ISE 216Question Hour

Chapter 3

Mar. 16th 2011

Q 3.12The personel department of A&M Co. wants to know how many workers will be

needed each month for the next six month production period. The following is a monthly demand d-forecasts. The inventory on hand at the end of june was 500 units. The company wants to maintain a min. inventory of 300 units each month and would like to have 400 units at the end of dec. Each unit requires five-employee hours to produce, there are 20 working days in a month and each employee works 8-hour day. The workforce at the end of june was 35 workers.

a) Determine a minimum inventory production plan. (one that allows arbitrary hiring and firing)

b) Determine the production plan that meets demand but does not hire or fire workers during six-month period.

Month Number of Patrons

July 1250

Aug. 1100

Sep. 950

Oct. 900

Nov. 1000

Dec. 1150

A 3.123.12 a) Month For Dem Net F Dem Ratio Prod Inv(-300) Ratio(adj) July 1250 (1250-500+300)1050 (1050/32)33 (33*32)1056 6 33 Aug 1100 1100 (110/32)35 (35*32)1120 26 35 Sept 950 950 30 960 36 29 Oct 900 900 28 896 32 28 Nov 1000 1000 32 1024 56 31 Dec 1150 (1150+400)1550 49 1568 74 48 The column labeled Adj gives the minimum number of workers without incurring negative inventory. (i.e., not dropping below 300 units). b) Month For Dem Net F Dem Cum N Dem Cum # Ratio Prod Inv(-300) July 1250 1050 1050 32 33 1120 70 Aug 1100 1100 2150 64 34 2240 90 Sept 950 950 3100 96 33 3360 260 Oct 900 900 4000 128 32 4480 480 Nov 1000 1000 5000 160 32 5600 600 Dec 1150 1550 6550 192 35 6720 170

The maximum of these numbers, 35, is the minimum constant work force required to guarantee that no shortages occur. Assuming 35 workers each month gives the production and inventory levels in the last two columns. The inventory levels given in the final column must be adjusted by adding 400 units for December initially netted out and the 300 units added to the Net F. demand for July.

Q 3.32The Paris Paint is in the process of planning labor force requirements and production

levels for the next four quarters. The following forecasts are made:There are 280 workers with the company. Hiring cost=1200 $/emp., firing cost=2500

$/emp.inverntory cost=1 $/gallon per quarter. It is estimated that one worker produces 1000 gallons of paint each quarter. Paris has 80000 gallons of paint in inventory now and would like to have 20000 gallons at the end of year.

a) Determine the min. workforce plan. Calculate the cost.b) Determine the zero inventory plan. Let hire and fire of employees. Calculate the

cost.c) If Paris were able to back-order access demand at a cost of 2 $/gallon per quarter,

determine a minimum constant workforce plan that holds less inventory than the plan you found in part a. Let stock-outs only in quarter 2. Calculate the cost.

Quarter Demand forecast (*1000)

1 380

2 630

3 220

4 160

A 3.323.32 a) Unit/Worker Net Net Cum. Cum. Min. Quarter (000) Demand Dem. Unit/worker Work Force 1 1 (380-300)300 300 1 (300/1)300 2 1 630 930 2 (930/2)465 3 1 220 1150 3 (1150/3)384 4 1 (160+20)180 1330 4 (1330/4)333

Min. constant workforce is 465 workers (take the maximum).

The cost of the resulting plan is: Cum Cum Ending Production Net Demand Inventory Quarter (000) (000) (000) 1 465 300 (465-300)165 2 (465*2)930 930 (930-930)0 3 (465*3)1395 1150 245 4 (465*4)1860 1330 530 Total 940 We must also add back in the 20,000 required to be on hand in the fourth quarter. Hence the total cost of this plan is: (1,200)(465 - 280) + (1000)(940 + 20) = $1,182,000.

A 3.323.32 b) Units/ Predicted Worker Net Ending Quarter (000) Demand Workers Hires Fires Inv. 1 1 300 300 20 0 0 2 1 630 630 330 0 0 3 1 220 220 0 410 0 4 1 180 180 0 40 0 Totals 350 450 Total cost = (1,200)(350) + (2500)(450) + 20,000 = 1,565,000

c) If we use the minimum number of workers required through period 3 of 1150/3 = 384, it will satisfy the conditions stated. (This does not work every time!!! You have to try for different number in a valid range.)

Quarter Cum Prod Cum Net Dem. Ending Inv. d) 1 384 300 84 2 768 930 -162 3 1152 1150 2 4 1536 1330 206 Total cost = (1,200)(384 - 280) + (1,000)(84+2+206+20) + (2,000)(162) = $760,800. Better than policies in parts (a) or (b).

Q 3.34a) Formulate Q.32 as a linear program.b) Solve the linear program. Round variables

and determine the cost of the plan.

A 3.343.34 a)

Min

Sub to: (A) W1 - W0 - H1 + F1 = 0 W2 – W1 – H2 + F2 = 0 W3 – W2 – H3 + F3 = 0 W4 - W3 - H4 + F4 = 0 (B) P1 - W1 = 0 P2 - W2 = 0 P3 - W3 = 0 P4 - W4 = 0 (C) I0 - I1 + P1 = 380 I1 - I2 + P2 = 630 I2 - I3 + P3 = 220 I3 - I4 + P4 = 160 (D) I0 = 80 W0 = 280 I4 20 Ht, Ft, It, Wt, Pt 0 1 t 4.

1.200 Ht 2,500 Ft 1, 000 I tt1

4

t1

4

t1

4

A 3.343.34 b) LP OPTIMUM FOUND AT STEP 1 OBJECTIVE FUNCTION VALUE= 1059500.00 VARIABLE VALUE REDUCED COST H1 185.000000 .000000 I3 .000000 500.000000 H2 .000000 1350.000000 I4 60.000000 .000000 H3 .000000 3700.000000 W1 465.000000 .000000 H4 .000000 2200.000000 W2 465.000000 .000000 F1 .000000 3700.000000 W3 220.000000 .000000 F2 .000000 2350.000000 W4 220.000000 .000000 F3 245.000000 .000000 P1 465.000000 .000000 F4 .000000 1500.000000 P2 465.000000 .000000 I1 165.000000 .000000 P3 220.000000 .000000 I2 .000000 4850.000000 P4 220.000000 .000000 Since rounding is unnecessary, this is the optimal solution. Writing the solution in tabular form we have:

Cum Cum Quarter #Workers Prod. Demand Inv.

1 465 465 300 165 2 465 930 930 0 3 220 1150 1150 0

4 220 1370 1310 60 Checking the cost: hires fires holding (185)(1,200) + (245)(2,500) + (225)(1000) = $1,059,500 as above.