Hillier and Lieberman Case 3.3 Solution

1. CUTTING CAFETERIA COSTS
Presented By:
Ankit Anupam (UM16190)
Ankit Mangla(UM16191)
Ankit Prusty (UM16192)
Arupananda Rout (UM16193)
Aravind Sarangi (UM16194)

2. PROBLEM
STATEMENT
SCENARIO I
SCENARIO II
SCENARIO III
SCENARIO IV
SCENARIO V
SCENARIO VI
 Maria Gonazalez, a cafeteria manager is looking to cut costs on a Casserole recipe whose main
ingredients are potatoes and green beans. She has to minimize the costs maintaining the
nutritional and taste requirements.
 Pricing of potatoes and green beans –
Potatoes – $ 8.818 x10-4 / 1 g
Green Beans - $ 2.205 x10-3 / 1 g
453.6 g in 1 lb
 University has mandated that a meal must contain 180 grams (g) of Protein, 80
milligrams (mg) of Iron, and 1,050 mg of Vitamin C
 She researched on the nutritional contents of Potatoes and Beans.
 The cook Edson, informs that for a casserole to taste good the potatoes and green beans ought to
be at least in the ratio of 6:5.
 Maria needs to prepare a minimum of 10 kgs of Casserole each week.
Assumptions
 Potatoes and green beans are taken to be main ingredients that determine the cost, taste and
nutrition in the meal.
 There is no upper limit on the amount to be prepared as it can always be served afterwards.
Potatoes [X1] in
grams
Green Beans [X2] in
grams
Protein (per gram) 0.015 0.02
Iron (per gram) 0.3*10-5 1.2*10-5
Vitamin C (per gram) 12*10-5 1*10-4

3. CONSTRAINTS
 Protein Constraint
0.015 X1 + 0.02 X2 ≥ 180
 Iron Constraint
0.3*10-5 X1 + 1.2*10-5 X2 ≥ 0.08
 Vitamin C
12 * 10-5 X1 + 10-4 X2 ≥ 1.05
 Taste Requirements Constraints
5 X1 ≥ 6 X2
5X1 – 6X2 ≥ 0
 Total Requirement Constraint
X1 + X2 ≥ 10000
 Non-negative Constraint
X1 , X2 ≥ 0
OBJECTIVE FUNCTION
Z = 8.818*10^(-4) X1 + 2.205*10^(-3) X2
PROBLEM
STATEMENT
SCENARIO I
SCENARIO
II
SCENARIO
III
SCENARIO
IV
SCENARIO
V
SCENARIO
VI
Optimum Solution:
X1 (potatoes) = 6.15 Kg;X2 (beans) = 5.128 Kg;
Z(Min price) = 16.73 $
 To determine the amount of Potatoes and Green Beans Maria would buy in order to
minimize cost as well as meet the nutritional, taste and total production requirements.

4.  Maria is not very concerned about the taste of the casserole; she is only concerned about meeting
nutritional requirements. Edson to change the recipe to allow for only at least a one to two ratio in the
weight of potatoes to green beans.
CONSTRAINTS
 Protein Constraint
0.015 X1 + 0.02 X2 ≥ 180
 Iron Constraint
0.3* 10-3 X1 + 1.2*10-5 X2 ≥ 1.05
 Vitamin C
12 * 10-5 X1 + 10-4 X2 ≥ 1.05
 Taste Requirements Constraints
 Total Requirement Constraint
X1 + X2 ≥ 10000
 Non-negative Constraint
X1 , X2 ≥ 0
OBJECTIVE FUNCTION
Z = 8.818*10^(-4) X1 + 2.205*10^(-3) X2
Scenario 1:
5 X1 ≥ 6 X2
5X1 – 6X2 ≥ 0
5 X1 ≥ 6 X2
5X1 – 6X2 ≥ 0
2 X1 ≥ X2
2X1 – X2 ≥ 0
Optimum Solution:
X1 (potatoes) = 4.7 Kg;X2 (beans) = 5.5 Kg;
Z(Min price) = 16.24$
PROBLEM
STATEMENT
SCENARIO I
SCENARIO
II
SCENARIO
III
SCENARIO
IV
SCENARIO
V
SCENARIO
VI

5.  Maria decides to lower the iron requirement to 65 mg. Determine the minimum amount of potatoes
and green beans Maria should purchase each week given this new iron requirement.
CONSTRAINTS
 Protein Constraint
0.015 X1 + 0.02 X2 ≥ 180
 Iron Constraint
 Vitamin C
12 * 10-5 X1 + 10-4 X2 ≥ 1.05
 Taste Requirements Constraints
5 X1 ≥ 6 X2
5X1 – 6X2 ≥ 0
 Total Requirement Constraint
X1 + X2 ≥ 10000
 Non-negative Constraint
X1 , X2 ≥ 0
OBJECTIVE FUNCTION
Z = 8.818*10^(-4) X1 + 2.205*10^(-3) X2
Scenario 2:
Optimum Solution:
X1 (potatoes) = 7.2 Kg;X2 (beans) = 3.625 Kg;
Z(Min price) = 14.312 $
PROBLEM
STATEMENT
SCENARIO
I
SCENARIO II
SCENARIO
III
SCENARIO
IV
SCENARIO
V
SCENARIO
VI
0.3* 10-3 X1 + 1.2*10-5 X2 ≥ 1.050.3* 10-3 X1 + 1.2*10-5 X2 ≥ 1.050.3* 10-3 X1 + 1.2*10-5 X2 ≥ 0.065

