Example I
We study an inventory model with the following characteristic.The unit purchase cost of raw material of supplier is Cr = $30, the unit selling price of raw material of supplier is ws = $35, the unit
urchase cost of raw material of manufacturer’s warehouse is C ′r = $40, setup cost of supplier As = $650, inventory holding cost of supplier
hs = $1.5 per unit per unit time, inventory holding cost of warehouse Chw = $2.5 per unit per unit time, inventory holding cost of man-facturer Chm = $3.5 per unit per unit time, the unit selling price of finished product of manufacturer is wm = $80, fixed cost like labour,echnology, energy, etc for manufacturing items per cycle L = $6000, unit shortage cost of manufacturer Sm = $12, corrective maintenanceost Cm$10 per cycle, production rate P =10,000, setup cost of manufacturer Am = $700, demand rate of retailer Dr = 6000, inventory holdingost of retailer Chr = $4.5 per unit per unit time, unit shortage cost of retailer Sr = $15, setup cost of retailer Ar = $800 and demand rate ofustomer Dc = 4000, b = 10, d = 12, c = 8. Using the data, we have the optimal order quantity of the supply chain R* = 19034.64 units, optimalxpected safety stock S = 1811.59 unit, optimal per unit expected cost of the supplier ESC = $31.16, optimal per unit expected cost of thearehouse EWC = $4.99, optimal per unit expected cost of the manufacturer EMC = $47.95, optimal per unit expected cost of the retailer
RC = $14.01, optimal per unit integrated expected total cost TEIC = $98.11 and optimal cycle length of the chain T1 = 4.76 unit. This solutions a global minimum for the Eq. (u.1) (see Fig. 5).
.2. Example II
When manufacturer are leader and other character are follower, then we study the model using above data: we have the optimal orderuantity R* =59,018.50 units, optimal expected safety stock S =17,415.90 unit, optimal per unit expected cost of the supplier ESC = $31.82,ptimal per unit expected cost of the warehouse EWC = $17.41, optimal per unit expected cost of the manufacturer EMC = $46.67, optimaler unit expected cost of the retailer ERC = $10.96, optimal per unit integrated expected total cost TEIC = $106.87 and optimal cycle lengthf the chain T1 = 14.75 unit.
From numerical results, we have observed that the integrated expected per unit cost is less than to the expected cost per unit of the
upply chain by Stakelberg approach. Also, the per unit expected cost of the supplier and warehouse is higher in Stakelberg approach thanntegrated case but the per unit expected cost of the manufacturer and the retailer is less in Stakelberg approach than integrated case.o, Stakelberg approach is less profitable for integrated approach, supplier and warehouse but more profitable for the manufacturer andetailer.
B. Pal et al. / Journal of Manufacturing Systems 33 (2014) 262–276 273
Chart 1. Flow chart of optimal solution.
274 B. Pal et al. / Journal of Manufacturing Systems 33 (2014) 262–276
Fig. 5. Expected integrated cost versus ordering lot-size.
Table 1Sensitivity of Example I.
