Integrated Supplier Selection and Inventory Management Problems Considering Discounts and Uncertainty: A Fuzzy-Probabilistic Approach

Decision-makers in manufacturing and retail industries have been continuously attempting to optimize all supply-chain activities in order to generate larger profit. Optimizing the whole supply-chain problems simultaneously from the upstream parties such as raw material suppliers up to the downstream parties such as buyers is impossible due to the extensive size of the problems. In many cases, decision-makers divide supply-chain problems into supply-chain activities and optimize them independently or in an integrated manner for some inter-connected activities.

Numerous approaches, mostly in mathematical model forms, have been proposed to optimize certain problems in supply chain independently. For supplier selection problems, some pioneering models can be found in the study [1]. Monteiro et al. [2] dealt with simpler problem specifications, e.g., there are no discounts, and all parameters are certain. Generally, more complex specifications lead to more complex models. For instance, more complex models have been proposed in Wicaksono [3] and Widowati [4] to address supplier selection problems under full truckload transportation schemes. Models with price discounts are built in the studies [5, 6], however, all involved parameters are certain. Cases with uncertain parameters in the form of probabilistic and fuzzy were constructed in this study to fill the prevalent research gap. Numerous studies have been carried out in dealing with supplier selection problems, indicating the need of decision-making support for such problems in many fields such as power management [7-9], healthcare [10, 11], automotive [12, 13], electrical equipment manufacturing [14, 15], garment industries [16], mega-construction [17], and others.

Several mathematical models have also been proposed to solve inventory management problems independently. For example, in the studies [18-20], three models have been proposed to solve inventory under sales dependent stochastic return flows, carbon emissions policies, and with varying perishability rate product, respectively. Another approach has been proposed using deep reinforcement learning in the study [21]. However, there were no integrations with other parties such as suppliers. Various studies that had been conducted emphasize the need of inventory models in real world problems such as pharmaceutical inventory [22], textile industries [23], grocery retail [24], and many more. In particular, case studies that contain uncertain parameters can be found in numerous reports from various fields such as supply chain management [25], mechanical devices [26, 27], and transportation [28], further demonstrating that uncertain parameters naturally occur in real world problems.

To address the aforementioned supplier selection and inventory management problems in an integrated manner, some models have been proposed in previous studies such as two models in the studies [5, 6]. However, all parameters were fully certain, and no model has been constructed under uncertainty, especially with both probabilistic and fuzzy variables; this is the main weakness of the existing studies and will be resolved in this paper. Furthermore, uncertain parameters naturally occur in real-world problems. Probabilistic parameters can be used to treat parameters with historical data. Nevertheless, the trend of the data may change, and when data do not represent the reality anymore, fuzzy parameters can be used to treat the corresponding parameters. This leads to the need of having both probabilistic and fuzzy parameters in the decision-making processes. Therefore, this study aimed to propose a joint decision-making support for integrated supplier selection and inventory management involving discounts, probabilistic parameters, and fuzzy parameters. The proposed model is in the form of fuzzy-probabilistic programming and is demonstrated via computational simulations. The discounts are in piecewise constant forms whereas the probabilistic parameters have normal probability distribution functions, and the fuzzy parameters have discrete membership functions; those approaches are commonly used in practice due to their simplicity in formulating the corresponding price or probability or membership functions.

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Figure 1. The supply chain flow

The notations used in the mathematical modeling process of the decision-making support are available in the Nomenclature section at the end of the paper. The decision variables, semi-decision variables (values that follow the decision variables), uncertain parameters, and the certain parameters are also listed in the Nomenclature section. First, we introduce in the following the discounted prices/costs functions, which are in piecewise constant functions:

$U P_{t s p}= \begin{cases}U P_{t s p}^{(1)} & \text { if } \mathcal{X}_{t s p} \leq \mathcal{X}_{t s p}^{(1)}, \\ U P_{t s p}^{(2)} & \text { if } \mathcal{X}_{t s p}^{(1)}<\mathcal{X}_{t s p} \leq \mathcal{X}_{t s p}^{(2)}, \\ \vdots & \\ U P_{t s p}^{(\hat{i})} & \text { if } \quad \mathcal{X}_{t s p}>\mathcal{X}_{t s p}^{(\hat{i}-1)} ;\end{cases}$         (1)

