1
A reformulation for the stochastic
multi-item capacitated lot sizing
problem with penalty costs
Master thesis
By
Detmer Boels
University of Groningen
Faculty of economics and business
Technology and operations management
2
Abstract
3
1 Introduction ... 4
2 Back ground and Literature Review ... 6
2.1 Background ... 6
2.1.1 Backorders ... 6
2.1.2 Strategies ... 7
2.2 Literature review ... 8
2.2.1 Insights literature review ... 10
3 Methodology ... 11
4 Model formulation ... 12
4.1 Deterministic single-item capacitated lot sizing model ... 12
4.1.1 PF deterministic single-item capacitated lot sizing model ... 12
4.2 Stochastic single-item capacitated lot sizing model ... 15
4.2.1 Expected backlog and expected inventory under demand uncertainty... 15
4.2.2 Piecewise linear approximation ... 16
4.3 Reformulated stochastic single-item capacitated lot sizing model ... 19
4.3.1 TF single-item capacitated lot sizing model ... 19
4.4 Stochastic Multi item capacitated lot sizing problem. ... 21
4.4.1 PF Stochastic multi item capacitated lot sizing problem ... 21
4.4.2 TF Reformulated Stochastic multi item capacitated lot sizing problem ... 22
5 Numerical study ... 24
5.1 Analysis of the numerical results ... 25
5.1.1 Overall results ... 26
5.1.2 Case 1: one product no capacity limitation ... 26
5.1.3 Case 2: one product with capacity limitations ... 27
5.1.4 Case 3: Multi products with capacity limitations ... 28
5.1.5 Effect of capacity limitations ... 29
5.1.6 Effect of Setup costs ... 30
5.1.7 Effect of coefficient of variation ... 30
6 Implications, Limitations and further research ... 31
6.1 Implications... 31
6.2 Limitations and further research ... 31
7 Conclusions ... 33
4
Chapter 1
1
Introduction
In this thesis the capacitated lot sizing problem (CLSP) is addressed. The CLSP occurs in industrial practices and a description of the problem is as follows. A company faces period demand of multiple items over a finite planning horizon. These items must be produced on the same resource which has a maximum amount of production capacity available. In addition to this, a production of one item can only start after a setup which is associated with setup costs. On the other hand, holding costs is incurred for each unit of inventory transferred from one period to the next. Therefore, the company aims to plan the timing and sizing of her production quantities (lot sizes) in such a way that the total cost over the planning horizon is minimized. Furthermore, the company must ensure that the production quantities of each period do not exceed the maximum available capacity of the resource. The majority of the literature about CLSP is confined to the situation where all data is deterministic. Thus, the period demand of each item is known in advance. Here, future demand uncertainty is taken into consideration by reserving a fixed amount of inventory as safety stock. However, the amount of this safety stock is usually computed with simple rules of thumb (Tempelmeier, 2013). Anyhow, literature is dominated by the deterministic lot sizing problems, Robinson, Narayanan & Sahin (2009) and Gicquel, Minoux & Dallery (2008) gives reviews of research approaches for the CLSP.
The stochastic counterpart of the well-known deterministic CLSP is the stochastic multi-item capacitated lot sizing problem (SCLSP). In the SCLSP demand uncertainty is taken into account, thus in each period the exact demand is not known in advance. However, the demand is known by its probability distribution function. To handle the randomness of the stochastic demand over time in the planning horizon, Bookbinder & Tan (1988) developed several strategies which define rules with respect to the timing and sizing of production quantities. One of these strategies is the so called “static-uncertainty strategy”. In this strategy the complete production planning is fixed at the beginning of the entire planning horizon. Thus, the timing of the production quantities and the size of the production quantities are determined at the beginning of the planning horizon. Most of the current literature about the SCLSP adapted this static-uncertainty strategy (e.g. Tempelmeier et al, 2010; Helber et al, 2010; Helber et al, 2013 and Tempelmeier & Hilger 2015.
5
the static-dynamic uncertainty strategy only fix the timing of the production quantity. The sizes of the production quantities are determined at each period after the actual inventory is observed.
This thesis contributed to the literature by presenting a reformulation for the static uncertainty strategy in the SCLSP. The reformulation of the SCLSP builds on the ideas of Tunc et al (2014). They reformulated the single-item uncapacitated lot sizing problem. In their reformulation they used the static-dynamic uncertainty strategy. Thus, the formulation of Tunc et al. (2014) will be implemented in a problem with a different strategy. In addition to this they formulated the model for the single-item lot sizing problem without capacity limitations. The model developed in this paper will be implemented in a more complicated problem which has multiple items which share capacity limitations. The reformulated SCLSP model will be compared against the existing SCLSP formulations. Therefore, the following research question is developed:
How could the stochastic multi item capacitated lot sizing problem be reformulated with the ideas of the formulation presented by Tunc et al. (2014) and what is the consequence of the reformulation on the performance of the model compared to existing models?
From a practical point of view the SCLSP with the static uncertainty strategy is worth studying for successful operational planning and production control. In this thesis we assume that demand is stochastic and normally distributed. Many organizations deal with products of which demand is normally distributed (Hopp & Spearman, 2008). In addition to this, organizations can deal with stochastic demand to determine or adjust the planned production after demand is realized. The drawback of this approach is that it induces planning nervousness and in supply chains it can lead the notorious bullwhip effect. Therefore, organizations fixed the entire planning and thus use the static uncertainty strategy. This approach leads to more stable planning and production systems and is especially used for product which demand is normally distributed (Hopp & Spearman, 2008).
6
Chapter 2
2
Background and Literature Review
This section is structured as follows. First the background of the SCLSP is described. The background consists of two important aspects of the lot sizing problem. It is discussed what happens when the demand is bigger than the inventory, thus the inventory cannot satisfy the demand. And the different strategies of Bookbinder & Tan (1988) are further explained. Furthermore, the related literature based on the introduction and the background is reviewed. Finally, insights from the literature review are summarized.
