ABCO manufactures two products X and Y$.$ ABCO's manager is interested in production and liquidity

planning for the next two weeks of operation. The first week has four working days and the second week has

five working days. Each working day has one eight$-$hour shift.

The firm has $200$ machines that can be used to produce both X and Y$.$ A unit of product X

takes $10$ hours of machine time, whereas a unit of Y takes $16$ hours. Sales of product Y cannot exceed $800$

units during the next two weeks whereas there is no such restriction on the sales of product X$.$ Within those

limits$,$ ABCO does not produce on inventory and sells immediately what it produces. Sales price, wages to

labor, and cost of raw material per unit of product X and product Y do not vary over time and are given in

Table $1.$

Wages are proportional to production volume and paid in the week when production occurs, raw materials

are bought on credit of one week and delivered immediately, i$.$e$.,$ ABCO uses "just$-$in$-$time" delivery on a

weekly basis and maintains no inventory of raw materials for its current products. All sales are on credit,

and payments are received one week from the date of sale. The salary of the manager is $$200$ per week; it

is not affected by the production volume, but it is paid weekly like the wages are. The balance sheet at the

beginning of the planning period is given in Table $2.$

From it we see that ABCO receives $$32000$ from accounts receivable and pays out $$9500$ at the start

of the first week. As a matter of policy, ABCO maintains inventories of $$2000$ in parts of some obsolete models. The current balance of marketable securities is earmarked for tax payments which are due later in the year: ABCO maintains a minimum cash balance of $$500$ and has an open line of credit of $$7500$ from a local bank. This line of credit is presently fully used, but repayment and interest payments on its loans are scheduled for later in the year. In order to preserve a favorable picture of the firm's financial situation, the manager has agreed with the local bank that ABCO's "quick ratio" should not drop below $2\mathrm{.}0.$

$1.$ Verify that at the beginning of the period ABCO has a quick ratio greater than or equal to $2\mathrm{.}0.$ Show your calculations. The "quick ratio" is defined as

Quick Ratio $=($Current Assets $$ Inventories$)/($Current Liabilities$)$

$2.$ Let

$$ x$1$: the number of units X produced and sold in week $1,$

$$ y$1$: the number of units Y produced and sold in week $1,$

$$ x$2$: the number of units X produced and sold in week $2,$

$$ y$2$: the number of units Y produced and sold in week $2.$

Formulate the problem of finding a profit optimal production mix for ABCO that is within the various

physical and policy restrictions as well as the limited scope of ABCO's financial capabilities as a

two$-$period linear program in the above four variables.

$3.$ Compute a profit optimal product mix for ABCO using an LP solver. What is the optimal profit

ABCO can obtain?

$4.$ To ensure a "smooth" production schedule ABCO's management wants that the two weekly production volume to schedule for product X and product Y$,$ respectively, do not fluctuate too widely. More precisely, management wants the absolute difference of the production volume of product X to not exceed $20\%$ of the total production volume for product X over the two periods and likewise for product Y$.$ Reformulate the linear program to include these considerations. What are the optimal production volume to be scheduled and what is the optimal profit for the changed linear program? How much does it cost ABCO to smooth its production this way? How does this answer change if management insists on a maximum fluctuation of $10\%$ rather than $20\%?$

$5.$ Summarize your answers in a brief report, which also includes a listing of your GAMS model.

in dollars per unit Product X Product Y

Sales Price $$55,00$ $$125,00$

Labor $$30,00$ $$48,00$

Raw materials $$10,00$ $$50,00$

Profit $$15,00$ $$27,00$

Table$1$: Cost and revenue data $($in dollars per unit$)$

Assets Liabilities and Equity

Cash $$2\mathrm{.}000$ Accounts payable $$95\mathrm{.}000$

Marketable securities $$600$ Bank loan $$7\mathrm{.}500$

Accounts receivable $$32\mathrm{.}000$ Current liabilities $$17\mathrm{.}000$

Inventories $$2\mathrm{.}000$ Long$-$term debt $$22\mathrm{.}000$

Current assets $$36\mathrm{.}000$ Net equity $$47\mathrm{.}100$

Fixed assets $$50\mathrm{.}000$

Total assets $$86\mathrm{.}000$ Total Liabilities & Equity $$86\mathrm{.}000$

Table$2$: ABCO's Balance sheet as of the beginning of the planning period