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Planning Production and Workforce in a Discrete-Time Financial Model Using Scenarios Modeling


Production planning interacts with a number of functional aspects within the entire system of a company. Two of those aspects involve workforce and financial planning, which are usually conflicting, in the sense that more workforce increases the financial costs but it may also increase production capability. In effect, there should exist an adequate balance on the production levels such that the cash-flows can generate the highest profits. This suggests that these three elements of the system (production, workforce and cash-flows) are tightly connected, and that they should be planned together in a single framework. This paper explores this interaction using a mixed integer linear programming formulation for a general outline of this three elements system. The model includes sequences of overlapping work-shifts and sequences of overlapping short-term loans. We discuss a number of scenarios concerning some of the uncertain parameters involved in the process, assuming that they do not follow any known distribution function, so handling uncertainty using a scenarios modeling methodology. Contrarily to usual lot-sizing problems, we consider demand as an upper limit for setting the production. So, we want to plan production and workforce in order to assist the cash-flows along the planning horizon, to the best convenience of the financial profits. We discuss a small fictitious example using a single-product with a homogeneous type of workers in the process. We then indicate a number of generalizations, including some real-world applications.


Production planning is a central theme of discussion within Management Science. It also concentrates the attention of the Operations Research community, namely on lot-sizing problems, to which an extensive number of scientific works and a large amount of real-world successful applications are available in the literature (see, e.g., [1, 9, 14, 18, 21]). Stochastic production planning and lot-sizing perspectives are analyzed in [8, 12, 22], formulations for a large list of lot-sizing problems are described in [4, 11, 19] and an extensive survey work on single-item lot-sizing problems can be found in [5].

The integration of cash-flows and workforce along the production process has also been discussed in [10, 13, 15,16,17]. The formulation proposed in [13] includes financial aspects to support labor costs and production. In addition, the model addresses a multi-item production process, also involving a multi-level system that incorporates material purchases to support production. Moreover, they consider hired workers, laid off workers, different types of workers and overtime hours for each class of workers. Other investments, apart from production, are also incorporated in the model. Mathematical formulations integrating these three elements system was also proposed in [16] and in [15], considering short-term loans to support labor costs with flexible amortization plans. The associated production process is single-product and single-level, using the given demand amounts as upper bounds for the effective sales, allowing the model to find the appropriate production quantities, to the best convenience of the company, in order to get the best financial profits.

Semi-continuous time-production perspectives were discussed in [2, 3], including financial planning and capital liquidity issues, all integrated in a single framework. As stressed by the authors, limited access to capital markets by small and medium sized companies (SMEs) requires bank loans to be used to initiate production and distribution, also stressing that the two systems (operations and finance) should be planned together in a short-term multi-day planning horizon, which also requires discrete-time modeling due to financial imbalances during short periods.

Uncertainty is incorporated in some of the mentioned works addressing production or lot-sizing problems, namely in [8] and [22]. In the first paper, the authors use scenario modeling techniques for discussing uncertainty in demand, while the second paper resorts to stochastic programming considering that the parameters involving costs and demand are uncertain. A minmax robust lot-sizing problem is also discussed in [12], exploring some lot-sizing variants using discrete scenarios with uncertain lead times.

In the present paper, we propose a mixed integer linear programming formulation that also attempts to handle these three planning processes in a single framework, acting together in a discrete-time stream. In our approach, the production process is not as comprising as the version discussed in [13], as it lies on a single-item and single-level basis. Our model combines the three mentioned processes, including short-term loans strategies, with the objective to maximize the final period cash-flows. In addition, we do not force the sales to entirely meet the demand, but using the demand level as an upper limit for the sales strategy, as in [16] and in [15]. Hiring strategies are modeled by a sequence of overlapping work-shifts. These work-shifts have different durations, bringing more flexibility to the hiring process than in the versions discussed in the former papers. We further consider uncertainty in three parameters involved in the model, namely on the demand, on the sales’ net profits and on the loans’ interest rates. Then, we use one of the scenarios modeling techniques described in [8] and in [7] for discussing these three elements’ uncertainty in the entire system. Another work in the literature that also integrates single lot-sizing, cash-flow constraints and loan strategies can be found in [6], addressing adequate balance on inventory level planning and customers goodwill on small online retailers acting on e-commerce platforms.

Contrarily to the approach described in [13] that discusses the financial process under high inflation, our model considers the problem in a financial environment that operates under no inflation. Actually, when looking to now-a-days Euro Area (or Eurozone) economic environment, it operates under very low inflation rates and it has been operating this way since the last decade ([23]).

The relevancy for discussing alternative and innovative methodologies to bring financial resources to SMEs is being stressed by the European Union institutions. In Europe, SMEs are the main source of economic growth and new jobs. These firms, however, often face problems finding money to finance their activity, develop new products or access new markets. In the present paper we propose a sequence of overlapping short-term loans, with different maturities, for covering the financial needs of companies when facing short cash availability. This sequence of overlapping loans can provide more flexibility to the companies’ needs and compromises, while delivering low shortage risks for the banks due to the short time maturities. This may constitute an additional contribution for bringing financial means to SMEs. We should recall that there are more then 25 million SMEs in Europe, by size ([24]). Although the vast majority of these enterprises represent micro-sized firms, the fact is that during 2017 SMEs employed over 94 million workers in the European Union, corresponding to 66% of the entire workforce, as observed by Daniel Clarke from Statista ([25]).

In the next section, we describe the integrated financial/workforce/production problem under discussion, to which a scenario based formulation is proposed in Section 3. A case oriented study is discussed in Section 4. The paper ends with a section on conclusions.

Financial/Workforce/Production Planning Problem

Given a stream of discrete time periods, ranging from period 1 to n, we want to describe production, workforce and cash-flows in each of these periods, along the entire time horizon. The production process is single-item and single-level, that is, it involves a single product on a single processing unit. However, the entire system can be seen on a two-level scheme. The first level involves a financial sequence, while the second one runs on a production stream. Capital to borrow and workforce are additional resources for sustaining the two processes. Figure 1 describes these three elements system in a general period t. The solid arrows represent cash-flows, while the dotted arrows represent production flows. The double line represents workforce coverage over production.

