Frontiers in Computing and Intelligent Systems 
ISSN: 2832-6024 | Vol. 6, No. 2, 2023 

 

63 

Optimization	of	Food	Distribution	System	Based	on	
Dynamic	Planning	Model	
Xunyang Li 1, Jin Chen 2, * 

1 School of Finance, Anhui University of Finance and Economics, Bengbu, Anhui, China 
2 School of Accountancy, Anhui University of Finance and Economics, Bengbu, Anhui, China 

* Corresponding author: Jin Chen (Email: 1730607338@qq.com) 

 

Abstract: Based on the data of per capita annual income, food production per hectare, and per capita consumption of 
consumers in different countries, this paper first establishes a dynamic planning model, starting from three indicators, namely, 
regional transportation time, purchasing capacity and actual demand, and using MATLAB, establishes a visual image to derive 
the changes of satisfaction, profitability, environment, and efficiency in the next ten years. In order to determine the priority of 
the four indicators, a hierarchical analysis model is established on the basis of the dynamic planning model to derive the 
calculated weights of each indicator. In order to verify the adaptability of the model, the model is applied to the developed 
country Britain and the developing country China, respectively, and the dynamic planning model is confirmed again according 
to the information returned by the visualization image, which illustrates the scalability of the model to the food distribution 
system in different countries and regions. 

Keywords: Food Distribution System; Dynamic Programming Model; Hierarchical Analysis Approach. 
 

1. Introduction 
The instability and insecurity of the global food system and 

the inability of the food supply to meet the needs of individual 
countries and regions are realities [1], and 821 million people 
worldwide suffer from hunger. The current food distribution 
system also causes greenhouse gas emissions, fresh water 
consumption and other problems. This paper establishes a 
dynamic planning model and hierarchical analysis method to 
optimize the food distribution system, and creates a 
sustainable and efficient food distribution system of practical 
significance. 

2. Description of Research Hypotheses 
and Variables 

2.1. Research Hypothesis 
The study in this paper is based on the following 

assumptions: (1) The data obtained are perfectly correct. (2) 
Only disasters caused by pollution resulting from food 
production are considered. (3) People's subjective emotions 
do not affect the quantity and rate of food production. For 
example, there will be no strikes, lockouts, etc. (4) The impact 
of the environment on food production and transportation 
production can be accurately quantified. (5) The population 
of each region remains constant, as does the demand for food. 
(6) All food has been distributed. (7) Relationships between 
elements in the same level of the hierarchical analysis are 
equal and independent of each other. (8) The weights in a 
hierarchical analysis can be measured in a certain way and 
can be relatively accurate. 

2.2. Description of Variables 
The variables applied in this study are as follows: total 

amount of food (M), total number of districts (N), length of 
time (T), factors of sustainability (β), satisfaction level (μ), 
operational efficiency (e), net profit (W), transportation time 
for each district (t0), sum of t0 (t), purchasing capacity (c), 

actual demand for each district (n), factors of food 
transportation (α), food allocated to each district quantity (x), 
transportation cost per unit of time (θ), environmental impact 
of production methods (k), impact of production methods on 
production (p), and cost per unit of food (m). 

3. Dynamic Planning Models 

3.1. Establishment of Dynamic Programming 
Model 

Dynamic programming is a branch of mathematical 
planning and an important branch of operations research. In 
1951, state mathematician R. Bellman and others proposed 
optimization principles for solving multi-stage decision 
problems. Since then, dynamic programming was created. So 
far, dynamic programming has been widely used in the fields 
of economic management, enterprise management, 
engineering technology and optimization and control, and has 
achieved remarkable results [2]. 

We need to create a model of the food system to reimagine 
and reprioritize our food system. Create this system requires 
considering the composition of the current food system in 
context. A review of the literature and data shows that the 
current food system can be considered to be composed of the 
main subsystems of food production, processing, distribution, 
and consumption and other subsidiary subsystems [3], so the 
corresponding food system model can be established through 
this. 

First, we consider the point in time. Suppose there are M 
units of food that need to be allocated to N regions, and for 
each region, allocating i units of food, there is a corresponding 
benefit function F(i) that corresponds to it respectively. By 
adjusting the distribution scheme, the optimal distribution can 
be found and the maximum benefit can be obtained. Based on 
the background characteristics of the food distribution system, 
we find that the factors to be considered are environmental 
factors, efficiency, profitability and distribution fairness. For 
these factors, we can supplement the modeling framework 



 

64 

established above. 
We assume that the environmental factor is β, which 

corresponds to the following equation: 
β β k                   (1) 

Where, β0 is the initial environmental factor with a value of 
1; k is the environmental impact caused by current food 
production, which ranges from -0.1 to 0.1. T is the year. 

