Production Management Application

by

Aparna

Asha. v

Saritha

Jinto Antony Kurian

Contents

• A Make-or-Buy Decision

• Production Scheduling

• Workforce Assignment

A Make-or-Buy Decision

QUESTION

• A company markets various business and engineering products.

• Currently it is preparing to introduce two new calculators: one for the business market called the Financial Manager and one for the engineering market called the Technician.

• Each calculator has three components: a base, an electronic cartridge, and a face plate or top. The same base is used for both calculators, but the cartridges and tops are different.

• All components can be manufactured by the company or purchased from the outside suppliers.

• 3000 Financial Manager calculators and 2000 Technician calculators will be needed.

Contd:

• Manufacturing capacity is limited.• The company has 200 hours of regular manufacturing

time and 50 hours of overtime that can be scheduled for the calculators.

• Overtime involves a premium at the additional cost of $9 per hour.

• The problem for the company- to determine how many units of each component to manufacture and how many units of each component to purchase.

OBJECTIVE FUNCTION

• Decision Variables

BM No. of bases manufactured BP No. of bases purchased FCM No. of Financial manager cartridges mfrd FCP No. of Financial manager cartridges prchd TCM No. of technician cartridges manufactured TCP No. of technician cartridges purchased FTM No. of financial manager tops manufactured FTP No. of financial manager tops purchased TTM No. of technician tops manufactured TTP No. of technician tops purchased OT No. of hours of overtime to be scheduled.

• Objective of the decision maker To minimize the total cost , including

manufacturing costs, purchase costs and overtime costs.

Hence the objective function is: Min

0.5BM+0.6BP+3.75FCM+4FCP+3.3TCM+3.9TCP+0.6FTM+0.65FTP+0.75TTM+0.78TTP+9 OT

CONSTRAINTS

• Number of each component needed to satisfy the demand for 3000 FM calculators and 2000 Technician calculators.

• The five demand constraints are

BM+BP =5000 Bases

FCM+FCP=3000 FM cartridges

TCM+TCP=2000 Technician cartridges

FTM+FTP=3000 FM tops

TTM+TTP=2000 Technician tops

• Manufacturing capacities for regular time and overtime cannot be exceeded

1) Limits overtime capacity to 50 hours

OT <= 50

2) Total manufacturing time required for all components must be less than or equal to the total manufacturing capacity ( regular time + overtime)

BM+3FCM+2.5TCM+FTM+1.5TTM<=12,000+60OT

ie, BM+3FCM+2.5TCM+FTM+1.5TTM-60OT<=12,000

Complete Formulation

• Min 0.5BM+0.6BP+3.75FCM+4FCP+3.3TCM+3.9TCP+0.6FTM+

0.65FTP+0.75TTM+0.78TTP+9OT

S.t.

BM + BP = 5000 Bases

FCM + FCP =3000 FC

TCM + TCP = 2000 TC

FTM + FTP = 3000 FT

TTM + TTP = 2000 TT

OT <= 50 Overtime hours

BM+3FCM+2.5TCM+FTM+1.5TTM-60 OT <= 12,000 Manuftrng cpcty

Production Scheduling

One of the most important applications of linear programming deals with multiperiod planning such as production scheduling. Let us consider the case of Bollinger Electronics Company, which produces two different electronic components for a major airplane engine manufacturer. The airplane engine manufacturer notifies the Bollinger sales office each quarter of their monthly requirements for components for each of the next 3 months. The monthly requirements for the components may vary considerably, depending on the type of engine the airplane engine manufacturer is producing.

The table below shows the order that has been received for the next 3 month period.

Three month demand schedule for Bollinger Electronics Company

Component April May June

322A802B

10001000

3000500

50003000

After the order is processed, a demand statement is sent to the production control department. The

production control department must then develop a 3 month production plan for the components. In arriving at the desired schedules, the production

manager will identify:

• Total production cost• Inventory holding cost• Change in production schedule cost

Let xim denote the production volume in units for product i in month m. Here i=1,2 and m=1,2,3; i=1 refers to component 322 A, i=2 refers to component 802B, m=1 refers to April, m=2 refers to May and m=3 refers to June.