Fig. 1

Cash-flows, workforce and production interactions in a general period t

Considering the production stream, we know the demand in each period, which is not required to be fully attained, acting just as an upper limit for the sales, while assuming that the unsatisfied demand is lost, being accomplished by other companies in the market. We also consider, for each period, the unitary net profit of the sales, the unitary inventory cost of keeping the product in stock and the fixed production cost. The capacity of production is also known for each period. We assume that there is no initial stock of the product.

Workforce is required for covering the production. Thus, we have a sequence of time intervals defining overlapping shifts. Each of these sequences is characterized by work-shifts with a fixed time duration, lasting mi periods, for \(i \in {\mathscr{M}}\), where \({\mathscr{M}}\) represents the set of work-shifts’ classes under consideration in the model. Thus, considering the work-shifts of class \(i \in {\mathscr{M}}\), the first shift starts in period 1 and ends in period mi, then the second shift starts in period 2 and ends in period mi + 1, and so on. The last shift starts in period nmi + 1 so that no worker is on duty after period n. We want to determine the number of workers to hire for each shift such that the production is covered. In this case, we know the cost of each worker in each period (for the associated work-shift) and the productivity rate of a worker, that is, the number of units that a worker can produce in each period. We assume that this productivity rate is the same along the time and also the same among the workers in the various work-shift classes. However, the salaries to pay are different concerning the shift’s duration, decreasing along with the work-shift’s duration increase. The reason for this variation is related with the stability with the labor contract, that is, shorter time work-shifts are less desirable concerning work stability so we turn them more attractive for the workers accepting them. At the same time, shorter duration work-shifts provide more flexibility for the global hiring strategy of the company. Figure 2 provides a schematic representation of these sequences of overlapping work-shifts in the discrete time planning horizon.

Fig. 2

Sequences of overlapping loans and sequences of overlapping work-shifts along the planning horizon

The cash-flows are expected to interact with the previously described processes. They should pay production and workforce costs, while being fed by the sales’ profits. We can assume an initial cash income supplied by the shareholders, being available at the beginning of the process. In addition, the company can borrow a loan in every period of the planning stream, from a set of available loans’ classes, each one with a specific maturity, lasting hi periods, for \(i \in {\mathscr{H}}\), with \({\mathscr{H}}\) the set of loans’ classes being available in each period. Figure 2 provides a schematic representation of these sequences of overlapping loans during the entire planning horizon. Amortizations are paid in equal amounts along the hi period’s interval, for \(i \in {\mathscr{H}}\). We assume that both contracts and amortizations are made at the beginning of the associated period, and that all loans are entirely paid at the end of the planning horizon. Thus, the last period for borrowing is period nhi. There is a given interest rate for each of the loans starting in each period, assuming that interest rates increase along with the maturities enlargement, due to credit risk. We also consider an upper limit for the entire debt in each period. Cash-flows released outwards are also set, in a fixed amount, in each period of the planning horizon. Shortfalls in cash-balance are not permitted in all periods of the planning stream. Note that, as observed in [16], to concede shortfalls is the same as getting interest-free loans.

The objective is to maximize the cash-balance at the end of the planning horizon (period n).

Uncertainty is assumed to be related with three classes of parameters involved in the problem, namely on the demand, on the unitary net profits of the sales and on the interest rates associated with the loans. We further assume that these three classes of parameters do not follow any known distribution function. All other parameters are considered to be deterministic. Recall that product’s demand acts as an upper limit for the effective sales strategy. A similar version of the present study was discussed in [15], involving a single sequence of overlapping work-shifts with a single time duration and a single sequence of overlapping loans, with a single maturity’s time length. In the mentioned paper, the author assumes that the unitary net profits of the sales and the interest rates associated with the loans follow a normal distribution, conducting the entire study on a single scenario based approach. The present work handles uncertainty using a scenarios modeling methodology, proposing a mixed integer linear programming formulation that integrates in a single model various versions of the financial/workforce/production problem proposed, where each version represents a given scenario, covering the most relevant variations of the uncertain parameters involved in the model. In this case, we start considering a central scenario, involving the most probable settings for the uncertain parameters. Then, we consider optimistic and pessimistic additional states for each of these uncertain parameters. So, uncertainty is going to be explicitly incorporated in the model, being represented by a finite and discrete set of future scenarios \(\mathcal {S}\), where each scenario characterizes the value of all uncertain parameters, following a scenarios modeling methodology described in [7] and in [8]. We also assign individual probabilities to each scenario, denoting by pbs the probability of scenario \(s \in \mathcal {S}\) to happen, such that \({\sum }_{s \in \mathcal {S}} pb^{s} = 1\). The various scenarios are anchored in two decision aspects, namely on the number of workers to hire in each work-shift and on the capital to borrow in each loan, because they cover more than a single period, involving decisions that are hard to revert in practice. All the remaining decision aspects are assumed to be scenario dependent. The objective, then, becomes the maximization of the expected cash-balance at the end of the planning horizon.

Mathematical Formulation

In order to model the problem we start defining the sets, the parameters, the variables and then introduce the mixed-integer linear programming formulation that aggregates all the problems’ scenarios defined in set \(\mathcal {S}\). We use two letters for representing the parameters (apart from index ranges and sets) and a single letter for the variables. As mentioned above, n denotes the planning horizon, mi represents the work-shifts’ durations, for all classes \(i \in {\mathscr{M}}\); and hi denotes the loan’s maturities, for all classes \(i \in {\mathscr{H}}\). We assume that \(n > \max \limits \{m,h\}\), with \(m = \max \limits _{i \in {\mathscr{M}}} \{m_{i}\}\) and \(h = \max \limits _{i \in {\mathscr{H}}} \{h_{i}\}\).

