A manufacturing facility uses two production lines to produce three products ove

Question

A manufacturing facility uses two production lines to produce three products over the next 6 months. Backlogged demand is not allowed. However, a product may be overstocked to meet demand in later months. The following table provides the data associated with the demand, production, and storage of the three products Demand in period Unit Initial inventory Product I 2 3 4 5 6 holding 1 50 30 40 60 20 4 2 40 60 50 30 30 5 3 30 40 20 70 40 30 .50 . 35 .45 75 60 There is a fixed cost for switching a line from one product to another. The following tables give the switching cost, the production rates, and the unit production cost for each line: Line switching cost ($) Product 1 Product 2 Product 3 180 Line 1 Line 2 300 174 250 200 Production rate (units/month) Unit production cost ($) Product 2 Product 3 Product 1 10 12 Product I Product 2 Product 3 Line 1 40 Line 2 90 15 10 70 60 Develop a model for determining the optimal production schedule.

Explanation / Answer

Let us first find constraints if there are any by analyzing the demand and production capacities of 2 lines. Product Months Demand 1 2 3 4 5 6 1 Units 50 30 40 60 20 4 2 Units 40 60 50 30 30 5 3 Units 30 40 20 70 40 30 Total Demand Units 120 130 110 160 90 39 Cumulative Demand Units 120 250 360 520 610 649 Production Capacity Unit Production Cost Line 1 Units 1 40 40 40 40 40 40 10 Units 2 60 60 60 60 60 60 8 Units 3 80 80 80 80 80 80 15 Line 2 Units 1 90 90 90 90 90 90 12 Units 2 70 70 70 70 70 70 6 Units 3 60 60 60 60 60 60 10 Total Capacity Units 400 400 400 400 400 400 Our observations are as follows : Product 1 should be produced on line 1 as cost is better than line 2 . Product 2 should be produced on line 2 as cost is better than line 1 . Product 3 should be produced on line 2 as cost is better than line 1 In totality monthly production capacity far exceeds the monthly total demand for all the products. In Month 1 against a demand of 50 units of product 1 on line 1 only 40 units can be produced. In Month 4 against a demand of 60 units of product 1 on line 1 capacity is 40 units. In Month 4 against a demand of 70 units of product 3 capacity at line is 60 units. Now we will go ahead of production of all three products 1 by 1. Product Months Demand 1 2 3 4 5 6 1 Units 50 30 40 60 20 4 Opening Stock Units 55 5 0 0 Net Production requirement Units -5 25 40 60 Production Units 0 25 40 40 Closing Stock Units 5 0 0 -20 Now here as we can see in month 4 our requirement is 60 but we can produce only 40 . Here options are: A. Produce it in earlier months where we can produce 15 units in month 2 and 5 units in month 1. Inventory holding cost incurred on 5 units to be produced in month 1 will be 0.5+0.5+0.5=1.5 $per unit Inventory holding cost incurred on 15 units to be produced in month 2 will be 0.5+0.5=1$ per unit. B. If we produce these on line 2 we will be incurring 12-10=2 $/unit extra. So out of these 2 options option A is better. Product Months Demand 1 2 3 4 5 6 1 Units A 50 30 40 60 20 4 Opening Stock Units B 55 10 20 20 0 0 Net Production requirement Units C=A-B -5 20 20 40 20 4 Production Units D 5 40 40 40 20 4 Closing Stock Units E=B+D-A 10 20 20 0 0 0 Production Cost $/Unit F 10 10 10 10 10 10 Inventory holding cost $/Unit G 0.5 0.5 0.5 0.5 0.5 0.5 Production Cost $ H=F*D 50 400 400 400 200 40 Inventory holding cost $ I=G*E 5 10 10 0 0 0 Total Cost $ J=H+I 55 410 410 400 200 40 For product 2 there is no issue as capacity on line 2 exceeds demand of product 2 on line 2. Product Months Demand 1 2 3 4 5 6 2 Units A 40 60 50 30 30 5 Opening Stock Units B 75 35 0 0 0 0 Net Production requirement Units C=A-B -35 25 50 30 30 5 Production Units D 0 25 50 30 30 5 Closing Stock Units E=B+D-A 35 0 0 0 0 0 Production Cost $/Unit F 6 6 6 6 6 6 Inventory holding cost $/Unit G 0.35 0.35 0.35 0.35 0.35 0.35 Production Cost $ H=F*D 0 150 300 180 180 30 Inventory holding cost $ I=G*E 12.25 0 0 0 0 0 Total Cost $ J=H+I 12.25 150 300 180 180 30 Now let us see if there is any issue of producing product 3 on line 2. Product Months Demand 1 2 3 4 5 6 3 Units A 30 40 20 70 40 30 Opening Stock Units B 60 30 0 10 0 0 Net Production requirement Units C=A-B -30 10 20 60 40 30 Production Units D 0 10 30 60 40 30 Closing Stock Units E=B+D-A 30 0 10 0 0 0 Production Cost $/Unit F 10 10 10 10 10 10 Inventory holding cost $/Unit G 0.45 0.45 0.45 0.45 0.45 0.45 Production Cost $ H=F*D 0 100 300 600 400 300 Inventory holding cost $ I=G*E 13.5 0 4.5 0 0 0 Total Cost $ J=H+I 13.5 100 304.5 600 400 300 Here we need 10 units extra in Month 4 . Here better option was to produce these in month 3 as inventory holding cost is 0.35 $/unit against a production cost difference of 5$/unit if we produce these on Line 1.

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