Tommy & Lefebvre is Ottawa's full-line authorized dealer  for Atomic snowboards for the 2020-2021 winter season.  T&L's sales projections for November, December and January are 550, 800, and 300 snowboards for each of these next three months. Atomic has agreed to provide T&L with up to 650 boards each month at a unit cost of $82. Using "rush orders", Atomic can also provide T&L with up to 50 additional boards per month at an increased unit cost of $97 each. Boards not sold at the end of the month in which they are in stock are stored in the T&L store at a cost of $20 per board per month. It takes the store clerks 0.5 hours per board to set up, sticker, and display the snowboards and clerk capacity is limited to 500, 600, and 400 hours in each of the 3 months respectively. Finally, T&L have 50 Atomic snowboards from last season (2020 model) that they can sell this year and they want to have at least anther 75 snowboards (2021 model) left at the end of January. Write down the algebraic/mathematical formulation of this problem as a linear programming problem to minimize the total cost to T&L of purchasing and stocking the snowboards. (Define the decision variables, objective function, and constraints). DO NOT SOLVE.

Answer;

Sales projection for November=550

Sales projection for December=800

Sales projection for January=300

650 boards available per month @ $82 per unit

Additional 50 boards avaialble per month @ $97 per unit

Storage cost=$20 per unit per month

Clerk capacity in November=500hours

Clerk capacity in December=600hours

Clerk capacity in January=400hours

Setup, etc., time per unit=0.5hours

Number of units avaialble from last season=50

Number of units T&L want left at the end of January=75

Objective:to minimise the total cost of purchasing and stocking

The above information can be depicted as a transportation model as below.

Note that the clerk capacity is enough to meet the setup, etc., requirement per month

∴it does not have any impact over the situation

Supply points:

Past season, November regular supply, November additional supply, December RS,

December AS, January RS, January AS

Demand points:

November, December, January, Next Season, Dummy (as total avaialbility is greater than

total requirement)

Avaialbility:

Past season=50

November, December, January RS=650

November, December, January AS=50

Requirement:

November=550

December=800

January=300

Next season=75

Dummy=total availability-total requirement

=50+650+50+650+50+650+50-550+800+300+75

=2150-1725=425

Costs for each cell include the cost of procurement and storage cost

Note that the costs in dummy column will be zero

Cost for S1D4 is taken as M, a very big positive quantity so that in the optimum solution,

the cell remains empty because in next season boards from current season are required,

not from the previous season

Costs in cells S4D1, S5D1, S6D1, S7D1, S6D2, S7D2 are also taken as M because the demand

for past months cannot be met through supply from future months.

		November	December	January	Next season	Dummy	Availability
		D1	D2	D3	D4	D5
Past season	S1	0+0	0+20	0+40	M	0	50
November (regular supply)	S2	82+0	82+20	82+20×2	82+20×12	0	650
November (additional supply)	S3	97+0	97+20	97+20×2	97+20×12	0	50
December (regular supply)	S4	M	82+0	82+20	82+20×11	0	650
December (additional supply)	S5	M	97	97+20	97+20×11	0	50
January (regular supply)	S6	M	M	82+0	82+20×10	0	650
January (additional supply)	S7	M	M	97+0	97+20×10	0	50
Requirement		550	800	300	75	425	2150

 

	November	December	January	Next season	Dummy	Availability
Past season	0	20	40	M	0	50
November (regular supply)	82	102	122	322	0	650
November (additional supply)	97	117	137	337	0	50
December (regular supply)	M	82	102	302	0	650
December (additional supply)	M	97	117	317	0	50
January (regular supply)	M	M	82	282	0	650
January (additional supply)	M	M	97	297	0	50
Requirement	550	800	300	75	425	2150

Decision variables:

The values in each of the above cells would be the decision variables. That is, the amount of

demand met for November through past season will be one decision variable say x11.

Similarly, the extent of demand for November, met through regular supply in November

will be another decision variable x21

So, let the past season supply utilised in November=x11

Let the past season supply utilised in December=x12

Let the past season supply utilised in January=x13

Let the past season supply utilised in next season=x14

Let the past season supply left unutilised=x15

Let the November (RS) utilised in November=x21

Let the November (RS) utilised in December=x22

Let the November (RS) utilised in January=x23

Let the November RSutilised in next seaosn=x24

Let the November RS left unutilised=x25

Let the November (AS) utilised in November=x31

Let the November (AS) utilised in December=x32

Let the November (AS) utilised in January=x33

Let the November ASutilised in next seaosn=x34

Let the November AS left unutilised=x35

Let the December (RS) utilised in November=x41

Let the December (RS) utilised in December=x42

Let the December (RS) utilised in January=x43

Let the December RSutilised in next seaosn=x44

Let the December RS left unutilised=x45

Let the December (AS) utilised in November=x51

Let the December (AS) utilised in December=x52

Let the December (AS) utilised in January=x53

No file attached.

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