The role of risk preferences and flexibility for risk management: lessons from a custom manufacturing environment

Abstract

In this paper, we study the management of financial risks of a custom manufacturer in the specialty chemicals industry arising from increased volatility of profits caused by market uncertainty and growing competitive pressure due to globalization. We argue that such risk management can be established either by creating and utilizing flexibility or by adjusting the risk preferences. Using a model-based approach, we operationalize flexibility and risk preferences to show their effects on the profit/risk and the decision making of a firm through a stylized example motivated by the specialty chemicals business.

1 Introduction

In many industries, the last decade has seen an increase in competitive pressure and market uncertainty due to globalization. These effects are particularly observed in—and hence our research is motivated by—the market of specialty chemicals, where highly specialized companies are entrusted with the manufacturing of intermediates and active ingredients for the life science industry. As the name implies this market is per se very customer oriented and the requested products are generally for one customer exclusively.

Furthermore, the customers are typically large pharmaceuticals companies which exhibit a strong market power. As a consequence the contracts between the manufacturer and the customers traditionally feature a very high flexibility for the latter in terms of both demand quantity and delivery date and significant penalties for the manufacturer for failing to meet demand. Undoubtedly this increases customers’ loyalty and satisfaction but exposes the manufacturer to uncertainty and risk. ¹

More precisely, the delivery terms are in general comprised of two deadlines, namely the due-date, specifying the point in time when the manufacturer has to deliver the product to the customer, and the reveal-date referring to the point in time when the customer has to specify his final demand. Clearly the manufacturer will try to start production of a product only after the reveal-date to avoid potential overproduction under uncertainty. However, given the long setup and production times in this market and the capacity competition between different products this is not always possible and in the past production under uncertainty before the reveal-date was common for manufacturers to avoid the hefty shortage penalties. Thus, the operational strategy to deal with the uncertainty stemming from contractual agreements was to employ a make-to-stock (MTS) type production approach.

While for a long time the manufacturer could handle this situation and absorb the financial risk through large profit margins, recently the entry of low-cost manufacturers from south-east Asia as well as inefficient utilization of manufacturing capacity due to increasing demand uncertainty in general have led to an erosion of profit margins. Consequently the volatility of profits has also become a more pronounced issue and results in the need for a more explicit management of financial risks stemming from the demand uncertainties.

In this paper, we will argue that such a risk management can be established in two ways. First, a company can try to enhance its risk absorbing capacity by creating and—more importantly—efficiently utilizing flexibility. Particularly, the creation of flexibility per se is not a success guarantee for companies. In fact, in Bish et al. (2005) it was shown in a dynamic make-to-order environment that the inappropriate use of flexibility may actually have overall negative effects on a supply chain through swings in production leading to increases in order variability and higher inventory levels upstream. Such effects may also be the cause for the results of the empirical study reported in Pagell and Krause (2004) where no support for the causal relationship between increased flexibility—as a response to increased uncertainty—and increased performance was found.

Second, a firm can set its performance indicators to appropriately take into account risk. Flexibility, which is one of the strategic objectives in operations management is increasingly recognized by researchers and practitioners as an important performance measure of a company in general (see e.g. De Toni and Tonchia 2001). Further, in Chung et al. (2008) it is argued that incorporating risk as a measure for outcome performance is necessary to correctly understand the implications of different corporate strategies.

Using a model-based approach we will operationalize flexibility and risk preferences and show their effects on the profit/risk and the decision making of a firm through a stylized example motivated by the specialty chemicals business. Particularly the contribution of this paper lies (1) in highlighting the close relationship between flexibility and risk-aversion as measures to lower the financial risk in a custom manufacturing environment and (2) in showing the implications of this relationship on the actual decision making.

The remainder of this paper is organized as follows. In the next section we position our contribution in the existing literature by discussing some related work. Section 3 presents the general model including the operationalization of the flexibility and risk concepts and the mathematical model formulation proposed to address our research questions. Numerical results for a stylized example are provided in Sect. 4 and managerial implications are drawn in Sect. 5. The paper is concluded with an outlook on possibilities for future research in Sect. 6.

2 Review of related work

There exists a large body of literature on quantitative models dealing with uncertainty and flexibility in supply chain management. A model that is very similar to ours has recently been presented in Chung et al. (2008). They study a multi-item newsvendor model for a portfolio of products where for all products production can occur under uncertainty prior to the beginning of a selling season. Additionally, once the selling season has started and demands are assumed to be known reactive capacity can be used to fulfill demands that exceed the preseason production. Using an extension of the well-known newsvendor model the authors show analytically the optimal capacity allocation before and during the selling season. However, for analytical tractability it is assumed that the allocation of reactive capacity to the different products is done in the first stage as well. Thus, in the second stage the real decision is how much of the allocated capacity to actually use for the production of a product. There is no possibility for re-allocating capacity. Furthermore, there are no setup costs such that the result is that all products are produced in both seasons. Finally, the objective is to maximize expected profits and there is no consideration of risk aversion.

Our approach differs from this model by incorporating the effect of risk-aversion, explicitly allowing for an optimal allocation of the second stage production through recourse actions and by considering setup costs and times. In fact in our setting a strategy where production of all products in all periods occurs would be extremely inefficient and expensive.

With respect to the MTO characteristic of our model, a closely related work has been presented in Bish et al. (2005). They study the management of flexible capacity in a dynamic make-to-order environment, with the main focus being on the allocation of different products to different production plants. Depending on the flexibility of these plants—measured in terms of the product portfolio they can handle—different allocation policies are compared and it is shown that the (wrong) utilization of the flexibility can lead to production and inventory swings along the supply chain. This paper differs from our model by assuming the product portfolio as given. Further, while products are made-to-order they exhibit demands in every period and there are no reveal dates, thus eliminating the possibility of recourse. Finally, they also focus on risk-neutral decision making by maximizing expected profits.

