The survey found that the order cycle of most fresh agricultural products in supermarket chains is once a day. Resupply fresh agricultural products every morning before the open of stores, and with supply schedule. Only the order amount of fresh agricultural products of the sub supermarket chains store surveyed reach 1 ? 20,000 Yuan. The suppliers of this supermarket chains are the surrounding two distribution centers. The supermarkets form strategic partnerships with them, fixed to supply the appropriate amount of fresh agricultural products to the supermarket chains every day.

The maximum time of daily sales of fresh agricultural products is from nine to ten o’clock in the morning and from five to nine five o’clock in the afternoon. The reason for this phenomenon is that in the morning, the elderly choose to in part to purchase the relatively high-quality fresh agricultural products considering the advantage and the freshness of it, but in the afternoon, it is time of coming off duty of white-collar workers, so that the fresh agricultural products is more sales in the period of time.

In addition, the order types of fresh agricultural products in supermarket chains will change with the seasons. The types of fresh agricultural products are with seasonal fruits and vegetables, supplemented by a small amount of off-season fruits and vegetables. Usually there will be shortage or surplus situation are these off-season fruits and vegetables.

Supermarket chains in different geographical reasons, traffic conditions, household type, consumption levels and the size of their size, is leading to the different selling degrees of fresh agricultural products. Some supermarket owns surplus of fresh agricultural products because there is less people to buy fresh agricultural products, some supermarkets occur shortage phenomenon because there are more people to buy fresh agricultural products. Fresh agricultural products are easy to spoilage due to its own characteristics, which is a special class of commodities with shortest ordered period in supermarkets. And because the order cycle of fresh agricultural products category is very short, so when there is shortage phenomenon, just waiting for the replenishment by suppliers or self-built distribution center. But such an approach will result in the loss of profit which could be got, and this will reduce the supermarket customer service levels, result in the decrease of customer satisfaction, the reduction of source, and decline the purchase price. In addition, in order to obtain a certain income, the supermarket chains will give the discount when sales the surplus of fresh agricultural products. However, this approach would result in the loss of supermarket chains profits. And if using such methods, in the remaining amount of supermarket fresh produce more, it would not sale all the surplus of fresh agricultural products at the time of the arrival of the next order cycle. Also, when new merchandise is reached, the surplus of fresh agricultural products is easy to become the backlog commodities that cannot be sold, resulting in greater losses.

Due to the certain small demand of fresh agricultural products not up to the required amount of the distribution center, the distribution center is not to carry out the distribution or because the distribution center is relatively far from the supermarket stores, resulting in the shortage situation of fresh agricultural products. When some supermarket stores predict the demand of fresh agricultural products, there is large deviation, which results in excessive accumulation of fresh agricultural products. For the cases above, the transverse replenishment will be applied to supermarket stores, which will be more rational to use the resources fresh agricultural products to further reduce costs and losses.

Transverse replenishment of the supermarket chains fresh agricultural products aims to make the optimal allocation of reasonable inventory of the supermarket fresh agricultural products, to minimize the total costs, and to get the entire optimal system. To solve the issue of the replenishment of the fresh agricultural products between the supermarket chains stores, the transverse replenishment inventory model can be established. Firstly, to make the whole system to the minimum inventory costs, meanwhile, to make optimal replenishment strategies of all stores to the lowest cost. And, finally, the total cost of the entire system will be the lowest.

We assume that the order lead time of fresh agricultural products is same and unchanged. All supermarket stores apply (s, S) storage strategy. The demand of fresh agricultural products is independent and is in keeping with poisson distributions. The operational status of each supermarket store is same. When the stock fresh agricultural products are less than s, then restocking to S. In the case of each supermarket stores appear out of fresh agricultural products, only when the existing inventory below the safety stock quantity, there is replenishment applications from the supermarket stores, which stock stores is greater than the safety stock. The symbols and meanings involved in the model are as follows: Each supermarket chains store is represented as i, j, and N is the total number. The fresh agricultural products inventory of the supermarket chains system is represented as S₀, the fresh agricultural products stocks of store i is represented as S_i. The ordering cost of fresh agricultural products of each supermarket chains stores is represented as C_r. The unit time fee of fresh agricultural products of each supermarket chains stores is represented as C_h. The Shortage Costs of fresh agricultural products of each supermarket chains stores is represented as C_l. The lateral transportation costs of fresh agricultural products of each supermarket chains stores is represented as C_ij, the average transportation cost of the supermarket chains stores is represented as. The probability of supermarket store i meeting the demand of fresh agricultural products with its own stocks is represented as a_i. The probability of supermarket store i meeting the demand of fresh agricultural products rely on transverse replenishment is represented as b_i. The probability that the fresh agricultural products of supermarket store i out of stock is represented as l_i. Because only when the fresh agricultural products of entire system is out of stock, then the fresh agricultural products of each supermarket store may is out of stock, therefore, the probability of the shortage of system fresh agricultural products: l = l_i. The expectation customer service levels of the supermarket chains system is represented as e. The demand of fresh agricultural products of unit time of supermarket stores i is represented as D_i; the demand of fresh agricultural products of system is represented as D₀; the average demand of each supermarket store is represented as m_i; the replenishment quantities from store i to stores j is represented as D_ij. The replenishment lead time of fresh agricultural products of each supermarket chains stores is represented as L_i. It is the time that from the issue replenishment orders to the commodity storage.