Table 1
Table 2
Table 3
Table 4
Table 5
$$ \begin{array}{@{}rcl@{}} \text{maximize} & \displaystyle \sum\limits_{s \in \mathcal{S}} pb^{s} \cdot {v_{n}^{s}} \end{array} $$
$$ \begin{array}{@{}rcl@{}} \text{subject to} & 0 \leq u_{t-1}^{s} + {p_{t}^{s}} - {u_{t}^{s}} \leq d{m_{t}^{s}} , \ t=1,\ldots,n , \ s \in \mathcal{S} \end{array} $$
$$ \begin{array}{@{}rcl@{}} & {p_{t}^{s}} \leq cp_{t} \cdot {y_{t}^{s}} , \ t=1,\ldots,n , \ s \in \mathcal{S} \end{array} $$
$$ \begin{array}{@{}rcl@{}} & {p_{t}^{s}} \leq \displaystyle \sum\limits_{i \in \mathcal{M}} \displaystyle \sum\limits_{j=\max\{1,t-m_{i}+1\}}^{\min\{n-m_{i}+1,t\}} pr\cdot w_{ji} , \ t=1,\ldots,n, \ s \in \mathcal{S} \end{array} $$
$$ \begin{array}{@{}rcl@{}} & \displaystyle \sum\limits_{i \in \mathcal{H}} \displaystyle \sum\limits_{j=\max\{1,t-h_{i}+1\}}^{\min\{t,n-h_{i}\}} \textstyle \frac{j-t+h_{i}}{h_{i}} \cdot b_{ji} \!\leq\! dl , \ t = 1,\ldots,\max_{i \in \mathcal{H}} \{n-h_{i}\} \end{array} $$
$$ \begin{array}{@{}rcl@{}} & ic_{0} + p{s_{1}^{s}} \cdot ({p_{1}^{s}} - {u_{1}^{s}}) + \displaystyle \sum\limits_{i \in \mathcal{H}} b_{1i} = \\ & = ro_{1} + cs_{1} \cdot {u_{1}^{s}} + fc_{1} \cdot {y_{1}^{s}} + \displaystyle \sum\limits_{i \in \mathcal{M}} cw_{1i} \cdot w_{1i} + {v_{1}^{s}} , \ s \in \mathcal{S} \end{array} $$
$$ \begin{array}{@{}rcl@{}} & v_{t-1}^{s} + p{s_{t}^{s}} \cdot (u_{t-1}^{s} + {p_{t}^{s}} - {u_{t}^{s}}) + \displaystyle \sum\limits_{i \in \mathcal{H}} b_{ti} = \\ & = ro_{t} + cs_{t} \cdot {u_{t}^{s}} + fc_{t} \cdot {y_{t}^{s}} + \displaystyle \sum\limits_{i \in \mathcal{M}} \displaystyle \sum\limits_{j=\max\{1,t-m_{i}+1\}}^{\min\{n-m_{i}+1,t\}} \left( cw_{ji} \cdot w_{ji} \right) + \\ & + \displaystyle \sum\limits_{i \in \mathcal{H}} \displaystyle \sum\limits_{j=\max\{1,t-h_{i}\}}^{\min\{t-1,n-h_{i}\}} \left( \textstyle \frac{1 + ir_{ji}^{s} \cdot (j-t+h_{i}+1)}{h_{i}} \cdot b_{ji} \right) \\&+ {v_{t}^{s}} , \ t=2,\ldots,n , \ s \in \mathcal{S} \end{array} $$
$$ \begin{array}{@{}rcl@{}} & {p_{t}^{s}}, {u_{t}^{s}}, {v_{t}^{s}} \geq 0 , t=1,\ldots,n , \ s \in \mathcal{S} \end{array} $$
$$ \begin{array}{@{}rcl@{}} & b_{ti} \geq 0 , t=1,\ldots,n-h_{i} , \ i \in \mathcal{H} \end{array} $$
$$ \begin{array}{@{}rcl@{}} & {y_{t}^{s}} \in \{0,1\} , t=1,\ldots,n , \ s \in \mathcal{S} \end{array} $$
$$ \begin{array}{@{}rcl@{}} & w_{ti} \in \mathbb{N}_{0} , t=1,\ldots,n-m_{i}+1 , \ i \in \mathcal{M} \end{array} $$

The variable vt should be ignored in the equalities (7) for t = nh + 1,…,n. We have left them in the model in order to simplify the exposition. The set of constraints (2) model the production stream, where the amount sold in period t, represented by (\(u_{t-1}^{s} + {p_{t}^{s}} - {u_{t}^{s}}\)), is bounded by the demand (\(d{m_{t}^{s}}\)). Inequalities (3) impose an upper limit on the production in each period and for each scenario s, whenever production is on, which will activate the associated fixed cost. Also, constraints (4) relate each scenario production to workforce availability in each period t, considering the various work-shifts running in period t. These constraints are also bounding the production in each period and for each scenario. Then, inequalities (5) impose an upper limit (dl) on the sum of the debt in each period t, considering the various loans running in that period. Note that we do not need to impose limitations on the periods \(t > \max \limits _{i \in {\mathscr{H}}} \{n-h_{i}\}\), because they are dominated by the inequality defined for \(t = \max \limits _{i \in {\mathscr{H}}} \{n-h_{i}\}\). Further, constraints (6) and (7) describe cash-flow conservation, where (6) involves period t = 1 and (7) characterizes the remaining periods. In these equalities, we set all the cash income in the left-hand side and the outgoing cash is placed in the right-hand side. Considering the general constraint (7), the cash-income terms involve the cash-balance released from the previous period (characterized by \(v_{t-1}^{s}\), and being represented by the initial capital released by the shareholders (ic0) in equality (6)), plus the profits from the sales (characterized by \(p{s_{t}^{s}} \cdot (u_{t-1}^{s} + {p_{t}^{s}} - {u_{t}^{s}})\)), plus the capital arriving from the loans starting in period t (characterized by \({\sum }_{i \in {\mathscr{H}}} b_{ti}\)). Then, the outgoing cash terms involve the cash-flows released outwards (which is a fixed amount represented by rot), plus the inventory costs (characterized by \(cs_{t} \cdot {u_{t}^{s}}\)), plus the fixed costs on the production (characterized by \(fc_{t} \cdot {y_{t}^{s}}\)), plus the workforce costs with the workers on duty in period t (characterized by the term \({\sum }_{i \in {\mathscr{M}}} {\sum }_{j=\max \limits \{1,t-m_{i}+1\}}^{\min \limits \{n-m_{i}+1,t\}} \left (cw_{ji} \cdot w_{ji} \right )\)), plus the payments to be made in period t concerning the amortizations and interests involving the various active loans in that period (characterized by the term \({\sum }_{i \in {\mathscr{H}}} {\sum }_{j=\max \limits \{1,t-h_{i}\}}^{\min \limits \{t-1,n-h_{i}\}} \left (\textstyle \frac {1 + ir_{ji}^{s} \cdot (j-t+h_{i}+1)}{h_{i}} \cdot b_{ji} \right )\)) and the final term, characterized by variable \({v_{t}^{s}}\) represents the follow-on cash-balance that will leave period t and entering into the very next period.

