Stochastic lot sizing under mean-MAD information

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Stochastic lot sizing under

mean-MAD information

Master thesis, Technology and Operations Management University of Groningen, Faculty of Economics and Business

June 26, 2017

Anje van der Wijngaard Studentnumber: 3014452

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ABSTRACT

In this thesis, we develop a mixed-integer programming model and a dynamic programming model for the static stochastic lot-sizing problem. Our models build on domain, mean, and mean absolute deviation of the demand instead of full distributional information. This approach is interesting to study because it shows that full distributional information is not needed to efficiently model and solve the problem. The first numerical study shows the dynamic programming based on the algorithm of Vargas (2009) outperforms the mixed-integer programming model in terms of computational times. The second numerical study shows the value of having full distributional information is relatively small when the planning horizon is short, setup costs are high, and the cost ratio between holding and backlogging is small.

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PREFACE

This thesis is the end product of a journey I started half a year ago. Due to my lack of programming skills at the time, it initially seemed an interesting endeavor. However, it quickly turned into many long days of staring to my computer screen full of frustration, looking for a solution for yet another bug. Looking back, I am certain that I have never learned so much about mathematics and programming as I did in those few months. I need to express my greatest gratitude to Dr. Kilic, who spent so many hours helping me.

Special thanks to all my friends and family for their support throughout this time.

Groningen, June 26th 2017

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1 INTRODUCTION

The deterministic lot-sizing problem is a well-known problem. It involves deciding how many products to produce and when to produce them to minimize the costs for a finite

planning horizon of discrete time periods. These costs consist of setup costs and holding

costs. Fewer setups decreases setup costs but increases holding costs due to keeping large quantities in stock for a longer period of time. On the other hand, regular setups decrease holding costs because only small quantities are kept in stock, but they increase setup costs. Therefore, the objective is to minimize the total costs over the whole planning horizon (e.g. Harris, 1913; Whitin & Wagner, 1958). To solve the deterministic lot-sizing problem, full knowledge of future demand is needed. When future demand is random, the lot-sizing problem is called stochastic.

If the lot-sizing problem is stochastic, the expected total costs are often minimized. To calculate the expected total costs, the balanced costs are extended with the expected backlogging costs when inventory is too low to fulfil demand (e.g. Beale, 1955; Dantzig, 1955). Due to uncertain future costs, the core problem of the stochastic lot-sizing problem is to make decisions under uncertainty.

To assign probabilities to stochastic demand, the demand is often assumed to be normally distributed in previous literature (Helber et al., 2013; Rossi et al., 2015; Tempelmeier & Hilger, 2015). Checking if demand is normally distributed can be done by using historical data. However, historical data can be unreliable due to a limited sample size or it can be known to be non-comparable in advance (Gallego, 1992; Scarf, 1958). When this occurs, we are dealing with an ambiguous distribution (Knight, 1921).

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6 towards optimizing order policies for lot-sizing problems with stochastic demand (e.g. Gabrel et al., 2014; Jans & Degraeve, 2008; Karimi et al., 2003).

Since many probability demand distributions are assumed to be normal, most research makes use of this element. Recent examples include papers by Helber et al. (2013) and Tempelmeier and Hilger (2015) on capacitated lot-sizing for products with a normal demand distribution. To apply the findings of these papers in practice, extensive distributional information about the demand is needed.

To solve stochastic optimization problems in general, three main modeling techniques have been developed. The first is stochastic programming (SP). SP assumes that the probability distribution is known or can be estimated by historical data (e.g. Birge & Louveaux, 1997; Prékopa, 1973). The goal of SP, according to Shapiro and Philpott (2007), is “to find some policy that is feasible for all (or almost all) the possible parameter realizations”. The possible parameter realizations can also be called scenarios. The second modeling technique is robust optimization (RO). RO uses computational tools and processes optimization problems in which the data are uncertain but belong to an uncertainty set (Ben-Tal et al., 2004). An uncertainty set is a set of all possible values of the uncertain parameters. The objective is to obtain solutions which are guaranteed to be feasible for all possible realistic realizations (e.g. Ben-Tal et al., 2004; Ben-Tal & Nemirovski, 1998; 2002). The third modeling technique is distributionally robust optimization (DRO). DRO closes the gap between SP and the conservatism of RO by using the minimax stochastic-programming approach introduced by Scarf (1958) and Prékopa (1973). The goal is to determine the worst-case expectation of a given objective function. With DRO, a set of values are defined with a true distribution or a set of probability distributions are defined with the limited distributional information treated as uncertain (e.g. Delage & Ye, 2010; Gabrel et al., 2014; Wiesemann et al., 2014).

Ben-Tal and Hochman (1972) proved the worst-case distribution can be equal to a distribution with a maximum of three possible realizations of the joint distribution of the n random parameters (3n). This is the case when the objective function is convex and the limited distributional information consists of at least a mean, mean absolute deviation and domain.