In terms of the operational background of the underlying problem our paper is closely related to the study reported in Bonfill et al. (2004). In that paper, a batch scheduling problem with uncertain demands arising in the chemical industry is modelled and analyzed to evaluate the effect of risk-aversion and the advantage of the stochastic modelling approach over its deterministic counterpart in general as well as the reduction of solution variability under a risk perspective. However, rather than looking at the problem at the level of batch scheduling [as done in Bonfill et al. (2004)] we take a broader perspective and model the operational problem as a capacity allocation problem for a portfolio of products over time. In this sense our model is a rather sophisticated extension of the classical newsvendor model as applicable in the specialty chemicals business. Of particular importance are the consideration of (long) setup times between production batches and the two-stage structure implied by reveal- and due-dates.

To account for this our approach is based on twostage stochastic optimization consisting of a irreversible first stage decision and a second stage recourse action that depends on the realization of the uncertain variables. One of the first approaches using stochastic optimization in manufacturing is described in Eppen et al. (1989), where a capacity planning problem at GM is analyzed in a multiperiod setting. A similar early stochastic optimization model is proposed in Gupta (1993) to determine the number of machines as well as their degree of flexibility in a general multi-period capacity planning model with stochastic demands for several products. The degree of flexibility is measured in terms of the number of different products a machine can process. Setup times are neglected such that in each period a linear program is solved to determine the optimal capacity allocation. In Ahmed and Sahinidis(1998) a capacity planning problem in chemical process networks is studied using robust planning. Finally, very recently a strategic model focusing on the design and use of process flexibility in the automotive industry is presented in Francas et al. (2009). Process flexibility is given by the possible assignments of products to plants and used to deal with the changing demands over the lifecycles of the different car models. A more extensive review of capacity flexibility models is also provided in Francas et al. (2009).

In all these approaches the first stage decision is how much capacity to setup in general, while the recourse decision is how to optimally allocate this capacity to different products over time. In contrast to that, in our model the first and second stage decisions both relate to capacity allocation to the different products. Further, these models mostly assume risk-neutrality [with the exception of Eppen et al. (1989) where the expected downside risk is utilized as an optimization criterion], focus exclusively on one specific type of flexibility and consider the demand side as given whereas we also include demand side flexibilities in the form of contractual design with the customers in our model. Thus, our model encompasses a broader perspective on the problem of matching supply with demand under uncertainty and allows us to study the interaction of both demand and supply-side flexibilities as suggested in Vickery et al. (1999).

From a technical point of view, the approaches discussed above as well as our model are difficult to solve due to the presence of integer variables (caused e.g. by setup considerations). In fact, while twostage stochastic optimization in a convex setting is well understood the inclusion of integer variables leads to much higher complexity (see e.g. Ahmed 2004; Schultz 2003). Thus, it is common practice to employ sample average approximation (SAA), where the uncertain data are discretized and the optimal solution maximizing the average profit over the finite number of resulting scenarios is sought as an approximation of the true expected value of the continuous probability distribution (c.f. Ahmed and Shapiro 2002). By using scenarios the resulting problem can then be modelled and solved as an (integer) linear program.

Finally, concerning the consideration of risk preferences other than risk-neutrality our paper follows a rather recent stream of literature incorporating risk-aversion in classical models. A generalized approach combining risk-averse and risk-taking aspects in the newsvendor model has been proposed in Jammernegg and Kischka (2007). In Gotoh and Takano (2007) the newsvendor problem is modified to account for the Conditional Value-at-Risk (CVaR) and it is shown that for the basic version of the newsvendor a solution can still be analytically found. Unfortunately, this is no longer true for more complex extensions like the one treated in this paper, such that approximations (through e.g. demand scenarios) have to be used. As mentioned above the capacity planning model at GM (Eppen et al. 1989) utilized the expected downside risk while in Van Landeghem and Vanmaele (2002) the well-known mean-variance framework from Markowitz (1952) was employed. The CVaR was used in Bonfill et al. (2004) to evaluate the effect of risk-aversion in the batch scheduling problem in chemical industry.

In order to address different risk preferences at all a measure has to be chosen and from our discussion above it shows that there is no general agreement on which measure to use. However, in the seminal work of Artzner et al. (1999) the concept of coherent risk measures was introduced based on a number of widely accepted properties that ‘good’ risk measures should fulfill. It was subsequently shown that e.g. variance and the value-at-risk (VaR) lack some of these propoerties and hence do not belong to the class of coherent risk measures. On the other hand, CVaR, which generally speaking maximizes the expected value over a certain percentage of the worst outcomes of a profit distribution induced by the choice of some decision variables, is coherent. Its inclusion into the general twostage stochastic optimization framework was recently studied in Schultz (2005). Further, under certain conditions CVaR can be approximated linearly and thus lends itself for linear programming (see Rockafellar and Uryasev 2002).

3 Problem description and mathematical model formulation

3.1 General problem setting

Following the basic description of the problem environment in custom manufacturing given in the introduction we set up the following general model shown in Fig. 1 to study the risk management of a manufacturer in this business.

Fig. 1

Model framework

The two main operational processes we model are the contracting of new product requests from customers and the subsequent allocation of manufacturing capacity to the different accepted requests. While the former is typically handled by key account managers in the marketing or sales department the latter is dealt with by the production unit. Jointly these two processes will determine the financial success of the business operations. A key factor complicating both processes is the underlying demand uncertainty of the products, which leads to variability in the actual profits and introduces considerable financial risk. From the manufacturer’s perspective two levers can be adjusted to influence financial performance, namely setting the appropriate level of risk tolerance and creating and utilizing flexibility. Our main focus is on analyzing the associated strategies and their interplay.

As shown in Fig. 1 and discussed in more detail below, the core of our approach is an optimization model for capacity allocation which focuses on the optimal utilization of flexibility. Its inputs are the flexibility as such, which in our model stems from the contract design and will enter as a constraint and the risk preferences, which guide the decision making and thus constitute the objective function.

3.2 Capacity allocation model, flexibility and risk preferences

In order to get general insights into the risk management strategies of the custom manufacturer we use a simplified twostage capacity allocation model, which derives its structure from the flexibility to be studied. More precisely, we will focus on contractual flexibility which is created by the mutual agreement between the manufacturer and its customers on some sharing of the demand uncertainty. Specifically, as mentioned in the introduction, for each accepted product i the associated contract set up with the customer features a due-date T ^d _i at which the (uncertain) demand for this product is to be satisfied. In addition to that, a customer may agree to share some of their underlying demand uncertainty by placing a firm order at the so-called reveal-date T ^r _i , where T ^r _i ≤ T ^d _i .

