Sales forecasts for a certain product for the first 6 months of 2012 are given i
ID: 346871 • Letter: S
Question
Sales forecasts for a certain product for the first 6 months of 2012 are given in the following:
January—2,500 February—5,000 March—7,500
April—10,000 May—9,000 June—6,000
Production can be increased from 1 month to the next at a cost of $5 per unit, by hiring new workers. Production can also be decreased at a cost of $3 per unit by laying off existing workers. Inventory is limited to 7000 units due to warehouse capacity. Storage cost is estimated at $3 per unit per month based on the end of the month inventory. December 2011 production is set at 2000 units and the expected inventory at the end of December 2011 is 1000 units. The company does not allow shortages and the demands must be met every month. The company would like to have an inventory of 3000 units at the end of June 2012. The problem is to determine the production schedule for the first 6 months of 2012 that will minimize the total cost of production changeovers and storage. Formulate this as a LP problem as follows:
(a) Define your variables clearly.
(b) Write out the linear constraints that must be satisfied, briefly explaining the significance of each.
(c) Write out the linear objective function that must be minimized.
Explanation / Answer
(a) Define your variables clearly.
Decision variables
The decision to be made is the amount of additional capacity hired or fired in each month so that the constraints are met
Let Hi be the amount hired during month and Li be the amount fired during each month
Let Pi be the amount produced during month and Ii be the ending inventory during each month
Decision variables
H1,H2,H3,H4,H5.H6
L1,L2,L3,L4,L5,L6
P1,P2,P3,P4,P5,P6
I1,I2,I3,I4,I5,I6
(b) Write out the linear constraints that must be satisfied, briefly explaining the significance of each.
Production capacity constraints
P1=2000+h1-l1
P2 =P1+h2-l2
P3= P2+h3-l3
P4=P3+h4-l4
P5=P4+h5-l5
P6= P5+h6-l6
Demand constraints (production/ total capacity must meet demand)
P1 2500
P2 5000
P3 7500
P410000
P5 9000
P6 6000
Storage constraints
I17000
I27000
I37000
I47000
I57000
I67000
June inventory constraint
I6=3000
Inventory balance constraints
I1 = P1-2500
I2=I1+P2-5000
I3=I2+P3-7500
I4=I3+P4-10000
I5=I4+P5-9000
I6 =I5+P6-6000
Non-negativity constraints
P1,P2,P3,P4,P5,P6 0
H1,H2,H3,H4,H5,H6 0
L1,L2,L3,L4 ,L5,L6 0
I1,I2,I3,I4,I5,I6 0
(c) Write out the linear objective function that must be minimized.
Minimize the total cost of production changeovers and storage.
Objective function
Minimize
5* (H1+H2+H3+H4+H5+H6) + 3*(L1+L2+L3+L4+ L5+L6)+3*( I1+I2+I3+I4+I5+I6)
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