Parameter value R* S* ESC* EWC* EMC* ERC* TEIC* Cycle time (T∗1 )
Chs
0.750 20357.20 2072.09 30.62 5.37 47.85 13.68 97.52 5.091.125 19703.00 1941.05 30.89 5.18 47.90 13.83 97.81 4.931.875 18353.30 1684.22 31.40 4.80 47.99 14.17 98.37 4.582.250 17660.60 1559.49 31.63 4.60 48.05 14.34 98.64 4.42
Chw
1.125 21816.70 2379.83 31.27 5.07 47.75 13.35 97.45 5.451.187 20348.90 2070.38 31.21 5.04 47.85 13.68 97.78 5.093.125 17852.02 1593.47 31.11 4.93 48.04 14.29 98.38 4.463.750 16783.60 1408.45 31.06 4.88 48.13 14.59 98.64 4.20
Chm
1.750 23632.70 2792.53 31.33 6.30 50.06 12.98 100.68 5.912.625 21280.90 2264.39 31.25 5.63 49.08 13.47 99.43 5.324.375 16912.70 1430.20 31.07 4.40 46.67 14.55 96.68 4.235.250 14941.10 1116.19 30.98 3.86 45.25 15.12 95.20 3.74
Chr
2.250 24924.60 3106.19 31.38 6.68 47.56 13.05 98.67 6.233.375 21918.50 2402.10 31.28 5.81 47.74 13.62 98.45 5.485.635 16303.60 1329.04 31.04 4.23 48.17 14.17 97.61 4.086.750 13791.10 950.97 30.92 3.54 48.40 14.14 97.00 3.45
Sm
6 12740.40 811.59 30.87 3.260 38.97 15.81 88.91 3.199 15708.80 1233.83 31.01 4.07 43.62 14.89 93.59 3.93
15 22554.10 2543.44 31.30 5.99 51.91 13.20 102.40 5.6418 26163.10 3422.54 31.42 7.04 55.53 12.52 106.52 6.54
Sr
7.50 7452.90 277.73 30.60 1.86 49.27 6.97 88.70 1.8611.25 12265.50 752.21 30.85 3.13 48.57 11.25 93.79 3.0618.75 25030.60 3132.65 31.39 6.71 47.55 16.04 101.69 6.2622.50 29930.50 4479.18 31.53 8.16 47.30 17.81 104.80 7.48
b
5.0 8937.21 798.74 30.66 4.10 49.23 17.18 101.17 2.237.5 14599.80 1421.03 30.94 4.792 48.44 15.22 99.40 3.65
12.5 22272.80 1984.31 31.31 4.95 47.61 13.26 97.12 5.5715.0 24703.50 2034.21 31.41 4.82 47.36 12.78 96.38 6.18
d
6 52739.90 13907.50 30.41 17.80 50.80 11.76 110.77 13.189 30766.00 4732.74 31.38 8.89 49.82 13.87 103.97 7.69
15 11959.40 715.13 30.85 2.94 46.33 13.32 93.44 2.9918 8554.61 365.91 30.67 2.02 45.05 12.12 89.86 2.14
c
4 5672.11 160.86 30.51 1.29 39.86 4.74 76.40 1.426 9319.77 434.29 30.69 2.25 43.37 9.44 85.76 2.33
10 34069.10 5803.53 31.64 9.77 53.03 17.67 112.10 8.5212 52073.20 13558.10 31.84 16.22 58.23 20.79 127.07 13.02
8
i
•
•
•
•
•
•
•
•
•
9
atppswsttpwahaoim
A
tS
R
B. Pal et al. / Journal of Manufacturing Systems 33 (2014) 262–276 275
. Sensitivity analysis and managerial insights
We study the changes of optimal values (Table 1) of the variables from changes of the parameters. The following features and managerialnsights are observed.
