Question

USE EXCEL. While we assume the principle of certainty, in reality, the coefficients of constraints or the objective function in a LP are subject to changes. Dried cranberries calories is assumed to be 3.08. However, this value can change for different batches of dried cranberries. Assume that dried cranberries calories is a uniform random variable in the interval [2.4, 3.4]. Generate 30 random numbers in the interval [2.4, 3.4]. For each random number generated solve the LP once considering the random number to be the dried cranberries calories. Out of 30 times that you solved the LP problem, what percentage of times were you not able to find an optimum solution due to infeasibility? Create a 95% confidence interval for this percentage. Note that the calculated infeasibility percentage is p_bar. What does this practice tell you about LP limitations? Make sure you include all related information such as test statistic in your answer.

Healthy Snacks Co. produces snack mixes. Recently, the company has decided to introduce a new snack mix that has peanuts, raisins, pretzels, dries cranberries, sunflower seeds and pistachios. Each bag of the new snack is designed in order to hold 250 grams of the snack. The company has decided to market the new product with a emphasis on its health benefits. After consulting nutritionists, Healthy Snacks decides to mix the ingredients so that the snack has the following specifications:

1 - Total calories <= 800  

2 - Fat <= 20

3 - Sodium <= 200

4 - Potassium <= 700

5 - Carb <= 140

6 - Protein >= 10

7 - Vitamin A >= 3

8 - Vitamin B6 >= 25

9 - Vitamin C >= 2

10 - Iron >= 15

11 - Calcium >= 5

Table below shows the amount of each nutrition in each snack's ingredient per 1 gram:

	peanut	raisin	pretzel	dried cranberries	sunflower seed	pistachio
Calories	5.67	3.00	1.00	3.08	5.84	5.62
Fat (g)	0.49	0.00	0.01	0.01	0.51	0.06
Sodium (mg)	0.18	0.12	3.32	0.03	0.09	0.01
Potassium (mg)	7.05	7.49	0.36	0.40	6.45	10.25
Carb (g)	0.16	0.79	0.21	0.83	0.20	0.28
Protein (g)	0.26	0.03	0.03	0.00	0.21	0.20
Vitamin A (%)	0.00	0.00	0.00	0.00	0.01	0.08
Vitamin B-6 (%)	0.17	0.12	0.00	0.00	0.68	0.85
Vitamin C (%)	0.00	0.02	0.00	0.00	0.02	0.09
Iron (%)	0.25	0.09	0.07	0.03	0.29	0.21
Calcium (%)	0.09	0.05	0.00	0.00	0.07	0.10

Table below shows the cost of each ingredients per 1 gram:

	peanut	raisin	pretzel	dried cranberries	sunflower seed	pistachio
cost per 1 gram ($)	$ 0.005	$ 0.020	$ 0.013	$                     0.013	$                  0.022	$     0.031

Per marketing department research, each bag of snack should have at least 10 grams of each ingredient.

Answer 1

Problem can be initialized with all zero values for quantity of the six ingredients as

Ingradient	peanut	raisin	pretzel	dried cranberries	sunflower seed	pistachio
Variable Name	X1	X2	X3	X4	X5	X6
Initialization	0	0	0	0	0	0
Minimum Quantity Constraints
Variable Quanity	X1	X2	X3	X4	X5	X6
Inequality	>=	>=	>=	>=	>=	>=
Value	10	10	10	10	10	10
Nutritional Value (Health Benefits) Constraints
Variable Quanity	5.67(X1)+3(X2)+(X3)+3.08(X4) +5.84(X5)+5.62(X6)	0.49(X1)+0.01(X3)+0.01(X4) +0.51(X5)+0.06(X6)	0.18(X1)+0.12(X2)+3.32(X3) +0.03(X4)+0.09(X5)+0.01(X6)	7.05(X1)+7.49(X2)+0.36(X3) +0.4(X4)+6.45(X5)+10.25(X6)	0.16(X1)+0.79(X2)+0.21(X3) +0.83(X4)+0.2(X5)+0.28(X6)	0.26(X1)+0.03(X2)+0.03(X3) +0.21(X5)+0.2(X6)
Initial Value	0	0	0	0	0	0
Inequality	<=	<=	<=	<=	<=	>=
Value	800	20	200	700	140	10
Variable Quanity	0.01(X5)+0.08(X6)	0.17(X1)+0.12(X2)+0.68(X5) +0.85(X6)	0.02(X2)+0.02(X5)+0.09(X6)	0.25(X1)+0.09(X2)+0.07(X3)+ 0.03(X4)+0.29(X5)+0.21(X6)	0.09(X1)+0.05(X2)+0.07(X5)+ 0.1(X6)
Initial Value	0	0	0	0	0
Inequality	>=	>=	>=	>=	>=
Value	3	25	2	15	5
Total Quantity Constraint
Total Quantity	X1+X2+X3+X4+X5+X6	=	250
Initial Value	0
Cost Optimization Function
Total Cost Function	0.005(X1)+0.02(X2)+0.013(X3) +0.013(X4)+0.022(X5)+0.031(X6)
Cost	0

The Solver can be initialized with all constraints outlined above as

which then gives the optimized solution as