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7 problems. Constraints for using the modeling approach from Postek et al. include having distributional information, as described by Ben-Tal and Hochman (1972), and convexity of the total-costs function.

These findings of Postek et al. (2016) are valuable to the stochastic lot-sizing literature because they can be used to solve the lot-sizing problem efficiently in a static uncertainty

setting given information about the domain and the mean of the demand. Within a static

uncertainty setting all decisions about timing and amount are made beforehand and no revisions are allowed (Bookbinder & Tan, 1988).

Applying the approach to the stochastic lot-sizing problem also means an exponential growth in possible realizations when the number of time periods in the planning horizon increases. This has as consequence that the computational effort required to calculate an optimal solution increases. Vargas (2009) introduced an algorithm to efficiently solve stochastic lot-sizing problems in a static uncertainty setting. The algorithm can be used to calculate the optimal order policy based on the cumulative demand between periods. Instead of processing all possible realizations individually, the algorithm of Vargas combines the same outcomes. This could have a decomposing effect and thus save computational effort.

It is important for multiple reasons to study using domain, mean, and MAD for the worst-case distribution in the DRO context for the static uncertain stochastic lot-sizing problem. First, as mentioned before, the historical data of the demand on which probability distributions are based is not always reliable (Gallego, 1992; Scarf, 1958). Second, when historical data is not reliable and a probability distribution cannot be assumed, it becomes hard to use the discussed modeling techniques due to the (almost) infinite number of possible realizations.

The aim of this study is to develop an efficient model for calculating an optimal order

policy (when to produce how many products) with minimal expected worst-case total costs for

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8 Because the lot-sizing problem involves a binary variable (setup performed or not in every time period), the logical way to solve the problem is using mixed-integer programming (MIP). MIP can be used to calculate each scenario individually. Another approach is dynamic programming (DP) as used by Vargas (2009) to decrease the amount of possible realizations. The research questions are as follows:

- What is the effect on computational efficiency when using DP instead of MIP for solving a static uncertain stochastic lot-sizing problem with mean-MAD information?

- What are the costs of having non-complete distributional demand information for a static uncertain stochastic lot-sizing problem?

A quantitative study is conducted to answer these questions. We have chosen to model the static uncertain stochastic lot-sizing problem using MIP and DP as used by Vargas (2009). The models are compared to provide an answer to the problem of computational efficiency. To analyze the effect on cost performance, the DP model is compared with a DP model which assumes full distributional information. Chosen is to assume a normal demand distribution.

The theoretical contribution of this study is to show that the distributionally robust stochastic lot-sizing problem can be efficiently modelled and solved by using mean-MAD information. The practical contribution of this study is a provision of managerial insights concerning the value of complete distributional information to the lot-sizing problem. In addition, managerial insights are obtained for how to deal with limited distributional information.

The structure of this thesis is as follows: Chapter 2 presents the theoretical background. Chapter 3 describes the methodology. The results of the numerical study can be found in Chapter 4 and the discussion in Chapter 5. Ultimately, a conclusion is drawn in Chapter 6.

2 BACKGROUND

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9 information for the circumstances in which future demand is unknown. First, the stochastic lot-sizing problem is discussed in detail. Next, the three main modeling methodologies are explained, which are currently used to solve optimization problems and applied to the stochastic lot-sizing problem. After the main modeling methodologies are described, the DRO methodology using mean-MAD information is explained. Finally, the reason the methodology is applicable to the stochastic lot-sizing problem is clarified.

2.1 Stochastic lot-sizing problem

The lot-sizing problem started with Harris’ (1913) recognition of the need to balance ordering costs and holding costs. After introducing the simple square-root formula for determining optimal economic order quantity, Wagner and Whitin also proposed a model to determine the optimal timing of ordering new inventory: the dynamic lot-size model (1958). Wagner and Whitin assumed a finite planning horizon and fixed variable demand for each future time period.

When demand for each period is not precisely known in real life, forecasts are used to determine the optimal order policy. An order policy consists of a decision concerning when to produce how many products. Early contributions were made by Askin (1981) and Silver (1978), who incorporated safety stocks if inventory is too low to meet demand.

However, when demand is stochastic and time-dependent due to seasonality and lifetime cycle, for example, it is called non-stationary demand (e.g. Morton & Pentico, 1995). Tunc et al. (2011) show that it is costly to use stationary policies given non-stationary demands. Therefore, forecasts could be expensive when the forecast and reality vary and thus possible solutions are non-optimal. This is supported by Ben-Tal et al. (2004), who show that small variations can have a large effect on the optimal order policy.