2.1 Background
2.1.1 Backorders
The SCLSP is a problem where there is a tradeoff between setup costs and holding costs. However, with demand uncertainty it is unavoidable that in some periods the inventory on hand is not sufficient to satisfy the complete expected demand. When the inventory cannot satisfies the complete expected demand a part of the demand will be fulfilled in a later period. This is denoted as backorders. The term backorder denotes the demand within a period that is satisfied with delay. Backorders are added to a backlog variable. Thus, backorders denote the demand within a time interval which cannot be satisfied. The backlog is the cumulated set of backorders which has not been satisfied yet. A standard approach to deal with backordering and backlog is to introduce penalty costs. Penalty costs are incurred for each unit demand in a period that will be fulfilled with delay. Then, the problem is to find the optimal trade-off between setup, holding and penalty costs. Defining an accurate value for backorder costs is difficult. Therefore, backorders can be set to a service level. In the literature there are several service levels, also known as performance measures (Schneider, 1981) which regulate the backorders. There are three well-known service levels. These are the α, β and γ service level.
The α service level is an event-oriented measure. It measures the probability that all customer demand will be fulfilled without any backorders and can be written as follows:
This criterion is also called the ready rate. It is the probability that the net inventory is non-negative.
7
The β service level does not only reflect the event that a stock-out occurs but also the quantity of backorders.
The third service level is the γ service level and is a time- and quantity-oriented performance criterion. It reflects not only the amount of backorders but also the waiting times of the backordered demand. The γ service level can be written as follows:
Due to the fact of demand uncertainty it is possible that the expected period demand will be smaller than the expected backlog or even be zero. Therefore the γ service level can become negative. For this reason Helber, Sahling, & Schimmelpfeng (2013) developed the δ service level which can be written as follows.
Here the expected total maximum backlog equals the expected cumulated demand including that of the considered period.
2.1.2 Strategies
8
The dynamic uncertainty strategy determines in each period, after the realized actual demand, if a production is needed and how large the production quantity should be. The dynamic uncertainty strategy is the most cost effective strategy. However finding the optimal values with this strategy is a complex task. In addition to this, significant planning nervousness for planners arises, because it is unknown at the beginning of each period what will happen and planners don’t know how to allocate available production capacity.
Therefore, to combine the positive features of both strategies there should be a compromise between the static uncertainty strategy and the dynamic uncertainty strategy Bookbinder & Tan (1988). The compromise is called the static dynamic strategy. This strategy fixed the periods in the planning horizon when a production occur but the actual production quantity is determined at the beginning of the production period. Compared to the dynamic uncertainty strategy there is no uncertainty in the timing of the production quantities.
2.2 Literature review
Harris (1913) is one of the founders of the lot sizing problem. Harris (1913) proposed the economic order quantity (EOQ) model. The EOQ model is a continuous time model with an infinite time horizon. The objective of the EOQ model is to minimize the sum of the setup and holding costs. Wagner (1958) first introduced the discrete lot sizing problem which assumes that the planning horizon is finite and divided into discrete periods for which the demand is given and deterministic and may very between periods. As mentioned in the introduction, the literature is dominated by the deterministic lot sizing problems. Robinson et al. (2009) provided a review of research about the deterministic lot sizing problems. They argued that due to the importance in industry and mathematical complexity the deterministic lot sizing problem is frequently studied.
A natural extension of the deterministic single-item lot sizing problem is the deterministic multi item capacitated lot sizing problem. The capacitated problem is first discussed by Florian, Lenstra, & Rinnooy Kan (1980) and the multi item capacitated lot sizing problem is first discussed by (Billington, McClain, & Thomas, 1983). In the CLSP the multi items are produced on the same resource. The link between the multi items is the limited capacity of the shared resource. Without capacity limitation the production schedules of the multiple items can be determined separately.
9
Tempelmeier & Herpers (2010) proposed a formulation of the SCLSP and applied a β service level for the backorders. They propose an ABC heuristic which is an extension of the A/B/C heuristic presented by Maes & Van Wassenhove (1986). Basically the ABC heuristic is a period-by-period approache which transforms the demand of each item in each period into a matrix of production quantities for each item for each period. When the production quantities for period is calculated several parameters, called ABC, are used to govern the sequence in which future product specific demands are considered. The ABC heuristic is developed for the deterministic capacitated lot sizing problem. Tempelmeier & Herpers adjusted the ABC principals in such a way it is able to deal with stochastic demand. Tempelmeier & Herpers, (2010) also argued that it is desirable to follow the static uncertainty strategy. This seems to be very important when production schedules have to be coordinated over multiple products with capacity limitations.
Tempelmeier, Helber, Sahling, & Buschk (2009) propose a solution approach that solves a series of MIP problems using a fix and optimize heuristic. The fix and optimize heuristic decompose the complex problem into smaller subproblems of more practical dimensions. For all subproblems the whole planning horizon is considered and the lot sizes are determined. In each subproblem a large number of binary setup variables are fixed whereas only a small subset of these variables is optimized. Another heuristic procedure is based on a deterministic multi item capacitated lot sizing problem of Manne & Science (1958). The heuristic approximate the lot sizing problem by a set of several models. For each item several production plans are determined. The problem then is to select for each item exactly one production plan such that in all periods the limited capacity is respected. This heuristic is originally proposed for a variant of the deterministic capacitated lot sizing problem. Tempelmeier (2011) proposed a heuristic which is a variant of the problem of Manne et al. (1958). To select the production plans Tempelmeier (2011) applied a column generation approach that define the final production plans. A column generation starts with a master plan which contains a few variables. New columns, which are new variables, are generated in a repeated procedure. Each procedure contains of two steps. First, the master plan is solved. Second, to find the most promising new variable to be introduced, a sub problem is solved with the objective to minimize the costs. If the costs are greater or equal than zero there is no improvement and the original problem is solved. In this heuristic the static uncertainty strategy is applied.