With the increasing value of the holding cost (Chs) of the supplier, the optimal ordering lot size of the supplier, safety stock level inmanufacturer’ warehouse, per unit expected cost of the warehouse and cycle time decreases. The per unit expected cost of the supplier,manufacturer, retailer and integrated cost increases with the higher value of Chs.The optimal ordering size for the supplier, per unit expected cost of the supplier and warehouse, safety stock level in manufacturer’warehouse and cycle time decreases but per unit expected cost of the manufacturer, retailer and integrated case increases with theincreasing value of the holding cost (Chw) of the warehouse.When the holding cost (Chm) of the manufacturer is increased, the optimal ordering size for the supplier, per unit expected cost of thesupplier, warehouse, manufacturer and integrated case, safety stock level in manufacturer’ warehouse and cycle time decreases but theper unit expected cost for the retailer increases. As, the decreasing rate of the optimal ordering size for the supplier and safety stock levelfor the warehouse is too higher, so, the expected cost of safety stock, the maintenance cost of machine, shortages cost decrease. Hence,the per unit expected cost of the manufacturer and integrated case are decreased with increasing value of Chm.With the increasing value of the holding cost (Chr) of the retailer, the optimal ordering size for the supplier, per unit expected cost ofthe supplier, warehouse and integrated case, safety stock level in manufacturer’ warehouse and cycle time decreases but the per unitexpected cost for the manufacturer and retailer increases.The optimal ordering size for the supplier, per unit expected cost of the supplier, warehouse, manufacturer and integrated case, safetystock level in manufacturer’ warehouse and cycle time increases but the per unit expected cost for the retailer decreases with the highervalue of per unit per unit time shortages cost (Sm) for the manufacturer.When the per unit per unit time shortages cost (Sr) for the retailer is increased, the optimal ordering size for the supplier, per unitexpected cost of the supplier, warehouse, retailer and integrated case, safety stock level in manufacturer’ warehouse and cycle timeincreases but the per unit expected cost for the manufacturer decreases.With the increasing value of the uniform distribution parameter (b) for the supplier, the optimal ordering size for the supplier, per unitexpected cost of the supplier and warehouse, safety stock level in manufacturer’ warehouse and cycle time increases but the per unitexpected cost for the manufacturer, retailer and integrated case decreases.The optimal ordering size for the supplier, per unit expected cost of the warehouse, manufacturer and integrated case, safety stocklevel in manufacturer’ warehouse and cycle time decreases with the higher value of the uniform distribution parameter (d) for machinebreakdown scheduling.When the uniform distribution parameter (c) for the time to perform corrective maintenance is increased, the optimal ordering sizefor the supplier, per unit expected cost of the supplier, warehouse, manufacturer, retailer and integrated case, safety stock level inmanufacturer’ warehouse and cycle time increases.
. Conclusion
In the multi-echelon production inventory model, we have investigated the optimal ordering lot-size for the integrated expected costnd manufacturer Stakelberg approach of the model where supplier, manufacturer’s warehouse, manufacturer and retailer are present inhe chain. In Stakelberg approach, manufacturer is the decision maker and other members are the followers of the manufacturer. In theresent model, supply disruption may be occurred after a random time by different reasons (transportation problem, labor strick, naturalroblem, etc.) and supply of raw materials by supplier be stopped by the disruption. At beginning, warehouse of manufacturer stores safetytock of raw material avoiding the effect of supply disruption within production run-time. Also, the warehouse stores the raw materialhen the production system breakdown be happened before supply disruption within actual regular production run-time. The production
ystem starts again after a random recovery time from the time of breakdown. For the effect of machine breakdown of production system,he inventory of finished good of manufacturer and retailer may fall into shortage. We compare the integrated expected cost model ofhe chain to the Stakelberg model where manufacturer is leader and other are follower. We also investigate the sensitivity of the keyarameters for the model. The major contribution of the supply chain are introducing supply disruption, safety stock of manufacturer’sare house and production system breakdown in a multi-echelon supply chain. Also, we consider shortage of inventory of manufacturer
nd retailer which is another important contribution. These new contributions fill up the gap of the existing literature. The present modelas many limitations. These are one time disruption of supply and machine break down, deterministic production rate and supply ordernd deterministic demand of the end customers. The model can be extended relaxing the above limitations in many ways: We could studyur model immediately with stochastic market demand or price dependent market. It might be interesting to consider trade credit policy,nflation or different contracts among the members. Also we can extend the model with multiple items. Moreover, multi-supplier and
ulti-retailer levels may also be introduced in our model.
cknowledgments
The authors would like to express their gratitude to the editors and referees for their valuable suggestions and corrections to enhancehe clarity of the present article. The first author also acknowledges the department of mathematics, Jadavpur University and Council ofcientific & Industrial Research, Govt. of India for financial assistance.
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