The researchers who started to investigate the stochastic lot-sizing problem started with the introduction of penalty costs (Beale, 1955; Dantzig, 1955). These costs, also called

backlogging costs, are incorporated in the model to include the situation in which expected

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10 introducing the use of service levels. One condition on the use of Bookbinder and Tan’s method is that the assumed demand distributions must be known. These distributions can be estimated by historical data.

Most previous studies assume a known demand distribution based on historical data and use the static uncertainty setting. Static uncertainty is a strategy in which the decision concerning when to produce what amount is fixed and made at the beginning of the finite planning horizon (Bookbinder & Tan, 1988). The two other strategies that could be used are the dynamic uncertainty strategy and the static-uncertainty strategy. The objective of the stochastic lot-sizing problem under static uncertainty is as follows: to minimize expected total costs by developing a fixed order policy that can be applied before the start of a finite planning horizon.

Recent examples where the demand distribution is assumed to be normal and the setting is static uncertain include papers by Vargas (2009), Helber et al. (2013), and Tempelmeier and Hilger (2015). Vargas uses an algorithm to find an optimal solution to the stochastic version of the Wagner-Whitin model. Helber et al. (2013) and Tempelmeier and Hilger (2015) focus on capacitated lot-sizing. Following these auteurs, we use the static uncertainty setting in this study.

2.2 Modeling methodologies

SP is described by Shapiro and Philpott as, “An approach for modeling optimization problems that involve uncertainty” (2007). When applying SP to the stochastic lot-sizing problem, the methodology assumes that the demand probability distribution is known or can be estimated by historical data (Birge & Louveaux, 1997; Prékopa, 1973) for all periods. The goal of stochastic programming, according to Shapiro and Philpott (2007), is “to find some policy that is feasible for all (or almost all) the possible parameter realizations”. The possible parameter realizations can also be called scenarios. Most stochastic programming models which are studied are two-stage problems according to Shapiro and Philpott (2007). In a two-stage problem, the decision is made in the first two-stage. After random events take place, actions are undertaken in a second stage to compensate for negative effects. Static uncertainty fits well in a two-stage problem.

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11 uncertain and known to belong to some uncertainty set”. The uncertainty set is a set of all possible values of the uncertain parameters. If all the constraints of the objective function hold for all values in the uncertainty set when a certain value is chosen for the design vector, a feasible solution is found (e.g. Ben-Tal et al., 2004; Ben-Tal & Nemirovski, 1998; 2002). To apply RO to an optimization problem, the uncertainty set must be established. If the uncertainty set is too big, the solution can be too conservative. Therefore, the drawback of the methodology is in the conservatism of the solution (Goh & Sim, 2010). Although a feasible solution can be found, it will probably not be the optimal solution.

DRO uses the minimax stochastic-programming approach. This approach was first introduced by Scarf (1958). While working on the inventory problem with unknown demand and known mean and variance, he stated that the optimal policy is a function of the total stock on hand and on order. When finding the minimum/maximum, the function is always convex/concave. This means that the optimal solution can be obtained by finding the minimum of the deviation of the function. Žáčková (1966) explains how optimal decisions can be found when worst-case probability distributions are used. According to Shapiro et al. (2009), the worst-case approach prepares the decision maker “for the worst possible outcome of the maximal cost”. Delage and Ye (2010) extend the approach by treating the mean and variance of the unknown parameter as further unknowns.

Applied to stochastic lot-sizing, solutions are derived from a worst-case demand probability distribution in which expected costs are maximized. The optimal solution is equal to the minimal worst-case expected costs over the whole planning horizon.

The advantage of this methodology is that it is possibile to use limited distributional information (e.g. Delage & Ye, 2010; Wiesemann et al., 2014).

2.3 Distributionally Robust Optimization Mean-MAD

Ben-Tal and Hochman (1972) proved the worst-case distribution can be equal to a distribution with three possible realizations with known probabilities. This is the case when the objective function is convex and the limited distributional information consists of at least mean, mean absolute deviation and domain.

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12 distribution can be equal to a distribution with a maximum of three possible realizations of the joint distribution of the n random parameters (3n_{). This is the case when mean-MAD}

information of the unknown parameter is known and the objective function convex. See Figure 2-1 for an example when the method is applied to the lot-sizing problem with weeks as parameter. The numbers on the bottom of the figure (3, 9, and 27) represent the total number of scenarios for the time periods.

Using this method for defining scenarios for DRO, the total scenarios are limited to a manageable number necessary for solving and are thus computationally trackable (Postek et al., 2016).

Figure 2-1: Three-point distribution

Applied to the stochastic lot-sizing problem, two conditions must be met to use the worst-case probability distribution as described by Ben-Tal and Hochman (1972).

The first condition requires knowledge of the mean-MAD information of the demand for every period in the planning horizon. The MAD can be estimated via the procedures described in, for example, Postek et al. (2015).

