# Supply Chain Design for Blending Technologies

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Methodology of the SLR

- definition of research questions;
- search process in Science Direct;
- inclusion and exclusion process;
- descriptive analyses of chosen articles;
- content analysis;
- identification of scientific gaps, bottlenecks, and limitations.

#### 2.2. Descriptive Analysis

#### 2.3. Content Analysis

#### 2.4. Consequences of Literature Review

## 3. Materials and Methods

**Input parameters**: in the basic model of the blending problems, the following parameters are assumed to be given:

- y
_{j}customer’s demand for product j, - p
_{ik}quality parameter k of raw material i, - ${b}_{kj}^{min}$ lower bound for parameter k of finished product j,
- ${b}_{kj}^{max}$ upper bound for parameter k of finished product j,
- ${c}_{i}^{B}$ price of raw material i.

**Decision variable**: the decision parameter of the blending problems is ${x}_{ij}$ which represents the amount of raw material i assigned to product j.

**Objective function**(

**profit maximization**): in the basic model of the blending problems, the objective function is the maximization of profit, which can be written as a function of the revenue from the finished products sold and the purchase price of the raw materials needed to produce the finished product, as follows:

_{j}is the specific purchasing price of product j (EUR/kg).

**Constraint 1**(

**quality of final product**): in the basic model of the blending problem, two basic constraints must be taken into consideration. For the first constraint, we can define the limits of the technological specifications for the quality of the final product. For these limit values, a constraint for the lower and upper limit values can be specified as a separate constraint in the following way:

**Constraint 2**(

**meet customer’s demand**)

**:**The second constraints for the basic model defines the need to obtain a quantity of raw materials that can meet customer demand:

**Constraint 3**(

**production capacity**)

**:**In general, the production capacity for blending problems can be considered as given. This condition modifies the basic model in the sense that, for this constraint, customer requirements are not given per product, but can be freely modified according to the production capacity. Due to the production capacity constraint, the limitation on the quantity of raw materials to be purchased can be written in the following form:

**Constraint 4**(

**built-in rate of raw materials**)

**:**for the basic model, we did not take into consideration the fact that, depending on the nature of the raw materials and the processing technology, not all of them are built into the finished product, and that the parameters affecting the quality of the finished product are not fully reflected in the finished product. Based on this condition, it is possible to modify the basic model in two directions and add a new constraint focusing on the build-in proportion. In the first case, the purchased raw material is not processed in its full quantity, and therefore its parameters influence the parameters characterizing the finished product according to the built-in rate. For example, if broccoli is used to prepare a dish, the stems and leaves of the broccoli purchased are not processed and therefore the nutrients and vitamins they contain are not built into this final dish. For this model, we can define the ${\alpha}_{i}$ percentage of the purchased raw material i that can be used to produce the final product. If we take this constraint into consideration, the objective function is still to maximize profit, but the revenue from the finished products sold is affected by the built-in rate of the raw materials. Since the full quantity is not built in, it is necessary to purchase a larger quantity depending on the built-in rate of the raw material, which increases the cost of the raw material associated with the finished product:

**Constraint 5**(

**limited raw material sources**)

**:**This constraint specifies that the maximum or minimum quantity of available raw materials must be taken into account when placing an order. As a consequence, a lower and an upper limit constraint are also included:

**Constraint 6**(

**marketing increases customer’s demand**)

**:**In this model of the blending problem, targeted, product-specific promotion can be used to increase demand for each finished product, resulting in increased demand for the finished product and thus higher profits. Assuming that the expenditure on advertising is directly proportional to the increase in demand for the finished product generated by the advertising, there is a change in our objective functions used earlier.

- y
_{j}customer’s demand for finished product j without advertising, - h
_{j}is an advertising cost for product j, - g
_{j}the rate of increase in demand for product j caused by advertisements, - z
_{j}number of advertisements for product j.

- y customer’s demand without advertising,
- h is the cost of an advertising,
- g the rate of increase in demand caused by advertisements,
- z number of advertisements.

