Hammoud is planning to start a diet program. Following are the main ingredients of his diet: milk, bread, rice, eggs, vegetables, chicken, lamb, fish, halawa, and ice-cream. The related information of the ingredients are given as follows: Ingredients Units Calories/ one unit Protein/one unit Fiber/one unit Cost/one unit milk bread rice glass slice bowl unit 45 250 200 155 8 15 0 2 0.5 0 1 2 4 1 2 1 vegetables chicken lamb fish bowl piece slice 120 300 3 1 5 piece 250 40 0 10 40 0 40 200 40 0 30 Following are the main restriction of his diet plan for a given meal: (i) The total calories intake per meal should be 800 +50. (ii) The total protein intake per meal should be 6010. (iii) The total fiber intake per meal should be 15 +5. (iv) He cannot eat more than three types of main ingredients in a meal. (v) If milk is taken in the meal, then bread and eggs must be part of the meal. (vi) If lamb or chicken is taken in the meal, then fish should not be taken in that meal. (vii) If bread and eggs are part of the meal, then milk must be part of that meal. (viii) Either halawa is included or ice-cream is included in the meal (but not both). (ix) Chicken, lamb and fish cannot be in one meal. (x) No main ingredient must be taken more than two time in a day. halawa scoop 250 3 1 10 ice-cream scoop 200 2 1 5 Now, answer the following: (a) Define a variable that captures the number of units of the main ingredients. (b) Using the above variable, build the constraints corresponding to restriction (i), (ii) & (iii). (e) Can you use the above variable to build the rest of the restrictions? If not, then define a new variable and relate it to the previous variable. (d) Using the new or old variable, build the rest of the constraints. (e) If the goal is find a meal with lowest total cost, then define the corresponding objective function. (f) Write the complete mathematical formulation with the following modifications: The diet plan is for a month (30 days), where each day should contain (3) meals. The goal is to reduce the total cost for one month diet plan. Restriction (i) to (viii) are applicable for each meal. A main ingredient should not be included in all the three meal of a day. Every main ingredient should be taken at-least once per 5 days. Note: You may need to update the variable definitions and define parameters, in-order to rewrite the entire model. Note: In Part (f), write an implicit model, that is, use summation (E) and enumeration (V) symbols wherever possible. For summation symbol, specify the limits of the summation. If you use enumeration symbol, make sure the index range is clearly defined in your solution. Hammoud is planning to start a diet program. Following are the main ingredients of his diet: milk, bread, rice, eggs, vegetables, chicken, lamb, fish, halawa, and ice-cream. The related information of the ingredients are given as follows: Ingredients Units Calories/ one unit Protein/one unit Fiber/one unit Cost/one unit milk bread rice glass slice bowl unit 45 250 200 155 8 15 0 2 0.5 0 1 2 4 1 2 1 vegetables chicken lamb fish bowl piece slice 120 300 3 1 5 piece 250 40 0 10 40 0 40 200 40 0 30 Following are the main restriction of his diet plan for a given meal: (i) The total calories intake per meal should be 800 +50. (ii) The total protein intake per meal should be 6010. (iii) The total fiber intake per meal should be 15 +5. (iv) He cannot eat more than three types of main ingredients in a meal. (v) If milk is taken in the meal, then bread and eggs must be part of the meal. (vi) If lamb or chicken is taken in the meal, then fish should not be taken in that meal. (vii) If bread and eggs are part of the meal, then milk must be part of that meal. (viii) Either halawa is included or ice-cream is included in the meal (but not both). (ix) Chicken, lamb and fish cannot be in one meal. (x) No main ingredient must be taken more than two time in a day. halawa scoop 250 3 1 10 ice-cream scoop 200 2 1 5 Now, answer the following: (a) Define a variable that captures the number of units of the main ingredients. (b) Using the above variable, build the constraints corresponding to restriction (i), (ii) & (iii). (e) Can you use the above variable to build the rest of the restrictions? If not, then define a new variable and relate it to the previous variable. (d) Using the new or old variable, build the rest of the constraints. (e) If the goal is find a meal with lowest total cost, then define the corresponding objective function. (f) Write the complete mathematical formulation with the following modifications: The diet plan is for a month (30 days), where each day should contain (3) meals. The goal is to reduce the total cost for one month diet plan. Restriction (i) to (viii) are applicable for each meal. A main ingredient should not be included in all the three meal of a day. Every main ingredient should be taken at-least once per 5 days. Note: You may need to update the variable definitions and define parameters, in-order to rewrite the entire model. Note: In Part (f), write an implicit model, that is, use summation (E) and enumeration (V) symbols wherever possible. For summation symbol, specify the limits of the summation. If you use enumeration symbol, make sure the index range is clearly defined in your solution.