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Multi-objective Model for Daily Diet Planning
1Mohd Arif Khan, 2,*Ahteshamul Haq, and 3Aquil Ahmed
Department of Statistics & Operations Research Aligarh Muslim University, Aligarh-202002, Uttar Pradesh, India 1mohdarifkhan4012@gmail.com, 2,*a.haq@myamu.ac.in, 3aquilstats@gmail.com
*Corresponding author
Abstract
In this paper, we present the development of a daily diet model using fuzzy multi-objective goal programming (GP) to satisfy daily nutrient. We have designed the objective function as minimize the cost of diet, Saturated Fat and carbohydrate. This paper consists of ten consumed foodstuffs as the decision variable. The daily diet's tolerable lower and upper intake level is given for the Protein, Vit. B6, Vit. C and Calcium. This paper aims to present a stepwise solution procedure based on fuzzy GP to obtain the compromise solution of the diet problem. Finally, a numerical example is illustrated to compare the daily diet plan with weighted GP, pre-emptive GP and fuzzy GP.
Keywords: Diet Planning, Multi-objective, Weighted goal programming, Preemptive Goal Programming, Fuzzy Goal Programming
I. Introduction
It is a severe problem throughout life to meet their health goals based on the daily diet. Researcher and scientists have been using different types of mathematical programming to solve this type of question. With the assist of operations research techniques, it is quite viable to discover a listing of foodstuffs in an appropriate quantity which can grant all nutrient pointers in a day. Firstly, the diet problem by using linear programming was solved by Smith [1]. Anderson and Earle [2] have done the comparative study of diet planning through linear programming and GP approach for daily nutritional requirements of Thais. Nutritionists are turning into extra conscious about the overdoses of vitamins and want for a balanced consumption of all nutrient. The essential traits of the real-world decision-making problems going through human beings at present are multidimensional and have multiple objectives which include economics, social, environmental and technical ones. Hence, it appears natural that the consideration of many objectives in the actual decision-making process requires multi-objective approaches rather than a single objective.
Linear GP is one of many techniques for dealing with the modelling, solution, and analysis of multiple and conflicting objective by reducing it to a single (or sequential) objective one. Since Charnes and Cooper [3] introduced the concept of GP. Pre-emptive GP is a particular case of GP in which the more critical (upper level) goals are optimized before lower-level goals. Once complication concerned the weighting of goals in the objective function, Ignizio [4] demonstrated the use of weighted GP in diet planning and presented the results of the problem involving the selection of foodstuffs for improvement in nutritional balance by minimizing the cost of foodstuffs. Many authors have been worked on diet planning, some of them are listed in Table 1.
Table 1. Research Review Summary
Authors Model Techniques used Remarks
Objective
Single Multi LP IP WGP FGP PGP
Eghbali et al. ✓ ✓ Multi-objective Fuzzy Linear
[5] Programming, Diet Model
Nath et al. [6] ✓ ✓ Trial and Error Method,
Nutritional model
Eghbali et al. ✓ ✓ Multi-objective Fuzzy
[7] Programming, Diet Problem
Mamat [8] ✓ ✓ Diet Planning, nutritional
requirements, Fuzzy
Programming
Eghbali- ✓ ✓ Mixed Integer Linear
Zarch et al. Programming, Diet Plan, Jimenez
[9] and Epsilon-constraint Method
Sheng and ✓ ✓ Diet Planning, Integer
Sufahani [10] Programming
Ali et al. [11] ✓ ✓ diet planning for boarding
schools, Zero One Integer
Programming
Bhargava et ✓ ✓ weighted GP, Diet planning
al. [12]
Proposed ✓ ✓ ✓ ✓ Diet plan, pre-empty GP,
model weighted GP, fuzzy programming
Decerle et al. [13] highlight the relationship between working time, quality of service and route balancing for the home health care problem by using Pareto based approach. The objectives of the model are the minimization of the total working time of the caregivers while maximizing the quality of service and minimizing the maximal working time difference among nurses and auxiliary nurses. Nguyen and Montemanni, [14] propose mixed linear programming to find the best schedule minimizing the costs due to the non-respect of patients' time windows and exceeded hours of caregivers. En-nahli et al. [15] develop a multi-objective optimization problem in which the model tries to satisfy the Home Health Care Services objectives. On the other hand, 'patients and caregivers' objectives that satisfy all patients by assigning their wished caregiver, help to get solutions taking into account the priority of a patient and the affinity patient-caregiver. Mutingi & Mbohwa [16] present a multi-agent architecture that facilitates decision making characterised with multiple objectives and the capabilities of a multi-agent system and Web services as to facilitate effective decisions for home healthcare services by using genetic algorithm. Niakan & Rahimi [17] presents a multi-objective mathematical model to address a Healthcare Inventory Routing Problem for medicinal drug distribution to healthcare facilities. The first part of objective function minimizes total inventory and transportation costs, while satisfaction is maximized by minimizing forecast error which caused by product shortage and the number of expired drugs; Greenhouse Gas emissions are also minimized. A hybridized possibilistic method is applied to cope with uncertainty, and an interactive fuzzy approach is considered to solve an auxiliary crisp multi-objective model and find optimal solutions. Othman et al. [18] composed two phases: the first one is an assignment procedure based on fuzzy logic and the second phase is based on an evolutionary method to solve the problem of medical staff scheduling which improves the performance of the scheduling system in order to help physicians to manage the organization better. Turgay & Ta§kin [19] presents fuzzy GP using exponential membership function, which uses the modelling, and solving of health care system for optimal, efficient management and prioritized for the strategic
planning and resource allocation. Zhang et al. [20] examine the health-care facilities that should be located to improve the equity of accessibility, reduce the population that falls outside the coverage range, raise the total accessibility for the entire population, and decrease the cost of building new facilities and use genetic algorithm-based multi-objective optimization approach to yield a set of Pareto solutions. The multi-objective optimization approach is used to optimize the location of new health-care facilities which provides a set of different plans that compare the values of the objectives and comparing the Pareto solutions with other solutions.
In this paper, we present the development of a multi-objective daily diet model using preemptive and fuzzy GP to satisfy daily nutrients with an example. The objective function is designed to minimize the cost of the diet, Saturated Fat and carbohydrate. The objective of this approach is to select diets to meet specific nutritional requirements. The comparison for the daily diet plan with weighted GP, pre-emptive GP and Fuzzy GP is also shown.
II. Formulation of the Diet Model
A diet is required to propose by the dietician for the special needs of the patient. An integer number of units of the diet can be composed as ten basic foodstuffs termed as Food 1, Food 2, ..., Food 10. The nutrients that are used in the model are Saturated Fat, Carbohydrate, Protein, Vitamin C, Vitamin B6 and Calcium. The lower and upper levels of Protein, Vitamin C, Vitamin B6 and Calcium are used as constraints.
In the diet model, x(j = 1,2,..., n) represents the different types of food items that work as a
decision variable and the cost of food, Saturated Fat, and Carbohydrates for each food are CDj,SPIj, C^ (j = 1,2,..., n). Then the objective function will be
Min Z - y C x , Min Z =y S x , Min Z =y C x
1 Djj 2 ¿^ Fj j 3 ¿^ dj j
j-1 j "I j "I
The constraints of the model satisfy the nutrients requirements. The nutrient contents of the food items in respect to the diet concerning nutrients are represented on the left-hand side of the constraints, and the right-hand side of the constraint is lower and upper demand of each nutrient (Protein, Vitamin B6, Vitamin C, and Calcium). The upper requirement of Saturated Fat also works as a constraint. Then the constraints are as follow:
y Px > P , y Px < P ; for Protein
^^^ j j Min ' ^^^ j j Max J
j-1 j-1
y Vx > V , y Vx < V ; for Vitamin B
Bj j BMtn ' Bj j BMax 6
j-1 j-1
y V x > V , y V x < V ; for Vitamin C
^^ Cj j CMn ' Cj j CMax '
j-1 j-1
ZC x >C , y C x < C ; for Calcium
lj j lMin lj j lMax
j-1 j-1
ZS x < S ; for Saturated Fat
Fj j FMax 7
j = 1
x > 0, x < 4, x e Integer Vj - 1,2,...,n
III. Procedure for Solving Multi-Objective Problem
Fuzzy GP is flexible and powerful techniques that can be applied to a variety of decision-making problems that have multiple objectives. Therefore, we can use this approach to obtain the optimal compromise solution for the formulated models. The stepwise solution procedure is given as follows:
Step 1: Solve the multiple objective problems by considering a single objective at a time and ignoring the others with the given set of constraints. The solution thus obtained is the idle solution. The payoff matrix constructs using idle solutions. Finally, the payoff matrix helps to construct the aspiration level to each objective function.