For the fairness of distribution, we use the sum of 
satisfaction of each region as an evaluation criterion. We 
assume that its sign is μ. For any region, we assume that it has 
three attributes, i.e., distribution time t0, food demand n, and 
real purchasing power c. In addition, we assume that the 
amount of food allocated to each region is x, which is given 
by the following equation: 

.                          (2) 

Where, t0 is based on 1, i.e., it can be delivered in one day; 
and the actual purchasing capacity c is also based on 1, where 
1 is just the ability to purchase, and greater than 1 indicates 
that people are willing to consume more money to spend on 
food. Taking reality into account, it is easy to see that for a 
region, the more food that is delivered, the higher the 
purchasing power, the shorter the delivery time, and the 
higher the corresponding satisfaction. Therefore, we put x in 
the numerator divided by n, which is used to reflect whether 
the delivered food meets the demand of the region; we put t 
in the denominator, where the shorter the time, the higher the 
satisfaction; and finally multiply it by c to get the formula. μ 
is the satisfaction of a certain region. In order to get the 
satisfaction of all regions, we need a new formula as follows: 

μ ∑                     (3) 

It is worth noting that the distribution time for each location 
is not static, it is related to the magnitude of the transportation 
factor α and the environmental factor β, which is given by the 
following equation: 

t                    (4) 

The base value of α is also assumed to be 1. We set the 
weights of transportation and environmental factors on 
delivery time to 0.5. When both α and β are 1, t equals t0. 

Next, we consider the efficiency of food distribution. In a 
broad sense, efficiency is the effect divided by the time. In our 
model, efficiency can be expressed as the total annual 
production divided by the distribution time at each location 
multiplied by the amount of food distributed. We use the 
symbol e for this. The formula is as follows: 

∑
                  (5) 

Where, ti denotes the actual distribution time in each region, 
xi denotes the amount of food distributed in each region. 

Finally, analyze the profit factor. Common sense suggests 
that the magnitude of net profit is related to expenses and 
benefits. The formula expressed in terms of W is as follows: 

W x. c x.m θ. t             (6) 
Where, θ is the transportation cost per unit of time, which 

is assumed to be a constant and multiplied by the 
transportation time as the total transportation cost, and m is 
the production cost per unit of food, which includes the food 
raw materials and the labor required for food processing. w is 
also the net profit of each region, which needs to be summed 
up to the total net profit. 

In explaining environmental factors, we mentioned that the 
food system can influence the environment. However, the 
environment in turn can affect the food system. In equation 
(4), environmental factors affect the distribution time of food. 

Applied to reality, disasters such as mudslides and acid rain 
that may be caused by food production can affect 
transportation. In addition, the environment also affects food 
production. For example, bad weather can affect rice growth 
and reduce production. From this, we establish a formula for 
the amount of production M on the environment: 

M 1                 (7) 
Where, p is the impact of innovations in production 

methods, with base 0. It is assumed that technological 
innovations in production are not necessarily linked to the 
environment. As p increases, β may not decrease; M0 is the 
base production. This equation implies that, over time, annual 
food production varies with the impact of innovations in 
production methods and changes in environmental factors. 

After the modeling framework is built, there is a specific 
problem; food distribution is not a simple allocation of 
resources; people in each region have a minimum standard of 
food needs, which cannot be based on evaluation indicators 
such as overall satisfaction and profitability. Distribute food 
to specific regions. Similarly, we cannot allocate most of the 
food to that region, because a large food allocation to a 
particular region is associated with a high increase in 
satisfaction. Therefore, we added upper and lower limits to 
the amount of food to be allocated to each region, with the 
minimum not to be less than half of the food demand in a 
particular region and the maximum not to be higher than 1.5 
times of the food demand in a particular region, and thus the 
model was initially established. 

The evaluation indicators are summarized in a table, as 
shown in Table 1: 

 
Table 1. Summary of evaluation indicators 

Name (of a thing) Formulas Note 

Level of 
satisfaction (μ) 2

1 1

 

M=∑  

Profit (W)  net profit 

Environment (β)  
Changing with 
the passage of 

time

Efficiency (e) 
1
∑

  

 
Once the model is established, it can be used to solve 

practical problems. For governments around the world, the 
only factors that should be considered in the food system are, 
for the time being, the impact on the environment, the 
satisfaction of the people and their own profitability, which 
we believe are the only three aspects to be taken into account. 