As mentioned, the last double summation in equalities (7) involves both amortizations and interests to be paid in period t. In effect, the term \(\left (\textstyle \frac {1 + ir_{ji}^{s} \cdot (j-t+h_{i}+1)}{h_{i}} \cdot b_{ji} \right )\) could have been decomposed into the amortizations part, represented by \(\left (\textstyle \frac {1}{h_{i}} \cdot b_{ji} \right )\); and by the interests part, represented by \(\left (\textstyle ir_{ji}^{s} \cdot \left (\frac {j-t+h_{i}+1}{h_{i}}\right ) \cdot b_{ji} \right )\).

The objective function is to maximize the expected profits at the end of the planning horizon, defined by the last period cash-balances. The model includes continuous, binary and integer variables, together with the inhibition of shortfalls on cash-balance (\({v_{t}^{s}} \geq 0\)), as assumed.

We also remark that in the double summations involved in constraints (4), (5) and (7), the ranges of variation in the second (inner) summation are taking into account that work-shifts and loans have an initial period for setting-up, and that they are not supposed to keep running after the end of the planning horizon (period n).

It is also important to note that the set of constraints (2) do not follow the usual equalities form of lot-sizing flow conservation conditions. Instead, by setting those constraints as inequalities, we are not forcing the sales (\(u_{t-1}^{s} + {p_{t}^{s}} - {u_{t}^{s}}\)) to entirely fulfill demand (\(d{m_{t}^{s}}\)), allowing to sale below the demand, if it is more profitable. This aspect was discussed in [16], showing that the version that uses constraints (2) as equalities may increase significantly the operational costs, as it may force the solution to extra hiring and further produce to stock in order to reach the entire demand. This way, the relaxed version lets the solution find the adequate balance among costs and profits in order to maximize the cash-balance at the end of the planning horizon.

Another aspect to detach in the proposed formulation is that dividends are paid in fixed amounts in each period (if assuming that parameters rot are just dividends), which represents a modeling simplification. This situation can occur if the shareholders are investors and the fixed income is previously assumed in a contract. The usual procedure, however, is to pay dividends on an annual basis, right after ascertaining the company’s annual accounts. This more realistic version can be answered by substituting parameters rot by a new set of variables rt, representing cash-flows released outwards (e.g., dividends) in period t, for t = 1,…,n. If these variables represent dividends released annually and the periods represent months, then we should only consider variables rt for \(t=12,24,{\dots } ,tr\), with trn, removing all other rt variables from constraints (7). In that case, to model the annual payment of dividends and if dividends are calculated as a proportion of the sum of the profits accumulated along the year, then we should add to the model the following set of inequalities: \(r_{t} \geq \alpha \cdot \left ({\sum }_{j=t-11}^{t} p{s_{t}^{s}} \cdot \left (y_{t-1}^{s} + {p_{t}^{s}} - {u_{t}^{s}} \right )\right ), \text { for } t=12, 24, {\ldots } , tr \text { and } s \in \mathcal {S}\), considering that the new parameter α characterizes the mentioned proportion, with α a non-negative value possibly lower than 0.05. An alternative approach could involve the cash-balance variables at the end of each year, instead of relating dividends with the profits. These more realistic versions will be considered in a future research work.

Discussing an Application

In this section we propose a fictitious example, in order to simulate some aspects of the three elements problem (financial/workforce/production) described in the previous sections.

We consider a stream of discrete-time periods where each period represents a month and setting a 50 months planning horizon, thus, n = 50. We also consider two classes of work-shifts in \({\mathscr{M}}\) where the shifts in the first class last 6 months and those in the second class last 12 months, that is, m1 = 6 and m2 = 12, respectively. We also define two classes of loans in \({\mathscr{H}}\) with maturities of 3 and 6 months, that is, h1 = 3 and h2 = 6, respectively. In addition, we assume that there is no significant inflationary effect during the entire time horizon. In effect, if inflation is a relevant issue in the process, it should be reflected on the parameters, namely on the production and workforce costs, as cash-flows should rigourously respect capital balance between the periods.

Following the usual life time stream of a product, we consider that the entire time horizon (n) includes the usual four stages for the demand: introduction, growth, maturity and decline. In the present example, we assume that the first period demand is equal to 1000 units in all scenarios (\(d{m_{1}^{s}}=1000\), for all \(s \in \mathcal {S}\)), and that the product will be off-line at the end of the planning horizon. In addition and for simplifying the exposition, we also assume that all demand states-of-nature have the same time durations on the four demand stages, considering that the introduction term lasts 5 months, the growth term lasts 15 months, the maturity term lasts 20 months and the decline period will last 10 months. The rates of variation of the demand during these terms is described in Table 1 for the 3 possible states-of-nature: pessimistic, normal and optimistic. Figure 3 outlines the demand evolution for the three proposed states-of-nature, during the entire planning horizon.

Table 1 Rates of variation of the demand during the four terms, for the three proposed states-of-nature
Fig. 3

Demand evolution during the entire life cycle of the product for the three states-of-nature

Uncertainty on the unitary net profit of the sales will also be defined on the same three states-of-nature, considering 10 euros in the pessimistic scenario, 12 euros in the normal scenario and 14 euros in the optimistic case.

Other values and alternative states-of-nature for both demand and unitary net profits could be considered to further explore this example. In real practice, these values and states depend on the economic sector, geographical location and with the size and economic structure of the company in hands.

In most cases, demand and profits are related to each other, depending on the economic sector and other specificities of the company. In a broad sense, profits are the result of the selling prices minus the production costs. According to the economic Law of Demand, buyers tend to buy less when the prices increase, and vice-versa (see, e.g., [20]). On the other hand, production costs tend to decrease for higher production volumes, due to economies of scale. Once again, this relationship depends on the economic sector and other specificities of the company in hands. In this particular example, however, we assume that profits are more influenced by selling prices, and thus consider that demand should likely increase when profits are lower, and vice-versa. Thus, Table 2 establishes the probabilities of occurrence of each particular combination of the three states-of-nature proposed for the demand and for the unitary net profits.

