Jay Co. is a contract manufacturer that produces two products, Prod A and Prod B

Question

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:

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

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

Explanation / Answer

Productivity of product A per worker per month

Regular working hours per month = 200

Direct labor to produce 1 unit = 2 hours

Unit per worker per month = 100 units per worker per month (200/2)

Productivity of product B per worker per month

Regular working hours per month = 200

Direct labor to produce 1 unit = 6 hours

Unit per worker per month = 33.3 units per worker per month (200/6)

Hiring cost (a worker gets $4,000 per month)

Hiring cost = 2 months’ salary as signing bonus = 2*$4,000 = $ 8000

Firing cost = half-month salary = 0.5*$4,000 = $ 2000

Formulae the linear programming model as follows:

Let h, f, w be the number of workers hired, fired and available per month repectively

rA, oA be the regular and overtime production of product A per month

rB, oB be the regular and overtime production of product B per month

iA and iB be the ending inventory of products A and B respectively per month

Objective function

Min Z =$4,000*(w1+w2+w3+w4) +$8,000*(h1+h2+h3+h4) +$2,000*(f1+f2+f3+f4) + 2*$30*(oA1+oA2+oA3+oA4)+6*$30*(oB1+OoB2+oB3+oB4)+$5*(iA1+iA2+iA3+iA4)+$20*(iB1+iB2+iB3+iB4)

s.t.

rA1+oA1-iA1 = 1000

rA2+oA2+iA1- iA2 = 4000

rA3+oA3+iA2-iA3 = 1000

rA4+oA4+iA3-iA4 = 3000

rB1+oB1-iB1 = 500

rB2+oB2+iB1-iB2 = 1500

rB3+oB3+iB2-iB3 = 250

rB4+oB4+iB3-iB4 = 1500

wj = wj-1 + hj - fj

2rAj + 6rj <= 200wj

2oAj + 6oBj <= 0.1*200wj (j=1,2,3,4)

h, f, w, oA, oB, rA, rB, iA and iB >= 0

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