The total inventory costs include order costs, storage costs, lateral transportation costs, the shortage loss costs.

For the order costs C_R: the order cost of fresh agricultural products can be represented a product by a single order costs and the total number of orders.

In the formula, represents the actual shortage amount of each time, represents the total number of orders.

For the storage costs C_H: the order cost of unit time of fresh agricultural products can be represented a product by the unit storage cost and the Average inventory quantities.

Because the model is based on the premise allowing out of stock, so the average inventory quantities do not include the demand quantities in the period of lead time. The average inventory quantities and the storage costs can be got as follows:

Lateral transportation costs C_T: the transverse transportation costs is related with the average transportation costs, the average demand rate of fresh agricultural products and the probability of meeting the demand of fresh agricultural products through transverse transportation.

Throughout the supermarket chains system, there are three cases about the stock demand of fresh agricultural products: 1) the fresh agricultural products of stores can be met through the original stock, then the probability is a_i; 2) the fresh agricultural products of stores can be met through lateral transportation, the probability is b_i; 3) the entire supermarket system is out of stock, the fresh agricultural products cannot be met, then the probability is l_i. And, a_i + b_i + l_i = 1.

According to independent demand inventory model, in the period of order lead time, the demand of store i is D_i L_i, then:

If the system is out of stock, it does not meet the conditions of transverse replenishment. Then, the store i is out of stock in the period of optimal ordering, while other stores j is in the lead time or in the period of order cycle. And the demand is always greater than the actual stocks of stores. Then, the probability is as follows:

Average demand of the supermarket store i mentioned in the parameter setting is m_i, but the actual average demand need to be further refined. Some of which are transported from other supermarket stores laterally, and they did not affect the overall system inventory, therefore, they need to be removed. After the transverse replenishment among other stores, there are still unresolved demand, and the demand in this part needs to be calculated in the store i. The actual average demand of the store i fresh agricultural products is derived as follows.

The lateral transportation costs of the whole supermarket chains system can be obtained from the above equation as follows:

The shortage loss costs of the whole supermarket chains system is as follows:

In summary, the inventory total cost model of fresh agricultural products of the supermarket chains system is as follows:

s.t.

If there is no lateral transportation between supermarket store, the model becomes the follows:

s.t.

Assume that eventually all fresh agricultural products of the supermarket chains system is distributed to all the sub stores of the supermarket chains system, the model will be the supply and demand balance transport model. m represents the stores that has Surplus, referred to as A₁, A₂, A₃, …, … Am. n represents the stores that has shortage, denoted by B₁, B₂, B₃, …, … Bn. The quantities of surplus is recorded as a₁, a₂, a₃, …, …, … am The quantities of shortage is recorded as b₁, b₂, b₃, …, …, … bn. x_ij is the transporting capacity of the supermarket chains which is transported from store A_i owning surplus to store B_j that has shortage (i = 1, 2, 3, …, ... m, j = 1, 2, 3, …, … n). C_ij represents the unit transportation costs from store A_i to store B_j d_ij is the distance from store A_i to store B_j, but it will be affected by traffic in the actual transportation process. If it is considered only in distance or unit transportation cost. There will be inaccurate to the distribution program of the whole system. And the traffic situation the day is not the same in each time. r_ij represents traffic situation coefficients from store A_i to store B_j. r_ij is only in 0 - 1. The more it is closer to 1, the better the traffic situation it is. On the contrary, it is worse. q_ij represents the quality factor of fresh agricultural products from store A_i to store B_j. q_ij is only in 0 - 1. The more it is closer to 1, the better the quality factor it is. On the contrary, if it is closer to 0, it is worse.

When there is shortage of store i, the store i need to make replenishment application to other stores that owning more fresh agricultural products. And, when store i has the surplus of fresh agricultural products goods stores, the surplus that has excess safety stock, needs to be transported to the stores is out of fresh agricultural products. Then, the transverse replenishment transportation model is established on the basis of the traditional transportation model as follows:

A supermarket is a national supermarket chains and is the large-scale retail enterprises. There are a total of seven regional chain store, these stores are replenish by the same fresh agricultural products distribution center. In recent years, in order to increase traffic, improve the market competitiveness of industry peers, the supermarket chains increase a significant operating proportion of fresh agricultural products in supermarkets stores, and regard the fresh agricultural products as highlights and features.