Helber et al. (2013) developed two approaches to solve the SCLSP under the static uncertainty strategy. To restrict the backorders the δ service level is used. The first approach is the scenario approach. This approach considers a set of demand scenarios. Each scenario represents one of several equally likely paths of realizations of the random demand. Than the production quantities of the set demands are optimized. In the second approach they developed a MIP model were the non-linear functions of the backlog and inventory are approximated by use of piecewise linear segments. The resulting piecewise linear model is then solved with a MIP-based heuristic.
10 2.2.1 Insights literature review
11
Chapter 3
3
Methodology
The aim of this research is to reformulate the stochastic multi item capacitated lot sizing problem with the ideas of Tunc et al, (2014). Furthermore, we want to investigate the consequence of the reformulation on the performance of the model compared to existing formulations. In order to come to this aim, the proposed methodology is quantitative modeling. Quantitative modeling aims on building objective quantitative models that explain part of the decision making problem that managers face in real-life operational processes and to test the performance of the proposed problem solution (Karlsson, 2016).
We reformulate the SCLSP under the static uncertainty strategy were the backorders are penalized. First we explain an existing SCLSP formulation. The existing formulation is mostly based on the models presented by Tempelmeier & Hilger (2015) and Helber et al. (2013) This model formulation has decision variable based on each period in the planning horizon. Therefore we call this model the Period formulation (PF). Second we reformulate the period formulation with the ideas of Tunc’s et al. (2014). Therefore we call this formulation the Tunc’s formulation (TF). The models are outlined and explained in the next chapter.
12
Chapter 4
4
Model formulation
In this chapter we first explain the existing deterministic single-item capacitated lot sizing problem. Then we extend this formulation with stochastic demand. Then we describe a reformulation for the stochastic single-item capacitated lot sizing problem. After that we explain the SCLSP formulation and the reformulated SCLSP. The current formulations are denoted as PF and the reformulations are denoted as TF.
4.1 Deterministic single-item capacitated lot sizing model
We begin by introducing the problem statement of the deterministic single-item capacitated lot sizing model and the notation that will be used.
In the deterministic lot sizing problem we assume that demands that arrive are known in advance. When the demand is bigger than the inventory it is backordered this causes a backorder cost. We consider a finite planning horizon of discrete time periods. There are three cost components which are and . Denotes the setup costs that are incurred when there is a production scheduled. refers to the holding costs that is incurred per unit that is transferred forward from one period the next period. Lastly, is the backorder cost per unit that is backordered. Following the static uncertainty strategy a production schedule is determined at the beginning of the entire planning horizon. Thus, it is known when a production quantity is produced and how large the production quantity will be. Each produced item requires a specific amount of time. In addition to this, in each period there is a maximum amount of production time available. It is assumed that the inventory at the beginning of the planning horizon is zero.
4.1.1 PF deterministic single-item capacitated lot sizing model
13 Table 1: Symbols used
Indices and index sets
Set of periods (
Parameters
Setup costs
Holding costs per unit
Big number
Backorder costs per unit
Variables
Production quantity (lot size) in period Cumulated production quantity in period Demand in period
Cumulated demand in period Binary setup variable in period Physical inventory in period Backlog in period
Approximated loss function value at period
Available capacity in period
In the model formulation we determine production quantities as cumulated production quantities. Likewise, the demand will be the cumulated demands. We introduce and as cumulated production quantities and cumulated demand for each period and are calculated as follows:
∑ ∑
14
The PF single-item capacitated lot sizing problem is as follows: Model 4.1: Single-item capacitated lot sizing model with back orders
∑( ) ( 4.1 ) ( 4.2 ) ( 4.3 ) ( 4.4 ) ( 4.5 ) ( 4.6 ) ( 4.7 ) ( 4.8 )
In model 4.1 we have setup costs , holding costs and backorder costs . For each time period the decision variables are the binary setup variable , the cumulated production quantity , the physical inventory and the backlog . For each time period we have the limited capacity . Furthermore, de cumulated demand for each period is .
Objective function 4.1 minimizes the total setup, holding and backorder costs. The production quantity in period is the diffrences between the cumulated production quantity of period and . We can write this condition as constraint 4.2. In addition to this, constraint 4.2 ensures that is non-decreasing.
The decision variable can only be 1 if we produce a production quantity. Constraint 4.3 relates the production quantity to the binary variable through the big method, thus, if there is a production quantity in period t the variable will be forced to 1. To be sure, the production quantity is the difference between the cumulated production quantity of period and . Constraint 4.4 limits the times spend on production each period to the maximum capacity value of .
Constraint 4.5 and 4.6 determines the value of and and are calculated as follows:
Thus, is either 0 or the positive difference between the cumulated production quantity minus cumulated demand. Which results in a positive inventory. In addition to this, is either 0 or the positive difference between the cumulated demand minus the cumulated production quantity. this results in a negative inventory i.e. backlog. The above two equations can also be written as
and .
15
4.2 Stochastic single-item capacitated lot sizing model
Now we have discussed the deterministic lot sizing model, we move to the lot sizing problem with demand uncertainty. In the deterministic formulation the demand for each period was known in advance at the beginning of the planning horizon. In this section we consider the cases were the demand is not known in advance. Here the demands are random variables with a known probability distribution. With the introduction of demand uncertainty with demand as random variables we move to the stochastic counterpart of the deterministic lot sizing problem.
4.2.1 Expected backlog and expected inventory under demand uncertainty
With demand as random variables we assume that demands for each period are normally distributed. When demand is normally distributed we know in each single period the demand with a mean
and a variance of . Furthermore, we know the probability density function and the cumulative
density function . To be clear, this is a single period demand and not a cumulative demand. When
the period demands are random variables, the inventory and backorders becomes also random variables. Therefore from now on we work with the expected values of the demand, inventory and backorders.