$T C_{t s}=\left\{\begin{array}{lll}T C_{t s}^{(1)} & \text { if } & \mathcal{T}_{t s} \leq \mathcal{T}_{t s}^{(1)}, \\ T C_{t s}^{(2)} & \text { if } & \mathcal{T}_{t s}^{(1)}<\mathcal{T}_{t s} \leq \mathcal{T}_{t s}^{(2)}, \\ \vdots & & \\ T C_{t s}^{(\hat{j})} & \text { if } & \mathcal{T}_{t s}>\mathcal{T}_{t s}^{(\hat{j}-1)}.\end{array}\right.$              (2)

$I C_{t p}= \begin{cases}I C_{t p}^{(1)} & \text { if } \mathcal{I}_{t p} \leq \mathcal{I}_{t p}^{(1)}, \\ I C_{t p}^{(2)} & \text { if } \mathcal{I}_{t p}^{(1)}<\mathcal{I}_{t p} \leq \mathcal{I}_{t p}^{(2)}, \\ \vdots & \\ I C_{t p}^{(\hat{k})} & \text { if } \mathcal{I}_{t p}>\mathcal{I}_{t p}^{(\hat{k}-1)}.\end{cases}$             (3)

The above piecewise constant functions of prices/costs represent the discount schemes in which more goods/services provide cheaper prices/costs, which are common in industries. Each price level is separated by breakpoints, which are determined by the goods/services provider.

The target of the optimization is the total operational cost, which includes the following cost components:

• The total order cost for the selected suppliers along the optimization time horizon $t=1,2, \ldots, T$:

$Z_1=\sum_{t=1}^T \sum_{s=1}^S\left[O C_{t s} \times \mathcal{Y}_{t s}\right].$         (4)

• The total purchasing cost to all suppliers along the optimization time horizon $t=1,2, \ldots, T$:

$Z_2=\left\{\begin{array}{l}\sum_{t=1}^T \sum_{s=1}^S \sum_{p=1}^P\left[U P_{t s p}^{(1)} \times \mathcal{X}_{t s p}\right] \\ \text { if } \mathcal{X}_{t s p} \leq \mathcal{X}_{t s p}^{(1)}, \\ \sum_{t=1}^T \sum_{s=1}^S \sum_{p=1}^P\left[U P_{t s p}^{(2)} \times \mathcal{X}_{t s p}\right] \\ \text { if } \mathcal{X}_{t s p}^{(1)}<\mathcal{X}_{t s p} \leq \mathcal{X}_{t s p}^{(2)}, \\ \vdots \\ \sum_{t=1}^T \sum_{s=1}^S \sum_{p=1}^P\left[U P_{t s p}^{(\hat{i})} \times \mathcal{X}_{t s p}\right] \\ \text { if } \mathcal{X}_{t s p}>\mathcal{X}_{t s p}^{(\hat{i}-1)} .\end{array}\right.$          (5)

• The total delivery cost for transporting goods from suppliers to the warehouse along the optimization time horizon $t=1,2, \ldots, T$:

$Z_3=\left\{\begin{array}{l}\sum_{t=1}^T \sum_{s=1}^S\left[T C_{t s}^{(1)} \times \mathcal{T}_{t s}\right] \text { if } \mathcal{T}_{t s}^{(0)} \leq \mathcal{T}_{t s} \leq \mathcal{T}_{t s}^{(1)}, \\ \sum_{t=1}^T \sum_{s=1}^S\left[T C_{t s}^{(2)} \times \mathcal{T}_{t s}\right] \text { if } \mathcal{T}_{t s}^{(1)} \leq \mathcal{T}_{t s} \leq \mathcal{T}_{t s}^{(2)}, \\ \vdots  \\ \sum_{t=1}^T \sum_{s=1}^S\left[T C_{t s}^{(\hat{j})} \times \mathcal{T}_{t s}\right] \text { if } \mathcal{T}_{t s}^{(\hat{j}-1)} \leq \mathcal{T}_{t s} \leq \mathcal{T}_{t s}^{(\hat{j})}.\end{array}\right.$           (6)