Of course we do not claim that this is the only flexibility available in this setting. On the contrary, in Reimann and Schiltknecht (2009) we analyze the interdependence of operational and contractual flexibilities in this setting under risk-neutrality. However, to enhance the interpretabilty of the results on the interplay between flexibility and risk preferences in the current paper we restrict ourselves to a single type of flexibility.

In our simplified capacity allocation model, we assume that all products need to be manufactured on a single bottleneck plant with capacity C between now, time T ₀ and the end of the planning horizon T ₂, which corresponds to the due-dates of all products. We further assume that the reveal-dates of all products also coincide at one point in time T ₁, which lies between T ₀ and T ₂. Given this structure, the time horizon of our model is split into two phases as shown in Fig. 2 and the production capacity C on the bottleneck plant can be divided in capacity C ₁ before T ₁ and capacity C ₂ between T ₁ and T ₂, i.e. C = C ₁ + C ₂.

Fig. 2

Structure of the 2-stage model

Given our setting the focus of the capacity allocation decisions is on the question what to produce now (under uncertainty) and what to postpone to the second stage, i.e. after the reveal-dates at T ₁. Through the latter option, the capacity planners can utilize the contractual flexibility created by the key account managers when designing the contracts with the customers. In the literature this type of flexibility is sometimes referred to as time or postponement flexibility (e.g. Chod et al. 2004).

We operationalize it by looking at the capacity C ₂ available in the second stage, which will serve as a proxy for the contractual agreed upon lead time between reveal- and due-date. Clearly, when C ₂ increases the manufacturer cannot only react better to realizations of the uncertain variables, but also pro-actively improve its capacity allocation policy through improved decisions in the first stage. On the other hand, when C ₂ decreases the manufacturer not only has limited reactive ability in the second stage, but also needs to produce more under uncertainty. Thus, an increase in C ₂ implying earlier information availability clearly increases the manufacturers’ flexibility.

For our numerical analysis we quantify contractual flexibility as a percentage value of the total capacity C, i.e.

$$ \hbox{Contractual flexibility}=\frac{C_2}{C}. $$

A contractual flexibility of 0% then specifies the case where no advance demand information is available, i.e. all customers reveal their true demand only at the due-date. This case, where T ₀ < T ₁ = T ₂ mimicks a pure MTS system, where demand is always satisfied from inventory. On the other hand, a contractual flexibility of 100% implies that all the information is available at the beginning of the planning horizon T ₀. Here T ₀ = T ₁ < T ₂ and the resulting problem resembles a pure MTO system, where production only takes place for known orders. In our numerical study, we will use these two special cases as benchmarks for the more general setting T ₀ < T ₁ < T ₂ (as found in the specialty chemicals industry).

Unless contractual flexibility is 100%, the demand uncertainty will lead to uncertainty about the actual profits of a capacity allocation strategy. In that case comparing different strategies and finding an optimal one needs to be based on the risk preferences of the decision maker. As mentioned in the last section, most of the existing work in operations management has focused on risk neutrality by employing the expected profit of a strategy as an optimization criterion. Only recently has the notion of risk aversion been incorporated into the research in this area acknowledging that especially in capital intensive businesses (e.g. the market of specialty chemicals) limiting the potential losses under extreme demand scenarios may be more important than maximizing expected profits. While a large number of models and measures for risk in general and risk aversion in particular have been proposed, especially in the community of financial mathematics there is a relatively high degree of agreement about the desirable properties risk measures should fulfill (c.f. Artzner et al. 1999). These properties define the class of so called coherent risk measures. Based on these theoretical results we will use two members of the class of coherent risk measures to analyze the effects of different risk tolerance on decision making and profit/risk. To model risk neutrality we will focus on Expected Profits, while for studying risk aversion we will employ the CVaR.

Further, in Lüthi and Doege (2005) it was shown that on the basis of evaluating performance with coherent risk measures the value of flexibility is simply given by the difference in the optimal value of the risk measure under different flexibility settings. First, this definition emphasizes the importance of the optimal exercising of flexibility to obtain the true value of flexibility. We take this issue into account by formulating an optimization model for flexibility utilization which will be presented in the next section. Second and equally important, in our setting this result from Lüthi and Doege (2005) serves as a tool for the key account managers in determining the value of an increase in C ₂. Specifically, for any given level of contractual flexibility we can compute the marginal value of contractual flexibility (MVoCF), i.e. the value of an additional unit of the flexibility as

$$ \hbox{MVoCF}=\left\{{ \begin{array}{ll} \bar{\pi}(C_2+1)-\bar{\pi}(C_2) &\quad \hbox{under risk neutrality}\\ \hbox{CVaR}_{\beta}(C_2)-\hbox{CVaR}_{\beta}(C_2+1) &\quad \hbox{under risk aversion}\\ \end{array}}\right. , $$

where $\bar{\pi}(C_2)$ is the sample average of profits as given in Eq. 4 below, and CVaR _β (C ₂) is the approximate CVaR defined in Eq. 19.

As mentioned above, earlier demand information is clearly beneficial for the manufacturer. However, as it shifts market risk to the customers the latter may not be willing to commit to an earlier reveal date unless they are compensated financially. Thus, the consideration of reveal dates and hence the manufacturers’ flexibility and prices are interlinked. Moreover, whether the customer accepts a reveal date–price combination (or whether such a combination even coordinates the manufacturer–customer supply chain) depends on internal characteristics and considerations of the customer. However, while these are interesting issues they are outside of the scope of the current paper. Thus, in computing the MVoCF we assume that product prices and demands remain unchanged as C ₂ changes. Still, from the manufacturers’ point of view we will provide first insights into this link between reveal dates and product prices by using the MVoCF to determine the maximum possible price reduction that can be agreed upon with the customer for an additional day of delivery time. Results for this will be shown in Sect. 4.2.2.

