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Abstract

Product allocation is one of the most important duties in warehousing operations. Based on the time and the number of products to put away and retrieve, storage allocation can directly affect the efficiency of a warehouse in terms of both response and operation time. Traditionally, the product storage allocation is done based on the less distance of shipment but focusing on distance may decrease the efficiency of a warehouse in terms of efficient utilization of resources such as warehouse inbound shipping fleet. This paper proposes a storage space allocation model that considers the availability of forklift fleet in a warehouse instead of product shipping distance. The numerical example shows that storing the products based on the proposed model can reduce the volume of forklift fleet idle hours on days with fewer volume of receiving and shipping outs. The proposed model also reduces the overtime working hours on days with higher volume of receiving and shipping out products.

Keywords

Unit-load warehouse storage allocation forklift fleet material handling plan distance-based allocation model

Introduction

Warehousing is an important part of a supply chain or any production plant, which enables the firm to manage the market demand fluctuations. Warehousing the products will not add any value to it (except specific products such as wine), and shipping/storing the products is a costly activity that the company operates to create the opportunity of making benefit by selling the product in the future.¹ Frazelle² believes that the warehousing systems determine the final efficiency or inefficiency of a whole supply chain. Warehousing is commonly used to achieve some of the most important goals of a company³:

benefit from the transportation economies,

benefit from the quality purchase discounts,

benefit from the production economies,

improve the customer service policies, and

deal with the varying demand and market uncertainties.

Unit-load warehouses are so common in industry such as the import distribution centers and third-party logistics providers’ warehouses. Moreover, unit-load warehouses are common in bulk storage areas warehousing operations. These warehouses receive, store, and ship the products in unit-load and generally transport them by forklifts. The increased volume of imported products in recent years necessitated the need to larger industrial warehouses and distribution centers and larger warehouses need more traveled distance by the forklift fleet.⁴

Current study is trying to find out the effect of product storage allocation in a warehouse on the forklift fleet utilization and reduction of the need to the forklift fleet drivers’ overtime working. More explicitly, when the volume of receiving products and shipping out products among different days of a week is not equal, we may have the forklift drivers to be idle on some days and very busy on the other days. As an attempt to deal with this problem, the current paper is developing a model that assigns the product location assignments in a warehouse to minimize the forklift drivers’ idleness and overtime working hours on various days of a week. Based on the specific limitations of the problem under study, two optimization models are developed and compared to determine the efficiency of the utilization-based allocation problem. The remainder of this paper is organized as follows. The next section includes the literature review. Then the problem description and mathematical formulation are explained. This is followed by a section in which the numerical example and sensitivity analysis are provided. Finally, some conclusions and directions for future research are presented.

Literature review

Cost reduction studies focused on different aspects of warehouse design and operations. Some of the studies focused on the racks layout to develop some configurations other than traditional parallel picking aisles layout to reduce the total traveled distance by the forklift fleet,^5–8 some other studies focused on applying the new technologies such as autonomous vehicle storage retrieval systems to more efficiently ship the products.^9–12 Moreover, there are studies that focused on reducing the overall time and cost of material handling in the warehouse,^13–16 (routing the forklifts to have both put-away and order-picking operations in a same cycle,¹⁷ and reducing the consumed energy and released emissions by the warehousing operations,¹⁸ and so on.

Gu et al.¹⁹ categorized the warehouse studies to two classes as design and operations studies. Warehouse design is mostly a strategic decision which can truly affect the success or failure of a business, but operational decisions are made according to shorter horizons (daily to monthly) which affect the operational cost of the system. Francis et al.²⁰ considered three approaches to assign the storage locations to the products: (1) random storage, (2) turn-over-based storage, and (3) class-based storage. Warehouse storage space allocation is studied by many other researchers as well. Gu et al.¹⁹ considered three important sections for the operational activities as: (1) receiving and shipping, (2) storage, and (3) order picking. Among the warehousing operations, order picking consumes the highest time and labor volume, and commonly is the main target to improve the productivity of a warehouse.²¹ Consequently, an important operational decision in unit-load warehouses is to determine the best product storage locations to minimize the total material handling costs.

