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Q: Linear Programing / Product Mix Problem (answerguru-ga, or anyone...) ( Answered ,   1 Comment )
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 Subject: Linear Programing / Product Mix Problem (answerguru-ga, or anyone...) Category: Science > Math Asked by: thanksmate-ga List Price: \$100.00 Posted: 21 Oct 2003 14:11 PDT Expires: 20 Nov 2003 13:11 PST Question ID: 268371
 ```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```
 thanksmate-ga rated this answer: and gave an additional tip of: \$50.00 `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```