Question

1)Jay Co. is a contract manufacturer that produces two products, Prod A and Prod B; and supply the two products to various retailers. The numbers of units for Prod A and Prod B that Jay Co is contracted to supply for next four months are given in the following table:

	Month 1	Month 2	Month 3	Month 4
PROD A	1000	4000	1000	3000
PROD B	500	1500	250	1500

The production processes for Prod A and Prod B are quite labour intensive. It takes two hours direct labour to produce one unit of Prod A and six hours of direct labour to produce one unit of Prod B. One factory worker gets paid $4000 per month. Each factory worker works up to 200 hours per month as regular working hours. If a factory worker works more than 200 hours per month, those “over-time” portion will get compensated at the rate of $30 per hour. However, due to the agreement made with the labour union, total hours of overtime work cannot exceed 10% of regular working hours.

Currently, the industry suffers from labour shortage. As such, every newly hired factory worker receives two-month salary as a signing bonus. On the other hand, every worker made being redundant (i.e. fired) by the company receives a half-month salary as a severance payment.

Another cost element that Jay Co must consider is the inventory holding cost. The monthly inventory holding cost for Prod A is $5 per unit and the monthly inventory holding cost for Prod B is $20 per unit.

Due to its labour intensive nature of production processes, Jay Co does not consider any other production related costs. Their main concerns are manpower cost and inventory holding cost.

Every single contracted unit of Prod A or Prod B must be delivered. Due to the severe penalty Jay Co must pay to the retailers, it is not an option for Jay Co not to meet the contracted units.

(a) Apply linear programming model to develop a formulation for Jay Co’s decision problem. Describe any assumptions you made for your formulation

(b) Use Microsoft Excel to solve the formulation in Question 1(a). Your answer must include the sensitivity report.

(c) Interpret the sensitivity report you obtained in Question 1(b) and answer the following questions.
Ci)After reading your sensitivity report, your production manager wants to renegotiate with the labour union on “over-time” clause. She would like to talk to the labour union whether they can agree to work more than 10% of regular working time as over-time. Discuss the reason(s) why she wants to have more overtime working.

cii)It turns out the labour union is quite open to the idea of allowing more over-time hours. They willing to increase the “10% cap” into 20% of regular working hours. However, they demander-time working hours will be paid at the rate of $40 per hour. Examine whether the request of the labour union is acceptable to Jay Co.

ciii) Suppose the labour shortage situation has improved a little bit. Now, Jay Co only needs to pay one-month salary as a signing bonus to newly hired workers. Analyse how it will affect your optimal solution.

Answer 1

(a)

Assumptions

• We already have 50 workers with us before the start of the first month.
• We don't have any beginning inventory (for both A and B) at the start of the first month.
• Inventory carrying cost is levied on ending inventory and not on the average inventory of a month.

Let H_j, L_j, R_aj, R_bj, O_aj, O_bj, E_aj, E_bj are defined as the decision variables such that -

H_j = # workers hired in the month-j
L_j = # workers laid off in the month-j
R_aj = Regular production hours for product A in the month-j
R_bj = Regular production hours for product B in the month-j
O_aj = Overtime production hours for product A in the month-j
O_bj = Overtime production hours for product B in the month-j
E_aj = Ending Inventory (in labor hours) for product A in the month-j
E_bj = Ending Inventory (in labor hours) for product B in the month-j; j=1,2,3,4

Also, define D_aj = Demand for product A (in labor hours) in the month-j. D_aj: [2000; 8000; 2000; 6000]
D_bj = Demand for product A (in labor hours) in the month-j. D_bj: [3000; 9000; 1500; 9000]

Objective Function: Minimize Z = total cost

Subject to, 1. Production balance constraints Regular Capacity Constraints 2. Raj Roj s 200*Wj; j-1,2,3,4 Here Wj stands for the worker level for the month-j and W-WH-L starting with Wo-50 3. Overtime Capacity Constraints OajObj 0.1(RRoj) S0 4. Non-negativeity Hi, Li, Raj, Rbj, Oaj, Obj, E, Ej 2 0; j-1,2,3,4