In most of the unit-load warehouses, the racks are single-deep parallel configured and separated with aisles. Each single-deep rack includes many bins/bays, which are called unit-load or store.¹² In some of the traditional unit-load allocation problems, there is a rule to assign the units of a same product type to a same area of the warehouse. But, Ho and Sarma¹³ showed that there is no need to store all units of a product type in a same area or near together. New technologies such as radio-frequency identification help us to find each specific product at the pickup time. Moreover, some new devices are used to remove the need of assigning the units of a product type to a same area, such as Amazon Kiva robot),²² which can pick up the rack of any specific product type based on the available data in its central database.

Öztürkoğlu et al.⁵ considered the operational costs of a warehouse and noted that the workforce cost (especially forklift drivers) is an important portion of a warehousing system and third-party logistics providers, but the focus of their study is on the racks layout and its effect on the traveled distance.

To the best of our knowledge, the effect of product location assignment in a warehouse on forklift fleet utilization is not studied in the literature so far. Authors believe that the fleet utilization is an important issue regarding to the energy and human resource (forklift drivers) costs. This paper shows that product location assignment in a warehouse based on the forklift fleet utilization can reduce the idle and overtime working hours of the forklift fleet.

Problem description and formulation

The following notations are used to mathematically model the problem:

Notations

$n$	Number of products	t	Needed time for a forklift to move a product per distance unit
$m$	Number of working days	$τ_{k}$	Working hours on day k
$z$	Products index	T	Number of in hand forklifts
$k$	Working days index	d_ij	Distance between receiving door to storage location (i,j)
$i$	Counter of racks $i = 1, …, I$	$l_{i j}$	Distance between storage location (i,j) to shipping door
$j$	Counter of bays $j = 1, …, J$	$x_{i j z k}$	Is 1 if product z on day k is stored in location (i,j), otherwise is 0
$E_{z k}$	Volume of product z entrance on day k	$y_{i j z k}$	Is 1 if product z on day k is shipped out from location (i,j), otherwise is 0
$S_{z k}$	Volume of product z shipped out on day k	$C_{O}$	Cost of overtime working per time unit

Problem description

There is a unit-load one-block (no cross aisle) warehouse with $i$ one-level racks, where each rack has $j$ bays to store the products. Receiving and shipping out the loads are operated from two different doors. There is a determined schedule for received and shipped out products on each day, which are put away and retrieved based on single command cycles. More specifically, when a load of products arrive to the warehouse, each unit is loaded to the forklift and put away in its assigned space, and the forklift comes back to its origin without any load. On the other hand, when a load of products is needed by a customer, the forklift goes empty to the location of each demanded product unit and retrieves it toward the shipping door of the warehouse.

As the main contribution of the current work is to compare the distance-based and forklift overtime cost-based storage allocation, the empty traveling distance is not considered in our calculations at all. Moreover, the needed time for picking up and dropping off the products is the same for both storage allocation approaches and is not considered in any of our mathematical models. The problem is to find out which product unit is going to be stored at which location at the receiving day and which product unit is going to be retrieved from a specific location at the shipping out day. There are also following restrictions on the operating processes of the warehouse that facilitates the modeling of the problem under study:

There is no backorder possibility, and all shipping outs must be performed on their specified day even with overtime working.

The warehouse does not have the possibility of serving like a cross-dock, and all products must stay at the warehouse for at least one day.

Each storage location is able to be occupied or emptied just once a day. It is not possible to empty a location on a day and occupy that on the same day.

There is no possibility of shipping the products between the storage locations. Each product only can be shipped from the receiving door or to the shipping out door.

According to the explanation of the problem, two modeling approaches are presented in this section. The first approach is based on the total traveled distance by the forklift fleet to put away and retrieve the planned products. The second approach is based on minimizing the cost of overtime working of the forklift fleet according to the available forklifts during the regular working hours. Application and computability of both of the approaches are shown on a variety of material handling plans and two layouts of door locations in “Numerical example” section.

