Question

The J. Mehta Company’s production manager is planning a series of one-month production periods for stainless steel sinks. The forecasted demand for the next four months is as follows:

Month	Demand for Stainless Steel Sinks
1	120
2	160
3	240
4	100

The Mehta firm can normally produce 100 stainless steel sinks in a month. This is done during regular production hours at a cost of $100 per sink. If demand in any one month cannot be satisfied by regular production, the production manager has three other choices:

he can produce up to 50 more sinks per month in overtime but at a cost of $130 per sink;

he can purchase a limited number of sinks from a friendly competitor for resale (the maximum number of outside purchases over the four-month period is 450 sinks, at a cost of $150 each);

Or, he can fill the demand from his on-hand inventory (i.e. beginning inventory). The inventory carrying cost is $10 per sink per month (i.e. the cost of holding a sink in inventory at the end of the month is $10 per sink).

Back orders are NOT permitted (e.g. order taken in period 3 to satisfy the demand in later period 2 is not permitted). Inventory on hand at the beginning of month 1 is 40 sinks (i.e. beginning inventory at month 1 is 40 sinks)

Set up and formulate algebraically the above “production scheduling” problem as a TRANSPORTATION Model to minimize cost.

Answer 1

Let Di be the demand from ith month. Thus D1 = 120, D2 = 160, D3 = 240, D4 = 100

Let Li be the on hand inventory after ith month. Lo is the on hand inventory in the start of firs month which is 40, so Lo = 40.

Let Si be the number of sinks manufactured by Mehta firm in normal time utilized to satisfy a portion of the demand in ith month.

Let Ci be the number of sinks outsourced from friendly competitor utilized to satisfy a portion of the demand in ith month

Let Oi be the number of sinks manufactured by Mehta firm in overtime utilized to satisfy a portion of the demand in ith month

Cost of Production at Normal time = 100

Cost of Production at Over time = 130

Cost of Procurement from Competitor = 150

Holding Cost of Inventory = 10

Our Objective is to minimize cost of demand fulfillment through Production in Normal & Over time and Procurement from Competitor. Thus Objective function is-

Min( 100 * (S1 + S2 + S3 + S4) + 130 * (O1 + O2 + O3 + O4)+ 150 * (C1 + C2 + C3 + C4) + 10 * (L1 + L2 + L3 + L4))

Constraints-

As Demand will be equal to summation of quantity on-hand from previous month, quantity prodced in Normal time, quantity produced in overtime, quantity outsourced from friendly competitor and quantity on hand for the next month. i.e

Di + Li = L_i-1 + Si + Oi + Ci where i = 1, 2 , 3 , 4

Maximum 50 sinks can be produced overtime,

Oi <= 50 where i = 1, 2 , 3 , 4

Maximum of 450 sinks can be outsourced from friendly competitor,

C1 + C2 + C3 + C4 <= 450

Maximum of 100 sinks can be produced at normal time,

Si <= 100

As all variables are quantities, hence they can't be negative.

Di, Si, Ci, Oi >= 0

The solver snaps are-

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