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13 minimum is equal to the optimal solution (Scarf, 1958). If the function is not convex, the minimum worst-case expected costs are more difficult to be found. Because the lot-sizing problem includes a binary variable (setup performed or not), the problem can be solved as a two-stage optimization problem. The binary variable is incorporated in the first stage. This leaves the remaining variables in the second stage continuous, thereby creating a convex total costs function.

Vargas (2009) introduced an algorithm for solving the static stochastic lot-sizing model. The algorithm consists of two stages. In the first stage, the optimal cumulative replenishment quantities are calculated for every possible sequence between setups. In the second stage, a sequence of setups that minimizes total cost is chosen.

2.4 Stochastic lot-sizing problem Mean-MAD

As stated previous, the aim of this research is to develop an efficient model for calculating an optimal order policy under mean-MAD information. For the stochastic lot-sizing problem in a static uncertainty setting with ambiguous distributions in every period, the three main modeling techniques are not favorable.

SP is difficult because the distributions are ambiguous. RO can be used because distributional information is not required, but the solution will be conservative. Due to a lack of information, a feasible solution will be found instead of an optimal solution. DRO can be used due its ability to handle uncertain distributional information, but because only the domain and mean of the demand in every period are known, many probability distributions are possible. Many probability distributions can cause long computational times.

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3 METHODOLOGY

This chapter is dived into two parts. The first part consists of an explanation of the models used for the numerical study. The second part describes how the models are used to collect data and perform the numerical study to analyze the performance of the models.

3.1 Mathematical models

Three models are built for the numerical study. The first model is a mean-MAD stochastic lot-sizing model which involves MIP. The second model is a DP model which can be based on either mean-MAD information or assumed normal demand distributions.

3.1.1 Mean-MAD mixed-integer programming model

To explain the stochastic lot-sizing model which is altered to a mean-MAD model in DRO context, we start from the beginning: the stochastic lot-sizing model. The objective of the static stochastic lot-size model derived from Shapiro et al. (2009), is as follows:

min 𝑥 ∑ E{𝑘 ∗ 𝑌𝑡+ ℎ(𝑥𝑡− 𝐷𝑡) +_{+ 𝑏(𝑥} 𝑡− 𝐷𝑡)−} 𝑇 𝑡∈𝑇 (1) s.t. 𝑥_𝑡 ≤ 𝑀 ∗ 𝑌_𝑡 𝑌_𝑡∈ {0,1} 𝑥_𝑡 ≥ 0. 𝑥+ _{= max {0, 𝑥}} 𝑥− = max {0, −𝑥} (2)

where k and h denote, respectively, the fixed setup costs and the holding costs. Furthermore,

Yt indicates whether a setup is performed in time period t and xt denotes the cumulative

produced quantity. Dt denotes the cumulative demand. E shows the uncertainty of the model.

ℎ(𝑥_𝑡− 𝐷_𝑡)+_{means that, if the quantity is higher than the expected demand in time period t,}

the holding costs must be paid. For 𝑏(𝑥_𝑡− 𝐷_𝑡)−_{counts, if quantity is lower than demand,}

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15 Postek et al. (2016) use mean-MAD information to find the minimal worst-case expected costs for multiple optimization problems. When their approach is applied to the static stochastic lot-sizing model with mean-MAD information, the objective is as follows:

min 𝑞 ∑ 𝑘 ∗ 𝑌𝑡+ ∑ 𝑃𝑠[∑ ℎ(𝑄𝑡− 𝐷𝑡 𝑠₎+_{+ 𝑏(𝑄} 𝑡− 𝐷𝑡𝑠)− 𝑇 𝑡∈𝑇 ] 𝑆 𝑠∈𝑆 𝑇 𝑡∈𝑇 (3) s.t. (2)

where s denotes scenario and P denotes the probability of each s. Qt and 𝐷𝑡𝑠 denote the

quantity and demand, respectively.

The model shows how the worst-case expected total costs are calculated for each scenario and multiplied with the probability of realizations. Minimizing the worst-case total costs can be achieved by changing the number of setups and the decision concerning how many products must be produced.

For the model, the mean-MAD information of each time period is needed to build distributions for the whole planning horizon. For a planning horizon of one period, the distribution consists of a maximum of three realizations. In this three-point distribution the possible realizations are equal to the minimum, the mean, or the maximum. When the planning horizon consists of multiple periods, every possible realization from the previous period can again have three possible realizations. For example, the distribution of a two-period planning horizon has 32_{possible realizations. This means that the number of maximum}

realizations is equal to 3n_{, where n is the number of time periods. Every possible realization is}

a scenario.