Minimizing the travel distance

Considering the notation and description of the problem in “Notations” and “Problem description” sections, we will have the following model to optimize the storage allocation of the products based on the objective of minimizing the total products travel distance:

M i n \sum_{k = 1}^{m} \sum_{z = 1}^{n} \sum_{j = 1}^{J} \sum_{i = 1}^{I} d_{i j} x_{i j z k} + \sum_{k = 1}^{m} \sum_{z = 1}^{n} \sum_{j = 1}^{J} \sum_{i = 1}^{I} l_{i j} y_{i j z k}

(1)

St:

\sum_{i = 1}^{I} \sum_{j = 1}^{J} x_{i j z k} = E_{z k} for k = 1, …, m and z = 1, …, n

(2)

\sum_{i = 1}^{I} \sum_{j = 1}^{J} y_{i j z k} = S_{z k} for k = 1, …, m and z = 1, …, n

(3)

x_{i j z_{1} k_{1}} \leq [1 - \sum_{z_{2} = 1}^{n} \sum_{k_{2} = 1}^{k_{1} - 1} (x_{i j z_{2} k_{2}} - y_{i j z_{2} k_{2}})] for i = 1, …, I and j = 1, …, J and z_{1} = 1, …, n and k_{1} = 1, …, m

(4)

\begin{array}{l} y_{i j z k_{1}} \leq \sum_{k_{2} = 1}^{k_{1} - 1} (x_{i j z k_{2}} - y_{i j z k_{2}}) for i = 1, …, I and \\ j = 1, …, J and z = 1, …, n and k_{1} = 1, …, m \end{array}

(5)

\begin{array}{l} x_{i j z k} + y_{i j z k} \leq 1 for k = 1, …, m and i = 1, …, I \\ and j = 1, …, J and z = 1, …, n \end{array}

(6)

\begin{array}{l} \sum_{z = 1}^{n} x_{i j z k} \leq 1 for k = 1, …, m and \\ i = 1, …, I and j = 1, …, J \end{array}

(7)

\begin{array}{l} \sum_{z = 1}^{n} y_{i j z k} \leq 1 for k = 1, …, m and \\ i = 1, …, I and j = 1, …, J \end{array}

(8)

x_{i j z k} = 0, 1, y_{i j z k} = 0, 1

(9)

Equation (1) minimizes the objective function integrating the distance from the receiving door to the storage location and the distance from the storage location to the shipping door for all product types in all working days of the study period. Equalities (2) and (3) ensure that the number of put-away and retrieved products is as same as the received and shipped-out products in each day. Inequalities (4) and (5) ensure that an item won’t be stored in an occupied place, and no item would be shipped out from a storage location that is empty. Inequality (6) shows that the system does not work such as a cross-dock, and all items must stay at the warehouse for at least one day. Inequalities (7) and (8) show that each storage location at each day can at last be occupied and retrieved by one product. Finally, constraint number (9) shows that our decision variables are binary allocation variables.

The proposed model is numerically solved by ILOG CPLEX solver in “Numerical example” section. It is notable that ILOG CPLEX is not the most efficient optimization software package for solving the large-scale models based on the CPU time. There are two reasons that authors used CPLEX: (1) current study is just trying to develop the modeling idea that the forklift fleet utilization can be considered as an important measure for product assignment activities in a warehouse. Accordingly, the solution time and developing an efficient solution procedure for large-scale models are not the major contributions of this work. (2) The very first models of this paper were developed by CPLEX, so the authors extended the models and sensitivity analyses in the same package as well.

The obtained allocation by the proposed model in “Minimizing the travel distance” section ensures that there is no other allocation which can put away and retrieve all of the assigned products with a lower traveled distance, and this is the minimum possible distance to handle all of the planned put away and retrievals.

Minimizing the overtime cost

The proposed model in “Minimizing the travel distance” section considers the traveled distance as the main objective to allocate the products to the storage locations. As in many cases, the volume of receiving and shipping the products is not as same in all working days, assigning the products only based on the distance may not be sufficient in terms of optimizing the resources utilization such as the forklift fleet. As all of the planned demand must be satisfied on their prescheduled day, there may be a need of overtime working hours, which can be so costly for the company. On the other hand, assigning the products only based on the distance may bring a lot of idle time for the forklift fleet on less busy days, which is paid by the company.

According to above explanations, following model is trying to allocate the products to the storage locations which minimizes both idle time and overtime working hours during the whole scheduled warehousing period. More explicitly, following model tries to minimize the difference between the needed working hours and available working hours of the forklift fleet in the whole scheduled period.