• The total cost to penalize late delivery along the optimization time horizon $t=1,2, \ldots, T$:

$Z_4=\sum_{t=1}^T \sum_{s=1}^S \sum_{p=1}^P\left[\mathcal{P}_{t s p}^{\mathcal{L}} \times \overline{\overline{\mathcal{L}}}_{t s p} \times \mathcal{X}_{t s p}\right].$            (7)

• The total cost to penalize rejected goods along the optimization time horizon $t=1,2, \ldots, T$:

$Z_5=\sum_{t=1}^T \sum_{s=1}^S \sum_{p=1}^P\left[\mathcal{P}_{t s p}^{\mathcal{D}} \times \overline{\overline{\mathcal{D}}}_{t s p} \times \mathcal{X}_{t s p}\right].$             (8)

• The total cost for purchasing alternative goods (if any and desired by decision-makers) needed to satisfy the demand along the optimization time horizon $t=1,2, \ldots, T$:

$Z_6=\sum_{t=1}^T \sum_{p=1}^P\left[R C_{t p} \times \mathcal{R}_{t p}\right].$           (9)

• The total inventory cost of all goods along the optimization time horizon $t=1,2, \ldots, T$:

$Z_7= \begin{cases}\sum_{t=1}^T \sum_{p=1}^P\left[I C_{t p}^{(1)} \times \mathcal{I}_{t p}\right] & \text { if } \mathcal{I}_{t p} \leq \mathcal{I}_{t p}^{(1)}, \\ \sum_{t=1}^T \sum_{p=1}^P\left[I C_{t p}^{(2)} \times \mathcal{I}_{t p}\right] & \text { if } \mathcal{I}_{t p}^{(1)}<\mathcal{I}_{t p} \leq \mathcal{I}_{t p}^{(2)}, \\ \vdots & \\ \sum_{t=1}^T \sum_{p=1}^P\left[I C_{t p}^{(\hat{k})} \times \mathcal{I}_{t s p}\right] & \text { if } \mathcal{I}_{t p}>\mathcal{I}_{t p}^{(\hat{k}-1)} .\end{cases}$         (10)

• The total contract cost for all selected suppliers:

$Z_8=\sum_{s=1}^S\left[C C_s \times \mathcal{Z}_s\right].$             (11)

The constraint functions that need to be satisfied were subsequently formulated based on the assumptions and conditions described in the problem statement. The constraint functions are formulated as follows:

• Available goods should be sufficient to satisfy the demand, i.e., for each goods type p at each time observation period t, goods in the inventory plus the arriving goods (goods that are ordered at the current time period plus the arriving goods from the previous time period minus late delivered goods minus defect goods) plus alternative goods minus goods decided to be stored in the inventory for future time periods should no less than the current demand; this is expressed as follows:

$\begin{gathered}\mathcal{I}_{(t-1) p}+\sum_{s=1}^S \mathcal{X}_{t s p}+\sum_{s=1}^S\left[\overline{\overline{\mathcal{L}}}_{(t-1) s p} \times \mathcal{X}_{(t-1) s p}\right] -\sum_{s=1}^S\left[\overline{\overline{\mathcal{L}}}_{t s p} \times \mathcal{X}_{t s p}\right]-\sum_{s=1}^S\left[\overline{\overline{\mathcal{D}}}_{t s p} \times \mathcal{X}_{t s p}\right]+\mathcal{R}_{t p} \geq \mathbb{D}_{t p}, \\ \forall t=1,2, \ldots, T, \forall p=1,2, \ldots, P;\end{gathered}$            (12)

where, $\mathcal{I}_{(0) p}$ is the initial inventory level for goods p; commonly, it is zero if no goods are available in the warehouse at the initial time, and $\overline{\overline{\mathcal{L}}}_{(0) s p}$ is zero as no goods was purchased before the initial time.  If the inventory is initially not empty, then $\mathcal{I}_{(0) p}$ can be set to be nonzero. This inequality is used to ensure that the demand is expected to be always satisfied.

• The number of delivery calculation for each supplier based on the trucks’ capacities used in the transportation and the amount of purchased goods to each supplier; this is expressed as follows:

$\sum_{p=1}^P \mathcal{X}_{t s p} \leq T R C \times \mathcal{T}_{t s}, \quad \forall t=1,2, \ldots, T, \forall s=1,2, \ldots, S.$            (13)

This is needed to calculate the number of deliveries which is used to calculate the transportation cost.

