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Frontiers of Engineering Management

Front. Eng    2020, Vol. 7 Issue (2) : 248-258     https://doi.org/10.1007/s42524-019-0069-5
RESEARCH ARTICLE
Two-level uncapacitated lot-sizing problem considering the financing cost of working capital requirement
Yuan BIAN1(), David LEMOINE2, Thomas G. YEUNG2, Nathalie BOSTEL3
1. School of Economics and Management, University of Chinese Academy of Sciences, Beijing 100049, China
2. LS2N UMR CNRS 6004, IMT Atlantique, Nantes 44300, France
3. LS2N UMR CNRS 6004, University of Nantes, Saint-Nazaire 44606, France
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Abstract

During financial crisis, companies constantly need free cash flows to efficiently react to any uncertainty, thus ensuring solvency. Working capital requirement (WCR) has been recognized as a key factor for releasing tied up cash in companies. However, in literatures related to lot-sizing problem, WCR has only been studied in the single-level supply chain context. In this paper, we initially adopt WCR model for a multi-level case. A two-level (supplier–customer) model is established on the basis of the classic multi-level lot-sizing model integrated with WCR financing cost. To tackle this problem, we propose sequential and centralized approaches to solve the two-level case with a serial chain structure. The ZIO (Zero Inventory Ordering) property is further confirmed valid in both cases. This property allows us to establish a dynamic programming-based algorithm, which solves the problem in O(T4). Finally, numerical tests show differences in optimal plans obtained by both approaches and the influence of varying delays in payment on the WCR of both actors.

Keywords two-level ULS problem      lot-sizing      working capital requirement      ZIO property      infinite production capacity     
Corresponding Author(s): Yuan BIAN   
Just Accepted Date: 17 December 2019   Online First Date: 17 January 2020    Issue Date: 27 May 2020
 Cite this article:   
Yuan BIAN,David LEMOINE,Thomas G. YEUNG, et al. Two-level uncapacitated lot-sizing problem considering the financing cost of working capital requirement[J]. Front. Eng, 2020, 7(2): 248-258.
 URL:  
http://journal.hep.com.cn/fem/EN/10.1007/s42524-019-0069-5
http://journal.hep.com.cn/fem/EN/Y2020/V7/I2/248
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Yuan BIAN
David LEMOINE
Thomas G. YEUNG
Nathalie BOSTEL
Fig.1  Physical  and financial flows in a two-level supply chain.
Parameters
T Number of periods
N
dit
Number of items
Customer’s demand for item i at period t
aij Gozinto coefficient
hi Inventory holding cost for item i
si Setup cost for item i
Ii0 Initial inventory for item i
M Big number
Decision variables
Xit Production quantity for item i at period t
Iit Inventory for item i at the end of period t
Yit Binary variable that indicates whether a setup for item i occurs at period t
Tab.1  Notation for MLLP 
Fig.2  Illustration  of WCR in the context of a lot-sizing problem (Bian et al., 2018).
Parameters
T Number of periods
dit Customer’s demand for site i at period t
vi Unit product price for item i
ai Unit raw material cost for item i
hi Inventory holding cost for item i
si Fixed setup cost for item i
pi Unit production cost for item i
ri Delay in payment from site i to site i-1
Li Delivery delay from site i-1 to site i
αi Discount rate per period of site i
bi Interest rate for financing WCR of site i
Decision variables
Qit Total production quantity at site i in period t
Xitk Production quantity in period t for satisfying (a part of) demand in period k of site i
Iit Inventory for item i at the end of period t
Yit Binary variable, which indicates whether a setup for item i occurs at period t
Tab.2  Notation for the 2ULS P(WCR)
Fig.3  A node  and associated arcs in Zangwill’s network presentation of the multi-level production problem.
Fig.4  Centralized  approach for 2ULSP(WCR).
Algorithm 1 Solving 2ULSP(WCR)
Require: All parameter values
for k-1 to T do
for t=0 to k—1 do
for l=t+1 to k do
for q=tto k—1 do
if????Op tkl 1> Optk q1+Et kql1??? ?then
Op tkl1>O ptkq1+ Etkql1
end if
end for
end for
if?????Cost[k]>Cost[t] +Etk0??? ?then
Cost [k] =Cost [t]+ Etk0
end if
end for
end for
Tab.3  
Parameter S M Parameter S M
vi 35 50 hi 1 2
pi 3 3 αi 0.05 0.01
si 800 600 bi 0.03 0.03
ai 3 35 r2 = r1 =2, r0 = 1
Tab.4  Parameter values for the 24-period instance 
Fig.5  Comparison  of the optimal production programs of the manufacturer with different approaches.
Fig.6  Comparison  of the optimal production programs of the supplier through different approaches.
Approaches Supplier Manufacturer
Total products held (in unit/period)
MLLP 670 3085
Sequential 1840 1230
Centralized 1005 2065
Number of setups
MLLP 5 6
Sequential 6 10
Centralized 6 8
Tab.5  Comparison of the total products held in inventory  over time and of the number of setups
Varying parameter DOV DTC1 DProfit0
r2 59979.8 60742.7 762.9
r1 -31035.1 -61904.4 -30869.3
Tab.6  Consequences of varying delays in payment on  financial terms
Fig.7  Comparison  of production plan with varying delays in payment in the centralized case.
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