Since the distribution center is relatively far from the supermarket stores, or due to the certain small demand of fresh agricultural products not up to the required amount of the distribution center, the distribution center is not to carry out the distribution, resulting in the shortage situation of fresh agricultural products. Because when some supermarket stores predict the demand of fresh agricultural products, there is large deviation, which results in excessive accumulation of fresh agricultural products. And the fresh agricultural products cannot be sold due to the rot, the profit margins of the fresh agricultural products will be reduced. For the cases above, the transverse replenishment will be applied to the supermarket, which will be used to solve the shortage and the surplus problems of fresh agricultural products.

The operational status of the seven chain stores is same. The demand of fresh agricultural products is independent and is in keeping with poisson distributions. The order lead time of fresh agricultural products is same and unchanged. All supermarket stores apply (s, S) storage strategy.

Related data are as follows:

1) The S_i, s_i of each supermarket store, the average transportation cost: is as shown in the Table 1.

2) The single ordering cost: C_r = 1000, the inventory holding costs of a single piece of fresh produce of unit time: C_h = 5, the shortage loss costs of a single piece of fresh agricultural products: C_l = 20.

3) The demand of fresh produce stores varies every day. Criteria to the demand of a month, the average daily demand m_i of the store is as shown in Table 1.

The above data are into the formula (1) and formula (2), then the order costs, storage costs can be obtained under the conditions of not allowing transverse replenishment. Figure 1 shows the calculation of order costs by using MATLIB. Figure 2 shows the calculation of storage costs by using MATLIB.

4) The demand of fresh agricultural products is in keeping with poisson distributions: D_i ~ P (s_i)；

,.

The fresh agricultural products of stores can be met through the original stock, and the probability is ai.

According to the Equation (3) and the probability formula of poisson distributions, the can

be got. And the The results are shown in Table 2.

When the supermarket chains system is out of stock, it does not meet the conditions of transverse replenishment. Then, supermarket stores at this time needs to place an order to suppliers or distribution center, the rate of shortage is l_i, b_i. According to Equation (4) and the formula a_i + b_i + l_i = 1, the results obtained are shown in Table 3.

5) Derive the probability of the supermarket stores using its own inventory to directly meet demand or the probability of the supermarket stores meet demand through replenishment indirectly. According to the formula 5, the stores actual average demand rate can be got.

And so on, , , , ,

6) The above data are into the formula (6) and formula (7), then the lateral transportation costs, the shortage loss costs can be obtained.

7) The order costs, storage costs can be obtained under the conditions of allowing transverse replenishment.

8) According to the inventory cost model, the total cost of inventory can be obtained under the conditions of allowing transverse replenishment.

9) When the supermarket chains system does not meet the needs of fresh agricultural products, supermarket stores at this time needs to place an order to suppliers or distribution center. Then, there will be no lateral transportation costs, and the the total cost of inventory is as follows.

With regard to the two inventory costs of allowing the transverse replenishment and not allowing the transverse replenishment, it can be got that the total inventory cost of allowing the transverse replenishment is lower than the total inventory cost of not allowing the transverse replenishment. When the supermarket adopt the transverse replenishment, even though the loss caused by shortages is much lower than ordering directly to the distribution centers, even though it needs bear the cost of transporting in the lateral transportation, and even though the storage costs are slightly higher than the cost of the replenishment adopted by distribution center or supplier, there but is a relatively small impact on the total cost of inventory. Therefore, the supermarket adopt the transverse replenishment can make the total inventory costs of the entire supermarket chains system’s more lower.

Because the transverse replenishment inventory model obtains the total inventory cost of allowing the transverse replenishment is lower than the total inventory cost of not allowing the transverse replenishment. So, it can be provided for the supermarket chains to reduce the loss of fresh agricultural products. After the analysis and calculation of inventory cost model, we can transport the fresh agricultural products from stores owning surplus to store owning shortage. Then, the best transportation plan should be designed in order to transport the fresh agricultural products with the lowest transportation costs.

Table 4 shows the unit transportation cost data. Table 5 shows the distance data of stores. Table 6 shows the traffic coefficient merits. Table 7 shows the quality coefficient table of fresh agricultural products. Related data of all stores of a supermarket chains are as follows:

According to the data, the transverse replenishment transportation model can be established.

According to the traditional method of solving the transport model, we first need to calculate the underlying table, which is listed as follows. Table 8 shows the underlying table of table-manipulation method.

For the accuracy of the calculation, Vogel method is used here to find the initial solution. Table 9 shows the initial solution of the transportation program.

Use the potential algorithm to distinguish the optimal initial solution, and the optimal transportation plan is obtained.

A₁ Þ B₁ (300), A₁ Þ B₄ (300);

A₂ Þ B₃ (800);

A₃ Þ B₂ (500), A₃ Þ B₃ (200), A₃ Þ B₄ (500).

From the above we can see that he transverse replenishment transport model above get the optimal dispatching scheme between stores, that means when we transport the fresh agricultural products from the surplus supermarket stores to the shortages supermarket stores, this strategy can be used to reduce the transportation costs.