First we take a look at the inventory and backorder level. The expected inventory ( [ ]) and expected backlog ( [ ]) level are determined by the expected demand. The cumulative expected demand equals the sum of independent and normally distributed random variables. Therefore the cumulative demand is normally distributed with a mean ∑ and a standard deviation
of √∑ with probability density function and cumulative density function . Thus,
the expected inventory and backlog can be written as:
[ ] ∫ ( 4.9 )
[ ] ∫ ( 4.10 )
These two equations are non-linear functions of the cumulated production in period 1 to . The values of [ ] and [ ] corresponding to a specific cumulated production quantity can be computed via
the “first order loss function” see Tempelmeier (2011, p. 338)
The first order losses function for the standardized normal distribution with a mean of 0, a standard deviation of 1 with a density function and cumulated distribution function . Is calculated as follows:
16
Here the value of is determined for the standardized normal distribution function with a mean of 0 and a standard deviation of 1. To use this equation for other values of the mean and standard deviation we must rewrite the equation. This is explained below.
Consider the case of a normally distributed cumulative demand with a mean and a standard
deviation . Than the expected inventory and expected backlog of equation 4.9 and 4.10 can be
calculated with the “first order loss function” as follows:
[ ] ( ) ( ) ( 4.11 ) [ ] ( ) ( ) ( 4.12 )
Keep in mind that 4.11 and 4.12 are non-linear functions. To use this in a mixed integer programming formulation we need to linearize these non-linear functions.
4.2.2 Piecewise linear approximation
In this section we describe how to linearize the non-linear function of the expected inventory and expected backlog described in the previous paragraph. In order to linearize the non-linear function we can approximate the non-linear function with an arbitrary precision using a sufficient number of linear line segments. To explain this we use a case of a single period with a normally distributed expected demand with a mean of 10 and a standard deviation of 5. Consider the case with a production quantity of zero, then all the demand will be backordered. In a case with infinity production quantity there will be no backorder at all. The non-linear function of the expected backlog with respect to a specific production quantity for the case with a mean of 10 and a standard deviation of 5 is presented in figure 4.1.
Figure 4.1: non-linear function of expected backlog
17
In order to approximate this non-linear function we opt for using a piecewise linear approximation. We approximate the non-linear function with a number of linear line segments of the standard normal loss function. It should be clear that more line segments provide a better approximation. However to many line segments contribute to a more complex model. Rossi, Tarim, Prestwich, & Hnich ( 2014) discussed lower and upper bounds of the number of line segments for the standard normal first order loss function.
Each line segment consists of an intercept and a slope . Thus there is a set of intercepts
and slope pairs which define the piecewise linear approximation of the standardized loss function .
Now, with a given set of standard piecewise linear segments we can define the piecewise
segments for our specific mean and standard deviation with the following formula.
, ( 4.13 )
For our case with a mean expected demand of 10 and a standard deviation of 5 the linear segments of a set with 11 pieces are presented in figure 2
Figure 4.2: approximated loss function
For each specific production quantity the upper line of the 11 segments is the approximated loss
function and the expected backlog. Thus we must ensure that the model always chooses the upper line. Therefore we introduce a new variable which is the approximate loss function value at period . To integrate this variable in the MIP model we use the following constraint:
( 4.14 )
To integrate the approximate loss function value in the objective function we must rewrite the earlier proposed objective function of model 4.1. The costs were written as follows:
18 After implementing the expected values we get:
To incorporate the approximate loss function value we get, after re-arranging the terms (Silver et al. 1998), the following equation:
With this equation we can develop the following objective function:
∑
( 4.15 )
The difference in 4.15 with 4.1 is that and are replaced respectively by and the variable.
Now we formulate the stochastic single-item capacitated lot sizing problem in model 4.2 Model 4.2:stochastic single-item capacitated lot sizing problem
∑ ( 4.16 ) ( 4.17 ) ( 4.18 ) ( 4.19 ) ) ( 4.20 ) ( 4.21 ) ( 4.22 )
Here, the objective function 4.16 minimizes the sum of total setup, holding and backorder costs. Constraint 4.17, 4.18 and 4.19 are the same as in model 4.1. Constraint 4.20 integrates the approximated loss function value at period with the given set of piecewise segment . Constraint
19
4.3 Reformulated stochastic single-item capacitated lot sizing model
The reformulated SCLSP is based on the ideas of Tunc, Kilic, Tarim, & Eksioglu, 2014. We consider again a finite planning horizon of periods which is divided in a set periods in . Thus let be the set of periods in . A production schedule is determined at the beginning of the planning horizon. Let be the production periods over the entire planning horizon. Here is the number of scheduled production periods, and is the period of the production scheduled. Than the interval between two successive production periods, let say ( and ), is defined as a
replenishment cycle. From now =1 and = +1. The planning horizon of periods can be
translated into a schedule of disjoint replenishment cycles.
Let us consider a replenishment cycle which starts at period and ends at period , and assume that the inventory level at the beginning of the cycle is . Then, the total cost of the replenishment cycle can be calculated as follows:
∑
( 4.23 )
Here, are the setup costs, are the holding costs and are the backorder costs. In 4.23 the costs of all the periods in a replenishment cycle will be calculated. Then, to determine a production schedule the idea is to choose replenishment cycles in such way that the costs are minimized and all the constraints are fulfilled. It should be clear that building a model based on replenishments cycles have a lot more decision variables than the PF lot sizing model formulations.