Table 2 Probabilities of the 9 scenarios involving the three states-of-nature for demand and for the unitary net profits

Considering the values proposed in Table 2, and as mentioned above, the most probable scenarios involve high demand when profits are low and low demand when profits are high. Assuming that selling prices prevail over production costs.

The third class of uncertain parameters (interest rates) are assumed to run independently from demand and unitary net profits. In this case, we consider only two states-of-nature: pessimistic and optimistic, both starting with interest rates of \(ir_{t1}^{s} = 0.40\%\) and \(ir_{t2}^{s} = 0.80\%\), for t = 1. Then, in each one-year cycle, the interest rates increase 0.1% in the pessimistic scenario, while decreasing in the same amount in the optimistic scenario. This means that the interest rates are kept unchanged during the first 12 months in both scenarios. Then, from the 13th to the 24th month they increase to \(ir_{t1}^{s} = 0.50\%\) and \(ir_{t2}^{s} = 0.90\%\) (for t = 13,…,24) in the pessimistic state and decrease to \(ir_{t1}^{s} = 0.30\%\) and \(ir_{t2}^{s} = 0.70\%\) (for \(t=13,\dots ,24\)) in the optimistic counterpart, and so on. From months 37 to 48, the interest rates become \(ir_{t1}^{s} = 0.70\%\) and \(ir_{t2}^{s} = 1.10\%\) (for \(t=37,\dots ,48\)) in the pessimistic scenario and \(ir_{t1}^{s} = 0.10\%\) and \(ir_{t2}^{s} = 0.50\%\) (for \(t=37,\dots ,48\)) in the optimistic case; and in months 49 and 50 (if borrowing is still possible) they become \(ir_{t1}^{s} = 0.80\%\) and \(ir_{t2}^{s} = 1.20\%\) (for t = 49,50) in the pessimistic state and \(ir_{t1}^{s} = 0.00\%\) and \(ir_{t2}^{s} = 0.40\%\) (for t = 49,50) in the optimistic counterpart.

Because uncertainty in loans’ interest rates are even harder to predict, especially in the medium and long terms, we propose using sensitivity analysis in these parameters, considering

$$ \lambda \equiv \text{p{\kern-.1pt}r{\kern-.1pt}o{\kern-.1pt}b{\kern-.1pt}a{\kern-.1pt}b{\kern-.1pt}i{\kern-.1pt}l{\kern-.1pt}i{\kern-.1pt}t{\kern-.1pt}y{\kern-.1pt} o{\kern-.1pt}f{\kern-.1pt} t{\kern-.1pt}h{\kern-.1pt}e{\kern-.1pt} p{\kern-.1pt}e{\kern-.1pt}s{\kern-.1pt}s{\kern-.1pt}i{\kern-.1pt}m{\kern-.1pt}i{\kern-.1pt}s{\kern-.1pt}t{\kern-.1pt}i{\kern-.1pt}c{\kern-.1pt} s{\kern-.1pt}t{\kern-.1pt}a{\kern-.1pt}t{\kern-.1pt}e{\kern-.1pt} t{\kern-.1pt}o{\kern-.1pt} h{\kern-.1pt}a{\kern-.1pt}p{\kern-.1pt}p{\kern-.1pt}e{\kern-.1pt}n{\kern-.1pt},{\kern-.1pt} o{\kern-.1pt}n{\kern-.1pt} l{\kern-.1pt}o{\kern-.1pt}a{\kern-.1pt}n's{\kern-.1pt} i{\kern-.1pt}n{\kern-.1pt}t{\kern-.1pt}e{\kern-.1pt}r{\kern-.1pt}e{\kern-.1pt}s{\kern-.1pt}t{\kern-.1pt} r{\kern-.1pt}a{\kern-.1pt}t{\kern-.1pt}e{\kern-.1pt}s{\kern-.1pt} } (\lambda \in [0,1]) $$

assuming that the states-of-nature for loans’ interest rates are made just by these two states (pessimistic and optimistic).

The remaining parameters take the following values:

Table 8

To simplify the identification of the scenarios, we use the notation S(a, b, c), where a, b and c take the letters “p,” “n” or “o,” for pessimistic, normal and optimistic, respectively, excluding the normal state on c. The first argument (a) is associated with the demand, (b) is related with the unitary net profits’ states-of-nature and the third argument (c) is associated with the interest rates of the loans’ states.

We will start with an individual discussion of each of the 18 scenarios, giving more emphasis to the normal states-of-nature on demand and unitary net profits, considering the pessimistic state on loans’ interest rates. Then, we show that any of the proposed scenarios on loans’ interest rates have very low influence on the final objective function values. As a result, we choose one specific value for probability λ and use the scenarios aggregated model proposed in Section 3 for discussing these scenarios in a single framework, anchored in common decisions involving hiring and loaning strategies.

All the models were solved using ILOG/CPLEX 11.2 and all experiments were performed under Microsoft Windows 10 operating system on an Intel Core i7-2600 with 3.40 GHz and 8 GB RAM. When running the mixed integer programming (MIP) algorithm of CPLEX we used most default settings, which involve an automatic procedure that uses the best rule for variable selection and the best-bound search strategy for node selection in the branch-and-bound tree. We have set an upper time limit of 10,800 seconds for each test. The times are reported in seconds.

Tests on Individual Scenarios

We have solved each of the individual scenarios using the formulation proposed in Section 3, considering set \(\mathcal {S}\) to include just a single scenario at a time, among the 18 scenarios previously introduced. Table 3 shows the optimum/best solution values returned by CPLEX and the associated execution times (in seconds and in brackets), for all states-of-nature on the uncertain parameters. Times below 1 second are indicated by “(< 1),” Recall that the optimum/best values represent the final period cash-balance (variable \({v_{n}^{s}}\), in euros) for each scenario s. This is the maximum capital generated by the financial process, in line with the workforce and production requirements, obtained at the end of the planning horizon. We should also remember that the system is forced to pay 1500 euros every month to the shareholders, generating a total of 75,000.00 euros of cash-flows released outwards along the 50 months time horizon.