**Constraint 7**(

**lot size of raw materials**)

**:**for the basic model, we did not take into consideration the fact that the lot sizes of raw materials to be ordered can be defined. Taking this lot size- or batch size-related constraint into consideration, we can define the following constraint:

**Constraint 8**(

**capacity of transportation**)

**:**for the basic model, we did not take into consideration the capacity of transportation vehicles from raw material suppliers to the production plant. If transportation processes are taken into consideration, then the capacities can be defined in two different ways. If the required raw materials are transported with the same vehicle from the same supplier, then the constraints can be written as follows:

**Constraint 9**(

**capacity of warehouses**)

**:**we can define a capacity of warehouses for raw materials and these limited capacities can be taken into consideration using the following constraint:

**Constraint 10**(

**capacity of loading and unloading operation**)

**:**for the basic model, we did not take into consideration the capacity of loading and unloading equipment. Loading and unloading operations must be performed at the suppliers (loading) and at the manufacturer (unloading). If loading and unloading processes are taken into consideration, then the capacities can be defined in two different ways. If the required raw materials are transported with the same vehicle from the same supplier, then the related loading and unloading operations are assigned to these supplies in the same way, therefore the loading and unloading constraint can be written as follows:

**Sign restrictions**: since in the course of solving the optimization problem it is possible that, based on the parameters of different types of raw materials, the optimal solution is obtained by assigning a negative quantity of certain raw materials, it is useful to formulate a sign restriction on the decision variables, which defines that no negative quantity of raw materials can be assigned to products:

## 4. Results

- impact of transportation cost on the optimal solution of blending problems,
- impact of inventory holding cost on the optimal solution of blending problems,
- impact of loading and unloading cost on the optimal solution of blending problems,
- impact of packaging cost on the optimal solution of blending problems,
- impact of storage capacities on the optimal solution of blending problems,
- impact of lot sizes of raw materials on the optimal solution of blending problems.

#### 4.1. Impact of Transportation Cost on the Optimal Solution of Blending Problems

- Phase 1: Optimization of the basic blending problem including the following constraints: quality of final products, meet customer’s demand.
- Phase 2: Computation of the resulting profit depending on the technological costs and incomes.
- Phase 3: Computation of the transportation cost of the optimal solution resulted by Phase 1.
- Phase 4: Computation of the total costs, by adding the resulted transportation costs to the technological costs computed in Phase 2.

- Phase 1: Optimization of the integrated blending problem focusing on both technological costs and transportation-related costs including the following constraints: quality of final products, meet customer’s demand.
- Phase 2: Computation of the resulted profit depending on the technological and transportation costs and incomes.

#### 4.2. Impact of Inventory Holding Cost on the Optimal Solution of Blending Problems

- Phase 1: Optimization of the basic blending problem taking only the quality of final products and the customer’s demand as constraints into consideration.
- Phase 2: Computation of the profit of the blending problem focusing on the incomes and technological costs.
- Phase 3: Computation of the inventory holding cost of the optimal solution resulted by Phase 1.
- Phase 4: Computation of the total costs, by adding the resulted inventory holding costs to the technological costs computed in Phase 2.

- Phase 1: Optimization of the integrated blending problem focusing on both technological costs and inventory holding costs including the following constraints: quality of final products, meet customer’s demand and the available warehouse capacity.
- Phase 2: Computation of the resulted profit depending on the technological costs, inventory holding costs and incomes.

#### 4.3. Impact of Loading and Unloading Cost on the Optimal Solution of Blending Problems

- Phase 1: Optimization of the basic blending problem taking only the quality of final products and the customer’s demand as constraints into consideration.
- Phase 2: Computation of the profit of the blending problem focusing on the incomes and technological costs.
- Phase 3: Computation of the costs of loading and unloading operations at the supplier and the manufacturing company from the optimal solution resulted by Phase 1.
- Phase 4: Computation of the total costs, by adding the resulted loading and unloading costs to the technological costs computed in Phase 2.

- Phase 1: Optimization of the integrated blending problem focusing on both technological costs, loading and unloading costs including the following constraints: quality of final products, meet customer’s demand and the available capacity of loading and unloading equipment of the supplier and the manufacturing company.
- Phase 2: Computation of the resulted profit depending on the technological costs, loading and unloading costs and incomes.

#### 4.4. Impact of Packaging Cost on the Optimal Solution of Blending Problems

- Phase 1: Optimization of the basic blending problem taking only the quality of final products and the customer’s demand as constraints into consideration.
- Phase 2: Computation of the profit of the blending problem focusing on the incomes and technological costs.
- Phase 3: Computation of the packaging costs resulted from the optimal solution in Phase 1.
- Phase 4: Computation of the total costs, by adding the resulted packaging costs to the technological costs computed in Phase 2.

- Phase 1: Optimization of the integrated blending problem focusing on both technological costs and packaging costs.
- Phase 2: Computation of the resulted profit depending on the technological costs, packaging costs and incomes.