If component 322A costs $20 per unit produced and component 802B costs $10 per unit produced, the total production cost part of the objective function is

Total production cost=20x11+20x12+20x13+10x21+10x22+10x23

In order to incorporate the relevant inventory holding cost into the model, let ‘Sim’denote the inventory level for the product ‘i ’ at the end of the month ‘m’. Bollinger has determined that the monthly inventory holding costs are 1.5% of the cost of the product,ie., (0.015)($20)=$0.30 per unit for the component 322A and (0.015)($10)=$0.15 per unit of component 802B..

A common assumption made in production scheduling is that the monthly inventories are an acceptable approximation of the average

inventory levels throughout the month. Making this assumption the inventory holding cost portion of the objective function will be as

follows:

Inventory holding cost= 0.30s11 +0.30s12 +0.30s13+0.15s21+0.15s22+0.15s23

To incorporate the costs of fluctuations in production levels from month to month, two more variables are defined:

Im =increase in the total production levels during the month m

Dm=decrease in the total production level during the month m

After estimating the effects of employee

layoffs,turnovers,reassignment training costs and other costs, Bollinger estimates that the cost of increase in

production level for any month is $.0.50 per unit increase .Similarly the cost of decrease in production

level for any month is estimated as $0.20 per unit decrease.

The third portion of the objective function will be:Change in production level costs- 0.50I1 + 0.50I2 +

0.50I3 + 0.20D1 + 0.20D2 + 0.20D3

Combining all the costs, the complete objective function becomes:

Min 20x11 + 20x12 +20x13 +10x21 + 10x22 + 10x23 + 0.30s11 + 0.30s12+ 0.30s13 + 0.15s21 + 0.15s22 + 0.15s23 + 0.50I1 + 0.50I2 + 0.50I3 +

0.20D1 + 0.20D2 + 0.20D3

Constraints

The units demanded can be expressed as:

Ending inventory from previous month + Current production-Ending inventory for this month=This month’s demand

Suppose the inventories at the beginning of the 3- month scheduling period were 500 units for component 322A and 200 units for component 802B. The demand for both products in the first month (April) was 1000 units, so the constraints for meeting demand in the first month becomes

500+x11-s11=1000

200+x21-s21=1000

Moving the constants to the right side we have:

X11-s11=500

X21-s21=800

Similarly demand constraints for second and third month:

Month 2S11+x12-s12=3000S21+x22-s22=500Month 3S12+x13-s13=5000S22+x23-s23=3000

If the company specifies a minimum inventory level at the end of the 3-month period of at least 400 units of component 322A and at least 200 units of component 802B,then

S13>=400

S23>=200

Suppose if additional information on machine, labour and storage capacity is given:

Month Machine Labour Storage capacity capacity capacity (hours) (hours) (square feet)April 400 300 10,000May 500 300 10,000June 600 300 10,000

Machine, Labour and storage requirements

Component Machine Labour Storage (hours/unit) (hours/unit) (square feet)

322A 0.10 0.05 2

802B 0.08 0.07 3

Then the constraints are:Machine capacity0.10x11+0.08x21<=400 Month 10.10x12+0.08x22<=500 Month 20.10x13+0.08x23<=600 Month 3

Labour capacity0.05x11+0.07x21<=300 Month 10.05x12+0.07x22<=300 Month 20.05x13+0.07x23<=300 Month 3

Storage capacity2s11+3s21<=10,000 Month 12s12+3s22<=10,000 Month 22s13+3s23<=10,000 Month 3

One final set of constraints must be added to guarantee that Im and Dm will reflect the increase or decrease in the the total production level for month m. Suppose that the production level for March, the month before the start of the current production scheduling period, had been 1500 units of component 322A and 1000 units of component 802B for a total production level of 1500+1000=2500

units. We can find the change in production for April from the relationship

April production-March production= Change

Using the April production variables, x11 and x21 and the March production of 2500 units, we have (x11 +x21) -2500 = Change

A positive change reflects an increase in the total production level, and a negative change reflects a

decrease in the total production level. We can use the increase in production for April, I1, and the decrease in production for April, D1, to specify the constraint for the

change in total production for the month of April: (x11 +x21) – 2500= I1 – D1

We cannot have an increase in production and a decrease in production during the same month; thus

either I1 OR D1 will be zero. This approach of denoting the change in production level as the difference between

two non negative variables, I1 and D1, permits both positive and negative changes in the production level. If a single variable (like cm) had been used, only positive

changes would be possible because of the nonnegativity requirement.