Table 3 Optimum/Best solutions of the 18 scenarios, taken individually

We have not been able to reach the optimum in scenarios Spop and Spoo, involving the pessimistic state of demand, the optimistic state on profits and in both pessimistic and optimistic states on loans’ interest rates. In these cases, the branch-and-bound stopped due to memory limitations, reporting the message “out-of-memory.” The duality gap reached was 0.08% in both cases. The correspondent values reported in Table 3 have an asterisk, representing the best lower bound obtained (best feasible solution objective function value) by the branch-and-bound.

Six brief observations before a detailed analysis indicate:


Differences among the two scenarios on the interest rates are not much significant, being even null in some cases;


Relevant differences are observed among the three states-of-nature involving demand;


Differences are even more significant among the states-of-nature involving profits;


The pessimistic scenario on the unitary net profits (with \(p{s_{t}^{s}}=10\)) was the only resorting to loans;


All these loans are class 1 (with 3 months maturities);


All the scenarios required the two classes of work-shifts, with more prevalence to class 2 shifts (cheaper and with 12 months durations).

We start observing the scenario involving the pessimistic state-of-nature on loans’ interest rates and the normal states on the demand and on the unitary net profit of the sales (with \(p{s_{t}^{s}}=12\)), represented in scenario S(n,n,p), reaching 254,919.00 euros cash-balance at the end of the planning horizon. The forthcoming figures describe the associated optimum solution. Figure 4 shows the cash balance and the sum of the cash-flows released outwards along the entire planning stream. Figure 5 represents the demand, the effective sales and the stocks along the same stream. Figure 6 compares the effective production amounts with the demand and the capacity of production, while Fig. 7 shows the number of workers arriving in each work-shift (in the bars), considering the two classes, and the line indicates the total number of workers in each period in the system, along the entire planning stream.

Fig. 4

Cash balance and sum of the cash-flows released outwards along the entire planning stream, in scenario S(n,n,p)

Fig. 5

Demand, sales and stocks along the entire planning stream, in scenario S(n,n,p)

Fig. 6

Demand, capacity of production and productivity along the entire planning stream, in scenario S(n,n,p)

Fig. 7

Number of workers arriving in each work-shift and the total number of workers in the system in each period, in scenario S(n,n,p)

Figure 4 shows a constant increase in cash-balance, especially after period 20, in which production reaches the highest stages. In addition, the graphics in Figs. 5 and 6 confirm that the sales are not entirely meeting the demand, while production and the stock is being planned to the best convenience of the company’s profits, regarding the capacity of production. In effect, the system could possibly produce more, namely during the first periods in which the capacity of production is much below the demand, using the stock to cover the needs between periods 21 and 46. However, that requires more workers to hire, which would possibly increase the costs, leading to a worse solution for the company. This loss would arise if constraints (2) are set as equalities. Actually, if constraints (2) are set as equalities in the model, then the problem becomes impossible for all the individual scenarios, except for the less restrictive cases (S(p,n,p), S(p,o,p), S(p,n,o) and S(p,o,o)), when demand is the lowest (pessimistic state) and the unitary net profits are on the normal or optimistic states-of-nature.

In addition, looking to the graphic in Fig. 7, we remark the shape of the line representing the total number of workers in each period. Actually, that line is much similar to the one representing the productivity in Fig. 6, suggesting that the proposed workforce truly covers the production needs. A close matching between these two lines can be seen if we multiply the workforce in each month by the productivity rate (parameter pr = 90). It is also interesting to observe that the work-shifts’ class 2 are preferred than those of class 1. Note that the class 1 shifts are shorter (6 months) than those of class 2 (12 months), providing higher flexibility for an adequate solution. However, the salaries of the longer duration shifts (class 2) are smaller, which may turn those shifts more attractive for the goal of the objective function in the model. Aside, this solution borrows no capital during the entire stream.

Now, alternatively to the previous case, we may observe one of the scenarios involving the pessimistic state-of-nature for the unitary net profits (this time with \({p_{t}^{s}}=10\)), being the only scenarios needing to borrow. They all start borrowing in the same month (8th) and stop borrowing in months 33, 22, and 16, for the pessimistic, normal and optimistic states of demand, respectively, closing the debts 2 months latter. To describe, we take one of those scenarios, say S(n,p,p), to stress some relevant aspects. Figure 8 shows the cash-balance and cash-flows released outwards along the planning stream; and Fig. 9 exposes the loaning strategy along the same stream, revealing the amounts borrowed in each loan (in the bars) and the total capital in debt in each period.

Fig. 8

Cash balance and sum of the cash-flows released outwards along the entire planning stream, in scenario S(n,p,p)

Fig. 9

Capital borrowed in each loan and the total capital in debt in each period, in scenario S(n,p,p)

A first observation shows that the solution prefers the shortest maturities’ loans, those in class 1. This is a natural option, because the shorter time lengths of those loans bring more flexibility for the solution to find the cheapest strategy, and also because their interest rates are more attractive. It is also worth to note that the solution resorts to the loans’ market only during the harder months in cash-balance, between periods 8 to 22. Recall that we have forced the solutions to avoid shortfalls. In effect, the process borrows during the harder months, pay all loans before period 25 and then starts growing along the remaining time stream, until reaching the highest profits at the end of the planning horizon, reaching \({v_{n}^{s}}\) = 15,018.57 euros, as shown in Fig. 8. This final result is higher than the initial capital released by the shareholders before starting the process (ic0 = 10,000 euros). This comparison should be corrected with any inflationary effect that may exist (excluded in our study).

It is also important to mention that other scenarios involving lower demand than that proposed for the pessimistic state may turn the problem infeasible, if all other parameters are kept unchanged. So, in this example, the pessimistic scenario of demand is close to the feasibility boundaries.

