Question $1(22$ marks$)$

Beluga caviar is one of the expensive foods that can be purchased from the market and sold to

restaurants and supermarkets. The CityTasty seafood distributor plans to buy Beluga caviar

and sell it later at an appropriate price. Sealed caviar can last up to $6$ weeks when stored

properly. The company has a large frozen storage facility which can store up to $20000$ kg of

caviar each week. Currently, it has $4500$ kg of fresh Beluga caviar in storage. On Friday every

week, CityTasty can buy Beluga caviar from the market and sell some or all its inventory at

market price in that week. Unsold caviar could be stored at a cost of $$600$ per kg per week. The

estimates of market prices of Beluga caviar for the next four weeks are $$21000,$ $$20790,$

$$22245$ and $$21133$ per kg$.$ As demand is anticipated to increase in the forthcoming festival

period, CityTasty would like to keep $5000$ kg of Beluga caviar in storage at the end of week $4.$

In order to determine the optimal strategy of buying, selling and storage for each of the next

four weeks to maximize the total profit, CityTasty has formulated the problem by a linear

program.

Let Bt $=$ amount $($kg$)$ of Beluga caviar to be bought in week t $($t $=1,2,3,4)$

St $=$ amount $($kg$)$ of Beluga caviar to be sold in week t $($t $=1,2,3,4)$

It $=$ remaining inventory $($kg$)$ of Beluga caviar to be stored in week t $($t $=1,2,3,4)$

Linear programming formulation:

$.=21000(11)+20790(22)+22245(33)+21133(44)600(1+2+3+4)$

subject to:

$1+1+1=4500$

$22+12=0$

$33+23=0$

$44+34=0$

$120000$

$220000$

$320000$

$5000420000$

All variables $0$

$2$

Question $1($continued$)$

When the linear program was solved using Solver, the following sensitivity report was obtained.

Use the linear program and the sensitivity report to answer the following parts $($a$)$ to $($f$).$ Each

part is independent.

$($a$)$ What is the optimal buying$-$selling$-$storing plan for the next four weeks? $(4$ marks$)$

$($b$)$ What is the expected maximum profit for the next four weeks? $(3$ marks$)$

$($c$)$ For each of the week$($s$)$ in which the storage facility is fully utilized, determine the impact

on the maximum profit by acquiring more storage capacity. Up to what storage limit would

this impact be valid? Alternatively, explain why it could not be determined from the

sensitivity report, assuming all other factors remain unchanged. $(4$ marks$)$

$($d$)$ Is there any alternative optimal plan? Explain. $(2$ marks$)$Variable Cells

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$B$$2$ Week $1$: Buy $00-2100001$E$+30$

$C$$2$ Week $1$: Sell $45000210000810$

$D$$2$ Week $1$: Store $0-810-6008101$E$+30$

$E$$2$ Week $2$: Buy $200000-207900810$

$F$$2$ Week $2$: Sell $002079001$E$+30$

$G$$2$ Week $2$: Store $200000-6001$E$+30855$

$H$$2$ Week $3$: Buy $00-2224501$E$+30$

$I$$2$ Week $3$: Sell $200000222450855$

$J$$2$ Week $3$: Store $0-1712-60017121$E$+30$

$K$$2$ Week $4$: Buy $50000-2113301712$

$L$$2$ Week $4$: Sell $002113301$E$+30$

$M$$2$ Week $4$: Store $50000-600217331$E$+30$

Constraints

Final Shadow Constraint Allowable Allowable

Cell Name Value Price R$.$H$.$ Side Increase Decrease

$N$$7$ Week $1$ balance $45002100045001$E$+304500$

$N$$8$ Week $2$ balance $0-2079001$E$+3020000$

$N$$9$ Week $3$ balance $0-222450200001$E$+30$

$N$$10$ Week $4$ balance $0-2113301$E$+305000$

$N$$11$ Week $1$ inventory $00200001$E$+3020000$

$N$$12$ Week $2$ inventory $20000855200001$E$+3020000$

$N$$13$ Week $3$ inventory $00200001$E$+3020000$

$N$$14$ Week $4$ inventory $50000200001$E$+3015000$

$N$$15$ Week $4$ min. inventory $5000-217335000150005000$

$3$

Question $1($continued$)$

$($e$)$ What is the maximum increase in storage cost $($$$600$ per kg per week$)$ in which the plan in

part $($a$)$ will remain optimal? Explain. Predict the limit of storage cost when there would be

no profit made in the optimal plan, assuming all other factors remain unchanged.

$(5$ marks$)$

$($f$)$ Determine how the change in the minimum inventory requirement $(5000$ kg$)$ of Beluga

caviar at the end of week $4$ would affect the maximum profit. For what range of values for

this requirement would the predicted impact remain valid? $(4$ marks$)$