Step 2: The aspiration level of the objective function is set as the goal value (gk,k = 1,2,3). Find X = (x2,■■■,xn) to optimize the following fuzzy goals
Z(J) Z(J) <g2,Z3(X)<g3
Subject to the constraint
10 10 10 10 y Px > P , y Px < P , y V x > V , y V x < V
j j Mm 7 j j Max ' ^^ Bj j BMin 7 Bj j BMax
j=1 j=1 j=1 j=1
10 10 10 10
y Vx > V , y V x < V , y Cx > C , y Cx < C
Cj j CMm ¿—i Cj j CMax ' lj j lMm ¿—i lj j l
j =1 j =1 j =1 j =1
y^S x < S x > 0, x < 4 Vj = 1,2,■■■,10
7 ' Fj j FMax j 3 j Jiii
j=1
where, (gi) = Min (zt(x)) (.-<,) \//»(x,(x)) and(:<) U/„ (z3(x>) • The symbol ' j' (the type of fuzzy-min) referring to that zt (X), z2 (X) and z3 (X) should be approximately less than or equal to the aspiration level g , g and g up to the specified tolerance limit.
Step 3: Construct the fuzzy linear membership function the membership function of the fuzzy goal of x, (_v) -<gi {i.e., fuzzy-min) as:
^l(Zl( X)) =\
1, if Zl(X) < g1
U - Z (X) 1 ^ if g1 <zi(x) <U
U: - g
.0, if Z1(X) > U
where, the upper tolerance limit for the fuzzy goal zl (x) is u ■
Similarly, the membership function for the fuzzy goal Z2 < g2 (i.e., fuzzy-min)
1 if Z 2 (X) < g 2
U 2 - Z 2 (X )
^(Z 2 (X))
if g 2 < Z 2 (X) < U 2
U2 - g2
[0, if Z2 (X) > U2
where, the upper tolerance limit for the fuzzy goal z2 (x) is u2 ■
Similarly, the membership function for the fuzzy goal Z3 ■< g3 (i.e., fuzzy-min)
M3( Z3( X )) H
1,
U 3 - Z,( X )
if z,(X) < g3
if g, < Z3(X ) < U3
U3 - ^3
[0, if Z3(X) > U3
Where the upper tolerance limit for the fuzzy goal z3 (x) is ul ■ Step 4: Finally, the mathematical form of all the above-given steps are summarised as:
Max D(M) = Mi (Zi (X)) + m2 (Z2 (X)) + m (Z3 (X)) subject to constraint
10 10 10 10
Y Px > P , Y Px < P ,y Vx > V , y Vx < V
^^^ j j Min ' ^^^ j j Max ~ ^^^ Bj j BMin ' ^^^ B j BMax
j=1 j=1 j=1 j=1
10 10 10 10 Yj > VCMn , Y VCjxj < VCMax , Y C/jxj > ClMin , Y j < ClMax
U - Z (X ) U - Z ( X ) ^,(Z,(X )) = -1-, 0 < Zi(X )) < 1, Z2(X )) = -2-2—,0 < Z2(X )) < 1
a( Z,( X )) =
U - g
11
U3 - Z ,(X )
U 3 - g 3
U - g
22
0 Z (X )) < 1, X. > 0, x. < 4, x. e Integer Vj = 1,2,..., n
D(m) is called the fuzzy achievement function. Finally, we have a single objective problem that can be solved by using a suitable classical optimization technique.
j=1
j=1
j=1
j=1
IV. Numerical case study
A diet is required to propose by the dietician for the special needs of the patient. An integer number of units of the diet can be composed as ten basic foodstuffs termed as Food 1, Food 2, ., Food 10. The values of protein, vitamin C, vitamin B6, saturated fat and calcium ideally fall between the bounds which are given in Table 2. The data is taken from Bhargava et al. [12].