Integrated Scenarios Modeling Tests

Table 4 Probabilities (pbs) of the 18 scenarios involving the uncertain parameters

The previous discussion on individual scenarios showed that the uncertain parameters involving loans’ interest rates have very low (or even null) influence on the final solution values, for any λ ∈ [0,1]. In effect, it can be shown using sensitivity analysis that loans’ interest rates’ parameters make no change on the dominance relationship of payoffs, considering the other two uncertain parameters, which are clearly dominant on this discussion. This result is also immediate by observing the optimum solution values reported in Table 3. Thus, we propose λ = 0.5 for the forthcoming tests. As a result, we can specify each scenario individual probability, which is indicated in Table 4. *****

Now, considering the global formulation that integrates all the 18 scenarios in a single framework, proposed in Section 3, CPLEX was unable to reach the optimum due to size memory limitations, stopping after 3015.69 seconds of running time. The best solution found is equal to 228,981.06 euros, with a duality gap of 0.88%. This solution was the best among a number of tests using different strategies on the branch-and-bound settings of CPLEX. Considering this solution, the expected gain at the end of the planning horizon should be, at least, 228,981.06 euros. The two common issues to the various scenarios, addressing a common hiring policy and a common loaning strategy, are depicted in Figs. 10 and 11.

Fig. 10

Number of workers arriving in each work-shift and the total number of workers in the system in each period, in the scenarios’ integrated model

Fig. 11

Capital borrowed in each loan and the total capital in debt in each period, in the scenarios’ integrated model

Comparing the solutions in Figs. 7 and 10, we may state that the integrated model hiring strategy has some differences compared to the strategy found on scenario S(n,n,p). While both have stronger hiring periods in 12 months intervals, using class 2 contracts, which are cheaper, the difference from the other periods is more pronounced on the individual scenario S(n,n,p). This aspect is quite common among all hiring strategies on the individual scenarios, except on the versions involving the optimistic state on unitary net profits. In addition, class 1 hiring periods appear occasionally, allowing the solution to benefit from shorter hiring periods, namely close to the end of the time stream, as observed in the integrated approach. In general, the integrated framework seems to prefer hiring fewer workers in each period in order to adjust better all scenarios’ needs, while the individual scenarios can be more regular on the hiring process, possibly because diversity in each of these cases is lower.

On the other hand, comparing the results in Figs. 9 and 11, we can observe that the debt weight and extent are larger in the scenarios’ integrated model, when compared with scenario S(n,p,p), and also different from any of the individual scenarios’ solutions needing to borrow. In fact, the integrated solution starts borrowing in the 8th month and ends borrowing in month 46 (with a few exceptions), closing the debt only in month 48. Instead, the individual scenarios (just those involving the pessimistic state on unitary net profits), start borrowing in the same month (8th) and stop borrowing in months 33, 22, and 16, for the pessimistic, normal and optimistic states of demand, respectively, closing the debts 2 months latter. Actually, the pessimistic scenario of demand is forced to a higher capital effort compared to the other individual scenarios. However, the effort imposed to the integrated framework is harder, in order to encompass the entire set of states.

It is also interesting to observe that the production variables (\({p_{t}^{s}}\)) present almost the same solution values among the various scenarios in the integrated model, as a result of the common workforce policy. Instead, stock variables’ values (\({u_{t}^{s}}\)) vary significantly depending on the demand states-of-nature. Besides, the other two uncertain parameters (unitary net profits and loans’ interest rates) have no influence on inventory strategies, when looking to the integrated model solution.

We also detach each scenario final solution value (defined by variables \({v_{t}^{s}}\)) in the integrated model best solution. Those solution values are reported in Table 5.

Table 5 Each scenario final solution value (defined by variables \({v_{t}^{s}}\)) in the integrated model best solution

These individual solution values are lower than those reported in Table 3, for the same scenarios, being penalized by the restrictions imposed by the common strategies concerning hiring and loaning.

Although uncertainty leads us to find a good integrated strategy covering a range of variations of the three selected parameters, we cannot ignore that when taken individually, the extreme scenarios may have quite different solutions when compared with the aggregated strategy, despite their low probabilities to occur. In effect, we may be hiding relevant information about each individual scenario’s best policy, if only the aggregated strategy is taken into account. So, we suggest looking for both lines of results in order to strengthen the decision making process.


We have proposed a financial/workforce/production planning problem that incorporates sequences of overlapping work-shifts and sequences of overlapping loans, with different time durations for the work-shifts and different maturities for the loans. The discussion was conducted using a scenarios modeling methodology that integrates in a single formulation different realizations of three uncertain parameters, ranging from pessimistic to optimistic states-of-nature. The three uncertain parameters involve the demand, the unitary net profits and the loans’ interest rates. The integrated model is anchored on the hiring and on the loans variables, so making the model to cover the various states while using common resources on the hiring and loans policies.

When looking to the various scenarios individually, the solutions showed how to produce, how to hire and how to borrow (if needed), in order to get the best profits at the end of the planning horizon. These solutions indicate that variations on the unitary net profits exert more influence than changes on the demand; and on its turn, changes on the demand influences much more the final gains than variations on the interest rates of the loans. In fact, the solutions required these loans only under the pessimistic scenarios for the unitary net profit of the sales. In addition, the hiring policies preferred work-shifts with longer time durations, probably due to their smaller salaries’ costs.

Then, following a different methodology, we have integrated in a single model all the scenarios’ formulations, sharing two common sets of variables involving hiring and borrowing strategies, handling the problem under a scenarios modeling perspective. In this case, we have obtained a global feasible picture of the entire system, when seen in an aggregated form, considering given pessimistic to optimistic realizations of the uncertain parameters under given probabilities to happen. This solution proposes the need to borrow during a longer time range and on a heavier amount, when compared with the borrowing strategies obtained for the individual states that also resort to loans. Also, the integrated solution proposes a smoother hiring policy, when compared with the individual scenarios’ solutions in which a large number of workers are hired in 12 months cycles, especially on the pessimistic and normal states-of-nature for the unitary net profits. In addition, due to the common workforce policy, the aggregated model solution proposes very similar production strategies for all the scenarios, although inventory (stock) quantities vary, depending on changes on the demand.

In this approach, we have assumed a given set of probable scenarios on three specific uncertain parameters involved in the problem, also assuming given probabilities for the various scenarios they produce. In fact, scenarios modeling methodologies are strongly dependent on these assumptions, which may limit the robustness of its answer if significant changes happen during the implementation of the results. In those cases, the initial assumptions should be revised and a new solution should be found in order to track the market behavior the best way. In practice, these assumptions are problem specific and they should reflect the decision maker perspective of the market in which the company is involved in.

