Question

A garden store prepares various grades of wood chips for mulch for sale in various tonnages for delivery to large garden construction sites around town. The grades are (a) fine, (b) standard and (c) course. The process requires red gum, machine time, labour time, and storage space.  

The garden store owner has identified that the store can generate $90 profit per storage bin for fine, $90 for standard but only $60 for course chips.

Each load of chips require inputs in the following quantities: Fine: 5 tonnes of material, 2 machine hours, 2 hours of labour and 1 storage bin Standard: 6 tonnes of material, 4 machine hours, 4 hours of labour and 1 storage bin Course: 3 tonnes of material, 5 machine hours, 3 hours of labour and 1 storage bin

Unfortunately, like every business, the garden store has limits in its production capacity. It is able to handle 600 tonnes of red gum at any one time, the machine can only operate for 600 hours before major maintenance must occur, it only has sufficient staff to provide 480 hours of labour time and it has 150 storage bins.

Required: (a) What is the marginal value of a tonne of red gum? Over what range is this price value appropriate? (b) What is the maximum price the store would be justified in paying for additional red gum? (c) What is the marginal value of labour? Over what range is this value in effect? (d) The manager obtained additional machine time through better scheduling. How much additional machine time can be effectively used for this operation? Why? (e) If the manager can obtain either additional red gum or additional storage space, which one should the manager choose and how much (assuming additional quantities cost the same as usual)? (f) If a change in the course chip operation increased the profit on course chips from $60 per bin to $70 per bin, would the optimal quantities change? Would the value of the objective function change? If so, what would the new value(s) be? (g) If profits on course chips increased to $70 per bin and profits on fine chips decreased by $6.00, would the optimal quantities change? Would the value of the objective function change? If so, what would the new value(s) be?

Note: In order to achieve full marks for this question it is essential that you fully explain what you are doing, why you are doing it and the steps involved in providing a final solution. Ensure your answer is not just a set of calculations as 25% of the marks for this question are set aside for your explanation.

Answer 1

Formulation

Let F, S, and C be the storage bins of Fine, Standard, and Course grades respectively.

Objective Function: Maximize Z = total profit
or, Z = 90F + 90S + 60C
Subject to,
5F + 6S + 3C <= 600 (tonnes of raw material availability)
2F + 4S + 5C <= 600 (machine hours availability)
2F + 4S + 3C <= 480 (labor hours availability)
1F + 1S + 1C <= 150 (storage bin availability)

F, S, C >= 0

Implementation and Sensitivity

Answer Report

Sensitivity Report

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(a)

Marginal value = shadow price; For a red gum (raw material), the marginal value is $15. This value is fixed for the range of feasibility which is [600-50, 600+150] i.e. [550, 750].

(b)

The store should be ready to pay any amount less than $15 (which is the shadow price) for one tonne of extra red gum because one unit of extra red gum will give rise to $15 extra profit.

(c)

Labor being a non-binding constraint, has a marginal value of zero. The range is [480-105, Infinity) i.e. [375, infinity).

(d)

Additional machine time will be of no use because it is a non-binding constraint with zero shadow price.

(e)

Since both the constraint have equal shadow price of $15, no one has the preference over the other. For the red gum, the increase can be up to 750 hours and for the storage bin, it can be up to 157 bins.

(f)

The allowable increase is $30. So, an increase of $10 will not change the optimal solution. However, the objective function will change and it will become = 75 x 90 + 0 x 90 + 75 x 70 = $12,000

(g)

In this case, two coefficients are changing simultaneously, so, we will use the 100% rule.

Product	Change	Allowable change	%
Course chips	+$10	+$30	33.33%
Fine chips	-$6	-$10	60.00%
		Total	93.33%

Since the total % is less than 100%, the optimal solution will not change. However, the objective function will change and it will become = 75 x 84 + 0 x 90 + 75 x 70 = $11,550.