6.  The wholesaler is now selling the green beans for a lower price of $0.50 per lb. Using the same iron
requirement from Scenario 2 and the new price of green beans, determine the amount of potatoes
and green beans Maria should purchase each week.
CONSTRAINTS
 Protein Constraint
0.015 X1 + 0.02 X2 ≥ 180
 Iron Constraint
 Vitamin C
12 * 10-5 X1 + 10-4 X2 ≥ 1.05
 Taste Requirements Constraints
5 X1 ≥ 6 X2
5X1 – 6X2 ≥ 0
 Total Requirement Constraint
X1 + X2 ≥ 10000
 Non-negative Constraint
X1 , X2 ≥ 0
OBJECTIVE FUNCTION
Z = 8.818*10^(-4) X1 + 2.205*10^(-3) X2
Scenario 3:
Optimum Solution:
X1 (potatoes) = 5.684 Kg;X2 (beans) = 4.736 Kg;
Z(Min price) = 10.233 $
0.3* 10-3 X1 + 1.2*10-5 X2 ≥ 0.065
PROBLEM
STATEMENT
SCENARIO
I
SCENARIO
II
SCENARIO
III
SCENARIO
IV
SCENARIO
V
SCENARIO
VI
OBJECTIVE FUNCTION
Z = 8.818*10^(-4) X1 + 2.205*10^(-3) X2
OBJECTIVE FUNCTION
Z = 8.818*10^(-4) X1 + 1.1023*10^(-3) X2

7.  Maria purchases lima beans instead of green beans since lima beans are less expensive and provide a
greater amount of protein and iron than green beans. Maria purchases lima beans for $0.60 per lb. Lima
beans contain protein: 22.68 g/10 ounces of lima beans, iron: 6.804 mg/10 ounces of lima beans, and 0g
vitamin C. The nutritional requirements includes reduced iron requirement from Scenario 2
CONSTRAINTS
 Protein Constraint
 Iron Constraint
 Vitamin C
 Taste Requirements Constraints
5 X1 ≥ 6 X2
5X1 – 6X2 ≥ 0
 Total Requirement Constraint
X1 + X2 ≥ 10000
 Non-negative Constraint
X1 , X2 ≥ 0
OBJECTIVE FUNCTION
Z = 8.818*10^(-4) X1 + 2.205*10^(-3) X2
Scenario 4:
Optimum Solution:
X1 (potatoes) = 8.75Kg;X2 (Lima beans) = 1.614Kg;
Z(Min price) = 9.851$
0.3* 10-3 X1 + 1.2*10-5 X2 ≥ 0.065
OBJECTIVE FUNCTION
Z = 8.818*10^(-4) X1 + 2.205*10^(-3) X2
OBJECTIVE FUNCTION
Z = 8.818*10^(-4) X1 + 1.323*10^(-3) X2
PROBLEM
STATEMENT
SCENARIO
I
SCENARIO
II
SCENARIO
III
SCENARIO
IV
SCENARIO
V
SCENARIO
VI
0.015 X1 + 0.08 X2 ≥ 1800.015 X1 + 0.02 X2 ≥ 1800.015 X1 + 0.02 X2 ≥ 180
0.3* 10-3 X1 + 1.2*10-5 X2 ≥ 0.0650.3* 10-3 X1 + 2.4*10-5 X2 ≥ 0.065
12 * 10-5 X1 + 10-4 X2 ≥ 1.0512 * 10-5 X1 + 10-4 X2 ≥ 1.0512 * 10-5 X1 + 0* X2 ≥ 1.05

8. PROBLEM
STATEMENT
SCENARIO
I
SCENARIO
II
SCENARIO
III
SCENARIO
IV
SCENARIO V
SCENARIO
VI
 Would Edson, the cook, be happy with the decision made in Scenario IV?
Edson would not be happy as the original recipe consisted potatoes and green, now the recipe
doesn’t take into consideration the taste requirements.
Scenario 5:

9.  The task force urges the university to adopt a policy that requires each serving of an entrée to contain
at least 120 mg of iron and at least 500 mg of vitamin C. Determine the amount of potatoes and lima
beans Maria should purchase each week and use data of Scenario 4
CONSTRAINTS
 Protein Constraint
 Iron Constraint
 Vitamin C
 Taste Requirements Constraints
5 X1 ≥ 6 X2
5X1 – 6X2 ≥ 0
 Total Requirement Constraint
X1 + X2 ≥ 10000
 Non-negative Constraint
X1 , X2 ≥ 0
OBJECTIVE FUNCTION
Z = 8.818*10^(-4) X1 + 1.323*10^(-3) X2
Scenario 6:
Optimum Solution:
X1 (potatoes) = 5.714Kg;X2 (Lima beans) = 4.28Kg;
Z(Min price) = 10.7$
0.3* 10-3 X1 + 2.4*10-5 X2 ≥ 0.12
0.015 X1 + 0.08 X2 ≥ 180
12 * 10-5 X1 + 0*X2 ≥ 1.05
PROBLEM
STATEMENT
SCENARIO I
SCENARIO
II
SCENARIO
III
SCENARIO
IV
SCENARIO
V
SCENARIO VI
0.3* 10-3 X1 + 2.4*10-5 X2 ≥ 0.0650.3* 10-3 X1 + 2.4*10-5 X2 ≥ 0.065
12 * 10-5 X1 + 0*X2 ≥ 1.0512 * 10-5 X1 + 0*X2 ≥ 0.5

10. THANK YOU