(b)

Ending Inv. (A) Month Demand A Demand B # of workers o/T hrs. o/Thrs. EndingReg. hrs. Inv. (B) Hire Layoff Reg. hrs. (hrs. (hrs.) 50 -1000 2 5006 E3+F4-G4 -4000 2 1500*6 E4+F5-G5 =1000"2-250"6 -E5+F6-G6 -3000 2 15006-E6+F7-G7 Totals SUM(E4:E7)SUM(F4:F7) SUM(G4:G7)SUM(H4:H7) SUM(14:17) SUM(J4:7) SUM(K4:K7) 9 Cost/unit 10 30 5/2 20/6 Total Cost Tota Total /T Regular capacity o/T capacity Output Output Output- (E.I- Output (E.I 4+M4G13-(J4-J3) H13-(K4-K3) -G15-(J6-J5) H15-(K6-K5) Month Regular -Reg+0/T =Reg+0/T (B) B.1.)-A B.I.)-B -SUMPRODUCT(E8:K8,E9:K9) 12 13 1 H4+14 200*E4 H5+5 200*ES -200 E6 L5+M5 15 3 16 I -H6+16 -16+M6 4 L7+M7 -10%*C16 -H74L7 -G16-(J7-J6 H16-(K7-K6)

Set Objective SKS12 Max Value Of: By Changing aiable Cells: SFS4:SMS7 Subject to the Constraints: SCS13:SDS16 SES13:SFS16 SIS13:SJS16 SCS4:SDS7 Add Change Delete Reset All Load/Save Make Unconstrained Variables Non-Negative Select a SolvingSimplex LP Method: Options Solving Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for linear Solver Problems, and select the Evolutionary engine for Solver problems that are non-smooth.

Final Result

Month Demand A Demand B #of Hire Layoff o/Thrs.o/Thrs.Ending Ending Reg. hrs. Reg. hrs. (hrs.) (hrs Workers Inv. (A)v. B (A) 0 0 0 8000 1000 7750 3000 1000 2000 1000 4 5750 1500 4 Totals Cost/unit 193 0 3.75 1000 1000 11750 0 $4,000$8,000 $2,000 $30 $30 Total Cost Month Tota Total O/T Regular o/T capacity Output Output Output Output $866,875 -RegtO/T (A) -Reg+O/T (B) (E.I B.I.)-A (E. B.I.) -B capacity 10000 10000 9250 9250 Regular 10000 10000 9250 9250 1000 1000 0 0 1000 1000 925 925 4

(c)

