$312$

CHAPTER $8-$ LINEAR PROGRAMMING APPLICATIONS

$8-3($Restaurant work scheduling problem$)$ The famous Y$.$ S$.$ Chang Restaurant is open $24$ hours a day. Waiters and busboys report for duty at $3$ A$.$M$.,7$ A$.$M$.,11$ A$.$M$.,3$ P$.$M$.,7$ P$.$M$.,$ or $11$ P$.$M$.,$ and each works an $8-$hour shift. The following table shows the minimum number of workers needed during the six periods into which the day is divided. Chang's scheduling problem is to determine how many waiters and busboys should report for work at the start of each time period to minimize the total staff required for one day's operation. $($Hint: Let $x_i$ equal the number of waiters and busboys beginning work in time period $i,$ where $i=1,2,3,4,5,6.)$

$\backslash$table$[[,,\backslash$table$[[$NUMBER OF WAITERS$],[$AND BUSBOYS REQUIRED$]]],[$PERIOD$,$TIME,$3],[1,3$ A$.$M$.-7$ A$.$M$.,12],[2,7$ A$.$M$.-11$ A$.$M$.,16],[3,11$ A$.$M$.-3$ P$.$M$.,9],[4,3$ P$.$M$.-7$ P$.$M$.,11],[5,7$ P$.$M$.-11$ P$.$M$.,4],[6,11$ P$.$M$.-3$ A$.$M$.,]]$

: $8-4($Animal feed mix problem$)$ The Battery Park Stable feeds and houses the horses used to pull tourist$-$filled carriages through the streets of Charleston's historic waterfront area. The stable owner, an ex$-$racehorse trainer, recognizes the need to set a nutritional diet for the horses in his care. At the same time, he would like to keep the overall daily cost of feed to a minimum.

The feed mixes available for the horses' diet are an oat product, a highly enriched grain, and a mineral product. Each of these mixes contains a certain amount of five ingredients needed daily to keep the average horse healthy. The table on this page shows these minimum requirements, units of each ingredient per pound of feed mix, and costs for the three mixes.

In addition, the stable owner is aware that an overfed horse is a sluggish worker. Consequently, he determines that a total of $6$ pounds of feed per day is the most that any horse needs to function properly. Formulate this problem and solve for the optimal daily mix of the three feeds.

: $8-5$ The Kleenglass Corporation makes a dishwasher that has excellent cleaning power. This dishwasher uses less water than most competitors, and it is extremely quiet. Orders have been received from several retail stores for delivery at the end of each of the next $3$ months, as shown below:

$\backslash$table$[[$MONTH$,\backslash$table$[[$NUMBER OF$],[$UNITS$]]],[$June$,195],[$July$,215],[$August$,205]]$

Due to limited capacity, only $200$ of these dishwashers can be made each month on regular time, and the cost is $$300$ each. However, an extra $15$ units per month can be produced if overtime is used, but the cost goes up to $$325$ each. Also, if there are any dishwashers produced in a month that are not sold in that month, there is a $$20$ cost to carry this item to the next month. Use linear programming to determine how many units to produce in each month on regular time and on overtime to minimize the total cost while meeting the demand.

: $8-6$ Eddie Kelly is running for reelection as mayor of a small town in Alabama. Jessica Martinez, Kelly's campaign manager during this election, is planning the marketing campaign, and there is some stiff competition. Martinez has selected four ways to advertise: television ads, radio ads, billboards, and newspaper ads. The costs of these, the audience reached by each type of ad$,$ and the maximum number available are shown in the following table:

Data for Problem $8\mathrm{.}4$

$\backslash$table$[[,$FEED MIX,$],[\backslash$table$[[$DIET$],[$REQUIREMENT$],[($INGREDIENTS$)]],\backslash$table$[[$OAT PRODUCT$],[($UNITS$/$LB$)]],\backslash$table$[[$ENRICHED$],[$GRAIN$],[($UNITS$/$LB$)]],\backslash$table$[[$MINERAL$],[$PRODUCT$],[($UNITS$/$LB$)]],\backslash$table$[[$MINIMUM DAILY$],[$REQUIREMENT$],[($UNITS$)]]],[$A$,2,3,1,6],[$B$,0\mathrm{.}5,1,0\mathrm{.}5,2],[$C$,3,5,6,9],[$D$,1,1\mathrm{.}5,2,8],[$E$,0\mathrm{.}5,0\mathrm{.}5,1\mathrm{.}5,5],[cos\frac{t}{l}b,$$$0\mathrm{.}09,$$$0\mathrm{.}14,$$$0\mathrm{.}17,]]$