For the optimization of fairness and sustainability, the 
distribution scheme of the masses and the environmental 
factors are optimized in the model we built. In terms of 
practical factors, food is the basic condition for human 
survival. If the distribution is more equitable, people will not 
have to go hungry, then the credibility of the government will 
increase and the people will be happier. That is, satisfaction 
as an evaluation indicator increases; when sustainability is 



 

65 

combined with reality, it is not difficult to find that the 
external environment affects all aspects of the food system, 
including food production and food transportation [4]. As far 
as these issues are concerned, dynamic programming 
algorithms can be better. 

Combining the model and analyzing the actual situation, 
the problem is simplified to the following form: M units of 
food are allocated to N regions, and each region is allocated 
at least half of the current demand, but not more than 1.5 times. 
It is assumed that the satisfaction of each region is mainly 
related to the amount of food allocated, and how the 
allocation of food is solved to find the optimal solution. 

For this multi-stage decision process to find the optimal 
decision problem, we can first understand the distribution of 
N regions as N stages, each stage allocates a certain amount 
of food to a region. The model is described as follows [5]: (1) 
The stage variable d represents region d, d=1,2,3,.... .N. (2) 
The decision variable ud represents the food provided to 
region d. (3) The state variable can be used to allocate food to 
each zone in the current and future phases, sd=∑  = 
1,2,3,...N. (4) The indicator function for stage d is the 
satisfaction of region d, i.e., equation (2). 

Suppose V(sd ,d) is the distribution of surplus food sd to the 
remaining d,d+1,d+2,... ,N regions to obtain the maximum 
profit, and suppose L(ud ,d) denotes the satisfaction of the 
region after receiving d units of food ud . Based on the above 
model, we can use the inverse order method to recursively 
solve. The basic equation is: 

, 	 , 1, 1 ,
1, … ,2,1                (8) 

, 	 ,               (9) 

3.2. Solving Dynamic Programming Models 
First, in order to test the validity of this model, we designed 

a set of test data: 
M 30, N 4, T 10												
k 0.01, p 0.01,m 0.3
α 1, 1, θ 1.2														

          (10) 

The three attributes of the four regions are obtained as 
shown in the table below: 

 
Table 2. Test data (regional attributes) 

Region 
Transportation 

time/t 0 
Purchasing 

power 
Actual 

requirement 

A 1.5 0.8 10 

B 1.1 1 8 

C 0.8 1 7 

D 0.7 1.1 6 

 
After programming and plotting, we get the following 

figure: 

Next we adjust k to 0.01, at which point the graph drawn is 
the exact opposite of it. Analyzing the causal relationship 
between the variables, we can get the following conclusions: 
when satisfaction with food distribution increases, fairness 
increases and the profitability of the food system increases; if 
sustainable development, i.e., the improvement of factors and 
environmental factors, the efficiency of food distribution 

increases, which ultimately leads to an increase in efficiency 
and a higher level of satisfaction of the people. 

 
Figure 1. Chart of changes in satisfaction over the next ten years 

 
Figure 2. Change in profitability over the next ten years 

 
Figure 3. Map of environmental changes over the next decade 

 
Figure 4. Chart of efficiency changes over the next decade 

 
The difference between a system optimized in terms of 

equity and sustainable development and the current system 
lies in the different priorities of the distribution schemes and 
systems, which can have a direct impact on people's 
satisfaction. Most of the food distribution in today's world is 



 

66 

controlled by businessmen, who are profit-oriented, so they 
mostly prioritize profit and efficiency, and are less concerned 
with people's satisfaction and the environment. The 
environment is mainly represented in the model. The 
optimization of sustainable development will improve the 
environmental factors, which will directly affect the 
production and transportation of food and indirectly affect 
people's satisfaction, profitability and efficiency, thus 
improving them all. 

4. Hierarchical Model of the Food 
Distribution System 

4.1. Establishment of the Model 
The Analytic Hierarchy Process (AHP) is an operations 

research theory that solves a range of problems related to 
decision making. The method divides the problem to be 
solved into three levels, namely the objective level, the 
indicator level and the program level. Next, qualitative and 
quantitative analyses are carried out on this basis [6] 

The model is constructed as shown in Figure 5: 

 
Figure 5. Hierarchical model of the food distribution system 

 
After constructing the hierarchical model, we need to 

compare the factors in the middle tier with the scenarios in the 
lowest tier. We can use aij to denote the comparison results of 
factor i and factor j relative to the factors in the previous tier. 
aij is a quantitative value that indicates the relative importance, 
and the quantitative value can be expressed in a nine-foot 
scale. 

