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
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.
. [J]. Frontiers of Engineering Management, 2020, 7(2): 248-258.
Yuan BIAN, David LEMOINE, Thomas G. YEUNG, Nathalie BOSTEL. Two-level uncapacitated lot-sizing problem considering the financing cost of working capital requirement. Front. Eng, 2020, 7(2): 248-258.
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
Fig.2
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
Fig.3
Fig.4
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
end if
end for
end for
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
Fig.5
Fig.6
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
Varying parameter
DOV
DTC1
DProfit0
r2
59979.8
60742.7
762.9
r1
-31035.1
-61904.4
-30869.3
Tab.6
Fig.7
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