In table 2 the parameters, sets and variables are presented which will be used in the reformulation. After the table the reformulated stochastic single-item capacitated lot sizing model is presented. Table 2: Symbols used
Indices and index sets
Set of periods (
Set of periods (
Set of periods (
Parameters
Setup costs
Holding costs per unit
Big number
Backorder costs per unit
Variables
Cumulated production quantity in period
Cumulated demand in period
Binary setup variable in period
Available capacity in period
Approximated loss function value 4.3.1 TF single-item capacitated lot sizing model
20
binary variable is used. It takes 1 if is a replenishment cycle and 0 otherwise. Is the
cumulative production quantity up to and including period if is a replenishment cycle. Model 4.3: reformulated single-item uncapacitated lot sizing model
∑ ∑ ( ∑ ) ( 4.24 ) ∑ ∑ ( 4.25 ) ∑ ( 4.26 ) ∑ ( 4.27 ) ( 4.28 ) ∑ ∑ ( 4.29 ) ( 4.30 ) ∑ ∑ ( 4.31 ) ( 4.32 ) ( 4.33 )
In model 4.3 the object function 4.24 minimizes the total costs. When =1, is a replenishment
cycle. Than the total costs of the replenishment cycles will be incurred.
21
The cumulative production quantity must be zero if a replenishment cycle is not part of the set of disjoint replenishment cycle. This is guaranteed by constraint 4.28. Here the big is a sufficient large number which is explained in the previous section.
Now, when the production quantities are cumulated production quantities we should make sure that the cumulative production quantities are non-decreasing. In addition to this, the production quantity in one cycle equals the difference between the cumulative production quantities of two consecutive cycles. This can be ensured by constraint 4.29.
Constraint 4.30 guarantees that takes the largest value of the approximated loss function which is
discussed in the previous section. Only If is a replenishment cycle should be calculated for
all periods in that cycle. This will be ensured trough the variable. Thus, when = 1 the values
of in periods are calculate.
Constraint 4.31 restrict the total capacity in period to the maximum capacity of . Because only one cycle ends at period , thus only one is larger than zero, we can sum all from the beginning
of the planning horizon till . In addition to this, only one cycle starts at period , so we can sum all
from till the end of the planning horizon. Then, the difference between these two is restricted to the maximum capacity .
Furthermore, constraint 4.32 and 4.33 set the variable domains.
4.4 Stochastic Multi item capacitated lot sizing problem.
Now we consider the SCLSP which is the extension of the stochastic single-item lot sizing problem with multiple items. The SCLSP is almost completely identical with the previous discussed models. The only difference is that we now have different products that are produced on the same resource.
In the SCLSP the limited capacity of the resource is the link between the production schedules of the different products. Without capacity limitations the schedules of the different products can be determine separately.
In the SCLSP we assume that demand is stochastic with a known distribution function. When demand is bigger than the inventory it is backordered and incur a backorder cost. We consider a finite planning horizon of discrete time periods and number of products.
4.4.1 PF Stochastic multi item capacitated lot sizing problem
For The PF SCLSP we use the same notation as presented in table 1. The only difference is that each parameter is used for all products . Based on the introduction of this section and the notation of table 1 we now formulate the SCLSP in model 4.4. In addition to this, the model explanation is the same as model 4.2. Therefore we explain the model briefly.
Model 4.4: PF Stochastic multi-item capacitated lot sizing problem
∑ ∑ ( ( ( )) )
22 ( 4.35 ) ( 4.36 ) ( 4.37 ) ∑ ( 4.38 ) ( 4.39 ) ( 4.40 )
In model 4.4 we have and which are the cumulated production quantities and cumulated
demand up to period for product . Is the approximated loss function value for product in
period .
Objective function 4.34 minimizes the total costs over the entire planning horizon. Constraint 4.35 regulates the binary variable. Constraint 4.36 ensures that the cumulated production quantity of
product in period is non-decreasing.
Constraint 4.37 integrates the approximated loss function value of at period for all products
with the given set of piecewise segment .
Constraint 4.38 limited the time spend on productions to the available capacity each period . Here the total time spending in period is the sum of production time of all products . Constraint 4.39 and 4.40 sets the variable domains.
4.4.2 TF Reformulated Stochastic multi item capacitated lot sizing problem
For the reformulated SCLSP we use the same notation as presented in table 2. Based on the introduction of this section and the work of tunc et al (2014) we reformulate the SCLSP in model 4.5. Again, we explain the model briefly.
Model 4.5: TF Stochastic multi-item capacitated lot sizing problem
23 ( 4.45 ) ∑ ∑ ( 4.46 ) ( ) ( ) ( 4.47 ) ∑ ∑ ∑ ( 4.48 ) ( 4.49 ) ( 4.50 ) ( 4.51 )
is the binary decision variable that takes value of 1 if of product is a replenishment cycle,
and 0 otherwise. Here, and are the cumulated production quantities and cumulated demand
of product up to and including if is a replenishment cycle. Is the approximated loss
function value of product at period of replenishment cycle .
Constraint 4.42 and 4.43 and 4.44 regulate the flow conservation. Constraint 4.45 regulates the
variable and constraint 4.46 ensures that is non-decreasing. Constraint 4.47 guarantee that for
each replenishment cycle is the approximated loss function value.
24
Chapter 5
5
Numerical study
This chapter deals with a comparison between the two models that we have presented in the previous chapter to solve the SCLSP. In order to test the performance and the behavior of the two models we conduct a numerical experiment under different parameters. The parameters are based on problem instances presented by Tempelmeier & Hilger (2015). For each combination of parameters we generate mean demand randomly from a uniform distribution on the interval [0,100]. This generated mean demand is the input for both models. To be ensuring that the output is reliable we perform 2 replications for each combination of parameters. This due to the fact than one particular random generated mean demand pattern could be in favor of one or other demand pattern.
For our numerical experiment we defined 162 test instances by varying different parameters. With the parameters we simulated different practical scenarios. Table 3 gives an overview of the parameters. Table 3: Parameters used in the numerical study
Number of products
Number of periods
Demand coefficient of variation
Time between orders
Capacity Holding costs Penalty costs
We generate problem instances with 1, 5 and 10 products and 10 and 20 periods. When the number of products or periods increases the complexity of the model also increases. The randomly generated mean demands are assumed to be normally distributed with a coefficient of variation of 0.1, 0.2 and 0.3.
