AN EFFICIENT EXACT METHOD FOR MULTILEVEL ...

AN EFFICIENT EXACT METHOD FOR MULTILEVEL SERIAL SYSTEMS UNDER UNCERTAINTIES OF LEAD TIMES AND FIXED DEMAND F. Hnaiena* and A. Dolguib ICD-LOSI, University of Technology of Troyes, UMR STMR-CNRS.12 rue Marie Curie, CS42060, 10004 Troyes Cedex, France [email protected]

a

b

Ecole Nationale Superieure des Mines, CNRS UMR 6158 LIMOS, 158, cours Fauriel, 42023 Saint-Etienne Cedex 2, France [email protected] ABSTRACT

The planned lead time analysis is considered for a multilevel serial supply chain with unlimited number of levels and random component lead times for each level. The demand of the finished product is assumed to be fixed and known. The lead time of each component (delivery time for the next level) is supposed to be a discrete random variable. No restrictive hypothesis is made on such random variables; only that the probability distributions are known. The holding as well as tardiness costs are considered. Tardiness costs can be due to the cost of revising a schedule. For the first level (level 1), corresponding to the finished product, tardiness means backlog, so for the finished product backlogging cost is introduced. Thus, the problem is to minimize the sum of expected holding, tardiness and backlogging costs. The decision variables are the planned lead times for components at each level. A similar multilevel production system (supply chain) was already studied in literature. However, in that case, the lead times of components were continuous random variables. In this paper we consider a discrete model with no restriction on the number of levels and our model offers the expression of objective function in a closed form. In addition, a lower bound of the objective function is provided and a Branch and Bound algorithm is developed to calculate the optimal planned lead times. The proposed Branch and Bound can solve until 10 levels. Two iterative heuristics based on the applying of the Branch and Bound iteratively are proposed and results show the effectiveness of theses heuristics compared to the solution given by Newsvendor model. Keywords: Inventory control, Stochastic lead time, Serial system, Nonlinear optimization 1

INTRODUCTION

In literature on supply planning, an extensive state of the art on the supply planning under uncertainties is given in (Dolgui and Prodhon [4] and Dolgui et al.[5]). The effects of lead time uncertainty are particularly problematic in multilevel production systems (see Bullwhip effect, Chatfield et al.,[1] and Chen et al. [3]). To minimize the influence of these random lead times, firms can implement safety lead times. The safety lead time is defined as the difference between the planned and expected lead

times. Nevertheless, excess the safety lead times creates stocks and stocks are expensive. Therefore, the problem is to optimize stock at each level by assigning adequate values of planned lead times. Weeks [9]) developed a one-stage model with tardiness (Tc) and holding costs (Hc) in which the processing time is stochastic and demand is deterministic. The author has proven that the problem is equivalent to the standard Newsvendor problem. Yano [9] used an analytic approach to determine optimal planned lead times in serial production systems in which the actual procurement and processing times may be stochastic and demand is deterministic and a lot-for-lot policy is used. The considered cost is the sum of Hc and job tardiness costs (Tc). The author only studied two and three stage serial systems due to the difficulties of the model. One of the main challenges for the model was to express the objective function in closed form when the number of stage exceeds two stages. To surmount the difficulty to express the closed form for the objective function to the problem presented by Yano [10], Elhafsi [6] developed a recursive scheme that evaluates the objective function efficiently for any number of stages in the production system without recurring to express it in closed form. Due to the convexity of objective function, any classical non-linear optimization algorithm can be used. However, for a large N the computer processing time CPU increases relatively quickly as the number of stages in the system increases. This is due to the polynomial complexity of numerical recursive scheme. To overcome this difficulty, author presents a heuristic based on using only a subset of adequate number of production stages for which the optimal planned lead time can be quickly obtained. Charharsooghi and Heydari [2] analyse the performance of a multi-level linear structure. To measure system performance studies, the authors used a simulation to model different combinations of input variables (lead time and supply variances) by indicators: storage, backlog, size, number of stock and the bullwhip effect. These indicators are dependent. Measuring the effect of all independent variable on all dependent variables is possible using the canonical correlation model, which eliminates the impact of intervening variables. According to the results, the authors showed that the effect of the variance of lead time performance is greater than the effect of mean of lead time. Wazed et al. [8] have investigated the effects of lead time and machine breakdown in a manufacturing environment. The authors considered the case of an electrical and electronic company located in Singapore. The production is characterized by multiple end items for multi-period and multi-stage dependent demand. A simulation is used to analyze the situations of the production lines. The authors observed that inclusion of common components in the manufacturing system is generally beneficial over the non-commonality environment in uncertain situations. Impacts of machine breakdown on system outcomes are higher than that of the lead time variation. The combination of uncertain factors has more impact on throughput and average production time compared to the individual factor. Apparently nowhere in literature has supply planning for the serial system under uncertainty of discrete lead times been studied before. Authors have considered continuous lead times where the distribution probability is either known or unknown. However, in industrial world, there are many manufacturing control systems that use discrete lead time data.

