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:

please give all calculations

(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?

Answer 1

            Let X, Y and Z be the number of loads of fine, standard and course          respectively produced

            Profit    =90x + 90y+     60z      

            Objective function

Maximize

Z=       90x +    90y+     60z      

Subject to

Capacity constraints

            5x+       6y+       3z?       600

            2x+       4y+       5z?       600

            2x+       4y+       3z?       480

            1x+       1y+       1z?       150

Non-negativity constraints

X,Y,Z ? 0

(a) What is the marginal value of a tonne of red gum? Over what range is this price value appropriate?

• The marginal value of a tonne of red gum -$ 15- as given by shadow price.

Constraints with a Shadow price-dual value (other than zero) are binding constraints (binding on optimality) . Shadow price is the marginal value of one additional unit of resource.

• The range of feasibility for each constraint is the range within which the shadow price is valid. ie the range of values for this coefficient in which the original dual price remains constant Beyond that the shadow price change

range of feasibility

(RHS value- Allowable Decrease) to (RHS value + Allowable Increase )

                650-750 tonnes

(b) What is the maximum price the store would be justified in paying for additional red gum?

The maximum price that one can pay to obtain 1 additional unit of binding constraint is equal to the shadow price .

So maximum price the store would be justified in paying for additional red gum =$ 15 for 1` tonne

(c) What is the marginal value of labour? Over what range is this value in effect?

Labour has shadow price of zero.           i.e. labour is not a binding constraint (not binding on optimality). As we see labour is already left in excess (surplus).

So marginal value of labour is $0 for the range - (375 hours to practically infinity)

(d) The manager obtained additional machine time through better scheduling. How much additional machine time can be effectively used for this operation? Why?

Machine time has shadow price of zero. i.e. Machine time is not a binding constraint (not binding on optimality). As we see, Machine time is already left in excess (surplus).

So additional machine time is only excess and will not increase effectiveness

(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)?

The manager should choose red gum for additional amount of 150 tons. Because increasing its capacity to 750 tonnes (600+ allowable increases) would give us greater profits than increasing the capacity of storage bins.

(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?

The range of optimality for each coefficient provides the range of values over which the current solution will remain optimal

For coefficient -Course chip profit, this is

(Objective value- Allowable Decrease) to (Objective value + Allowable Increase )

( 60-6) to (60+30)

= $ 54- $90

70$ is within the range of optimality. So the optimal solution does not change ie value of decisions variables (quantities produced)

Remains the same.

However due to increase in profit per tonnes, the overall profit changes

The value of optimal solution (value of objective function c) increases to $12000

(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?

For coefficient -Course chip profit, this is

(Objective value- Allowable Decrease) to (Objective value + Allowable Increase )

( 90-10) to (90+10)

= $ 80- $100

For fine chips the allowable decrease is $10 .So decreasing only by $6 to $84 will not change optimal solution. It is within the range of optimality.

For course chips ( as in question(f) ) 70$ is within the range of optimality. So the optimal solution does not change ie value of decisions variables (quantities produced)

However due to increase in profit per tonnes, the overall profit changes

The value of optimal solution (value of objective function c) increases to $11550