Table 2: Nutritional and cost of the foodstuffs
Nutritions Food Types Daily
1 2 3 4 5 6 7 8 9 10 Demand
Protein (g) 3.3 25.5 2.5 11.0 27.3 3.3 1.2 1.2 2.6 0.4 40.0 15.0
Vit B6 (mg) 0.06 0.10 0.02 0.22 0.29 0.11 0.29 0.05 0.07 0.06 1.0 2.0
Vit C (mg) 1 0 0 0 0 44 11 7 10 6 50 100
Calcium(mg) 120 720 11 35 7 40 6 20 18 4 700 1000
Sat. Fat (g) 1.0 21.7 0.6 0.4 5.2 0.2 0.1 0.1 0.3 0.0 - 15
Carbohydrat 5.0 0.1 0.0 75.7 0.0 1.1 23.2 2.6 30.9 11.8
es (g)
Cost (Rs.) 2.5 15.0 3.3 1.8 20.0 2.5 1.5 6.0 1.5 1.6
The bounds for the three objective functions are as: 29.9 < Z < 54.5, 5.7 < Z2 < 7.8, and
161.3 < Z3 < 366.7. Using these bounds, the corresponding linear membership functions for the three objective functions are constructed as follows:
X )) =
1,
54.5 - Z (X) 54.5 - 29.9 :
0,
M2(Z2 (X)) H
1,
7.8 - Z2 ( X ) 7.8 - 5.7 1
MZA X )) H
1,
366.7 - Z3 ( X ) 366.7 -161.3 0,
if Zl(X) < 29.9 if 29.9 < Z (X) < 54.5
if Z(X) > 54.5 if Z2(X) < 5.7
if 5.7 < Z(X) < 7.8
if Z2(X) > 7.8 if Z (X ) < 161.3
if 161.3 < Z (X) < 366.7 if Z (X) > 366.7
Using the method defines in Section 3; we calculate the compromise solution for the model, which is given in table 3.
Objective Values The optimal number of quantities of each Foodstuff
Zî=33.0, Z2=6.9, Z3=251.9 xi=4, X2=0, X3=2, X4=3, %5=0, %6=2, xz=0, X8=1, X9=0, xio=0
0
This solution is accepted by the DM, which belongs to the preferred compromise solution of fuzzy acceptance rate 0.620488. The membership values with the percentile contribution of each objective are shown in Fig. 1.
Fig. 1: Membership values of the objective with percentile contribution Table 4: Comparison with weighted, pre-emptive and fuzzy gp
Units of Foodstuff Nutrition Quantity
Method 123456789 10 P c S V V C Cost
d Fl B C l
Weighted GP 4032010330 57.4 273.0 8.0 1.21 99 737 48.5
Pre-emptive GP 4 0 0 3 0 1 0 4 0 0 54.3 258.6 5.8 1.21 76 705 41.9
Fuzzy GP 4 0 2 3 0 2 0 1 0 0 59.0 251.9 6.9 1.21 99 707 33.0
The optimal compromise values of each foodstuffs with Weighted, Pre-emptive and Fuzzy GP are shown in Fig. 2.
Fig 2: Compromise optimal unit for foodstuff
The comparison of membership values of each objective solved by Pre-emptive and Fuzzy GP are shown in Fig. 3.
Membership Values
1 0,8 0,6 0,4 0,2 0
PGP
FGP
Carbohydrate Saturated Fat Total Cost
0,5121951 0,952381 0,5262902
0,8739837 0,4285714 0,5589094
Fig 3: Graphical representation of membership values
V. Conclusion
The human body needs foods with a low content of Saturated Fat and Carbohydrate, although high in Protein, Vitamin B6, Vitamin C, and Calcium. The multi-objective daily diet model is solved by using pre-emptive and fuzzy GP to satisfy daily nutrients through an example. We have designed the objective function as minimizing the total cost of the diet, Saturated Fat and carbohydrate. We use this approach to select diets and meet precise nutritional requirements. The comparison for the daily diet plan with weighted GP, pre-empty GP and fuzzy GP is also shown. The finding obtained in the fuzzy programming approach has been contrasted with the weighted GP approach and the pre-emptive GP approach, and it demonstrates that the fuzzy GP approach gives a more precise and accurate solution and is a useful technique.
References
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