For every individual realization, the probability can be calculated with the formulas given by Ben-Tal and Hochman (1972). These are as follows:

𝑃𝑟𝑜𝑏_𝑚𝑖𝑛𝑖𝑚𝑢𝑚_𝑡 = 𝑀𝐴𝐷𝑡

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16 𝑃𝑟𝑜𝑏_𝑚𝑎𝑥𝑖𝑚𝑢𝑚_𝑡 = 𝑀𝐴𝐷𝑡 2(𝑚𝑎𝑥𝑖𝑚𝑢𝑚𝑡− 𝑚𝑒𝑎𝑛𝑡) (5) 𝑃𝑟𝑜𝑏_𝑚𝑒𝑎𝑛_𝑡 = 1 − 𝑃𝑟𝑜𝑏_𝑚𝑖𝑛𝑖𝑚𝑢𝑚_𝑡− 𝑃𝑟𝑜𝑏_𝑚𝑎𝑥𝑖𝑚𝑢𝑚_𝑡 (6) 0 ≤ 𝑀𝐴𝐷𝑡 ≤ 2(𝑚𝑎𝑥𝑖𝑚𝑢𝑚_𝑡− 𝑚𝑒𝑎𝑛_𝑡)(𝑚𝑒𝑎𝑛_𝑡− 𝑚𝑖𝑛𝑖𝑚𝑢𝑚_𝑡) 𝑚𝑎𝑥𝑖𝑚𝑢𝑚𝑡− 𝑚𝑖𝑛𝑖𝑚𝑢𝑚𝑡 (7)

To calculate the probability of realizing a scenario with multiple periods, the probabilities should be multiplied. For example, a lot-sizing problem with two periods where the probability of demand is equal to the minimum in the first period and equal to the maximum in the second period; the total probability of this scenario is 𝑃𝑟𝑜𝑏_𝑚𝑖𝑛𝑖𝑚𝑢𝑚₁∗ 𝑃𝑟𝑜𝑏_𝑚𝑎𝑥𝑖𝑚𝑢𝑚2. This distribution will be called the mean-MAD distribution further on.

Due to the binary variable Y, the problem is not a linear problem. For the numerical study, MIP is therefore chosen by using the commercial solver Gurobi. The model is called the MIP model further on.

3.1.2 Dynamic programming

The MIP model is capable of calculating an optimal order policy to minimize worst-case expected total costs. However, the algorithm of Vargas (2009) could have an effect on the computational time. The algorithm consists of two stages. In the first stage, the optimal cumulative replenishment quantities are calculated for every possible sequence between setups. In the second stage, a sequence of setups is chosen to minimize total cost.

To find the optimal replenishment quantities, the so called arc costs C must be optimal. The formula Vargas uses is as follows:

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17 which can be rewritten as,

𝐶(𝑆, 𝑖, 𝑗) = ∑ ℎ(𝑆 − 𝐷𝑡) + (ℎ + 𝑏) ∗ E(𝑆 − 𝐷𝑡)−, 𝑗−1

𝑡=𝑖

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where I is the replenishment period and j the last period before a new replenishment. S denotes the amount of production quantity received to period i.

Because the equation is convex and can be differentiated, the optimal solution for the optimal production amount S*, when assuming a normal distribution, is equal to

𝑆∗ 𝑛𝑜𝑟𝑚𝑎𝑙 = {𝑆: 𝐶′(𝑆, 𝑖, 𝑗) = 0}. (10)

The formula for C’ is as follows:

𝐶′(𝑆, 𝑖, 𝑗) = ∑(ℎ + 𝑏) ∗ F_𝑡(𝑆) − 𝑏

𝑗−1 𝑡=𝑖

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F(S) can be calculated with the cumulative distribution function for the normal distribution Ф. The formula is as follows with cumulative mean µ and cumulative standard deviation σ:

F(𝑆)_{𝑛𝑜𝑟𝑚𝑎𝑙} = Ф (𝑆−µ

𝜎 ). (12)

When the distribution is discrete, as in the case of the mean-MAD distribution, C’ could be non-zero and therefore another equation must be used. The optimal production amount is then equal to

𝑆∗

𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒 = 𝑚𝑖𝑛{𝑆: 𝐶′(𝑆, 𝑖, 𝑗) ≥ 0}. (13)

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18 𝑚₃(𝐷) = ∑ 𝑚₁(𝑣) ∗ 𝑚2(𝐷 − 𝑣)

∞ 𝑣=−∞

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with v as some value. By repeating this process for all possible realizations of cumulative demand and for every time period in the planning horizon, a distribution function is created for every possible realization of demand under mean-MAD information.