TBO stands for time between orders. With a higher TBO value we want the model to have fewer periods where there is a production. Higher setup costs encourage the model to have fewer periods were there is a production. Therefore, with the TBO values we determine different values of the setup costs. In addition to this, the setup costs are related to the holding costs. Because higher holding cost forces the model to have more periods where there is a production. Lastly, holding costs is incurred for a single unit, thus the total holding costs is related to the average mean demand. Therefore with the following equation we determine the setup costs, based on the values of TBO, holding costs and average mean demand .
25
Then, with a mean demand of 50 and a fixed holding cost of 1 the setup costs for the different values of TBO are respectively 25, 100 and 400.
With capacity we limit the maximum production in one period. Here, Low, medium and high stand respectively for 1, 2 and 4. A low capacity value of 1 result in tight capacity limitations and a high capacity value of 4 results in wider capacity limitations. The capacity is calculated as follows
∑ ∑
More especially the capacity is determined by the times the sum of all demand of all periods of all products divided by the number of periods.
As the result of the 162 test instances and 2 replications we perform in total 324 experiments. We run all experiments on a 3.7 GHz I3 Intel CPU with 8 GB RAM. As a MIP solver, we use Gurobi 7.0 and restrict the computation time to 600 seconds. Both models are programmed in Python 2.7.
5.1 Analysis of the numerical results
In this section we discuss the results of the experiments. First we present the overall results in table 4 and then provide results for specific cases. The cases are based on practical situations. Under table 4 we explain the different computational performance statistics applied to determine the performance of the two models.
Table 4: The solution statistics of PF and TF
Parameters Periods formulation Tunc’s Formulation
26
In table 4 the columns reflect upon computational performance by means of four solution statistics, which are time, s-gap, e-gap and nodes. All solution statistics are averaged of all problem instances characterized by the same parameter.
More specially, in time we provide the solution time in seconds. As already mentioned, not all the experiments are solved to optimality. There is a time limit of 600 second of each combination of parameters. In s-gap we report the average integrality gap. This is the percentage difference of the costs of the best integer solution and the optimal cost of the relaxed solution of the model. In the relaxed solution all the integers are relaxed thus the values of the integer variables can be any value between 0 and 1. In e-gap we report the gap at termination. This is the difference in percentage between the best solution and the best lower bound available. When an experiment is solved to optimality the best solution and the best lower bound are the same. When the experiment hits the time limit the best solution is not the optimal solution. Then the model provides the value of the solution at that moment and the best lower bound. The best lower bound is a value which at the model thinks it is possible to find as a solution. Then the optimal solution lies between the lower bound and the given solution at that moment. In nodes we give the number of nodes explored by the model.
5.1.1 Overall results
In table 4 it is visible that the overall performance statistics of TF are lower than the performance statistics of the PF. This indicates that the performance of TF is better than PF. Furthermore, for all instances with the same parameters the e-gap of PF is larger than TF, thus TF could solve a lot more instances to optimality. It is also visible that PF performed better than TF with instances characterized by 1 product. The s-gap and explored nodes for PF are larger than TF. This indicated that the linear relaxation of TF is stronger than PF. But this is based on numerical evidence only. We now explore the performance of both formulations in different cases.
5.1.2 Case 1: one product no capacity limitation
The first case consists of one product without capacity limitations. This case is the classic single-item uncapacitated lot sizing problem and is particular interesting because Tunc et al (2014) reformulated this problem and used the static-dynamic strategy. We are interested in what will happen when we use the reformulation in the static uncertainty strategy with a single product and without capacity limitations. In our original experiment we do not implement an experiment without capacity limitations, only with loose capacity limitations. Therefore we setup and experiment without capacity limitations and the following parameters:
Table 5 : Parameters used in case 1
Number of products
Number of periods
Demand coefficient of variation
Time between orders
Capacity
27 Table 6: solution statisctics case 1
Periods formulation Tunc’s Formulation
periods time s-gap e-gap Nodes time s-gap e-gap Nodes
10 0,10 46,30 0,00 79,00 0,08 0,00 0,00 0,00
20 0,25 43,66 0,00 1573,22 0,38 0,00 0,00 0,00
The results provided in table 6 demonstrate that that with 10 periods TF is slightly better than PF. When the numbers of periods increase to 20 PF performs better than TF. All instances are solved to optimality, therefor the e-gap for both formulations are zero. This experiment already shows the stronger linear relaxation of TF against PF based on numerical evidence. The S-gap for TF is zero where the S-gap of PF is around 45%. The explored nodes for TF are zero and for PF the explored nodes increases when the number of periods increases.
We were surprised that that PF performed better than TF when the number of periods increased. Tunc et al (2014) formulated their model for the stochastic uncapacitated lot sizing problem under the static dynamic strategy. Their reformulation was compared against the formulation of Tarim & Kingsman (2004). This formulation used the same structure of decision variables as our PF model. Tunc et al demonstrated that their reformulation out-performs the formulation of Tarim & Kingsman (2004). Therefore we expected that that TF outperforms PF in the case of a single-item without capacity limitations. However, PF performed better than TF. This can be explained due to the fact that Tunc et al used the static dynamic uncertainty strategy. A single-item model under the static uncertainty strategy is less complex and therefore it could be that a less complex model such as PF performs better than a more complex model such as TF.
5.1.3 Case 2: one product with capacity limitations
We now consider the case of one product with capacity limitations. Table 4 already shows that PF performs better than TF in instances characterized by 1 product. To better understand the performance of TF and PF with one product and different capacity limitations the following table present results of instance characterized by one product and different capacity limitations.
Table 7: solution statistics case 2
Period formulation Tunc’s Formulation
Capacity time s-gap e-gap Nodes time s-gap e-gap Nodes
1 0,05 31,81 0,00 0,22 1,40 13,01 0,00 0,00
2 0,09 50,14 0,00 93,00 3,62 9,99 0,00 319,72
4 0,11 45,10 0,00 326,33 0,52 1,03 0,00 5,64
Results provided in table 7 clearly demonstrate that PF outperforms TF on computational time. Again, PF and TF solved all instances to optimality. S-gap for PF is worse than for TF. It is interesting to see that for PF the computation time decrease when the capacity limitation becomes tighter. For TF the computation time and the explored node’s is the worse for a capacity limitation of 2.