The main motivation of the present paper is to bring an additional mathematical programming based tool for planning production, workforce and some relevant aspects involving cash-flows, exploring their interaction in a single framework. Using a small fictitious example, we have made a few steps pursuing the discussion of the entire system. Naturally, the model can be extended to larger dimensional and more complex problems, including additional features in the various streams involved. Concerning the production stream, we may think about multi-item, multi-level and backlogging issues, while in the workforce stream we may consider different types/skills of workers; and in the financial stream we may also include inflationary effects, Governmental financial support to promote employment, payment delays, more realistic approach for the payments of dividends, among others.

The option to handle the production process on a single-item and single-level basis is only to simplify the discussion, in order to emphasize the trade-off relationship among the three processes. As mentioned before, the entire system can be enlarged with the various features stressed in the previous paragraph.

In this context, we were involved in two real-world applications comprising production, workforce and financial planning. One was on an industrial company from the wiredraw sector. It involved different production lines (multi-item) and an homogeneous type of workers in the production process. The second application involved a new company from the tourism sector with different activities and different types of workers, requiring an initial investment of about 5 million euros. These cases were crucial to understand the real impact of the model in practice and help decision makers to better understand the steps ahead, while providing additional information to further complement the decision making process.


  1. 1.

    Absi N, Detienne B, Dauzère-Pérès S (2012) Heuristics for the multi-item capacitated lot-sizing problem with lost sales. Comput Oper Res 40 (1):264–272.

    Article  Google Scholar 

  2. 2.

    Albrecht W, Steinrücke M (2017) Continuous-time production, distribution and financial planning with periodic liquidity balancing. J Sched 20 (3):219–237.

    Article  Google Scholar 

  3. 3.

    Albrecht W, Steinrücke M (2020) Assessing site integration into semi-continuous production, distribution and liquidity planning of supply chain networks. EURO J Transp Logist 9(1):100002.

    Article  Google Scholar 

  4. 4.

    Belvaux G, Wolsey LA (2001) Modelling practical lot-sizing problems as mixed-integer programs. Manag Sci 47(7):993–1007.

    Article  Google Scholar 

  5. 5.

    Brahimi N, Absi N, Dauzère-Pérès S, Nordli A (2017) Single-item dynamic lot-sizing problems: An updated survey. Eur J Oper Res 263 (3):838–863.

    Article  Google Scholar 

  6. 6.

    Chen Z, Zhang RQ (2019) A cash-constrained dynamic lot-sizing problem with loss of goodwill and credit-based loan. International Transactions in Operational Research, early view.

  7. 7.

    Dembo RS (1991) Scenario optimization. Ann Oper Res 30:63–80.

    Article  Google Scholar 

  8. 8.

    Escudero LF, Kamesam PV, King AJ, Wets RJ-B (1993) Production planning via scenario modelling. Ann Oper Res 43:311–335.

    Article  Google Scholar 

  9. 9.

    Figueira G, Santos MO, Almada-Lobo B (2013) A hybrid VNS approach for the short-term production planning and scheduling: A case study in the pulp and paper industry. Comput Oper Res 40(7):1804–1818.

    Article  Google Scholar 

  10. 10.

    Fisk J (1980) An interactive game for production and financial planning. Comput Oper Res 7(3):157–168.

    Article  Google Scholar 

  11. 11.

    Graves SC (2002) Manufacturing planning control. In: Pardalos P., Resende M. (eds) Handbook of Applied Optimization. Oxford University Press, New York, pp 728–746

  12. 12.

    Hnaien F, Afsa HM (2017) Robust single-item lot-sizing problems with discrete-scenario lead time. Int J Prod Econ 185:223–229.

    Article  Google Scholar 

  13. 13.

    Kirka Ö, Köksalan M (1996) An integrated production and financial planning model and an application. IIE Trans 28:677–686

    Article  Google Scholar 

  14. 14.

    Loparic M, Pochet Y, Wolsey LA (2001) The uncapacitated lot-sizing problem with sales and safety stocks. Math Program 89:487–504.

    Article  Google Scholar 

  15. 15.

    Martins P (2016) Computational Management Science LNEMS 682. In: Fonseca RJ, Weber G-W, Telhada J (eds).

  16. 16.

    Martins P, Quelhas AP (2016) Workforce planning and financing on a production/capital discrete-time model. Int Trans Oper Res 23 (3):507–538.

    Article  Google Scholar 

  17. 17.

    Muñoz M.A., Ruiz-Usano R, Framiñán JM, et al (2000) A mathematical programming model for the integration of aggregate production planning with short-term financial planning. In: Proceedings of the First World Conference on Operations Management (POM 2000), Sevilla, Spain

  18. 18.

    Pastor R, Altimiras J, Mateo M (2009) Planning production using mathematical programming: the case of a woodturning company. Comput Oper Res 36 (7):2173–2178.

    Article  Google Scholar 

  19. 19.

    Pochet Y (2001) Mathematical programming models and formulations for deterministic production planning problems. In: Jünger M, Naddef D (eds) Computational Combinatorial Optimization LNCS 2241. Springer, Berlin, pp 57–111

  20. 20.

    Samuelson PA, Nordhaus WD (2010) Economics, 19th edn. McGraw-Hill International Edition, New York

    Google Scholar 

  21. 21.

    Toledo CFM, de Oliveira RRR, França P M (2013) A hybrid multi-population genetic algorithm applied to solve the multi-level capacitated lot sizing problem with backlogging. Comput Oper Res 40(4):910–919.

    Article  Google Scholar 

  22. 22.

    Zhao C, Guan Y (2014) Extended formulations for stochastic lot-sizing problems. Oper Res Lett 42:278–283.

    Article  Google Scholar 

  23. 23. Accessed 23 June 2020

  24. 24. Accessed 23 June 2020

  25. 25. Accessed 23 June 2020

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I would like to thank the reviewers for the very constructive comments and suggestions that helped improving the paper.


This work has been partially supported by the Portuguese National Funding: Fundação para a Ciência e a Tecnologia - FCT, under the project UIDB/04561/2020.

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Martins, P. Planning Production and Workforce in a Discrete-Time Financial Model Using Scenarios Modeling. SN Oper. Res. Forum 1, 35 (2020).

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  • Production planning
  • Financial planning
  • Workforce planning
  • Scenarios optimization