• Selected supplier indicator functions at each time observation period t for each supplier s; 1 means selected, while 0 means not selected; this is expressed as follows:

$\mathcal{Y}_{t s}=\left\{\begin{array}{ll}1 & \text { if } \sum_{p=1}^P \mathcal{X}_{t s p}>0, \\ 0 & \text { otherwise; }\end{array} \quad \forall t=1,2, \ldots, T, \forall s=1,2, \ldots, S.\right.$          (14)

This constraint function is used to determine whether a supplier is selected or not at a particular time observation period and is used to calculate the order cost at every time observation period in the objective function.

• Selected supplier indicator along the time horizon $t=1,2 ., \ldots, T$ to calculate contract costs; 1 means selected, 0 means not selected; this is expressed as follows:

$\mathcal{Z}_s=\left\{\begin{array}{ll}1 & \text { if } \sum_{t=1}^T \sum_{p=1}^P \mathcal{X}_{t s p}>0, \\ 0 & \text { otherwise; }\end{array} \quad \forall s=1,2, \ldots, S ;\right.$             (15)

or, alternatively,

$\mathcal{Z}_s=\left\{\begin{array}{ll}1 & \text { if } \sum_{t=1}^T \mathcal{Y}_{t s}>0, \\ 0 & \text { otherwise; }\end{array} \quad \forall s=1,2, \ldots, S.\right.$               (16)

This binary variable is utilized to determine whether a supplier is selected or not for the whole planning time horizon and is used to calculate the contract cost in the objective function.

• Suppliers’ capacity bounds in supplying goods at each time observation period t; this is expressed as follows:

$\begin{gathered}\mathcal{X}_{t s p} \leq S C_{t s p}, \quad \forall t=1,2, \ldots, T, \forall s=1,2, \ldots, S, \\ \forall p=1,2, \ldots, P .\end{gathered}$            (17)

This bound is rather obvious as suppliers have maximum capacity limits in supplying goods, and this constraint is used to ensure that the decision in ordering goods does not excess the maximum number of goods can be provided by suppliers. If a supplier does not have a capacity limit, the upper bound can be simply set as a sufficiently big number; this will imply that the feasible region of the optimization problem is bounded.

• Warehouse capacity to store goods at each time observation period t; this is expressed as follows:

$\mathcal{I}_{t p} \leq M I_{t p}, \quad \forall t=1,2, \ldots, T, \forall p=1,2, \ldots, P;$          (18)

Similar to upper bounds for suppliers, this is used to make sure that the number of goods stored in the warehouses does not excess its maximum capacity limits.

• Nonnegativity and integer assignments for all decision variables; this is expressed as follows:

$\mathcal{X}_{t s p}, \mathcal{T}_{t s}, \mathcal{I}_{t p} \geq 0$ and integer, $\forall t=1,2, \ldots, T, \forall s=1,2, \ldots, S, \forall p=1,2, \ldots, P.$              (19)

The nonnegativity constraints are obvious since all decision variables have to be nonnegative. The integer constraints are optional; some variables related to particular goods may be non-integer if their measurements follow real numbers.

Since the problem contains uncertain parameters, the expectation value of the total operational cost was minimized instead of its actual value. This is due to the presence of probabilistic and fuzzy parameters, and thus the actual value cannot be known before those uncertain parameters are revealed; and the decision-maker can only expect the operational cost. Subsequently, as a complete mathematical optimization problem, the proposed model was rewritten a`s follows:

$\min _{\mathcal{X}_{t s p}, \,\mathcal{I}_{t p}, \mathcal{R}_{t p}} Z=\mathcal{E}\left[\sum_{l=1}^8 Z_l\right]$           (20)

subject to:

