A firm wishes to produce a single product at one or more locations so that the t

Question

A firm wishes to produce a single product at one or more locations so that the total monthly cost is minimized subject to demand being satisfied. At each location there is a fixed charge to be paid if any are produced (but is nil otherwise), and a variable cost which depends on whether the units are produced on regular time or on overtime. Each location has capacity restrictions on regular and overtime production. The relevant data are: Demand is for 5100 units per month. Formulate an algebraic model for this problem. [There is more space on the next page.]

Explanation / Answer

Cost of producing a unit = fixed cost + variable cost + overtimme cost.

lets assume the units produced in Plant 1 to its full capacity. As per given table Plant 1 can produce 1000 units at normal and 400 units by doing overtime. So the total units that can be produced is 1400 units .

So cost of production = 2000 + 3.8 ( unit variable cost ) * 1000 + 4.6 ( overtime cost per unit ) * 400 ( overtime capacity) = 7640

Similarly below table shows the cost of production at plants when produced at their full capacity

Now as per above table if we consider , plant 1, 2 & 4 has less cost compared to Plant 3 & 5. The total units that can be produced by these plants = 1400 + 1750 + 1750 = 4900 .

But total Demand = 5100 units. So balance 200 Units can be produced by plant 5 which has cost lesser than plant 3.

Cost of balance 200 units produced in plant 5 = 2700 + 3.6 * 200 = 3420

So the minimum cost incured by firm will be = 7640 + 8735 + 8710 + 3420 = 28505.

Plant Fixed Regular time Overtime Max QTY with Overtime Variable Cost at capacity variable cost overtime Total cost if worked at full capacity with overtime Cost unit cost Capacity unit cost Capacity 1 2000 3.8 1000 4.6 400 1400 3800 1840 7640 2 3000 2.9 1200 4.1 550 1750 3480 2255 8735 3 1500 4.2 1500 5.6 600 2100 6300 3360 11160 4 2400 3.4 1300 4.2 450 1750 4420 1890 8710 5 2700 3.6 1400 5.1 500 1900 5040 2550 10290

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