M i n \sum_{k = 1}^{m} (| \sum_{z = 1}^{n} \sum_{j = 1}^{J} \sum_{i = 1}^{I} d_{i j} x_{i j z k} t + \sum_{z = 1}^{n} \sum_{j = 1}^{J} \sum_{i = 1}^{I} l_{i j} y_{i j z k} t - T τ_{k} |)

(10)

St:

\begin{array}{l} \sum_{i = 1}^{I} \sum_{j = 1}^{J} x_{i j z k} = E_{z k} for k = 1, …, m and \\ z = 1, …, n \end{array}

(11)

\begin{array}{l} \sum_{i = 1}^{I} \sum_{j = 1}^{J} y_{i j z k} = S_{z k} for k = 1, …, m and \\ z = 1, …, n \end{array}

(12)

\begin{array}{l} \begin{array}{l} x_{i j z_{1} k_{1}} \leq [1 - \sum_{z_{2} = 1}^{n} \sum_{k_{2} = 1}^{k_{1} - 1} (x_{i j z_{2} k_{2}} - y_{i j z_{2} k_{2}})] \\ for i = 1, …, I and j = 1, …, J and \end{array} \\ z_{1} = 1, …, n and k_{1} = 1, …, m \end{array}

(13)

\begin{array}{l} y_{i j z k_{1}} \leq \sum_{k_{2} = 1}^{k_{1} - 1} (x_{i j z k_{2}} - y_{i j z k_{2}}) for i = 1, …, I and \\ j = 1, …, J and z = 1, …, n and k_{1} = 1, …, m \end{array}

(14)

\begin{array}{l} \begin{array}{l} x_{i j z k} + y_{i j z k} \leq 1 for k = 1, …, m and i = 1, …, I \\ and j = 1, …, J and z = 1, …, n \end{array} \end{array}

(15)

\begin{array}{l} \sum_{z = 1}^{n} x_{i j z k} \leq 1 for k = 1, …, m and \\ i = 1, …, I and j = 1, …, J \end{array}

(16)

\begin{array}{l} \sum_{z = 1}^{n} y_{i j z k} \leq 1 for k = 1, …, m and \\ i = 1, …, I and j = 1, …, J \end{array}

(17)

x_{i j z k} = 0, 1, y_{i j z k} = 0,1

(18)

Equation (10) represents that the difference between the working hours of the forklift fleet and the available working hours of the fleet based on the number of in hand forklifts and working hours on each day must be minimized. This objective function considers the utilization of the forklift fleet instead of the traveled distance in the scheduled period. Constraints (11) to (18) are as same as constraints (2) to (9), which represents the operational limitations of the under study problem.

Applicability of the proposed model is shown in “Problem description and formulation” section with a numerical example, and the results are compared for both of the proposed models.

Numerical example

There is a unit-load one-block warehouse with 20 racks and 20 bays in each rack, which is used for storing three product types. Two single racks of the whole 20 racks are stuck to the side walls of the warehouse and the other 18 racks are stuck together. There are two possible layouts for positioning the receiving and shipping outdoors: (1) adjacent doors layout such as Figure 1(a) and (2) opposite doors layout such as Figure 1(b).

Figure 1.

(a) Adjacent doors and (b) opposite doors.

Each storage location is a square with the dimensions of 4 m to 4 m. The aisles between the racks have the width of 10 m to give a comfortable space for the forklifts to operate. The width of the horizontal aisles at the top and the bottom of the warehouse is 20 m. The bottom left corner of the layout is considered as the coordinate origin for calculating the distances. Therefore, the receiving door center at the adjacent doors layout is located at the coordination of (45,0), and the shipping door center is located at the coordination of (135,0). Moreover, the receiving door center at the opposite doors layout is located at the coordination of (90,0), and the shipping door center is located at the coordination of (90,120).

According to the physical constraints of the forklifts and overhead time for lifting each product, the carrying time of the product unit per meter is considered to be 0.01 of an hour. There are 10 forklifts in the fleet and each forklift works 8 h per day from Monday to Friday; therefore, the warehouse has 80 h of available forklifts per day. The warehouse is able to have overtime working hours of forklift fleet by the cost of $12 per hour per forklift. Therefore, the performance of the distance-based storage allocation and fleet utilization-based allocation are compared by the amount of money that must be paid for overtime working per week. It is notable as the products must stay at the warehouse for at least one day; there is not a shipping out order on the first day. The schedule of receiving and shipping out the products in the planning week is shown in Table 1.