Sensitivity Report

Variable Cells
			Final	Reduced	Objective	Allowable	Allowable
	Cell	Name	Value	Cost	Coefficient	Increase	Decrease
	$F$4	Hire	0	0	24000	9450	5450
	$G$4	Layoff	0	10000	-14000	1E+30	10000
	$H$4	O/T hrs. (A)	0	3.55271E-15	30	1E+30	3.55271E-15
	$I$4	O/T hrs. (B)	1000	0	30	3.55271E-15	327.5
	$J$4	Ending Inv. (A)	6000	0	2.5	0.833333333	29.77272727
	$K$4	Ending Inv. (B)	0	0.833333333	3.333333333	1E+30	0.833333333
	$L$4	Reg. hrs. (A)	8000	0	0	3.55271E-15	29.77272727
	$M$4	Reg. hrs. (B)	2000	0	0	327.5	3.55271E-15
	$F$5	Hire	0	4725	20000	1E+30	4725
	$G$5	Layoff	0	5275	-10000	1E+30	5275
	$H$5	O/T hrs. (A)	1000	0	30	0	272.5
	$I$5	O/T hrs. (B)	0	0	30	1E+30	0
	$J$5	Ending Inv. (A)	0	33.63636364	2.5	1E+30	33.63636364
	$K$5	Ending Inv. (B)	0	34.46969697	3.333333333	1E+30	34.46969697
	$L$5	Reg. hrs. (A)	1000	0	0	52.75	0
	$M$5	Reg. hrs. (B)	9000	0	0	0	34.46969697
	$F$6	Hire	0	10000	16000	1E+30	10000
	$G$6	Layoff	3.75	0	-6000	2500	5450
	$H$6	O/T hrs. (A)	0	16.25	30	1E+30	16.25
	$I$6	O/T hrs. (B)	0	16.25	30	1E+30	16.25
	$J$6	Ending Inv. (A)	5750	0	2.5	0.833333333	12.5
	$K$6	Ending Inv. (B)	0	0.833333333	3.333333333	1E+30	0.833333333
	$L$6	Reg. hrs. (A)	7750	0	0	0.833333333	12.5
	$M$6	Reg. hrs. (B)	1500	0	0	16.25	0.833333333
	$F$7	Hire	0	8750	12000	1E+30	8750
	$G$7	Layoff	0	1250	-2000	1E+30	1250
	$H$7	O/T hrs. (A)	0	13.75	30	1E+30	13.75
	$I$7	O/T hrs. (B)	0	13.75	30	1E+30	13.75
	$J$7	Ending Inv. (A)	0	18.75	2.5	1E+30	18.75
	$K$7	Ending Inv. (B)	0	19.58333333	3.333333333	1E+30	19.58333333
	$L$7	Reg. hrs. (A)	250	0	0	12.5	0.833333333
	$M$7	Reg. hrs. (B)	9000	0	0	0.833333333	19.58333333
Constraints
			Final	Shadow	Constraint	Allowable	Allowable
	Cell	Name	Value	Price	R.H. Side	Increase	Decrease
	$C$13	Total Regular	10000	-43.625	0	5.32907E-13	10909.09091
	$D$13	Total O/T	1000	-12.38636364	0	5.86198E-13	1047.619048
	$C$14	Total Regular	10000	-46.375	0	5.32907E-13	2000
	$D$14	Total O/T	1000	-14.88636364	0	5.86198E-13	1047.619048
	$C$15	Total Regular	9250	-13.75	0	500	1500
	$D$15	Total O/T	0	0	0	1E+30	925
	$C$16	Total Regular	9250	-16.25	0	11500	500
	$D$16	Total O/T	0	0	0	1E+30	925
	$I$13	Output - (E.I - B.I.) -A	2000	42.38636364	2000	12000	5.86198E-13
	$J$13	Output - (E.I - B.I.) -B	3000	42.38636364	3000	12000	5.86198E-13
	$I$14	Output - (E.I - B.I.) -A	8000	44.88636364	8000	44000	5.86198E-13
	$J$14	Output - (E.I - B.I.) -B	9000	44.88636364	9000	1833.333333	5.86198E-13
	$I$15	Output - (E.I - B.I.) -A	2000	13.75	2000	1500	500
	$J$15	Output - (E.I - B.I.) -B	1500	13.75	1500	1500	500
	$I$16	Output - (E.I - B.I.) -A	6000	16.25	6000	1500	500
	$J$16	Output - (E.I - B.I.) -B	9000	16.25	9000	500	9000

• The overtime capacity constraint shows negative shadow price. So, if we increase the overtime %, there is a chance that the total cost can further be reduced. This is the reason, she wants to have more overtime working.
• The shadow prices for the four months are -12.4, -14.9, 0, and 0 per hour. So, the total cost can reduce to 1000*(-12.4)+1000*(-14.9) = -27,300 and the rate increase is ($40 - $30)=$10 per hour. So, extra cost incurred will be 4000*10 = 40,000. So, this offer may not be worth considering.
• The decrease is $8,000 - $4,000 = $4,000. The allowable decrease is, however, $5,450. So, there should be any effect is the optimal solution.