Consequently it is necessary to investigate this research void and we invite any of our challenge to join us in this endeavour. 2

PROBLEM DESCRIPTION

We consider a serial production system with m levels (see Figure 1). We suppose that the demand D of the finished product is fixed and its due date is known. To satisfy this demand, we need to launch the production processes composed of m serial stages (m levels) for a lot of D items. The level numbers are enumerated as follows: level m corresponds to the first production stage, level m−1 to the second stage, and so on. The raw materials are released at level m, semi- finished products are processed at levels m−1, m−2,...,2 and finally, finished product is produced at level 1. After these m levels, the lot of D finished products is delivered to the customer. We assume also that the lead time at each level (delivery time for next level) is a discrete random variable. No restrictive hypothesis is made on such random variables; we only suppose that the distribution probabilities are known. The policy is the lot-for-lot for all levels. Level m delivers the semi-finished products to level m−1 within a random lead time

Lm , level j delivers the semi-finished products to level j−1 within a random lead time L j , j=2,...,m. When the items arrive at the end of level 1, the customer demand D of finished products is satisfied. There are stocks at each intermediate level (from level m to level 1). If total lead time exceeds the planned lead time of the component at level j, j=1,2,...,m, tardiness is incurred and therefore the corresponding tardiness costs for level m to level 2. For level 1 this is called backlogging cost and corresponds to the finished product backlog. Otherwise, we obtain stocks and therefore corresponding holding cost for each level. Hence, the objective is to minimize the total cost composed of the holding, tardiness and backlogging costs. The following notations are introduced:

Table 1: Notations Notations

Definitions

j

level index

m

number of levels

cj

component at level j

D

fixed and known demand of finished products per period

dj

number of components j needed at level j  1

Qj  d jD

lot size for components j

bj

unit backlogging cost for finished product per period

bj

unit tardiness cost for component j , j  2,..., m , per period

hj

unit holding cost for component j per period

Lj

lead time of the component j (discrete random variable)

Lj ( x j 1 , x j  2 ,..., xm )

cumulative lead time of level

j (proper lead time plus

delays due to level j  1 )

ul j

upper limit of L j , i.e., L j  ul j

ll j

lower limit of L j , i.e., ll j  L j

F j (k )  Pr( L j  k )

cumulative distribution function of L j

 j ( k , x j 1 ,..., xm )  Pr( Lj  k )

cumulative distribution function of Lj ( x j 1 , x j  2 ,..., xm )

E (.)

mathematical expectation operator

xj

planned lead time for component j (integer decision variable)

Let the lead time for components of level j be a random discrete variable with known distribution: Pr( L j  k ) , k  1,2,..., ul j , where ul j is the maximum planned lead time value, for j  1,..., m . The lead time takes into account all processing times at the level j plus transportation time between level j and j  1 . The actual cumulative lead time of the level j is given in (1):

Lj ( x j1 , x j 2 ,..., xm )  L j  (Lj 1 ( x j 2 , x j 3 ,..., xm )  x j 1 )  , for j  1,2,...,m - 1   L  L  m m

3

(1)

MATHEMATICAL MODEL

Proposition 1: The nonlinear integrated model can be written as follows: m

Min C ( X , L ) 

 Q j h j ( x j  Lj ( x j 1, x j  2 ,..., xm ))  j 1 m



  Q j (b j  h j )( Lj ( x j 1, x j  2 ,..., xm )  x j ) 



(2)

j 1

where,

X  ( x1 ,..., x m ) ;

 ) L'  ( L1 ,..., Lm

Proof. The cost is equal to the sum of the component holding, tardiness as well as finished product backlogging costs. If at a certain level, a job is completed before its planned due date, i.e. ( x j  Lj ( x j 1, x j  2 ,..., xm ))  0 , then a holding cost is incurring. There is a tardiness (respectively backlog) of components j (respectively finished product) if the total lead time exceeds the planned lead time of the component at level j, j = 1,2,..., m, i.e. ( Lj ( x j 1 , x j  2 ,..., xm )  x j )  0 .