28
observation because in literature the capacitated problem is assumed to be more difficult. It could be that the capacity was such tight that the model was forced to produce production quantities each period.
5.1.4 Case 3: Multi products with capacity limitations
Now we confine to the case of multiple products. Solving multiple products without capacity restriction is the same as solving schedules of the products separately. Therefore there is no experiment with multiple products without capacity limitations. This case is also the case of the SCLSP. In this case we interested in the effect of multiple products under different number of periods.
Table 8: solution statistics case 3
Period formulation Tunc’s Formulation
Products Periods time s-gap e-gap Nodes time s-gap e-gap Nodes
5 10 100,34 43,03 0,24 266542 1,39 0,34 0,00 99,28
20 400,87 40,66 8,43 709413 62,09 0,38 0,15 1145,44
10 10 400,39 42,36 5,40 949810 2,70 0,08 0,00 59,89
20 419,64 40,56 11,91 478082 76,95 0,08 0,02 912,48
Both formulations, TF and PF, could not solve all instances and hits the time limit of 600 seconds. Anyhow, in the e-gap column of table 8 it is visible that TF could solve a lot more instances than PF. In table 8 it is visible that TF outperforms PF in a case with multiple products. For TF the computation time is mostly dependent on the number of periods. And for PF the computation time is dependent on both, number of products and number of periods. The differences in computation time of PF of instances characterized by 10 products and 10 or 20 periods are barely visible. This is due to the fact that PF hits the time limit very often.
29 5.1.5 Effect of capacity limitations
Now we confine to the effect of different levels of capacity limitations. In this test we are interested of the effect of multiple products under different levels of capacity limitations.
Table 9: Effect capacity limitations
Period formulation Tunc’s Formulation
Products Cap time s-gap e-gap Nodes time s-gap e-gap Nodes
5 1 287,27 34,42 4,24 256454 92,05 0,95 0,22 1858,31 2 243,44 45,99 4,39 527336 2,10 0,12 0,00 8,53 4 221,10 45,13 4,38 680143 1,07 0,01 0,00 0,25 10 1 427,33 32,70 6,38 220038 114,84 0,21 0,03 1456,69 2 401,17 46,08 10,28 872535 2,88 0,02 0,00 1,86 4 401,53 45,59 9,31 1049265 1,75 0,00 0,00 0,00
The computation time of PF related to instances of 5 and 10 products increases slightly when the capacity becomes tighter. The computation time for TF increase dramatic when there is a capacity value of 1. In addition to this, the computation time of TF nearly double when the capacity becomes tighter. Thus TF is far more sensitive against changes in capacity limitations than PF.
30 5.1.6 Effect of Setup costs
Table 10 illustrates the effect of setup costs on the performance statistics of botch models. For this experiment we determined the effect of setup costs on cases with multiple products.
Table 10: effect steup costs
Setup costs determine the frequency of orders. Higher setup costs encourage the model to have fewer periods were there is a production. In table 10 it visible that the computation time of PF increased dramatic when the setup costs increases. In addition to this, PF could not solve any instances characterized by 10 products and a setup cost of 100 and 400. TF solved almost all instances with this combination of parameters. Both models solved instances with an ordering cost of 25 easily. It is interesting to see that PF solved instances, characterized by 5 products and a setup cost of 25, better than TF based on time performance. However, TF performed better than PF with instances of 10 products and setup costs of 25. Thus, for PF and TF the performance are strongly related to the variation in setup costs.
5.1.7 Effect of coefficient of variation
Table 11 shows the average performance for instances characterized with 5 or 10 products and different values of CV. For PF and TF the performance based on time isn’t very sensitive against the effect of CV. For TF the S-gap performance related to CV is more stable than for PF. The S-gap for PF decreases when the CV increases. For TF the overall performance decreases when the CV increases. Table 11: effect coefficient of variation
Period formulation Tunc’s Formulation
Products CV time s-gap e-gap Nodes time s-gap e-gap Nodes
5 0,10 250,82 52,80 5,25 516384,56 18,40 0,33 0,00 519,31 0,20 251,58 40,03 4,25 469624,83 37,45 0,38 0,11 716,28 0,30 249,41 32,71 3,51 477923,31 39,37 0,36 0,11 631,50 10 0,10 408,46 51,71 10,25 724059,61 36,09 0,09 0,00 544,89 0,20 418,66 40,06 8,69 715396,58 40,92 0,06 0,01 460,64 0,30 402,92 32,60 7,03 702381,14 42,47 0,08 0,02 453,03
The effect of the coefficient of variation lies in line with the results of our benchmark paper of Tunc et al. (2014). It can be concluded that the influence of the coefficient of variation does not much affect the performance of PF and TF.
Period formulation Tunc’s Formulation
Products Setup costs time s-gap e-gap Nodes time s-gap e-gap Nodes
31
Chapter 6
6
Implications, Limitations and further
research
All findings have been discussed in the numerical study. In this chapter the practical implications, limitations and ideas for further research are presented.
6.1 Implications
From a practical point of view this thesis focused on the case were a production company must produce different items over a finite planning horizon. Each period there is a maximum capacity available. Tempelmeier & Herpers (2010) argued that it seems particularly important to use the static uncertainty strategy when there are capacity restrictions. Furthermore, we assumed that the demand is normally distributed. Many companies operate in the context of this description. As mentioned in the literature review most research focused on the application of heuristic to solve the SCLSP. With heuristics the problem is not solved to optimality. We prove that our reformulation of the SCLSP is far more time efficient than the formulation presented in the literature. Furthermore, the reformulation makes It possible to solve the SCLSP to optimality.
