I would like this question answered by Answerguru-ga, but if it is
still available 24 hours from the time I post it, anyone can answer it
and the $50 tip still stands.

Question:
Thanksmate makes three nut mixes for sale to grocery chains. The three
mixes, referred to as the Regular Mix, the Deluxe Mix, and the Holiday
Mix, are made by mixing different percentages of five types of nuts.

In preparation for the fall season, Thanksmate has just purchased the
following shipments of nuts at the shown prices:

Type of Nut / Shipment Amount (pounds) / Cost per Shipment ($)

Almond / 6,000 / 7,500
Brazil / 7,500 / 7,125
Filbert / 7,500 / 6,750
Pecan / 6,000 / 7,200
Walnut / 7,500 / 7,875

The Regular Mix consists of 15% almonds, 25% Brazil nuts, 25%
filberts, 10% pecans, and 25% walnuts.
The Deluxe Mix consists of 20% of each nut.
The Holiday Mix consists of 25% almonds, 15% Brazil nuts, 15%
filberts, 25% pecans, and 20% walnuts.

The profit contribution per pound is:
Regular Mix: $1.65
Deluxe Mix: $2.00
Holiday Mix: $2.25

Orders are summarized here:
Type of Mix / Orders (pounds)
Regular / 10,000
Deluxe / 3,000
Holiday / 5,000

Because demand is running high, Thanksmate expects to receive more
orders than can be satisfied.
Thanksmate is committed to using the available nuts to maximize profit
over the fall season; nuts not used will be given to the homeless. But
even if it is not profitable to do so, Thanksmate has indicated that
the orders already received must be satisfied.


How to Answer:
Perform an analysis of Thanksmate's product mix, and prepare a report
for me that summarizes your findings. Be sure to include information
and analysis on the following:
1. The cost per pound of the nuts included in the Regular, Deluxe, and
Holiday mixes
2. The optimal product mix and the total profit contribution
3. Recommendations regarding how the profit contribution can be
increased if additional nuts can be purchased
4. A recommendation as to whether Thanksmate should purchase an
additional 1,000 pounds of almonds for $1,000
5. Recommendations on how profit contribution could be increased (if
at all) if Thanksmate does not satisfy all existing orders

note:
1. if answered within 3 days there will be a $US50 tip.
2. when you take this question, please estimate when you will have it
finished.
3. a written description as in the linear progam model whereby the
decision variables, constraints and the objective function is defined
should be included and please copy and paste the output (results) from
whatever decision analysis software or spread sheet you use to answer
my question

Thanks!

---

Request for Question Clarification by answerguru-ga on 21 Oct 2003 18:25 PDT

Hi again thanksmate-ga,

I appreciate you directing this question to me - I will get started on
it for you shortly and have it for you either tomorrow afternoon or
evening.

answerguru-ga

---

Clarification of Question by thanksmate-ga on 22 Oct 2003 14:18 PDT

Much appreciated :)

Hi thanksmate-ga,

I have completed the design and analysis as outline in the question
and have embedded my responses in the questions:


1. The cost per pound of the nuts included in the Regular, Deluxe, and
Holiday mixes 

This can be calculated  quite easily by first calculating the
(wholesale) price per pound for each type of nut:

Almond = 1.25
Brazil = 0.95
Filbert = 0.90
Pecan = 1.20
Walnut = 1.05

Since we are given the proportions of nuts to be included in each mix
we can sum up the partial cost of each nut within a mix to obtain the
cost of the mix in total:

Regular = 1.25*0.15 + 0.95*0.25 + 0.90*0.25 + 1.20*0.1 + 1.05*0.25
	= $1.03

Deluxe = 1.25*0.2 + 0.95*0.2 + 0.90*0.2 + 1.20*0.2 + 1.05*0.2
	= $1.07

Holiday = 1.25*0.25 + 0.95*0.15 + 0.90*0.15 + 1.20*0.25 + 1.05*0.2
	= $1.10

Based on the figures above, we know that in order to achieve the
specified profit margins, our selling prices for each mix type is:

Regular = $1.03 + 1.65 = $2.68/lb
Deluxe = $1.07 + $2.00 = $3.07/lb
Holiday = $1.10 + $2.25 = $3.35/lb


2. The optimal product mix and the total profit contribution 

The analysis was done using Excel with the following LP model:

Objective: 
Maximize function 1.65*(Quantity Regular) + 2*(Quantity Deluxe) +
2.25*(Quantity Holiday)

Subject to constraints:

(measured in pounds)

Quantity Regular <= 10000
Quantity Deluxe <= 3000
Quantity Holiday <= 5000

Almonds used <= 6000
Brazils used <= 7500
Filberts used <= 7500
Pecans used <= 6000
Walnuts used <= 7500

Almonds used = 0.15Qreg + 0.2Qdel + 0.25Qhol
Brazils used = 0.25Qreg + 0.2Qdel + 0.15Qhol
Filberts used = 0.25Qreg + 0.2Qdel + 0.15Qhol
Pecans used = 0.10Qreg + 0.2Qdel + 0.25Qhol
Walnuts used = 0.25Qreg + 0.2Qdel + 0.20Qhol

Based on this model, we can obtain these results:

Optimal profit contribution = $33 750

Product mix:

Regular	10000
Deluxe	3000
Holiday	5000

Nut usage:

Almond	3350
Brazil	3850
Filbert	3850
Pecan	2850
Walnut	4100

3. Recommendations regarding how the profit contribution can be 
increased if additional nuts can be purchased 

Currently there is an extreme over-supply of all types of nuts, and
for that reason the problem was constrained by the number of orders
placed. At this point no particular type of nut is preventing a higher
profit contribution. In fact, if we were to remove the constraints
relating to the number of orders received, the profit contribution
would skyrocket to $62250, and at that point we can see that pecans
would be the only type with excess 'slack' (given the current mixes).

4. A recommendation as to whether Thanksmate should purchase an 
additional 1,000 pounds of almonds for $1,000 

The correct solution to this question can really only be found by
plugging in the change into the model. The first thing to note is that
you are able to obtain these almonds cheaper that before ($0.25/lb
discount). In addition, if no constraints are placed on order levels,
we find that the profit contribution rises from $62250 (as we saw from
the previous question) to $66500. This is a marginal profit of $4250,
and when you offset the $1000 cost for the almonds it is $3250 in pure
profit. Therefore it is recommended that this purchase be made
(assuming orders are anticipated to consume the increased availability
that will result).

5. Recommendations on how profit contribution could be increased (if 
at all) if Thanksmate does not satisfy all existing orders 

Through sensitivity analysis in the Excel model, the holiday mix
appears to be the least profitable in terms of maximizing the
objective function. This is due to the mix and proportion of nuts used
in this mix; not only is it more costly (since it is using higher
price nuts), but the profit contribution in comparison to the other
two mixes is smaller compared to the costs of the raw nuts.

Therefore if Thanksmate were to focus on making regular and deluxe
mixes he could become more profitable!


Hopefully this answers your question - if you have any questions or
concerns regarding the information above please do post a
clarification and I'll respond promptly :)

Cheers!

answerguru-ga

Good answer! You make it look so easy! Thanks for your help!

Hi thanksmate-ga,

Once again, your kind words and generous tip are much appreciated :)

answerguru-ga