3.3 Mathematical formulation

By exploiting the special decision structure of our simplified capacity allocation model and following the stream of literature within stochastic optimization employing SAA (Ahmed and Shapiro 2002) based on discrete approximations of the underlying distributions (see e.g. Engell et al. 2002) we can formulate the problem as a twostage stochastic mixed integer programming model.

Hence, in this paper, we assume that for each product i from the portfolio I = 1,...,N the uncertain demand D _i is given by a sample J = 1,...,M drawn from the underlying distribution with probability density $f_{D_i}$ and cumulative density function $F_{D_i}(\cdot)$. More precisely, for every scenario j ∈ J the tuple of the product-wise Demands D = (D ₁,...,D _N ) is sampled. Each of the M scenarios is then characterized by a tuple d _j  = (d _1j ,...,d _Nj ) of N demands and the demand for product i in scenario j is given by d _ij .

The per-unit price for product i is denoted by p _i , while the per-unit production costs are c ^p _i and shortages incur per-unit penalties given by c ^u _i .² Before production can start, each product requires a (sequence-independent) setup of length t ^s _i incurring setup costs of c ^s _i .

The decision variables in this model are then q _i denoting the quantities to be produced of a product i in the first phase before the revelation of new demand information (one decision for each product independent of any scenario) and q _ij denoting the quantities to be produced in the second phase when the exact demands are known (one decision for each product and scenario, i.e. N·M decisions). Further let z _i and z _ij be binary variables with the following interpretation:

$$ z_i=\left\{{ \begin{array}{ll} 1 &\quad \hbox{if a setup for product $i$ has been}\ {{\mathbf{completed}}}\ \hbox{in the first stage}\\ 0 &\quad \hbox{otherwise}, \end{array}}\right. $$

$$ z_{ij}=\left\{{ \begin{array}{ll} 1 &\quad \hbox{if a setup for product $i$ has been}\ {{\mathbf{completed}}}\ \hbox{in the second stage in scenario}j\\ 0 &\quad \hbox{otherwise}, \end{array}}\right. $$

Note that with respect to the twostage model depicted in Fig. 2 we assume that upon the arrival of new information at T ₁ the planned capacity allocation for the second phase can be adjusted with no additional costs. In particular any process (setup or production) that is currently assigned to the plant can be interrupted and a new task (setup) can begin immediately. On the other hand, if the information update requires no adjustment of the production plan, our model ensures a smooth continuation of any process at T ₁ by monitoring the status of the plant at the end of the first phase.

More precisely, $y_h,\ h \in I \cup$ {0} are binary variables, where y _h  = 1 if product h has been the last product either (partly) setup or produced in the first stage, and y _h  = 0 otherwise. Here the dummy product h = 0 is used to model that the plant may be left completely idle in the first stage.

For the product i, where y _i  = 1 two cases may occur. First, the setup has been completed (i.e. z _i  = 1) and production can start (if q _i  = 0) or continue (if q _i  > 0) without interruption in the second phase. Alternatively, the setup of product i was started but not completed at T ₁ (i.e. z _i  = 0). In this case the setup can be resumed at the beginning of phase 2 without interruption. To account for the partial completion of setups we need the following two variables. For the first phase, s _i denotes the degree of completion of a setup for product i, i.e. 0 ≤ s _i ≤ t ^s _i , while s _ij has the same interpretation for each scenario j in the second phase.

Before we can formulate our mathematical model we need to define some more auxiliary variables. Let w _ij denote the quantity which is actually sold of a product i in scenario j, i.e.

$$ w_{ij}=\hbox{min}[d_{ij},q_i+q_{ij}] $$

(1)

and let w ^u _ij denote the amount of shortage of a product i in scenario j, i.e.

$$ w^u_{ij}=\hbox{max}[0, d_{ij}-(q_i+q_{ij})]. $$

(2)

Finally, let x correspond to a setting of our variables (q _i , q _ij , s _i , s _ij , z _i , z _ij , y _h ) yielding a feasible solution with respect to the constraints (5)–(18), while X denotes the set of all those feasible variable settings.

Based on these data and variables the profit function Π(d _j ,x) of an arbitrary scenario j is in our model given by

$$ \Uppi(d_j,x)=\sum_{i \in I}\left[w_{ij}{\cdot}p_i-(q_i+q_{ij}){\cdot}c^{p}_i-\left(\frac{{s_i+s_{ij}}}{{t^s_i}}\right){\cdot}c^{s}_i-w^{u}_{ij}{\cdot}c^{u}_i\right] $$

(3)

which accounts for the portfolio revenue w _ij ·p _i minus production and setup costs as well as the penalties incurred in the case of shortages.

We will now expose our formal optimization models for risk-neutral and risk-averse decision making by looking at approximations of the expected profit and the CVaR, respectively.

3.3.1 Optimization under risk-neutrality

To account for risk neutral decision making, the optimization problem is given by

$$ \bar{\pi}(C_2)=\hbox{max}\frac{{1}}{{M}}\sum_{j \in J} \Uppi(d_j,x) $$

(4)

subject to

$$ \sum_{i \in I} \left(q_i+s_i \right) \leq {C}_1 $$

(5)

$$ \sum_{i \in I} \left(q_{ij}+s_{ij}\right) \leq {C}_2\quad\forall j \in J $$

(6)

$$ z_i{\cdot}{C}_1 \geq q_i\quad\forall i \in I $$

(7)

$$ z_{ij}{\cdot}{C}_2 \geq q_{ij}\quad \forall i \in I,j \in J $$

(8)

$$ z_i{\cdot}t^s_i \leq s_i\quad\forall i \in I $$

(9)

$$ z_{ij}{\cdot}t^s_i \leq s_i+s_{ij}\quad\forall i \in I,j \in J $$

(10)

$$ (z_{ij}-y_{i}){\cdot}t^s_i \leq s_{ij}\quad\forall i \in I,j \in J $$

(11)

$$ \sum_{h \in I \cup \{0\}} y_h=1 $$

(12)

$$ q_i+q_{ij} \geq w_{ij}\quad\forall i \in I,j \in J $$

(13)

$$ d_{ij} \geq w_{ij}\quad\forall i \in I, j \in J $$

(14)

$$ d_{ij}-(q_i+q_{ij}) \leq w^{u}_{ij}\quad\forall i \in I, j\in J $$

(15)

$$ y_0, y_i, z_i, z_{ij} \in \{0,1\}\quad\forall i \in I, j\in J $$

(16)

$$ q_i \geq 0, q_{ij} \geq 0, w_{ij} \geq 0, w^{u}_{ij} \geq 0\quad\forall i \in I, j \in J $$

(17)

$$ s_i \geq 0, s_{ij} \geq 0\quad\forall i \in I, j \in J $$

(18)

In this formulation the objective function (4) is given by the maximization of the sample average of profits which is used as an approximation for the true expected profit and accounts for risk-neutrality.

