A good problem is the one that helps students understand concepts very well. As requested, I am posting first problem in Decision making process (more problems with solutions will follow). This problem involves finding an optimal sales mix within the given constraints. Though on one side the existence of multiple constraints makes it a challenging problem, on the other side it is a very interesting problem with a real time scenario involved.
In all it helps a student understand the concept of ‘key factors’ in decision making problems. If you need problems with solutions in any particular topic, leave a note in comment.
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Gold Farming Inc has put under cultivation a farm of 126 hectares to grow cherries of four varieties: Rainier, Wild, Bing and Sour. Of the total, 70 hectares are suitable for all four varieties, but the remaining 56 hectares are suitable for growing only Wild Cherry and Bing Cherry. Labor is available for all kinds of farm work and there is no constraint.
The market requirement is that all four varieties of cherries must be produced and with a minimum of 2,000 boxes of each variety. However, the limitation is that not more than 45,500 boxes of any one variety should be produced.
Gold Farming has decided that the area allotted to any variety of cherry should be in terms of complete hectares and not in fraction of a hectare. The following data are relevant:
Particulars | Rainer | Wild | Bing | Sour |
Annual Yield (Boxes per hectare) | 700 | 200 | 140 | 360 |
Costs | ||||
Direct Materials per hectare | 952.00 | 432.00 | 392.00 | 624.00 |
Labour : Growing per hectare | 1792.00 | 1216.00 | 742.00 | 1056.00 |
Harvesting per box | 3.60 | 3.28 | 4.40 | 5.20 |
Packing per box | 5.20 | 5.00 | 4.00 | 9.60 |
Selling price per box | 15.38 | 15.87 | 18.38 | 22.27 |
Fixed overheads per annum – $ 50,000
Find out:
Within the given constraints, the area to be cultivated with each variety of tomatoes, if the largest total profit has to achieved and the amount of such profit.
Solution:
Step I: Product prioritization base on the contribution per hectare
Since some of the cost data is available per box, we need to convert them to per hectare
Statement showing contribution per hectare |
Particulars | Rainier | Wild | Bing | Sour |
Annual Yield (Boxes per hectare) | 700 | 200 | 140 | 360 |
Selling price per box | $ 15.38 | $ 15.87 | $ 18.38 | $ 22.27 |
Revenue per hectare | $ 10,766.00 | $ 3,174.00 | $ 2,573.20 | $ 8,017.20 |
Costs per hectare | ||||
Direct Materials per hectare | $ (952.00) | $ (432.00) | $ (392.00) | $ (624.00) |
Labour : Growing per hectare | $ (1,792.00) | $ (1,216.00) | $ (742.00) | $ (1,056.00) |
Harvesting per hectare | $ (2,520.00) | $ (656.00) | $ (616.00) | $ (1,872.00) |
Packing per hectare | $ (3,640.00) | $ (1,000.00) | $ (560.00) | $ (3,456.00) |
Conribution per hectare | $ 1,862.00 | $ (130.00) | $ 263.20 | $ 1,009.20 |
Rank | I | IV | III | II |
Step II: Determine the optimal product mix
Here, we need to consider the following key factors / constraints
a) Total land available is limited to 126 hectares
b) 56 hectares are suitable to grow only Wild and Bing varieties
c) Minimum production is 2000 boxes
d) Maximum production is 45,500 boxes
Observation
- 56 hectares are suitable to grow only ‘Wild’ and ‘Bing’ varieties.
It should be noted that they are ranked IV and III respectively. Also that ‘Wild’ is a loss making product. - Therefore, we need to allocate the 56 hectares (suitable only to grow Wild and Bing varieties) first.
- In case the minimum boxes of Wild and Bing can be achieved within these 56 hectares, the balance 70 hectares though suitable to all varieties should be allocated only to Rainer and Sour (Ranked I & II respectively)
A) Allocation of 56 hectares
Since Wild is a loss making product, minimum units of the same need to be produced and the balance hectares should be allotted to Bing.
Product | Boxes | Hectares | |
Wild | 2000 | 10 | |
Bing | 6440 | 46 | (56 – 10) |
B) Allocation of 70 shares
Product | Boxes | Hectares | |
Sour | 2160 | 6 | |
Rainier | 44800 | 64 | (70 – 6) |
To produce 2000 boxes of Sour Cherry required area is 5.56 hectares. However, since Gold farming has decided not to allot fraction of hectares, sour cherry need to be allotted 6 hectares.
Step III: Calculation of profitability for optimal sales mix
Particulars | Rainier | Wild | Bing | Sour | Total | |
Hectares alloted | 64 | 10 | 46 | 6 | – | |
Contrubution per hectare | $ 1,862.00 | $ (130.00) | $ 263.20 | $ 1,009.20 | – | |
Total Contribution | $ 119,168.00 | $ (1,300.00) | $ 12,107.20 | $ 6,055.20 | $ 136,030.40 | |
Fixed costs | $ 50,000.00 | |||||
Profit | $ 86,030.40 |
Dear,
I greatly appreciate this wonderful piece of work.
I am soon siting for a cpa and this piece has helped elaborate some technical aspects. But one thing still bothers me, what is the easiest logical approach of sharing out the limited resource especially after ranking and then the clear way of determining the optimal sales or output. I will be very greatful if you helped me out especially if you included yet another example of optimal sales and probably optimal output mix.
Thanks
Patrick Ikileng