Table 1.

Plan of the receiving and shipping products in a week.

	Product 1		Product 2		Product 3
	Receiving	Shipping	Receiving	Shipping	Receiving	Shipping
Monday	20	0	10	0	15	0
Tuesday	30	10	15	5	20	10
Wednesday	10	20	20	15	20	25
Thursday	15	15	30	25	15	15
Friday	15	30	15	30	30	20

Current numerical example is solved for both adjacent doors and opposite doors layouts, to find out the proposed fleet utilization approach works better in which one of the two mentioned layouts, which are popular layouts for positioning the doors. It is notable that the rectangular distance is used to calculate the distance between the doors and the storage locations. Therefore, “Comparing the models in a layout with adjacent doors” section is assigned to the application of the proposed models in the adjacent doors layout, and “Comparing the models in a layout with opposite doors” section is assigned to the application of the proposed models in the opposite doors layout.

Comparing the models in a layout with adjacent doors

By running the ILOG CPLEX code for both the models, the optimal allocations based on the distance-based objective function and fleet utilization-based objective function are produced with the results as shown in Table 2:

Table 2.

Comparing the cost and distance for the proposed models (adjacent doors).

Traveled distance of the distance-based model	Overtime working cost of the distance-based model	Traveled distance of the fleet utilization-based model	Overtime working cost of the fleet utilization-based model	Percentage of more traveled distance	Percentage of saved cost in overtime fleet working
39,810	681	43,818	459.72	0.1	0.325

The results of the implemented models show that the proposed distance-based model is more effective in storing and retrieving the products in terms of traveled distance. On the other hand, the distance based model is less effective in terms of the overtime working costs in comparison to the fleet utilization-based model. As it is shown in Table 2, the fleet utilization-based model output handles the storage allocations with 10% more traveled distance but 32.5% of lower costs on overtime working payments.

As our problem is using a predefined plan for putting away and retrieving the products in a week, the applicability of the presented models must be investigated in some other cases as well. It would be interesting to find out the performance of the proposed model under a material handling plan with shipping outs with higher variance and lower variance than the current plan and also at the same variance but with higher volume of received and shipped out products. Different varieties of a week schedule are presented in Appendix 1 and the objective functions of these different schedules are shown in Table 3:

Table 3.

Comparing the cost and distance for different volumes (adjacent doors).

	Traveled distance of the distance-based model	Overtime working cost of the distance-based model	Traveled distance of the fleet utilization-based model	Overtime working cost of the fleet utilization-based model	Percentage of more traveled distance	Percentage of aved cost in overtime fleet working
Less shipping out variance	39,800	859.68	43,984	478.68	0.105	0.443
More shipping out variance	39,940	626.88	43,784	456.36	0.096	0.272
Higher receiving and shipping out volumes	52,919	2021.88	54,109	1693.08	0.022	0.162
Much higher receiving and shipping out volumes	66,908	3604.44	67,438	3292.56	0.008	0.086

The results of Table 3 show that the increasing variance will decrease the percentage of saved expenditures on overtime fleet cost. The reason is not related to the performance of the fleet utilization-based model, but the reason is less overtime cost of distance-based model in higher variances. More explicitly, the overtime cost of the distance-based model decreases in higher variances of shipping out volumes in the scheduling period. This could be the result of the trade-off between the storage allocation possibilities and disperse volumes of the needed shipping outs.

Moreover, it is seen that in schedules with the same variance but higher volumes of received and shipped out products, the percentage of saved cost on overtime fleet working is decreased as well. The reason could be the less available empty storage locations in schedules with higher volume of operated products. When more products are received and shipped out, most of the storage locations are occupied and less empty locations are available to maneuver for different allocations. Therefore, the optimal allocations resulted by both distance-based and fleet utilization-based models are getting closer, and both the portion of differences between the traveled distance and the saved overtime fleet working cost are getting smaller.