The feasibility of a solution is safeguarded by the following constraints. First of all constraints (5) and (6) guarantee that the production in the first and second stage and the corresponding (partial) setups respect the available capacity. Constraints (7) and (8) ensure that production of a product i may only take place when the associated setup has been completed.

It is easy to see that in the first stage production of a product i can start as soon as s _i ≥ t ^s _i , see Eq. 9, while for the second stage the situation is a little more difficult. Here two cases—modelled by constraints (10) and (11)—can occur: (1) either it is possible to take benefit of the last setup started in stage 1 (i.e. y _i  = 1) and it is not necessary to perform the full setup, or (2) if the last setup of stage 1 cannot be credited to product i it is necessary to perform the full setup in the second stage, i.e. s _ij = t ^s _i (see constraints (11) where y _i  = 0). For case (1) constraints (10) ensure that the sum of the setup times allocated in the first and the second stage equal the total setup time required for a product i. Obviously only one last setup can exist for the first stage and consequently Eq. 12 has to hold.

Constraints (13) and (14) model the minimum function for the sold quantities (see Eq. 1), while constraints (15) deal with the accounting of the shortages (see Eq. 2). Finally, constraints (16) represent the binary condition, while (17) and (18) enforce the non-negativity of the decision variables.

In the remainder of this paper we will use the term risk neutrality when referring to results obtained with this model.

3.3.2 Optimization under risk-aversion

The model presented in the last section finds the solution that maximizes the mean profit regardless of its variance and in particular the magnitude of losses incurred in some extreme scenarios. We will now show how the above mixed integer program (MIP) can be extended towards risk-aversion by introducing the minimization of the CVaR as objective for the optimization.

Conditional Value-at-Risk is defined as the expected value over the right tail of the loss function as visualized in Fig. 3. We stick to this convention and reverse the sign in our profit function (3) to obtain the loss function −Π(D,x).

Fig. 3

CVaR in profit and loss functions

As shown in (Rockafellar and Uryasev 2002), the CVaR _β of a loss function can be approximated by minimizing the convex and piecewise linear (in α) function

$$ \tilde{F}_\beta(x,\alpha)=\alpha+\frac{{1}}{{(1-\beta)}}\frac{{1}}{ {M}}\sum_{j \in J}\left[-\Uppi(d_j,x) -\alpha\right]^+, $$

where (1 − β) corresponds to the fraction of scenarios one wishes to take into account and α approximates the associated VaR of the loss function (see Fig. 3).³ Thus, $\tilde{F}_\beta(x,\alpha)$ is the mean value of all losses equal to or greater than α. Note that by varying β one obtains different levels of risk aversion. Particularly, if β → 1 decision making will tend to resemble a maximin-Strategy, i.e. a worst case optimization, whereas as β → 0 a more and more risk-neutral strategy is followed.

Using the auxiliary variables $\nu_1,\ldots,\nu_{M} \in {{\mathbb{R}}}$ we can rewrite the function $\tilde{F}_\beta(x,\alpha)$ and obtain a reformulation of our optimization problem (4)–(18) as follows

$$ \begin{aligned} \hbox{CVaR}_{\beta}(C_2)=&\hbox{min}\,\alpha+\frac{{1}}{{(1- \beta)}}\frac{{1}}{{M}} \sum_{j \in J} \nu_{j}\\ &\hbox{s.t. constraints (5)}{\rm --}\hbox{(18)} \end{aligned} $$

(19)

$$ \nu_{j} \geq -\Uppi(d_j,x)-\alpha,\quad j\in J $$

(20)

$$ \begin{aligned} &\nu_{j} \geq 0,\quad j \in J\\ &\alpha \in {{\mathbb{R}}}. \end{aligned} $$

(21)

Note that the minimization in this model is not only over our vector of decision variables x but also over α. As mentioned above α is an approximation of the VaR _β and the second part of the objective accounts for the mean excess of the losses over α in the (1 − β)·M worst scenarios. This is modelled through constraints (20) and (21), which ensure that ν _j ≥ 0 only if −Π(d _j ,x) ≥ α. Apart from that, the original constraints (5)–(18) are unchanged and α can take any real value.

Note finally that optimization of this model will provide an approximate CVaR _β for the loss function induced by reversing the sign in our profit function (3). As also visualized in Fig. 3 the associated risk measure for our profit function is simply the negative CVaR _β . Thus, we will refer to this negative CVaR _β in the remainder of this paper. Further, when discussing results of the model presented in this section we will use the term risk aversion to reflect the underlying risk preferences.

4 Numerical analysis

All the computational experiments reported on in this section were done using CPLEX as a solver for the MIP. The test case investigated is a stylized example taken from some real-data provided by our industrial partner to reflect the main properties of the real-world problem. In particular it is given by N = 5 products and M = 100 scenario-tuples. The contractual and operational parameters for the products are summarized in Table 1. ⁴

Table 1 Operational and financial parameters of the products

The product portfolio is characterized by two types of products differing with respect to their demand characteristics. Particularly, product 1 is modelled to be a new product. These products reflect a particularity of the specialty chemicals business, where customers place orders for intermediates needed for their own products without yet having them authorized by the regulatory authorities, like the Food and Drug Administration (FDA) in the USA. If the necessary authorization fails, production has to be stopped immediately and consequently the intermediates are no longer needed. The contract between manufacturer and customer thus features a cancellation clause, which allows the customer to withdraw the order at the reveal-date. In our model the realization of demand thus depends on two consecutive decisions: (1) In a first step it is determined whether or not any demand is placed and according to a given probability of cancellation (PoC) the demand is set to 0. (2) If an order is placed, in a second step the effectively realized demand is determined by sampling from an uniform distribution on the interval [d _min, d _max] (see Table 1). By default, the probability of cancellation is set to PoC = 0.5 for product 1. In exchange for such cancellation clauses the financial terms for these products are generally very favorable for the manufacturer and this is reflected in the high profit margin for product 1 as shown in Table 1. ⁵

All the other products are assumed to be established products modelled by a normal distribution, whereby mean μ and deviation σ vary with the different products and the associated distributions are truncated at zero. The demand scenarios are then obtained by generating random numbers from the corresponding distributions.