 ) is random variable. The cost C ( X , L ) is a random variable because L'  ( L1 ,..., Lm Therefore, we will optimize the mathematical expectation of C ( X , L ) noted EC (X ) .

Proposition 2: The expected cost can be expressed as follows: m   EC( X )  Q j h j ( j (x j  k 1, x j 1..., xm ))  b j (1  j (x j  k, x j 1..., xm )) (3) j 1  k 0 k 0 

where, k   j (k , x j 1,..., xm )   Pr(L j  s)  j 1( x j 1  k  s, x j  2 ,..., xm ), for j  1,..., m  1  s 1   Fm (k ) m (k )

(4)

Proof. From relation (2), we derive the total expected cost: m

EC ( X )  E (C ( X , L' )) 

 Q j h j E Y   b j E Z  j 1

Where:

Z  ( Lj ( x j 1 , x j  2 ,..., xm )  x j )  Y  ( x j  Lj ( x j 1, x j  2 ,..., xm ))  Z is a positive discrete random variable with a finite number of possible values, its mathematical expectation is: i 1

E ( Z )   i Pr( Z  i)   i 0

 Pr(Z  i)    Pr(Z  i)   Pr( Z  k )  1  Pr(Z  k )

i 0 k 0

k  0i  k

k 0

k 0

Thus, we obtain:

E (Z ) 

 1  Pr( Lj ( x j 1 , x j 2 ,..., xm )  x j  k )   1   j ( x j  k , x j 1 ,..., x m ) k 0

k 0

where,

 j ( k , x j 1..., xm )  Pr( Lj ( x j 1 , x j  2 ,..., xm )  k )  Pr( L j  ( L j 1 ( x j  2 ,..., x m )  x j 1 )   k ) k



  Pr(L j  s)  Pr ( L j 1 ( x j  2 ,..., x m )  x j 1 )   k  s s 1



k



 Pr(L j





 s )  Pr L j 1 ( x j  2 ,..., xm )  x j 1  k  s  Pr k  s  0 

s 1

But, k  s  0 , thus: k

 j (k , x j 1 ..., xm ))   Pr(L j  s) j 1 ( x j 1  k  s, x j  2 ,..., xm ) s 1

Y is a positive discrete random variable with a finite number of possible values, its mathematical expectation is: E (Y ) 

 x j  Pr( Lj ( x j 1, x j  2 ,..., xm )  k     j ( x j  k  1, x j 1,..., xm ) k 0

k 0

Finally: m

EC ( X ) 

j 1

4





 Q j h j  ( j (x j  k  1, x j 1..., xm ))  b j  (1   j (x j  k , x j 1..., xm )) 

k 0



k 0

OPTIMIZATION METHOD

4.1 Solution space of decision variables Proposition 3: The optimal values of decision variables are in the following m  1  x j  ul j   (ul s 1), solution space:  s  j 1  1  xm  ulm

 j  1,..., m  1

(5)

Proof. Given that the lead time of level m varies in 1 and ulm then: for all Thus, for all xm  ulm , Fm ( xm )  1 . s  0, y  xm  s xm  ulm : EC ( x1,..., x j ,..., xm 1, y)  EC ( x1 ,..., x j ,..., xm 1, xm )  s  hm  0 .

and

This proves that for all xm  ulm , we obtain an additional stock. In the same way we can m

easily prove by recurrence that:  j ( x j , x j 1..., xm )  1 if x j  ul j 

 (ul s 1) . Thus, we s  j 1

m

obtain an additional stock for all x j  ul j 

 (ul s 1) . s  j 1

4.2 Branch and Bound In the following, we present a branch-and-bound method for solving the problem based on upper bound and lower bounds in order to reduce the search space. 4.2.1 Newsvendor Upper Bound (UB) Newsvendor algorithm is known at optimal solution for one level serial system (Weeks(1981)). So apply a Newsvendor at each level starting by level m until level 1. At each level a Newsvendor is applied as follows: at each level i the planed lead time xi is equal to the smallest integer k verifying:

bi  F (k ) . bi  hi

Proof. The proof is obvious. The newsboy solution is a feasible solution and each feasible solution is an upper bound. 4.2.2 Lower bounds We use two lower bounds to stop the branch and bound execution if a solution with a total cost value equal or inferior to one of these. They are defined as follows: m   LB1  Q j h j ( j (ll j  k 1, ll j 1...,llm))  b j (1  j (ul j  k, ul j 1...,ulm ))  j 1  k 0 k 0 LB2  EC(FV)  Newsboy_ cost( NFV)