$\begin{gathered}\mathcal{E}\left[\mathcal{I}_{(t-1) p}+\sum_{s=1}^S \mathcal{X}_{t s p}+\sum_{s=1}^S\left[\overline{\overline{\mathcal{L}}}_{(t-1) s p} \times \mathcal{X}_{(t-1) s p}\right]\right. \left.-\sum_{s=1}^S\left[\overline{\overline{\mathcal{L}}}_{t s p} \times \mathcal{X}_{t s p}\right]-\sum_{s=1}^S\left[\overline{\overline{\mathcal{D}}}_{t s p} \times \mathcal{X}_{s p}\right]+\mathcal{R}_{t p}-\mathbb{D}_{t p}\right] \geq 0 \\ \forall t=1,2, \ldots, T, \forall p=1,2, \ldots, P;\end{gathered}$            (21)

$\sum_{p=1}^P \mathcal{X}_{t s p}-T R C \times \mathcal{T}_{t s} \leq 0, \quad \forall t=1,2, \ldots, T, \forall s=1,2, \ldots, S;$                (22)

$\mathcal{Y}_{t s}=\left\{\begin{array}{ll}1 & \text { if } \sum_{p=1}^P \mathcal{X}_{t s p}>0, \\ 0 & \text { otherwise; }\end{array} \forall t=1,2, \ldots, T, \forall s=1,2, \ldots, S;\right.$         (23)

$\mathcal{Z}_s=\left\{\begin{array}{ll}1 & \text { if } \sum_{t=1}^T \mathcal{Y}_{t s}>0, \\ 0 & \text { otherwise; }\end{array} \quad \forall s=1,2, \ldots, S;\right.$           (24)

$\begin{gathered}\mathcal{X}_{t s p} \leq S C_{t s p}, \quad \forall t=1,2, \ldots, T, \forall s=1,2, \ldots, S, \forall p=1,2, \ldots, P;\end{gathered}$            (25)       

$\mathcal{I}_{t p} \leq M I_{t p}, \quad \forall t=1,2, \ldots, T, \forall p=1,2, \ldots, P;$           (26)

$\mathcal{X}_{t s p}, \mathcal{T}_{t s}, \mathcal{I}_{t p} \geq 0$ and integer, $\forall t=1,2, \ldots, T, \forall s=1,2, \ldots, S, \forall p=1,2, \ldots, P.$            (27)

Note that this optimization problem involves uncertain parameters, some of them are probabilistic with some known probability distribution functions such as normal distribution and some others are fuzzy with some known membership functions such as discrete membership function. This optimization problem belongs to uncertain programming, and the technique used to solve it in this paper is based on the uncertain programming method provided in the study [33]. The general steps to solve are is first, the uncertain optimization problem is converted as a deterministic-equivalent optimization model by utilizing the probability distribution functions of the probabilistic parameters and the membership functions of the fuzzy parameters. Then, the optimization algorithms are employed to calculate the optimal decision from the deterministic equivalent model. For technical details, one may refer to [29, 33]. Furthermore, it should be also noted that the objective function is in a piecewise function since it is a piecewise function of discounted prices/costs. Therefore, the above optimization problem is in a piecewise programming class. From all constraint functions, the decision variables’ feasible region is closed as bounded. Provided that it is not empty, then an optimal decision is guaranteed to exist. Hence, the mathematical optimization in Eq. (20) is well defined.

The integrated supplier selection and inventory management problem was solved by using a piecewise fuzzy-probabilistic programming approach. The problem was considered with discounted prices/costs and both probabilistic and fuzzy parameters. The proposed model can handle those discounts and both uncertainty types, and it was tested in a laboratory with randomly generated data. It successfully solved the given problem, and therefore can be utilized by decision-makers in relevant manufacturing/retail companies.

Nevertheless, the proposed model still have some limitations, for example, the capacity of trucks used in the delivery processes is equal where in practice, various trucks with various capacities might be used. Another example is that all types of goods were treated to have same sizes whereas in practice, they may vary. This will imply the formula of calculating the number of deliveries. One may consider such issues as possible future research directions. Furthermore, for other possible future research, the model can be developed by integrating other parties in the supply chain such as carrier agents, distributors, and production units. Some recently developed technologies such as blockchain and internet of things could also be integrated to create a better decision-making support.  Furthermore, sustainability aspect, such as carbon emission and circular economy, could also be taken into account to build green supply chain. One possible approach to integrate those aspects with the current model is by adding a new objective function such as carbon emission measurement produced by the procurement and storing activities, and minimize it.