From Table 1 one can see that the products have very different operational and financial characteristics and it is not a priori clear how to compose the portfolio or how to resolve capacity allocation conflicts. The computational experiments which we report on in this section were designed to shed light into this issue and to show the effects and importance of risk management strategies in a custom manufacturing context. As mentioned in the introduction we distinguish between risk management through the use of flexibility and the adjustment of risk preferences. Concerning the latter we will distinguish between and compare the cases of mean profit maximization (as mentioned above termed risk neutrality) and CVaR minimization (termed risk aversion). To mimick risk aversion we set the parameter β = 0.9 in our mathematical model. This means that we look for an optimization of the CVaR _β over the 10% worst scenarios. As we will keep this value of β fixed we will omit the subscript and simply refer to the negative CVaR in what follows. With respect to flexibility we will consider contractual flexibility as mentioned in Sect. 2. To exclusively focus on the relationship between the two strategies and eliminate any potential capacity effects, we assume throughout our numerical study that the total capacity C of the plant is in general sufficient to produce all products. ⁶

4.1 Profit and risk implications of risk management strategies

Let us first investigate the profit and risk implications of the two risk management strategies. Figure 4 shows mean profit (solid lines) and negative CVaR (dotted lines) under risk neutrality (bold lines) and risk aversion (thin lines) for different levels of contractual flexibility.

Fig. 4

Comparison of mean profit and negative CVaR under risk neutrality, risk aversion and different levels of contractual flexibility

Comparing the dotted lines in Fig. 4 we observe a considerable risk reduction (measured in terms of negative CVaR) under risk aversion when contractual flexibility is below approximately 70%. The solid lines show the negative side effect of this risk reduction in the form of a decrease in mean profits. Both effects seem to be rather independent from the level of contractual flexibility, such that the ‘cost’ of risk reduction in terms of reduced mean profits is roughly constant. On the other hand, when contractual flexibility exceeds 70% there is little and eventually no difference in the performance under risk neutrality and risk aversion. This implies that under high levels of flexibility demand uncertainty is completely resolved by appropriate decision making and there is no risk associated with the capacity allocation decisions. This will also be shown and discussed below when we focus on the implications of the risk management strategies for decision making.

Focusing on the effect of flexibility on profit and risk for fixed risk preferences we observe from Fig. 4 that regardless of the risk preferences an increase in flexibility initially improves both the mean profit and the CVaR. When contractual flexibility exceeds 70% the maximum mean profit and minimum risk are reached and there is no further improvement. Comparing risk aversion with risk neutrality it seems that the effect of increased flexibility is somewhat stronger under risk aversion. This effect will also be discussed in more detail in the next section when we focus on the value of flexibility.

4.2 Decision making implications of risk management strategies

In this section we will focus on the effects of flexibility and different risk preferences on decision making in custom manufacturing. First, we will look at capacity allocation decisions in production. Then we will analyze the value of flexibility under different risk preferences. Finally, we will consider contract acceptance decisions.

4.2.1 Capacity allocation

We will start by looking at the optimal production quantities for the different products under both risk neutrality and risk aversion and three different levels of contractual flexibility. The corresponding results are shown in Fig. 5a–f. Decisions based on risk neutrality are shown on the left (Fig. 5a–c), while risk aversion is dealt with in Fig. 5d–f on the right. Moreover, the level of contractual flexibility is varied between 100 (in Fig. 5a, d), 0 (in Fig. 5b, e) and 30% (in Fig. 5c, f).

Fig. 5

Production decisions under risk neutrality, risk aversion and different levels of contractual flexibility

The bars in the Fig. 5a–f show for each product the setup (light grey) and average production times (dark grey) and the average idle time of the plant. The lengths of these bars are in correct proportions with each other and with the total capacity available (which is given by the length of the box in each Figure). For each product the percentage value beside the bar corresponds to the percentage of scenarios where production occurs. For the idle time the percentage value beside the bar shows the percentage of scenarios for which the plant is not fully utilized. Additionally we report the average idle time of the plant for each of the six cases (a)–(f) beside the corresponding figures.

Comparing Fig. 5a and d we observe that there is no difference. This is clear, as under a contractual flexibility of 100% all production takes place under complete certainty and there is no trade-off between risk neutrality and risk aversion. While product 1 with PoC = 0.5 is produced only in 49 of the 100 scenarios—i.e. whenever there is a positive demand—the other products are always produced (as shown by the 100% values next to the bars). As the total capacity is chosen such that it is possible to satisfy all demands in all scenarios, the produced quantities q _ij exactly correspond to the individual demands and consequently the means over the produced quantities (corresponding to the length of the bars) match with the means μ given in Table 1. The mean profit in this case is approximately 120 a.u. while the negative CVaR is nearly 70 a.u. This case exhibits characteristics of a MTO system, where each product is only produced upon explicit demand. Thus, there is never overproduction. Further, capacity utilization (or idle time) is different for each scenario and active strategies to improve capacity utilization may be taken in given scenarios with low demands for all products. Finally, note that based on our observation in the last section these effects occur not only for a contractual flexibility of 100% but as soon as the level of contractual flexibility exceeds approximately 70%.