#### 4.5. Impact of Storage Capacities on the Optimal Solution of Blending Problems

#### 4.6. Impact of Lot Sizes of Ram Materials on the Optimal Solution of Blending Problems

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Final Product | |||||||
---|---|---|---|---|---|---|---|

FP1 | FP2 | FP3 | FP4 | FP5 | FP6 | FP7 | |

Demand [pcs] | 100 | 150 | 200 | 150 | 80 | 110 | 210 |

Selling price [EUR] | 20 | 30 | 40 | 50 | 60 | 70 | 50 |

Final Product | |||||||
---|---|---|---|---|---|---|---|

FP1 | FP2 | FP3 | FP4 | FP5 | FP6 | FP7 | |

Quality parameter 01 [%] | |||||||

Min | 9 | 10 | 15 | 14 | 12 | 9 | 11 |

11Max | 15 | 16 | 17 | 18 | 15 | 14 | 20 |

Quality parameter 02 [%] | |||||||

Min | 42 | 52 | 51 | 45 | 65 | 55 | 70 |

Max | 75 | 80 | 66 | 77 | 80 | 80 | 80 |

Quality parameter 03 [%] | |||||||

Min | 1 | 2 | 5 | 2 | 2 | 2 | 3 |

Max | 3 | 4 | 6 | 5 | 6 | 4 | 5 |

Quality parameter 04 [%] | |||||||

Min | 21 | 22 | 21 | 20 | 21 | 22 | 23 |

Max | 25 | 26 | 27 | 26 | 25 | 26 | 27 |

Parameter | ID of Raw Materials | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

A | B | C | D | E | F | G | H | I | J | |

Quality 01 [%] | 10 | 20 | 15 | 14 | 18 | 16 | 9 | 21 | 12 | 14 |

Quality 02 [%] | 40 | 50 | 60 | 70 | 80 | 45 | 55 | 65 | 52 | 47 |

Quality 03 [%] | 2 | 3 | 4 | 5 | 6 | 5 | 4 | 2 | 1 | 2 |

Quality 04 [%] | 22 | 25 | 24 | 26 | 27 | 20 | 26 | 23 | 24 | 22 |

Price [EUR] | 25 | 14 | 15 | 16 | 17 | 30 | 20 | 10 | 23 | 18 |

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**Figure 1.**Methodology of the systematic literature review [3].

**Figure 2.**Classification of blending processes related articles considering subject areas in Science Direct.

**Figure 3.**Classification of blending processes-related articles considering the year of publication, based on the search in Science Direct.

**Figure 4.**Classification of blending processes related articles considering the title and topic of the journal, based on the search in Science Direct.

**Figure 5.**Comparison of the structure of raw material portfolio in the case of conventional and integrated blending model.

**Figure 6.**Changes in the structure of raw material portfolio depending on the predefined lot-sizes in the case of Scenario.

**Table 1.**Comparison of results of conventional blending model and integrated blending model focusing on transportation costs.

𝜛 * | Conventional Model | Integrated Model | η^{TR} | ||||
---|---|---|---|---|---|---|---|

C^{TE} | C^{TR} | P | C^{TE} * | C^{TR} * | P^{TR} * | ||

1 | 14,739 | 10,145 | 20,116 | 16,839 | 4099 | 24,062 | 1.19 |

1.2 | 14,739 | 12,174 | 18,087 | 17,231 | 4502 | 23,266 | 1.29 |

1.4 | 14,739 | 14,203 | 16,058 | 17,324 | 5149 | 22,526 | 1.40 |

1.6 | 14,739 | 16,232 | 14,029 | 17,822 | 5373 | 21,805 | 1.55 |

1.8 | 14,739 | 18,261 | 12,000 | 18,191 | 5627 | 21,182 | 1.76 |

2.0 | 14,739 | 20,290 | 9971 | 18,199 | 6246 | 20,555 | 2.06 |

**Table 2.**Comparison of results of conventional blending model and integrated blending model focusing on inventory holding costs.

𝜛 * | Conventional Model | Integrated Model | η^{INV} | ||||
---|---|---|---|---|---|---|---|

C^{TE} | C^{INV} | P | C^{TE} * | C^{INV} * | P^{INV} * | ||

1 | 14,739 | 3640 | 26,621 | 15,142 | 2768 | 27,090 | 1.02 |

1.2 | 14,739 | 4368 | 25,893 | 15,577 | 2838 | 26,585 | 1.03 |

1.4 | 14,739 | 5096 | 25,165 | 15,695 | 3191 | 26,114 | 1.04 |

1.6 | 14,739 | 5824 | 24,437 | 16,464 | 2869 | 25,667 | 1.05 |

1.8 | 14,739 | 6552 | 23,709 | 16,502 | 3173 | 25,325 | 1.06 |

2.0 | 14,739 | 7280 | 22,981 | 16,538 | 3508 | 24,954 | 1.09 |

**Table 3.**Comparison of results of conventional blending model and integrated blending model focusing on loading and unloading costs.