Comparing the models in a layout with opposite doors

Different positions of the receiving and shipping doors can change the quality of the storage locations in terms of being close to or far from the doors. As the proposed models are compared together in the adjacent doors layout, it would be interesting to find out the performance quality of these models in the other popular layout of door positioning, which is the opposite doors layout. The same calculating procedure for the adjacent doors layout is applied for the opposite doors layout as well. Table 4 shows the comparison of the objective functions of the base case and different variations of received and shipped out products following the volumes presented in Appendix 1.

Table 4.

Comparing the cost and distance for different volumes (opposite doors).

	Traveled distance ofthe distance-based model	Overtime working cost of the distance-based model	Traveled distance of the fleet utilization-based model	Overtime working cost of the fleet utilization-based model	Percentage of more traveled distance	Percentage of saved cost in overtime fleet working
Less shipping out variance	35,320	352.08	40,408	148.56	0.144	0.578
Base example (Table 1)	35,404	296.64	40,276	133.68	0.138	0.549
More shipping out variance	35,548	304.08	40,444	149.76	0.138	0.507
Higher receiving and shipping out volumes	46,956	1353.36	49834	1180.8	0.061	0.127
Much higher receiving and shipping out volumes	59,448	2738.88	61156	2538.72	0.029	0.073

Table 4 shows the same results as Table 3 for different variances of shipped out products and different volumes of operation for both received and shipped out products. The main focus in this section is to compare the opposite doors layout with the adjacent doors layout. As it is seen in Table 4, the improvement of overtime fleet working cost in opposite doors layout is better than the adjacent doors layout. This could be due to the higher distance quality of the storage locations in the opposite doors layout. More explicitly, in opposite doors layout, the integrated distance between the furthest storage location from both receiving and shipping doors is better than the same distance in adjacent doors layout. Therefore, the fleet utilization-based model has more high-quality storage locations available which can be considered in the optimal allocation of the scheduled products. This availability of more options to store the products enables the fleet utilization-based model to better save the overtime fleet working costs in the opposite doors layout.

Following the above explanations, it is interestingly seen in Table 4 that when the material handling volume increases, the improvement of the saving cost of overtime working in opposite doors layout is less than the improved percentage in adjacent doors layout. The reason again goes back to the distance quality of the storage locations and also less availability of the empty storage locations in higher volumes of received and shipped out products. Because the distance-based quality of the storage locations in the opposite doors layout is less variated and dispersed than the adjacent doors layout, the improved performance achieved by using the fleet utilization-based model is not so distinguishable, as there is not much options to variate the final storage allocations.

The modeling approaches that are used in this paper are limited to some assumptions. Because the model is just trying to mention the importance of forklifts availability in logistics processes in a warehouse, the inventory priority is considered to be equal for all of the products in the warehouse. Moreover, the empty travel of the forklifts is not studied in the paper, and it is considered to be equal for both of the studied frameworks.

Conclusion

Depositing and picking up the products in unit-load warehouses are the most important activities in the operational planning of a warehouse in terms of reducing the labor and time consumption. Most of the previous studies considered the objective of minimizing the traveled distance in a warehouse to find the optimal storage allocation of the products. Current study considered the cost of forklift fleet drivers’ overtime working as another important objective in assigning the products to a warehouse. The numerical example shows that the proposed model can reduce the cost of drivers’ overtime working in exchange for a little more traveled distance. Moreover, the numerical results show that the proposed model has better cost reductions in warehouses with opposite doors layout in comparison to the warehouses with adjacent doors.

Future research can consider larger warehouses with other racks and door layouts to find out whether the proposed model can improve the operational costs in other configurations as well. Considering the shipping energy costs besides the labor costs can be another extension to the current work. Finally, studying the current storage allocation problem under uncertain demands can make a good contribution to the field in the future.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Appendix 1. Different schedules of receiving and shipping out products in a week

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De Koster

Le-Duc

Roodbergen

KJ.

Design and control of warehouse order picking: a literature review. Eur J Oper Res 2007; 182: 481–501.

Storage allocation in a warehouse based on the forklifts fleet availability

Abstract

Keywords

Introduction

Literature review

Problem description and formulation

Notations

Problem description

Minimizing the travel distance

Minimizing the overtime cost

Numerical example

Comparing the models in a layout with adjacent doors

Comparing the models in a layout with opposite doors

Conclusion

Footnotes

Declaration of conflicting interests

Funding

Appendix 1. Different schedules of receiving and shipping out products in a week

References