To calculate the cumulative distribution function F(S) for the discrete distribution, the formula is

F(S)𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒 = P{D ≤ 𝑆}. (15)

When an optimal S is established by differentiating the arc costs C for every order

cycle (i,j), the costs K can be calculated. K includes the setup costs k. The formula for K is:

K(𝑆, 𝑖, 𝑗) = 𝑘 + ∑ ℎ(𝑆 − 𝐷_𝑡) + (ℎ + 𝑏) ∗ 𝐸(𝑆 − 𝐷_𝑡)−

𝑗−1 𝑡=𝑖

. (16)

Assuming a normal distribution, the loss (E(S-Dt)-) can be calculated with the

probability density function φ and cumulative distribution function Ф for the normal distribution. With zt as variable equal to (s- µt)/ σt, the formula is as follows:

𝐸(𝑆 − 𝐷𝑡)−_{𝑛𝑜𝑟𝑚𝑎𝑙} = 𝜑(𝑧𝑡) − 𝑧𝑡∗ (1 − Ф(𝑧𝑡)). (17)

If the distribution is discrete, as in the case of the mean-MAD distribution, the loss in Equation 16 can be calculated as

𝐸(𝑆 − 𝐷𝑡)−_{𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒} = ∑ max{𝐷𝑡− 𝑆, 0} ∗ 𝑃𝑡 𝑊

𝑡=𝑤

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where w denotes the amount of possible realizations within the discrete distribution.

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19 While the MIP model has 3n_{scenarios (with n as the number of time periods) to be}

calculated individually, the DP model of Vargas looks at the cumulative production amounts rather than at each individual scenario. If multiple scenarios have the same cumulative production amount, the scenarios are combined and treated as one. This can cause a decomposition of the number of scenarios.

Vargas (2009) showed S(i,j) will always be smaller than S(i,j+1). This means the amount of production quantities received up to period i is always smaller when in period i products are produced for the periods till j instead of till period j+1. Therefore, the shortest path algorithm can be applied such that the sum of the costs are minimized.

The model based on a normal distribution is called the normal DP model further on. The model based on the mean-MAD distribution is called the mean-MAD DP model further on.

3.2 Numerical study

This sections describes how the models are used to collect data and perform a numerical study to analyze the performance of the models.

3.2.1 Design

Two sets are chosen for the numerical study. The first study compares the mean-MAD DP model with the MIP model. The study focuses on the difference in computation time due to a reduced number of possible realizations.

The second study compares the mean-MAD DP model with the normal DP model. The study focuses on cost performance. Differences in cost performance are believed to be found because the mean-MAD DP model minimizes the worst-case expected total costs and the normal DP model minimizes the expected total costs.

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20 Study 1 Mean-MAD DP - MIP Study 2 Mean-MAD DP- Normal DP Time periods T 1,2,3,4,5,6,7,8,9,10 5,10,15,20,25 Setup costs k 25,625,2500 25,625,2500 Backlogging costs b 2,5,10 2,5,10 Variance CV 0.1, 0.2, 0.3 0.1, 0.2, 0.3 Holding costs h 1 1

Table 3-1: Parameters numerical study

3.2.2 Data-input

To run the numerical studies, input is needed. As mentioned in the discussion of the theoretical background, the mean-MAD based models require a minimum demand, maximum demand, mean, and MAD for every time period (Ben-Tal & Hochman, 1972).

This information can be generated randomly. However, for the second study the mean-MAD DP model must be compared with the normal DP model. The normal DP model does not require a minimum and maximum, but a mean and standard deviation as data-input for every time period. To generate input which gives a comparable distribution for both models, the means for the normal DP model are extracted from the generated minimums and maximums. Table 3-2 reflects how the input is generated.

Input every time period Generating

Minimum Uniform distributed between 0 and 20

Maximum Uniform distributed between Minimum and 100

Mean µ Maximum + Minimum

2

Standard deviation σ µ ∗ 𝐶𝑉

MAD √2

𝜋𝜎 Table 3-2: Data-input

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4 NUMERICAL STUDY

This chapter provides an overview of the results of the numerical study. The first part shows the study in which the MIP and the mean-MAD DP models are compared. The second part shows the study where the mean-MAD DP and the normal DP models are compared.

4.1 Computational time

We built a MIP model. For the analysis of computational times and cost performance, the parameter settings are set as described in 3.2.2. To determine whether a decomposing algorithm has an effect on computational times, we built a mean-MAD DP model based on the Vargas algorithm. The parameter settings are set the same as for the MIP model. The results are presented in the Appendix (A). Table 4-1 shows a compact representation of the results where mmDP is an abbreviation for mean-MAD DP.

