Question

PRODUCTION INVENTORY. The Mobile Cabinet Company produces cabinets used in mobile and motor homes. Cabinets produced for motor homes are smaller and made from less expensive materials than are those for mobile homes. The home office in Ames, Iowa, has just distributed to its individual manufacturing centers the production quotas required during the upcoming summer quarter. The scheduled production requirements for the Lexington, Kentucky, plant are given in the following table.

	July	August	September
Motor home	250	250	150
Mobile home	100	300	400

Each motor home cabinet requires three man-hours to produce, whereas each mobile home cabinet requires five man-hours. Labor rates normally average $18 per hour. During July and August, however, when Mobile employs many part-time workers, labor rates average only $14 and $16 per hour, respectively. A total of 2100 man-hours are available in July, 1500 in August, and 1200 in September. During any given month, management at the Lexington plant can schedule up to 50% additional man-hours, using overtime at the standard rate of time and a half. Material costs for motor home cabinets are $146; for mobile home cabinets they are $210.

The Lexington plant expects to have 25 motor home and 20 mobile home assembled cabinets in stock at the beginning of July. The home office wants the Lexington plant to have at least 10 motor home and 25 mobile cabinet assemblies in stock at the beginning of October to cover possible shortages in production from other plants.

The Lexington plant has storage facilities capable of holding up to 300 cabinets in any one month. The costs for storing motor home and mobile home cabinets from one month to the next are estimated at $6 and $9 per cabinet, respectively. Devise a monthly production schedule that will minimize the costs at the Lexington plant over the quarter.

Hint: Define variables so that you can fill in the following charts.

Quarterly Production Schedule of Motor Home Cabinets

	Regular Time	Overtime
July
August
September

Quarterly Production Schedule of Mobile Home Cabinets

	Regular Time	Overtime
July
August
September

Quarterly Storage Schedule

	Motor Home	Mobile Home
July
August
September

Answer 1

14.

Formulate LP model as follows

Let XR1, XR2, XR3 be the units of Motor Home cabinets to be produced in Regular time in month of July, August and September respectively

XO1, XO2, XO3 be the units of Motor Home cabinets to be produced in Overtime in month of July, August and September respectively

YR1, YR2, YR3 be the units of Mobile Home cabinets to be produced in Regular time in month of July, August and September respectively

YO1, YO2, YO3 be the units of Mobile Home cabinets to be produced in Overtime in month of July, August and September respectively

XV1, XV2, XV3 be the ending inventory of Motor Home cabinets in month of July, August and September respectively

YV1, YV2, YV3 be the ending inventory of Mobile Home cabinets in month of July, August and September respectively

Min 146*(XR1+XR2+XR3+XO1+XO2+XO3)+210*(YR1+YR2+YR3+YO1+YO2+YO3)+6*(XV1+XV2+XV3)+9*(YV1+YV2+YV3)+14*(3XR1+5YR1)+16*(3XR2+5YR2)+18*(3XR3+5YR3)+21*(3XO1+5YO1)+24*(3XO2+5YO2)+27*(3XO3+5YO3)

s.t.

3XR1+5YR1 <= 2100

3XR2+5YR2 <= 1500

3XR3+5YR3 <= 1200

3XO1+5YO1 <= 1050

3XO2+5YO2 <= 750

3XO3+5YO3 <= 600

XR1+XO1-XV1 = 250-25

XR2+XO2-XV2+XV1 = 250

XR3+XO3-XV3+XV2 = 150

YR1+YO1-YV1 = 100-20

YR2+YO2-YV2+YV1 = 300

YR3+YO3-YV3+YV2 = 400

XV3 >= 10

YV3 >= 25

XV1+YV1 <= 300

XV2+YV2 <= 300

XV3+YV3 <= 300

All variables >= 0

Solution of LP model using Excel Solver follows:

Formula:

G3 =B9*3+B15*5 copy to G3:H5

D9 =SUM(B9:C9) copy to D9:D11, D15:D17, D22:D24

B22 =B21-B3+D9   copy to B22:B24

C22 =C21-C3+D15   copy to C22:C24

K2 =SUM(B9:C11)*G21+SUM(B15:C17)*H21

K3 =SUM(B22:B24)*G22+SUM(C22:C24)*H22

K4 =SUMPRODUCT(G3:H5,G15:H17)

K5 =SUM(K2:K4)