The case where contractual flexibility is 0%, i.e. production has to be planned under complete uncertainty (i.e. Fig. 5b, e) is equivalent to an unconstrained multi-product newsvendor setting and thus can be solved by determining the optimal production quantities for each product separately. ⁷ In other words, for each product a fixed production decision is taken and the resulting capacity utilization (and hence idle time) is identical over all scenarios. We observe that (1) regardless of risk preferences the total production of each product is less than under complete certainty and (2) under risk aversion the production quantities are reduced slightly more than under risk neutrality. Both effects are due to the fact that in our setting overproduction is more costly than shortages and thus to reduce risk it is reasonable to produce less under uncertainty. Consequently, the capacity utilization in Fig. 5b and e is worse than in a and d as shown by the increase in the idle time. Also the financial performance is clearly worse in this situation when compared to the case of 100% contractual flexibility. Under risk neutrality the mean profit is −30 a.u. and the negative CVaR is less than −150 a.u., while under risk aversion the negative CVaR is slightly more favorable at −130 a.u. at the cost of a mean profit that is even worse at around −50 a.u. This case of 0% contractual flexibility resembles an MTS-like system, where all production occurs prior to the demand realization. Thus, the idle time only shows that the system capacity is underutilized in general, while at the same time there may be production shortages for some products and overproduction for other products. Thus, adding a new product (or order) to the portfolio will definitely improve capacity utilization at the cost of a potential increase in shortages, while average overproduction remains unchanged.

The practically most relevant and interesting case is shown in Fig. 5c and f where contractual flexibility is 30%. By comparing Fig. 5c and f with b and e we observe the following structural results regardless of the risk preferences:

• First stage production of product 2 is reduced and shifted into the second stage, when demands are known. ⁸

• In the second stage, product 1 is always produced when it was not cancelled (i.e. has positive demand).

In order to have this possibility for product 1, it is necessary to reserve capacity even though in half of the scenarios the corresponding capacity will have to be filled with production of other products (if possible/necessary). In principle, such a reservation is achieved by a proactive planning of the first stage. Here the risk neutral and the risk averse approach differ quite drastically. Under risk neutrality (Fig. 5c) product 5 is only and products 3 and 4 are mainly produced under uncertainty. A second campaign for product 3 or 4 does only pay off if demands are large enough and products 1 and 2 do not occupy too much capacity, which obviously only occurs in a limited number of scenarios. More precisely, in stage two products 3 and 4 are only produced in 7% of the scenarios. Incidentally the occurrence of these second campaigns and the required additional setups are the reason why the mean over the idle times is smaller in Fig. 5c than for the case when all information is known, i.e. Fig. 5a. The mean profit in this case is 80 a.u., while the CVaR is around 0 a.u.

Under risk aversion (Fig. 5f), we observe a different phenomenon. Product 5 is no longer produced under uncertainty and even in the second stage it is only produced when product 1 is cancelled or the demands of the other products are sufficiently low. As shown this occurs in 54% of the cases. Here capacity reservation for product 1 is achieved by a capacity sharing between two products where the low-margin product is only produced when the risky, high-margin product is not requested. This is also related to our observation about the introduction of a new product in scenarios with low demand under 0% contractual flexibility. Clearly, the capacity sharing effect between products 1 and 5 that occurs here shows that such a strategy may be beneficial. Product 5 is actively used as a capacity filling product and shortages of this product are accepted in high demand scenarios of the high-profit products. Clearly this is beneficial with respect to risk reduction as shown by the CVaR of around 20 a.u., while the mean profit in this setting is approximately 60 a.u.

4.2.2 Creation of flexibility

Having analyzed the effects of risk management on the production planning decisions let us now turn to decisions concerning the creation of flexibility. As shown in the previous section the increase in flexibility leads to a more or less pronounced risk reduction. In the following we will analyze how the willingness to create this flexibility depends on the risk preferences. To that end, Fig. 6 shows the MVoCF under both risk neutrality and risk aversion. The MVoCF, as defined in Sect. 3.2, can be regarded as the maximum price the manufacturer should be willing to pay for an additional unit of the flexibility, i.e. the maximum price reduction that can be granted to the customers for committing to their final demands earlier.

Fig. 6

Marginal value of contractual flexibility under risk neutrality and risk aversion

From Fig. 6 we first observe that this MVoCF decreases with increasing levels of contractual flexibility, i.e. there are diminishing returns for flexibility. While this is intuitively clear Fig. 6 also shows that unless contractual flexibility is already very high (>50%), the MVoCF is higher under risk aversion than under risk neutrality. The reason for this is strongly related to the production planning decisions discussed above. When comparing Fig. 5c and f one observes that under risk aversion it is tried to reduce the risk of overproduction, which is more expensive than shortages, leading to smaller production quantities under uncertainty. In return this rather conservative production strategy leads to a higher necessity to adjust production in the second stage, wherefore every additional recourse possibility—given by an increase of contractual flexibility—results to be more valuable under risk aversion than under risk neutrality.

Note that the increase in the MVoCF for contractual flexibilities between 30 and 50% is due to the setups in our model. More precisely, it suddenly becomes feasible to move the setup for product 1 into the second stage (for levels of contractual flexibility below 30% it had to be at least started under uncertainty in the first stage), thus in particular reducing the potential waste of a setup in case product 1 is cancelled and improving capacity utilization in the first stage in general.

4.2.3 Acceptance of contracts

To conclude this section we will show another decision making implication of risk aversion and once again contrast this strategy with risk neutrality. In particular we now relax the assumption that the product portfolio is fixed and consider the possibility of rejecting the new product 1. More precisely, for different levels of contractual flexibility we study the interplay between the risk associated with the standard portfolio (consisting of products 2, 3, 4 and 5) and the uncertainty associated with product 1 by looking at the impact of product 1’s cancellation probability PoC. These results are shown in Fig. 7.

