Question

Zesto’s Zinging Smoothies

Zesto’s, Omaha’s College World Series provider of malts and shakes, has expanded into

healthy smoothies at the same time they have moved downtown near TD Ameritrade

Stadium (away from their historic location on South 13

th

by the parking lot formerly

known as Rosenblatt Stadium).

They use three fruits and a variety of other ingredients to make their smoothies. For

simplicity, we are only analyzing their fruit and “end product” requirements. Zesto’s uses

bananas, blueberries and strawberries (N,B and S abbreviations, respectively), and

produces “wildFruit”, “RedYellowNBlue” and “BerryMania” smoothies (WF, RYB,

BM). Our units are in ‘pounds’, and we assume that fruit weight is additive in the

smoothie products: 1 lb of bananas blended with 1 lb. of blueberries and 1 lb of

strawberries would result in 3 lbs of smoothie.

For the next production period, we want to purchase fruit such that we produce at least

50 lbs. of each smoothie product. We can produce more – plenty of freezer space exists at

the new downtown store.

Our suppliers have available 250 lbs of bananas, 100 lbs. of blueberries and 200 lbs. of

strawberries. Cost per pound is ($6, $10.5, $8.5) for (N,B,S) respectively. This is the most we can use.

On a per pound basis, we sell WF for $12, RYB for $15 and BM for $18.

Each of our smoothie products (in aggregate) have some mix requirements as well as average ORAC (Oxygen Radical Absorbance Capacity) requirements.

wildFruit – cannot be made up of more than 40% of any one kind of fruit. Average

ORAC must be at least 1000.

RedYellowNBlue – Must consist of at least 25% of each of the three kinds of fruit.

Average ORAC must be at least 1100.

BerryMania – cannot consist of more than 10% bananas, and each of the two other fruits

(berries) must be at least 30% of the total amount made. Average ORAC must be at least

1800.

Note that the individual ORAC ratings for bananas are 210, for blueberries 2400 and for

strawberries 1540.

Find the optimal way of producing smoothies for Zesto that satisfies all requirements and

maximizes profit. Whole numbers (and whole fruit) not required.

Add the following to “Zesto” problem and solve

. Whole smoothies (integers)

. At least 50% of each of the available fruits (250, 100, 200 maximum available) must be used in the production of smoothies.

. Bananas

. Blueberries

. Strawberries

Smoothie products (by weight)

. Individual smoothie products cannot make up more than 50% of the total smoothie production.

. Individual smoothie products must be at least 20% of the total smoothie production.

. WF, RYB, BM products

Answer 1

Original problem

	N	B	S	Total	S. price	Demand
WF	20.0	10.0	20.0	50.0	$12	50
RYB	42.7	21.7	22.3	86.7	$15	50
BM	0.0	68.3	157.7	226.0	$18	50
Total	62.7	100.0	200.0
Capacity	250.0	100.0	200.0
Cost per lb	$6.0	$10.5	$8.5
ORAC	210	2400	1540
N in WF	12.0	-4.0	-8.0	0.0	<=	0
B in WF	-8.0	6.0	-8.0	-10.0	<=	0
S in WF	-8.0	-4.0	12.0	0.0	<=	0
N in RYB	-32.0	5.4	5.6	-21.0	<=	0
B in RYB	10.7	-16.3	5.6	0.0	<=	0
S in RYB	10.7	5.4	-16.7	-0.7	<=	0
N in BM	0.0	-6.8	-15.8	-22.6	<=	0
B in BM	0.0	-47.8	47.3	-0.5	<=	0
S in BM	0.0	20.5	-110.4	-89.9	<=	0
WF - ORAC	-15800.0	14000.0	10800.0	9000.0	>=	0
RYB - ORAC	-38001.3	28177.5	9823.8	0.0	>=	0
BM - ORAC	0.0	40995.0	-40995.0	0.0	>=	0
		Revenue	$ 5,968
		Cost	$ 3,126
		Profit	$ 2,842

Additional constraints

The amendments in the model are highlighted with 'yellow'

	N	B	S	Total	S. price	Demand	Individual products <= 50%			Individual products >= 20%
WF	32.0	16.0	32.0	80.0	$12	50	0.5	-0.5	-0.5	-0.8	0.2	0.2
RYB	95.0	49.0	51.0	195.0	$15	50	-0.5	0.5	-0.5	0.2	-0.8	0.2
BM	0.0	35.0	80.0	115.0	$18	50	-0.5	-0.5	0.5	0.2	0.2	-0.8
Total	127.0	100.0	163.0				-115.0	0.0	-80.0	-2.0	-117.0	-37.0
Capacity	250.0	100.0	200.0				<=	<=	<=	<=	<=	<=
Cost per lb	$6.0	$10.5	$8.5				0	0	0	0	0	0
Min. cosumed	125.0	50.0	100.0
ORAC	210	2400	1540
N in WF	19.2	-6.4	-12.8	0.0	<=	0
B in WF	-12.8	9.6	-12.8	-16.0	<=	0
S in WF	-12.8	-6.4	19.2	0.0	<=	0
N in RYB	-71.3	12.3	12.8	-46.3	<=	0
B in RYB	23.8	-36.8	12.8	-0.3	<=	0
S in RYB	23.8	12.3	-38.3	-2.3	<=	0
N in BM	0.0	-3.5	-8.0	-11.5	<=	0
B in BM	0.0	-24.5	24.0	-0.5	<=	0
S in BM	0.0	10.5	-56.0	-45.5	<=	0
WF - ORAC	-25280.0	22400.0	17280.0	14400.0	>=	0
RYB - ORAC	-84550.0	63700.0	22440.0	1590.0	>=	0
BM - ORAC	0.0	21000.0	-20800.0	200.0	>=	0
		Revenue	$ 5,955
		Cost	$ 3,198
		Profit	$ 2,758