 
Table 3. Nine-foot scale method 

Comparison of factor i and 
factor j 

quantitative value 

equally important 1 

It's kind of important. 3 

critical 5 

very important 7 

extremely important 9 

The middle value of two adjacent 
judgments 

2,4,6,8 

in contrast 
Comparison of the 

importance of factor i and 
factor j Aij = 1/Aij

 
Using the nine-scale method, a judgment matrix of the 

following form can be constructed: 

⋯
⋯

⋮ ⋮ ⋱ ⋮
⋯

              (11) 

When constructing a judgment matrix according to the 
nine-scale method, the five conditions listed in the following 
equation must also be satisfied to maintain the consistency of 
the matrix, and the elements of the matrix are greater than 0, 
while the main diagonal is 1 There should be multiple 
relationships between each column and each row. 

0																																																											
1																																																												

	 	 	 	

	 	 	
																							

	 	 	 	

	 	 	 	
																		

	 	 	 	

	 	 	 	

    (12) 

 

4.2. Solving the Model 
By reviewing the data on the relevant factors in recent years, 

we can use the information in the following four tables to 
roughly represent the data in the judgment matrix that needs 
to be constructed: 

 
Table 4. Judgment Matrix for Target and Indexing Layers 

Optimal Food 
Distribution 
System (A)

level of 
satisfaction 

serve (in 
some 

capacity)

net 
profit

level of 
satisfaction 

1 2 5 

serve (in some 
capacity) 

1/2 1 2 

net profit 1/5 1/2 1 

 
Table 5. Satisfaction and Program Level Judgment Matrix 

Satisfaction 
(C1) 

Best 
satisfaction 

Optimal 
efficiency 

Optimal 
profitability 

Best 
satisfaction 

1 7 2 

Optimal 
efficiency 

1/7 1 1/3 

Optimal 
profitability 

1/2 3 1 

 
Table 6. Judgment Matrix for Efficiency and Program Levels 

Validity (C2) 
Best 

satisfaction 
Optimal 

efficiency 
Optimal 

profitability 
Best 

satisfaction 
1 1/6 1 

Optimal 
efficiency 

6 1 6 

Optimal 
profitability

1 1/6 1 

Four judgment matrices can be obtained using the matrices 
in the four tables above: 

 
 
 
 



 

67 

Table 7. Net Profit and Program Level Judgment Matrix 
Net profit 

(C3) 
Best 

satisfaction 
Optimal 

efficiency 
Optimal 

profitability 
Best 

satisfaction 
1 1 1/2 

Optimal 
efficiency 

1 1 1/3 

Optimal 
profitability 

2 3 1 

 
1 2 5
1
2

1 2

1
5

1
2

1

				 1

1 7 2
1
7

1
1
3

1
2

3 1

 

2

1
1
6

1

6 1 6

1
1
6

1

				 3

1 1
1
2

1 1
1
3

2 3 1

 

Initial processing of the judgment matrix using the eig 
function in Matlab yields the eigenvectors of the judgment 
matrix and the column vectors consisting of the eigenvalues 
of the matrix to form a diagonal matrix, and the eigenvectors 
of the largest eigenvalues are normalized. 

                     (13) 

Normalizing the eigenvectors according to this formula, i.e., 
summing a column vector by rows and then dividing that 
vector by the sum of each row, achieves the normalization of 
the eigenvector with the largest eigenvalue. We end up with a 
column vector between 0 and 1 as follows: 

, , ⋯ ,                (14) 
Find the maximum eigenvalue, and since they are all 

column vectors after multiplication, the expression can be 
expressed as: 

              (15) 

The maximum eigenvalue of the matrix can be found 
according to the above equation. The main idea is to use the 
judgment matrix A and the normalized maximum eigenvalue 
corresponding to the eigenvector w for matrix multiplication, 
and then the result and w for bitwise division, at this time, a 
vector will be obtained, which the data on the vector to sum 
the average of the maximum eigenvalue of the matrix can be 
obtained. 

Consistency test for judgment matrices. The constructed 
matrix satisfies the strict equation (12), then it is a consistency 
matrix, consistency matrix does not need to perform 
consistency test, otherwise it is a non-consistency matrix, for 
consistency matrix needs to be tested to verify the correctness 
of the results. Consistency index CI: 

                (16) 

The CI value is calculated from the maximum eigenvalue 
λmax and the order n of the judgment matrix. When CI=0, it 
means that the judgment matrix is perfectly consistent. the 
larger the value of CI, the worse the consistency of the 
judgment matrix. 

