Q3. Linear programming

(4 marks for presentation and explanation)

a. [6 marks]

Solve the following linear program using Simplex Method:

Maximize = 40 + 30 Subject to: + 2 16 +9

3 + 2 24 , 0

b. [36 marks]

Inadequate intake of micronutrients may result in micronutrient deficiency which will cause impaired immune and visual function. Evidence suggests that increased intake of micronutrient-dense foods with low energy density can to prevent micronutrient deficiency. The following table suggests the daily intake of four common food micronutrients for adults.

Table 1 Suggested daily intake for micronutrients

However, micronutrient-dense foods are relatively expensive. Jack would like to create a low-cost shopping list that are nutritionally adequate, health promoting, and culturally acceptable. Although these micronutrients can be obtained from all kinds of foods, Jack creates three food groups (FGs) based on his every day dietary habits (must have) to simplify the question. These groups are: milk and milk product, vegetables and fruits, and meat and meat products. Using a food composition database he found online, he make his own food composition table for per serving of each FG listed below.

Table 2 Food composition for three FGs

Furthermore, he surmises the average cost for each FG per serving as follow: $2.5 for milk and milk product, $2.8 for vegetables and fruits, and $4.6 for meat and meat products. Can you help Jack to create a low-cost shopping list?

i. Create a linear programming model for the question Your solution should provide:

[46 marks]

Micronutrient	Minimum	Maximum
Fibre (g/MJ)	3	n/a
Potassium (mg)	3500	n/a
Magnesium (mg)	350	n/a
Iron (mg)	9	25

Food group	Fibre (g/MJ)	Potassium (mg)	Magnesium (mg)	Iron (mg)
milk and milk product	0	150	11	0.4
vegetables and fruits	0.6	169	97	3.5
meat and meat products	0	420	86	2.5

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1. 1) Definitions for all the decision variables;

2. 2) The objective and the objective function;

3. 3) All constraint equations;

ii. Determine the optimal mix of three FGs using Excel. Your solution should provide:

1) an Excel model; 2) an answer report and a sensitivity report; 3) detailed the total cost and output for each FG;

(3 marks) (2 marks) (6 marks)

(6 marks) (4 marks) (4 marks)

iii. Answer the following questions using the reports 1) Find the range of optimality for the cost coefficient of each decision variable 2) Explainthemeaningof1E+30forMagnesium

3. 3) If possible, which micronutrient intake should be reduced to save the cost? Why? (2 marks)

4. 4) Find the new minimal cost if the average price for meat and meat products increased by $2.5.

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5)

(2 marks)

What will happen to the optimal solution if the suggest intake for Potassium is decreased to 2500 mg? Find the new minimal cost.