Solving time (sec) Total time (sec)

periods setup cv MIP mmDP MIP mmDP 2 25 0.1 0.09 0.05 0.72 0.61 0.2 0.09 0.05 0.72 0.63 0.3 0.09 0.05 0.73 0.60 625 0.1 0.07 0.06 0.68 0.58 0.2 0.08 0.05 0.69 0.59 0.3 0.08 0.05 0.71 0.58 2500 0.1 0.06 0.06 0.69 0.60 0.2 0.08 0.06 0.70 0.63 0.3 0.08 0.06 0.70 0.60 5 25 0.1 0.24 0.48 0.90 1.03 0.2 0.31 0.48 0.92 1.01 0.3 0.37 0.47 0.95 1.01 625 0.1 0.11 0.48 0.74 1.03 0.2 0.12 0.47 0.76 0.99 0.3 0.12 0.46 0.78 0.99 2500 0.1 0.08 0.48 0.72 1.01 0.2 0.08 0.47 0.73 1.00 0.3 0.08 0.46 0.74 1.01 10 25 0.1 15.67 2.89 75.74 3.16 0.2 19.49 2.87 78.61 3.12 0.3 20.86 2.85 80.38 3.11 625 0.1 19.67 2.94 78.92 3.18 0.2 25.11 2.91 83.28 3.14 0.3 27.73 2.89 85.83 3.16 2500 0.1 9.46 2.89 69.86 3.16 0.2 13.04 2.90 72.56 3.14 0.3 16.51 2.88 75.73 3.12

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22 The table differentiates between solving time and total time. Solving time represents the time how long it takes to calculate in which period a setup must take place and the quantities to produce to minimize the worst-case expected total costs. Total time represents the processing of the input data into scenarios and/or probability distributions, and the solving time.

As the table shows, the number of periods has an influence on solving time and total time. Figures 4-1 and 4-2 presents the computational times in graphs. Both figures with average solving time (Figure 4-1) and average total time (Figure 4-2) show how the computational time of the MIP model increases strongly when the number of periods increases. The computational time of mean-MAD DP model, on the other hand, increases slowly when the amount of periods increase. The figures shows a clear difference in computational effort between the use of the scenarios required to solve the problem (MIP model) and the use of cumulative demand to solve the problem (mean-MAD DP model).

4.2 Cost performance

We built a stochastic lot-sizing model which solves problems based on full distributional information. Parameter settings are based on the data-input described in 3.2.2. The model is a DP model based on the algorithm of Vargas (2009). The normal DP model. The model is needed to analyze the cost performance of the mean-MAD DP model. The solitary difference is the probability distribution itself.

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23 We present the results in the Appendix (B). Table 4-2 shows a compact representation of the results. The results show that different parameter settings have an effect on cost performance. The table excludes the parameter variance, because no effect on cost performance is found. See also Paragraph 4.2.1, which discusses the effects of the different parameters settings on cost performance. In the subsequent paragraph, the insights are presented.

Solving time (sec) Expected costs (€) Cost difference (%) periods setup b Mean-MAD DP Normal DP Mean-MAD DP Normal DP 5 25 2 0.5 0.1 206.7 177.2 16.6 10 0.5 0.1 309.5 229.1 35.1 625 2 0.5 0.1 959.2 942.8 1.7 10 0.4 0.1 1131.5 1080.2 4.7 2500 2 0.5 0.1 2834.2 2817.8 0.6 10 0.4 0.1 3006.5 2955.2 1.7 15 25 2 8.4 2.1 823.4 633.9 29.9 10 8.0 2.3 1268.2 883.8 43.5 625 2 8.5 2.1 2738.5 2650.1 3.3 10 7.9 2.3 3479.3 3218.8 8.1 2500 2 8.6 2.2 5344.7 5292.8 1.0 10 7.9 2.3 6597.0 6392.1 3.2 25 25 2 32.1 8.2 1626.4 1196.4 35.9 10 29.3 8.7 2533.4 1722.1 47.1 625 2 32.2 8.3 4689.6 4477.8 4.7 10 29.5 8.7 6045.1 5461.9 10.7 2500 2 31.9 8.3 9069.9 8923.2 1.6 10 29.4 8.6 10988.2 10521.6 4.4

Table 4-2: Results performance three-point Vargas and Vargas with normal distribution-50 replications

4.2.1 Effects of parameter settings

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24 We look first at the effect of a changing length of planning horizon on the reliability of the worst-case expected costs. The results are presented in Figure 4-3. When the difference between the costs of a planning horizon of five periods is 3.0 percent, the averages diverge when the planning horizon is extended. The difference at 25 periods is 8.2 percent. This means that the worst-case expected costs are closer to the expected costs when the planning horizon is shorter.

Figure 4-3: Average cost difference-periods Figure 4-4: Average cost difference-setup 0 10 20 30 40 50 0 1000 2000 3000 4000 5000 6000 7000 5 10 15 20 25 Co st diff er ence in % E x pect ed co st s in € Periods

Effect length planning

horizon

% Mean-MAD DP Normal DP 0 10 20 30 40 50 0 1000 2000 3000 4000 5000 6000 7000 25 625 2500 Co st diff er ence in % E x pect ed co st s in € Setup costs

Effect setup cost

% Mean-MAD DP Normal DP 0 5 10 15 20 25 30 35 40 45 50 0 1000 2000 3000 4000 5000 6000 7000 0.1 0.2 0.3 C o st diff er ence in % E x pect ed co st s in € Variance

Effect variance

% Mean-MAD DP Normal DP 0 5 10 15 20 25 30 35 40 45 50 0 1000 2000 3000 4000 5000 6000 7000 5 10 15 Co st diff er ence in % E x pect ed co st s in € Backlogging costs

Effect cost ratio

holding/backlogging

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25 The effect on cost differences of changing the setup costs is displayed in Figure 4-4. The graph shows that the cost differences decreases rapidly when the setup costs are set higher. Where a cost difference exists of 39.3 percent when setup costs are set at 25 euro, the difference is almost eliminated to 2.5 percent when setup costs are set at 2500 euro. This means the worst-case expected costs are closer to the expected costs when the setup costs are high.