Consistency test RI. The value of RI becomes the average 
random consistency index and its value can be obtained by 
checking the following table [7]: 

Consistency ratio CR: 

                   (17) 

 

Table 8. Mean random consistency index RI 

sequences 1 2 3 4 5 6 7

RI value 0 0 0.52 0.89 1.12 1.26 1.36

sequences 8 9 10 11 12 13 14 

RI value 1.41 1.46 0.94 0.52 1.54 1.56 1.58

 
The results of the CR test are as follows: 

	 	 	 	 				 	 0.1
	 	 	 	 	 0.1

   (18) 

When, CR=0, we consider that the judgment matrix has full 
consistency; when CR<0.1, the judgment matrix can be 
considered to have satisfactory consistency; when CR≥0.1, 
the judgment matrix needs to be adjusted to modify or discard 
elements. 

Finally, the combined weight of the unit on the target layer 
is calculated and the program is evaluated. 

Assume that the weight vectors of the target and indicator 
layers are: 

, …                    (19) 
Assume that the weight vectors for the program and 

indicator layers are: 
, , …             (20) 

The final weights of the programs can be expressed as 
follows: 

                   (21) 
By calculating the above process, we can get the maximum 

eigenvalue and its corresponding eigenvector, value CI, value 
RI. 

 
Table 9. Results of the evaluation of relevant parameters 

variant
values  

    

 
0.5954
0.2764
0.1283

 
0.6153
0.0925
0.2922

 
0.1250
0.7500
0.1250

 
0.2402
0.2098
0.5499

 

 3.0055 3.0026 3.0000 3.0183 

The 
Value of 

CI
0.0028 0.0013 0 0.0091 

CR 
value 

0.0053 0.0025 0 0.0176 

From this, the weights of the three options at the program 
level can be calculated: 

0.4294
0.2977
0.2715

                 (22) 

Visualize it using MATLAB: 

 
Figure 6. AHP-based solution analysis 



 

68 

5. Application of the Models 
We support our conclusions by applying the models 

developed to a developed country and a developing country 
respectively. The developed country chosen is the United 
Kingdom and the developing country is China, and six cities 
in each are chosen as test areas. For the developing country, 
we ranked the transportation capacity of its six cities, their 
actual demand based on their total population, and their 
purchasing power based on their annual per capita income, 
resulting in the table below: 

 
Table 10. Data for developing countries 

City name t 0 C n 

A 0.9 1.5 6 

B 0.9 1.4 5 

C 0.7 1.2 14 

D 0.8 1.15 4 

E 0.75 1.1 28 

F 0.75 1 20 

For China, production and environment are synchronized 
in conjunction with current Chinese policies, then set the 
corresponding k and p and use MATLAB to calculate its 
satisfaction, efficiency, environment and profitability for the 
next ten years. The results are shown in Figure7: 

 
Figure 7. Line graph of changes in evaluation indicators for 

developing countries over the next decade 

The overall trend of the image is consistent with the test 
data. When environmental factors and factors of production 
are increased simultaneously, all indicators improve, but 
focusing on environmental improvements reduces 
profitability and satisfaction. This supports the conclusions 
we reached earlier. 

For developed countries, the six cities are also ranked in 
terms of their transportation capacity, with actual demand 
based on total population and purchasing power based on 
annual per capita income. The table is shown below: 

 
 
 
 
 
 

Table 11. Developed country data 

City name t 0 C n 

A 0.7 1.2 51 

B 0.75 0.96 12 

C 0.8 0.96 13 

D 0.9 1.08 6 

E 1 0.7 4 

F 1.2 0.65 3 

 
Considering the real situation in the UK, we set p=0.005, 

k=--0.0005 and calculate the change over the next ten years. 
The MATLAB image is shown below: 

 

 
Figure 8. Line graph of changes in evaluation indicators in 

developed countries over the next decade 
 
In this graph, we find that although k < 0, k is not very small. 

Profitability, efficiency and satisfaction are still rising for a 
short period of time, and then we stretch the time and find that 
it's 19 years old. Profitability starts to fall, and for almost 100 
years, efficiency and satisfaction are down from the first year, 
as shown in the graph below: 

 
Figure 9. Line graph of changes in evaluation indicators in 

developed countries over the next 100 years 

It is not difficult to conclude that sacrificing the 
environment for productivity can indeed be profitable in the 
short term, but in the long term it will not only have an 
indelible impact on the environment, but will also reduce final 
profits and allocative efficiency. Crowd satisfaction will also 
decline. Therefore, this practice is not desirable. 



 

69 

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