𝜛 * | Conventional Model | Integrated Model | η^{LO} | ||||
---|---|---|---|---|---|---|---|

C^{TE} | C^{LO} | P | C^{TE} * | C^{LO} * | P^{LO} * | ||

1 | 14,739 | 14,555 | 15,706 | 17,564 | 9059 | 18,377 | 1.17 |

1.2 | 14,739 | 17,466 | 12,795 | 17,845 | 10,589 | 16,566 | 1.29 |

1.4 | 14,739 | 20,377 | 9884 | 17,886 | 12,281 | 14,833 | 2.50 |

1.6 | 14,739 | 23,288 | 6973 | 17,940 | 14,042 | 13,018 | 2.86 |

1.8 | 14,739 | 26,199 | 4062 | 17,961 | 15,775 | 11,264 | 3.77 |

2.0 | 14,739 | 29,110 | 1151 | 18,048 | 17,350 | 9602 | 9.34 |

**Table 4.**Comparison of results of conventional blending model and integrated blending model focusing on packaging costs.

𝜛 * | Conventional Model | Integrated Model | η^{PA} | ||||
---|---|---|---|---|---|---|---|

C^{TE} | C^{PA} | P | C^{TE} * | C^{PA} * | P^{PA} * | ||

1 | 14,739 | 5971 | 24,290 | 14,766 | 5732 | 24,502 | 1.008 |

1.2 | 14,739 | 7165 | 23,096 | 14,765 | 6869 | 23,366 | 1.011 |

1.4 | 14,739 | 8359 | 21,902 | 14,772 | 8007 | 22,221 | 1.014 |

1.6 | 14,739 | 9554 | 20,707 | 15,080 | 8808 | 21,112 | 1.019 |

1.8 | 14,739 | 10,748 | 19,513 | 15,095 | 9559 | 20,016 | 1.026 |

2.0 | 14,739 | 11,942 | 18,319 | 15,097 | 10,986 | 18,917 | 1.033 |

**Table 5.**Comparison of results of conventional blending model and integrated blending model focusing on storage capacities.

𝜛 * | Conventional Model | Integrated Model | η^{SC} | ||||
---|---|---|---|---|---|---|---|

C^{TE} | C^{SC} | P | C^{TE} * | C^{SC} * | P^{SC} * | ||

1 | 14,739 | 16,909 | 13,352 | 16,003 | 12,265 | 16,732 | 1.253 |

ψ * | Profit | ID of Raw Materials | Number of Raw Material Types | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

A | B | C | D | E | F | G | H | I | J | |||

1 | 55,492 | 0 | 11 | 472 | 292 | 113 | 41 | 272 | 589 | 1 | 1 | 9 |

2 | 55,440 | 0 | 12 | 548 | 288 | 114 | 44 | 234 | 550 | 2 | 0 | 8 |

4 | 55,432 | 0 | 12 | 544 | 296 | 112 | 44 | 236 | 548 | 0 | 0 | 7 |

8 | 55,144 | 0 | 24 | 464 | 256 | 144 | 48 | 264 | 560 | 8 | 24 | 9 |

16 | 54,976 | 0 | 16 | 496 | 160 | 208 | 48 | 256 | 560 | 32 | 16 | 9 |

32 | 54,688 | 32 | 64 | 320 | 224 | 256 | 32 | 320 | 544 | 0 | 0 | 8 |

64 | 54,528 | 0 | 64 | 512 | 320 | 128 | 64 | 256 | 448 | 0 | 0 | 7 |

128 | 53,248 | 0 | 0 | 384 | 256 | 128 | 128 | 384 | 512 | 0 | 0 | 6 |

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**MDPI and ACS Style**

Bányai, T.; Veres, P.
Supply Chain Design for Blending Technologies. *Sustainability* **2022**, *14*, 8760.
https://doi.org/10.3390/su14148760

**AMA Style**

Bányai T, Veres P.
Supply Chain Design for Blending Technologies. *Sustainability*. 2022; 14(14):8760.
https://doi.org/10.3390/su14148760

**Chicago/Turabian Style**

Bányai, Tamás, and Péter Veres.
2022. "Supply Chain Design for Blending Technologies" *Sustainability* 14, no. 14: 8760.
https://doi.org/10.3390/su14148760