The effect of changing backlogging costs is displayed in Figure 4-5. As the graph shows, the backlogging costs have an impact on the difference between the expected costs of both models. If the cost ratio between holding costs and backlogging costs increases, the difference becomes larger. With holding costs fixed at one euro per product and backlogging costs at two euro, the cost difference is 4.2 percent. Enlarging backlogging costs to ten euro per product shows a difference of 8.8 percent. This means the worst-case expected costs are closer to the expected costs when the ratio between holding and backlogging is small.

The graph in Figure 4-6 shows no trend on the effect of different variances on the cost performance of the worst-case expected costs.

4.2.2 Insights effects cost performance

The results show that different parameter settings can have an effect on the cost performance of mean-MAD stochastic lot-sizing models. The value of having complete distributional information decreases when the planning horizon is short, setup cost is high, and the cost ratio between holding and backlogging is small. For variance, no trend is found. This means variance is not influencing the value of having complete distributional information.

5 DISCUSSION

This study applies the three-point distribution approach to the stochastic lot-sizing problem to identify the consequences of using mean-MAD information. An important observation concerns the better computational performance obtained by using DP instead of MIP.

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26 DP model, on the other hand, shown faster computation times than the MIP model when the number of time periods in the planning horizon increases. Thus, computational efficiency becomes better when the number of possible realizations is limited.

In terms of solving capacity, the mean-MAD DP model outperforms the MIP. The maximum length of the time periods in a planning horizon the MIP model could solve is eleven. The mean-MAD DP model, on the other hand, could reach a planning horizon of 25 time periods with ease.

In the second part of the numerical study, the cost performance was shown to be dependent on different parameter settings. Differences between the worst-case expected costs and the expected costs are affected by the number of time periods in the planning horizon, setup costs, and the cost ratio of holding and backlogging costs. Because mean-MAD stochastic lot-size models use the robust MAD rather than the standard deviation, the indifference with respect to variance was expected.

5.1 Limitations

To simplify this study, some aspects of the lot-sizing problem have been excluded from it. First, the production capacity was assumed to be infinite. This means that there are no bounds on how many products could be produced in one period. An overview of models and algorithms for the stochastic lot-sizing model with limited capacity is given by Karimi et al. (2003). Second, we assume that setup, holding, and backlogging costs are known and are the same for every time period in a planning horizon.

The study can only be used to identify trends concerning the effects of the different parameters on the worst-case expected costs. The precise impact cannot be determined. The generated data-input consists of demand distributions with an absolute minimum between zero and 20 and an absolute maximum that is not bigger than the minimum plus 100. In real life, the demand distributions could be more alike or exactly the opposite. This could change the precise level of effect of the different parameter settings.

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27 5.2 Implications

The contribution to the theory of stochastic lot-sizing are two-fold. First, the use of mean-MAD in stochastic lot sizing is a computationally viable approach, as the mean-MAD DP can be used to solve extensive long-planning horizons in a short time. The mean-MAD DP model outperforms the MIP model, which uses large-scale scenarios. Second, the cost of using mean-MAD information rather than the full distributional information is increased when the length of the planning horizon includes more time periods, when setup costs are relatively low and when the cost ratio of holding and backlogging is relatively large.

From a managerial perspective, the findings are useful because they show that there is no need for complete distributional information. Using mean-MAD information provides a way to design an order policy. The costs of using limited information are small when the planning horizon is short, setup costs are high, and the cost ratio of holding and backlogging is small.

An example to apply the findings could be the production of spare parts. Due to limited production, full distributional information could be hard to obtain. The value of having full distribution is very small for this example when: the production schedule is planned ahead for a month and divided in weeks (4 time periods), setup times to prepare the manufacturing process are long and thus expensive, and due dates are flexible to avoid high backlogging costs.

6 CONCLUSION

The aim of this study was to develop an efficient model for calculating an optimal order policy with minimal expected worst-case total costs for circumstances in which future demand is unknown and information about the distribution limited.

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APPENDIX

A. Results computational times

Solving time Total time

periods setup b cv MIP mmDP MIP mmDP

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34 B. Results cost performance

Total time Expected costs Cost difference %

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Stochastic lot sizing under mean-MAD information