(6) (7)

Where:

FV : Fixed variable in branch and bound NFV : Non fixed variable in Branch and Bound Newsboy _ cos t    F j (x j  k  1)  b j j in NFV k  0

 (1  F j (x j )

(8)

k 0

Proof.  j ( x j  k , x j 1..., xm ) is an increasing function of x j , thus from equation (3) the first lower bound LB1 is obtained. The second lower bound is composed of two parts. The first part is the cost for the fixed variable in the Branch and bound using equation (3). The second part is the sum of newsboy cost for all non-fixed planned lead times (8). The second part is a lower bound for the non-fixed values because we relaxed the dependence between the levels. 4.2.3 Main procedure of Branch and Bound The planed lead times are arranged from level m until level 1. Therefore, each node of the branch and bound search tree is characterized by the following elements:  a level k: representing the decision variable xm k 1  a partial solution X k  ( xm ,..., xm  k 1 )

 a lower bound LB  max( LB1, LB2 ) The main steps of the Branch and Bound algorithm are provided as follows:  Initialization: The planned lead times take the best solution obtained from upper bound as the initial solution.  Branching: The exploration strategy used in the algorithm is a Depth Exploration. At this step, we have ulk  llk  1 possibilities, i.e. each planned lead time

xk can take value from llk to ulk .  Lower bounding: Compute the lower bound for the partial solution LB  max( LB1, LB2 ) . If the lower bound is greater than the initial solution, eliminate that node and all nodes beyond in the branch.  Upper bounding: If the value of the completed sequence is less than the initial solution, update it as the new solution. Otherwise, eliminate it.  Stopping rule: The branching is ended until all nodes have been visited. 5

COMPUTATIONAL EXPERIMENTS AND RESULTS

In this subsection, we present the results of our computational experiments for the Branch and Bound algorithm. The Branch and Bound algorithm was coded in the C++ language. The test problems were solved on an Intel Core i5-2520M with 2.50 GHz and 4GB of memory running on Linux operating system.

5.1 Problem data For the experiments, we generated 10 instances for every combination of parameters as follows:  The number of levels m was chosen in 5,10,15,20,25,30 .  The

lead

time

of



each

level j ,

L j is

a

discrete

variable

in

the



interval ll j  1; ul j  5 .  The inventory holding and tardiness (backlog) costs at each level were derived according to the following formulas: (8) hm  100 and hm  j  1.5hm  j 1 forall j  1,..., m  1

b j  10 hm forall j  1,..., m

(9)

5.2 Results and discussions

The different results are summarized in Tables 2-4. The following metrics are provided for each method: The computation time required to find the best solution, the number of nodes generated and the Gap from the UB (10). The program is terminated when the optimum is found or when the time limit of 1 hour (3600 seconds) was reached.

Gap_ UB 

UB EC( X *)

(10)

EC( X *)

Table2: Computational results for the branch-and-bound method Average usage of lower bound

LB1 (%)

LB2 (%)

Gap _UB (%)

Nodes

CPU (seconds)

5

33.37

66.63

0.51

10070

0

10

16.85

83.15

38.09

645364560

356

15

8.32

91.68

1.52

3407857960

3600

20

0.00

100.00

0.00

2062475070

3600

25

0.00

100.00

0.00

1407033180

3600

30

0.00

100.00

0.00

1030931390

3600

m

Results clearly indicate that the Branch and Bound method can solve instances with up to 10 levels. As we can see, the lower bound LB1 is almost responsible for retrieving the best solution. Table 3 summarizes the comparison result with two proposed heuristics based on the works of Elhafsi [6]:  K-Iterative Heuristic: we apply the Branch and Bound method iteratively each k level. We start by apply the Branch and Bound for level m until level m-k+1. After that we apply the Branch and bound for level m-k until m-2k+1 and taking into account the optimal planned lead time for the level m until level m-k+1,…  K-Elhafsi Heuristic: we apply the Branch and Bound method iteratively each k level. We start by apply the Branch and Bound for level m until level m-k+1. After that we apply the Branch and bound for level m-1 until m-k+2 and taking into account the optimal planned lead time for the level m. Then, m-2 until mk+3,… In the results proposed in table 3, k is chosen equal to 5 because the branch and bound can solve the problem of 5 levels very quickly (~=0 seconds).