Fig. 7

Portfolio profitability as a function of product 1’s cancellation probability

First observe that PoC = 1 corresponds to the case that there is never a demand for product 1, i.e. the product is definitely cancelled at its reveal-date. Of course such a contract would not make sense, however this case can be interpreted as the situation where product 1 is not at all in the portfolio and thus provides the benchmark negative CVaR associated with the standard portfolio. On the other hand, PoC = 0 implies that product 1 has already been approved and there will always be a positive demand. Between these two extreme cases it is not clear whether a demand is placed or not and Fig. 7 shows the risk implications of that. Starting from PoC = 1 a decrease of PoC first leads to a risk increase (a fall in -CVaR) before the risk decreases steadily (and -CVaR increases) as the PoC approaches PoC = 0. This effect stems from the capacity reservation for product 1 in the second stage between T ₁ and T ₂. If PoC is low the product will often be requested and the capacity reservation pays off. On the other hand, when PoC is high, less second stage capacity is reserved for product 1 and consequently the standard products benefit from the valuable capacity after the information update.

Figure 7 finally shows the break-even thresholds indicating when it starts to be favorable to accept product 1. Clearly, this is only the case when the -CVaR associated with the portfolio including product 1 is larger than the -CVaR of the standard portfolio as depicted by the horizontal break-even lines. Obviously the acceptance of product 1 is strongly tied to the contractual flexibility of the standard portfolio, in that an increased flexibility enables the manufacturer to accept contracts with a higher PoC.

Finally, let us now look at the acceptance decision of product 1 depending on the risk preferences of the manufacturer. Figure 8 summarizes these decisions and compares risk neutrality with the case of risk aversion just illustrated.

Fig. 8

Acceptance scheme for product 1 under different risk preferences

From Fig. 8 we observe that for any level of contractual flexibility product 1 is less likely to be accepted under risk aversion than when considering risk neutrality. Thus, regardless of the risk preferences a contract can always be accepted if it lies below the frontier induced by risk aversion, whereas it always has to be rejected if it lies above the frontier resulting from risk neutrality. However, between the two lines (white area) the acceptance of a contract crucially depends on the risk preferences of the manufacturer and the trade-off he is willing to make between risk and profit. This trade-off is particularly pronounced for medium levels of contractual flexibility (e.g. 30%) where the acceptance thresholds under risk aversion and risk neutrality lie almost at the opposite ends of the PoC spectrum.

5 Managerial implications

From our results presented in the last section we can draw two main managerial insights.

5.1 Incentives and decision making procedures in contracting and capacity allocation need to focus on flexibility

In the current practice, incentives for key account managers at our industrial partner lead to an emphasis on securing either profitable contracts for new products or follow-up contracts for established products. While the former is meant to increase the customer base or product portfolio in general, the latter focuses mainly on capacity utilization in manufacturing. In both cases there is some (informal) process of checking for production viability, it is however limited to average capacity availability. On the other hand, the introduction of contractual flexibility in terms of negotiating earlier demand reveal dates has not been an issue pursued actively. Particularly when the economy goes up and all customers request high quantities of the products this leads to severe capacity shortages and pressure on the capacity planning. The main incentives for the capacity planners, which center around a balanced capacity utilization thus lead to a capacity allocation procedure that tends to ignore reveal dates as well.

Our results suggest that while some capacity reservation for new products is optimal, the general procedure should be to exploit (contractual) flexibility and postpone production until (more) demand information is available, i.e. after the reveal dates. Clearly, this increases the likelihood of shortages such that for this strategy to be viable contractual flexibility needs to be actively dealt with by the key account managers. Moreover, the management of these contractual flexibilites has two desired effects. On the one hand, production planning is easier leading directly to positive implications for profit and risk. On the other hand, we have shown that given sufficient contractual flexibilities adding a risky product to the product portfolio may be feasible and lead to a further favorable financial impact.

5.2 Flexibility investments should be evaluated under a risk averse perspective

As mentioned above the negotiation of contractual flexibility constitutes an important strategy to enhance performance both in terms of profit and risk. Clearly, an increase in contractual flexibility will shift demand uncertainty back to the customer. In order for the customer to accept this, the level of flexibility has to be considered jointly with the price of the product. Our results provide the key account managers of our industrial partner with a tool to trade-off additional flexibility against price reductions. Moreover, they suggest that employing a risk averse approach the acceptable price reduction for earlier reveal dates may be larger as the value of flexibility is larger.

In general this implies that while risk aversion leads to a reduction in mean profits due to changes in decision making, it makes flexibility investments more likely and as a consequence the increased flexibility both improves mean profits and reduces risk.

6 Conclusion and outlook

In this paper, we have studied two management alternatives for dealing with financial risk stemming from market uncertainties in a custom manufacturing environment. These two alternatives, namely the creation and utilization of flexibility and the adjustment of risk preferences, were operationalized in a quantitative model to study their effects on both risk/profit and decision making. Through numerical results, based on a stylized example inspired by some real data from the market of specialty chemicals, it was shown that the two alternatives, while exhibiting positive returns individually, are strongly interdependent and unfold their full potential when employed jointly. Moreover, our results have helped our industrial partner to better understand the financial and operational implications of the trade-off between overproduction and shortages as well as to reconsider the organizational processes associated with managing their key accounts. On the other hand, our results have confirmed the validity of the established practice to aim at contractual flexibility of around 30%.

Clearly, the model presented in this paper is highly stylized and only presents a first step towards providing a decision guidance for risk management in a manufacturing context. However, it already covers aspects from a wide range of industries ranging from MTO to MTS environments. While these issues have not been studied in detail in our numerical example, we saw that all the effects described occur also in those two extreme cases although at different levels. For example, our results on contract acceptance suggest that in a MTS-like system companies need to be more prudent with respect to accepting new contracts. To properly answer these questions a more specific analysis of these two extreme cases is required. Our model provides a good basis for such a type of analysis, particularly it allows to also study mixed systems, where some products are MTO while others are MTS. We are currently researching in this direction.

Further, it was assumed that the operational process—in our case the capacity allocation—is guided purely by a financial objective. Hence, the financial risk could be directly measured. A possible extension of this work could deal with the hierarchical nature of performance measurement in companies, where strategic goals are operationalized and broken down into key performance indicators (KPI) applicable for the different business units. In this context it should be interesting how risk preferences can and should be reflected in non-financial KPI’s.

Finally, the interplay between reveal dates and product prices can be studied e.g. in a game-theoretic setting to analyze the implications of this relationship for the manufacturer-customer supply chain.