More especially, when companies operate in a production environment were the setup costs and capacity limitations are relative normal the reformulation outperform existing formulations. In addition to this the model could determine a production schedule in a reasonable amount of time. In situations with high setup costs or very tight capacity limitations the computation time increase, but it is still possible to determine a production schedule.
However, in production situations where the company must produce a single-item the reformulation perform worse than existing formulations. Based on time performance it is in this case better to use the existing formulations.
6.2 Limitations and further research
32
We have some advice for further research. First of all, in our experiment we used penalty costs to restrict the backorders. In most related papers backorders are restricted to a service level. Therefore the behavior of the models could be investigated under the different mentioned service levels.
33
Chapter 7
7
Conclusions
The purpose of this thesis was to reformulate the SCLSP formulation. In the reformulation we applied the ideas of Tunc et al (2014). We wanted to investigate the performance of the reformulation and to compare the performance against an existing formulation. The research question was stated as follow:
How could the stochastic multi-item capacitated lot sizing problem be reformulated with the ideas of the formulation presented by Tunc et al. And what is the consequence of the reformulation on the performance of the model compared to existing models?
It is found that the overall performance of the reformulated SCLSP is better than the existing SCLSP formulation. Since the formulation of Tunc et al (2014) is formulated for the stochastic single-item lot sizing problem we also investigated the behavior of the reformulation under these circumstances. It is found that the current formulation under the static uncertainty strategy with a single product performed better than the reformulation. The performance of the reformulated SCLSP performed better than the current SCLSP formulation in all circumstance except with an instance of a low number of products and low setup costs. The behavior of the reformulation is very sensitive against capacity limitations and setup costs. However, the current SCLSP formulation is less sensitive against capacity limitations and setup costs. From the results it is revealed that the reformulated SCLSP has a stronger linear relaxation based on numerical evidence than the existing SCLSP formulation.
34
8
References
Billington, P. J., McClain, J. O., & Thomas, L. J. 1983. Mathematical Programming Approaches to Capacity-Constrained MRP Systems: Review, Formulation and Problem Reduction. Management
Science, 29(10): 1126–1141.
Bookbinder, J. H., & Tan, J.-Y. 1988. Strategies for the Probabilistic Lot-Sizing Problem with Service-Level Constraints. Management Science, 34(9): 1096–1108.
Brandimarte, P. 2006. Multi-item capacitated lot-sizing with demand uncertainty. International Journal
of Production Research, 44(15): 2997–3022.
Buschkühl, L., Sahling, F., Helber, S., & Tempelmeier, H. 2010. Dynamic capacitated lot-sizing problems: A classification and review of solution approaches. OR Spectrum, vol. 32. https://doi.org/10.1007/s00291-008-0150-7.
Florian, M., Lenstra, J. K., & Rinnooy Kan, a. H. G. 1980. Deterministic Production Planning: Algorithms and Complexity. Management Science, 26(7): 669–679.
Gicquel, C., Minoux, M., & Dallery, Y. 2008. Capacitated lot sizing models: a literature review. Hal, 1– 23.
Harris, F. W. 1913. How Many Parts to Make at Once. JSTOR, 38(6): 947–950.
Helber, S., & Sahling, F. 2010. A fix-and-optimize approach for the multi-level capacitated lot sizing problem. International Journal of Production Economics, 123(2): 247–256.
Helber, S., Sahling, F., & Schimmelpfeng, K. 2013. Dynamic capacitated lot sizing with random demand and dynamic safety stocks. OR Spectrum, 35(1): 75–105.
Hopp, & Spearman. 2008. Factory Physics (3rd ed.). Waveland press.
Karlsson, C. 2016. Research methods for operations management (2nd ed.). New York.
Maes, J., & Van Wassenhove, L. N. 1986. A simple heuristic for the multi item single level capacitated lotsizing problem. Operations Research Letters, 4(6): 265–273.
Manne, A. S., & Science, S. M. 1958. Programming of Economic Lot Sizes. Management, 4(2): 115–135. Robinson, P., Narayanan, A., & Sahin, F. 2009. Coordinated deterministic dynamic demand lot-sizing
problem: A review of models and algorithms. Omega, 37(1): 3–15.
Rossi, R., Tarim, S. A., Prestwich, S., & Hnich, B. 2014. Piecewise linear lower and upper bounds for the standard normal first order loss function. Applied Mathematics and Computation, 231: 489–502. Schneider. 1981. the effect of service levels on order points or order levels in inventory models. Journal
of operational research.
Sox, C. R., & Muckstadt, J. a. 1997. Optimization-based planning for the stochastic lot-scheduling problem. IIE Transactions, 29(5): 349–357.
35
Tempelmeier, H. 2011a. A column generation heuristic for dynamic capacitated lot sizing with random demand under a fill rate constraint. Omega, 39(6): 627–633.
Tempelmeier, H. 2011b. No Title. Inventory management in supply networks, Problem, Models,
Solutions: 345.
Tempelmeier, H. 2013. Handbook of Stochastic Models and Analysis of Manufacturing System
Operations, vol. 192..
Tempelmeier, H., Helber, S., Sahling, F., & Buschk, L. 2009. Computers & Operations Research Solving a multi-level capacitated lot sizing problem with multi-period setup carry-over via a fix-and-optimize heuristic. Production, 36(May 2016): 2546–2553.
Tempelmeier, H., & Herpers. 2010. ABC A heuristic for dynamic capacitated lot sizing with random demand under a fill rate constraint. Omega, 48(6): 5181–5193.
Tempelmeier, H., & Hilger, T. 2015. Linear programming models for a stochastic dynamic capacitated lot sizing problem. Computers & Operations Research, 59: 119–125.
Tunc, H., Kilic, O. A., Tarim, S. A., & Eksioglu, B. 2014. A reformulation for the stochastic lot sizing problem with service-level constraints. Operations Research Letters, 42(2): 161–165.