Table3: Computational results for the two heuristics methods

IIterative Heuristic

Elhafsi Heuristic

Gap _UB (%)

Nodes

Gap _UB (%)

Nodes

10

37.58

1889820

38.00

1276840

15

94.59

4611840

97.91

8652870

20

112.46

8134820

117.98

18702220

25

112.54

12432920

117.53

29565840

30

98.75

15128829

104.80

38775900

m

Results clearly indicate that the Elhafsi Heuristic is better than the Iterative Heuristic in term of gap of upper bound (UB). This result is expected because the dependency between levels is better integrated into Elhafsi Heuristic. Another interest result is the Gap between the obtained solutions and the upper bound given by the Newsvendor model reached 117% for 20 levels. The results show the effectiveness of theses heuristics compared to the solution given by Newsvendor model. In addition, this shows the interest of this work and the Newsvendor solution is a so far from optimal. The CPU times, as illustrated in figure 1, the iterative algorithm has the minimum computation time due to the number to apply the Branch and Bound algorithm.

Figure 1: CPU times (seconds) for two heuristics

6

CONCLUSIONS AND PERSPECTIVES

The multilevel production systems under uncertainties of lead times have not been sufficiently studied. That was the motivation of this paper. Here, random discrete lead times for multilevel supply planning were studied. In contrast with the known approaches which are usually based on continuous time models, the suggested method uses discrete optimization techniques with integer decision variables. This is more appropriate for MRP environment where the planning horizon is divided into discrete periods (bucket times).The objective is to minimize the total cost introduced by the random lead times. The cost function was the sum of finished product backlogging, component holding and tardiness costs for all levels. A mathematical model for performance evaluation was suggested. A Branch and Bound algorithm was proposed to solve the small size problem and two heuristics are also proposed to solve large size problems. The results shows the effectiveness of heuristics compared to a well Known Newsvendor problem applied at each level.

Further research will be dedicated the extensions of this approach for multilevel assembly systems. In this perspective, the models of this paper may be useful for an approximate approach which consists in cutting the bill of material tree of the finished product into multi-level linear branches. 7

REFERENCES

[1] Chatfield, D. Kim, J. Harrison T. Hayya, J. 2004. The bullwhip effect in supply chains- impact of stochastic lead times, information quality, and information sharing: A simulation study. Production and Operations Management, 13 (4): 340353. [2] Chaharasooghi, S. and Heydari, J. 2010. LT variance or LT mean reduction in supply chain management: Which one has a higher impact on SC performance? European Journal of Operational Research, 204(1):86–95. [3] Chen, F. Drezner, Z. Ryan, J. Simchi-Levi, D. 2000. Quantifying the bullwhip effect in assembly supply chain: the impact of forecasting, lead times and information. Management Science, 46 (3), pp 436-443. [4] Dolgui A., Prodhon C. 2007. Supply planning under uncertainties in MRP environments: A state of the art, Annual Reviews in Control, 31(2): 269-279. [5] Dolgui A., Ben Ammar, O. Hnaien, F. Ould Louly, M.-A. 2013. A state of the art on supply planning and inventory control under lead time uncertainty, Studies in Informatics and Control, 22(3): 255-268. [6] Elhafsi M. 2002. Optimal leadtimes planning in serial production systems with earliness and tardiness costs. IIE Transactions, 34, pp 233-243. [7] Hnaien, F. Dolgui, A. Marian, H. Louly, M.A. 2007. Planning order release dates for multilevel linear supply chain with random lead times. Systems Science, 31 (1), pp 19-25. [8] Wazed, M., Shamsuddin, A., and Yusoff, N. 2009. Uncertainty factors in real manufacturing environment. Australian Journal of Basic and Applied Sciences, 3(2):342–351. [9] Weeks, J.K. 1981. Optimizing Planned Lead Times and Delivery Dates, 21st annual Conference Procceding, Americain Production and Inventory Control Society, pp 177-188. [10] Yano, C.A. 1987. Planned leadtimes for serial production systems. IIE Transactions, 19(3), pp 300-307.