Using the same approach in May and June, we obtain the constraints for the second and third months of the

production scheduling period:

(x12 + x22) – (x11 + x21) = I2 –D2 (x13 +x23) – (x12 + x22) = I3 – D3

Placing the variables on the left-hand side and the constraints on the right-hand side yields the complete set of what is commonly referred to as production-smoothing

constraints.

x11 +x12 – I1 + D1 = 2500 -x11 –x21 +x12 +x22 –I2 +D2 = 0

-x12 – x22 + x13 + x23 – I3 + D3 =0

Minimum-cost production schedule information forActivity April May June

Production 500 3200 5200

Comp 322A

Comp 802B

2500 2000 0

Ending inventory-322A

802B

0

1700

200

3200

400

200

Machine usage-scheduled hrs

Slack capacity

250

150

480

20

520

80

Labour usage-scheduled hrs

Slack capacity

200

100

300

0

260

40

Storage usage-scheduled storage

Slack capacity

5100

4900

10000

0

1400

8600

Workforce Assignment

Workforce assignment

• Important for managers to make decisions regarding staffing.

• Reduces the cost of labor if employees can be cross trained in two or more jobs.

• Not only optimal product mix, but also optimal workforce assignment.

Case overview

• McCormick manufacturing company produces two products with contributions to profit per unit of $10 and $9 respectively. The labor requirement per unit produced and the total hours of labor available from personnel assigned to each of four department are shown

Labor-Hours per unit

Department Product1 Product2 Total Available Hours

1 0.65 0.95 6500

2 0.45 0.85 6000

3 1.00 0.70 7000

4 0.15 0.30 1400

• Decision variableP1 = units of products 1

P2 = units of products 2

• Objective function : To maximize profitMax Z = 10P1+9P2

• Subjected to constraints0.65P1+0.95P2 <= 6500

0.45P1+0.85P2 <= 6000

1.00P1+0.70P2 <= 7000

0.15P1+0.30P2 <= 1400

• Non-negativity constraints

P1,P2 >= 0

• Optimal solutionP1 = 5744

P2 = 1795

Z = $ 73,590

Slack/Surplus

• Department 3 & 4 are on max operation at capacity.

• Department 1 & 2 have a slack of appx 1062 and 1890 hours.

• Feasible solutionTransfer labor hours from the departments which have slack to one’s which need more hours to increase the profit.

Cross training ability and capacity information

From Department

1 2 3 4 Max hours transferable

1 -- Yes Yes -- 400

2 -- -- Yes Yes 800

3 -- -- -- Yes 100

4 Yes Yes -- -- 200

Additional variables• bi= the labor hours allocated to department i (i = 1,2,3,4)

• tij = the labor hours transferred from department i to department j

Writing the capacity constraints in terms of b0.65P1 + 0.95P2 <= b1

0.45P1 + 0.85P2 <= b2

1.00P1 + 0.70P2 <= b3

0.15P1 + 0.30P3 <= b4

Bringing bi on to the left side of the inequalities for all the equations.

0.65P1 + 0.95P2 – b1<= 0 …………

• b1 = Hours initially in department 1 +

hours transferred into department 1 –

hours transferred out of department 2

b1 = 6500+t41-t12-t13

rewriting …..

b1-t41+t12+t13 = 6500

b2-t12+t42+t23+t24= 6000

b3-t13-t23+t34 = 7000

b4-t24-t34+t41+t42 = 1400

• Transfer capacity

t12+t13<= 400

t23+t24<= 800

t34 <= 100

t41+t42 <= 200

Profits maximized from $73,590 to $ 84,011

P1 = 6825

P2 = 1751

• Conclusion

Not only the profits are maximized, but also the labor workforce is also optimized.

THANK YOU