Question

You are the General Manager of a US owned plant located in Northern Mexico. The firm manufactures semiconductors and uses temporary employees from the local area to fulfill production labor requirements. Each quarter employees are hired and released depending upon production requirements and needs. The firm has a maximum storage capacity of 1,000,000 semiconductors. Agreements with the Mexican Government require that at least 325 workers must be on the payroll each quarter. There are currently 375 employees on the payroll and 120,000 semiconductors in storage. Sales forecast provided by the marketing department are estimated as follows for the upcoming year:
QUARTER PRODUCTION DAYS PREDICTED SALES FORECAST
First 59 940,000
Second 62 1,215,000
Third 55 860,000
Fourth 58 430,000

Inventory holding cost are $.25 per semiconductor per quarter. The holding cost applies to excess inventory after demand is fulfilled. Thus, assume inventory produced in the quarter to fulfill demand in that quarter is not subjected to holding cost.

The firm would like to have at least 100,000 semiconductors in stock at the end of the year. Each employee produces an average of 30 semiconductors per day. The cost of hiring a new worker is $200, and the cost of releasing a worker is $400. Determine a production plan that minimizes total cost and meets demand forecasts. This production plan must indicate the amount of inventory to produce, store and the amount of workers to hire and release each quarter. Use LP to determine a solution for this problem.

Answer 1

Answer:

tables to display the required information:

$\left[\begin{array}{ccccccc} quarter& beg workers& hired& lay-off& total& working days& production\\1st&375&174&0&549&59&971730\\2nd&549&23&0&572&62&1063920\\3rd&572&0&50&522&55&861300\\4th&522&0&196&326&58&567240\\\end{array}\right]$

$\left[\begin{array}{ccccc} quarter& beginning& production& demand& ending\\1st&120000&971730&940000&151730\\2nd&151730&1063920&1215000&650\\3rd&650&861300&860000&1950\\4th&1950&567240&430000&139190\\\end{array}\right]$

Total cost:

hiring cost: (174 + 23)*200 = 39,400

lay-off cost: (50 + 196) * 400 = 98,400

inventory cost: ending inventory * $0.25 = 73,380

Total cost: $211,180

Explanation:

We solve this using Excel SOLVER which uses linear programming.

First, build a table for workers and production.

$\left[\begin{array}{ccccccc}\\A&B&C&D&E&F&G\\ quarter& beg workers& hired& lay-off& total& working days& production\\1st&375&&&B + C - D &59&E \times 30 \times F\\2nd&E_1&&&B + C - D&62&E \times 30 \times F\\3rd&E_2&&&B + C - D&62&E \times 30 \times F\\4th&E_3&&&B + C - D&62&E \times 30 \times F\\\end{array}\right]$

Then we build a table for inventory:

$\left[\begin{array}{ccccc}\\A&B&C&D&E\\quarter&beginning&production&demand&ending\\1st&120000&&&B+C-D\\2nd&E_1&&&B+C-D\\3rd&E_2&&&B+C-D\\4th&E_3&&&B+C-D\\\end{array}\right]$

Last the cell for total cost we want to minimize:

=SUM(hired) x 200 + SUM(lay-off) x 400) + SUM(ending inventory) x 0.25

Our requirement is:

1)ending inventory of 100,000 or more

2) total worker of 325 or above

3) fulfill all demand so ending must be zero or higher than zero

4) hiring and lay-off are Natural numbers (we can't hire 0.5 employees)